FALL 2019 Volume 60 Issue 2 By understanding their experiences with finances and giving them opportunities to practice financial skills, I hope to empower my students through this curriculum and give them the confidence they need to be financially capable citizens.

Alisha Parashar-Choukulkar Contents Fall 2019 | Volume 60 | Issue 2

08 Math Catcher: Indigenizing Through Culturally-Based Storytelling By Alana Underwood

15 Rek enreks By Stephanie Turner

19 Why ? By Glen Van Brummelen

24 A Public Math Gathering in Chicago By Janice Novakowski 37 33 28 NCTM Annual Meeting and Exposition, San Diego By Alex Sabell

30 A Culturally-Relevant Financial Literacy Curriculum By Alisha Parashar-Choukulkar 38 Connecting Our Colleagues: 27 A Book Club Study Using Open Questions By Tamara DeFord and Marie Fanshaw

41 Lego Robotics CoLab in SD73 By Elizabeth deVries

45 BCAMT Award Winners

47 Using Self-Determination Theory to Engage Learners IN EVERY ISSUE Errata • Spring Vector 2019 By Jeff Irvine 07 President’s Message 52 Math Challengers The editors would like to apologize to Chris Hunter, 44 Cartoon By Matt Branch whose article “A Function of Freedom and Constraints” 56 Book Review was printed with the draft version instead of the final 57 Problem Set 53 Social Justice and edited version in the Spring 2019 issue of Vector. For an Mathematics updated version of his article, please visit our website at 59 Math Links By Janice Novakowski https://www.bcamt.ca. 60 BCAMT BCAMT EXECUTIVE The views expressed in each Vector article are those of its author(s), and not necessarily those of the editors or of the British Deanna Brajcich, President Jen Carter, Treasurer Sooke School District Vernon School District Columbia Association of Mathematics Teachers. [email protected] [email protected] Articles appearing in Vector may not be reprinted without the explicit written permission of the editors. Once written permission Susan Robinson, Vice President Brad Epp, Membership Chair is obtained, credit must be given to the author(s) and to Vector, Gulf Islands School District Kamloops School District citing the year, volume number, issue number and page numbers. [email protected] [email protected] Notice to Contributors Colin McLellan, Secretary and Listserve Manager We invite contributions to Vector from all members of the Richmond School District mathematics education community in British Columbia. We will [email protected] give priority to suitable materials written by BC authors on BC curriculum items. In some instances, we may publish articles written by persons outside the province if the materials are of ELEMENTARY particular interest in BC. REPRESENTATIVES Submit articles by email to the editors. Authors must also include Adam Fox Alex Sabell a short biographical statement of 40 words or less. North Vancouver School District Surrey School District [email protected] [email protected] Articles must be in Microsoft Word (Mac or Windows). All diagrams must be in TIFF, GIF, JPEG, BMP, or PICT formats. Debbie Nelson Photographs must be high print quality (min. 300 dpi). Comox Valley School District [email protected] The editors reserve the right to edit for clarity, brevity and grammar. SECONDARY Notice to Advertisers REPRESENTATIVES Vector is published two times a year: spring and fall. Circulation is Ron Coleborn Chris Hunter approximately 1400 members in BC, across Canada and in other Surrey School District Surrey School District countries around the world. [email protected] [email protected] Advertising printed in Vector may be of various sizes, and all Josh Giesbrecht materials must be camera ready. Abbotsford School District Usable page size is 6.75 x 10 inches. [email protected] Deadline for Submissions Spring: FEB 1 (for peer review, December 1) POST-SECONDARY NCTM AND Fall: SEP 1 (for peer review, July 1) REPRESENTATIVES NCSM REPRESENTATIVES Peter Liljedahl Marc Garneau Advertising Rates Per Issue Simon Fraser University Surrey School District $300 Full Page [email protected] [email protected] $160 Half Page $90 Quarter Page INDEPENDENT SCHOOLS Membership Enquiries REPRESENTATIVES If you have questions regarding membership status or have a Darian Allan Richard de Merchant change of address, please contact Brad Epp, Membership Chair: Collingwood School St. Michaels University School [email protected] West Vancouver Victoria [email protected] [email protected] 2019/20 Membership Rates $40 + GST (BCTF Member) $20 + GST Student (full time university only) $65.52 + GST Subscription (non-BCTF) VECTOR EDITORS Sean Chorney Susan Robinson Cover Art: This diagram was created by Eden Weiss, April 2019, for Donal Duncan's Foundations of Mathematics 12 class at Gulf Simon Fraser University Gulf Islands School District Islands Secondary School. The assigned task was to create a Voronoi [email protected] [email protected] diagram using GeoGebra, and then to four-colour it according to the Four Colour Map Theorem. Eden's artistic inspiration for this diagram was to create an abstract representation of a flower.

Photos taken by Sean Chorney and Susan Robinson Contributors

Ray Appel Elizabeth deVries his portfolio was Mathematics a passion for developing Grades 7–12. He has taught Thinking Classrooms. Michael Ray has taught elementary Elizabeth is the Coordinator at three faculties of education believes that mathematics is a school, been a mathematics/ of Technology in School (Brock University, Queen's social endeavour and is best science helping teacher, spoken District 73. Elizabeth has University, Daemen College) learned through collaborative throughout Canada and the US, a Bachelor of Science in and at Sheridan College. Jeff activity. and has authored mathematics animal biology, a Masters is coauthor or contributing curriculum. He is currently of Education in Leadership author for 11 high school Alex Sabell retired, delving deep into and Administration and has mathematics textbooks. He is reading, painting, drawing, taught French Immersion in currently a PhD candidate at Alex has been an elementary walking, website design, a bit of elementary school. Elizabeth is Brock University. school teacher in the Surrey mathematicsconsulting, home always eager to share her love School District for nine years building and more! His website, of learning, critical thinking, Janice Novakowski during which time she earned zapple.ca is used by thousands outdoors, adventure, and play. a masters of education in of educators worldwide. She loves to connect with other Janice is currently a district Curriculum and Instruction, educators to share ideas and teacher consultant working with with a focus on Numeracy. Matt Branch stories. K-12 educators and students in This degree has fostered Alex’s the areas of mathematics and passion for mathematics which Matt joined Math Challengers, Marie Fanshaw numeracy in the Richmond is now an integral part of her then known as Math Counts, in School District. She is also a teaching practice. In 2018, 2004. In 2016 Matt was elected Marie is in her third year member of the BC Numeracy Alex received the Outstanding to the executive as Vice-Chair, working in the role of Network and facilitates the BC Elementary Teacher Award, and was elected into his current Numeracy Resource Teacher Reggio-Inspired Mathematics as well as the Ivan. L Johnson Chair position in 2017. Matt for the Prince George School Project. Janice is interested Memorial Award. She is Branch is graduate of the uni- District. Prior to this, she was in curricular structures and currently on the executive versity of Waterloo Engineer- a teacher for 25 years with a pedagogy that is inclusive and of the BC Association of ing, has been a Professional focus on primary grades. She responsive to all students. Mathematics Teachers Engineer since 2007. Matt is a is passionate about helping devoted to his family and two teachers build capacity in the Alisha Parashar- Brian Taylor children. Matt believes strongly area of numeracy. She has been Choukulkar in community and support- involved in helping facilitate Brian is a mathematics teacher ing tomorrow's math leaders books clubs and professional Alisha is a Business Education at Point Grey Secondary in through Math Challengers. development both locally and teacher at the Vancouver Vancouver. He earned his provincially. School Board. She completed mathematics degree and Tamara DeFord her Bachelors of Arts at the teaching degree at UBC. When Adam Fox University of British Columbia not in the classroom he can be Tamara DeFord is an with a specialization in found swimming in the ocean. Adam teaches kindergarten in intermediate teacher in Prince international relations. She the North Vancouver School George School District. She has also received her masters in Stephanie Turner District. Prior to becoming been involved with the district Educational Administration a teacher, he was an IT mathematics enhancement and Leadership with a focus Stephanie has been teaching consultant in England. He team since 2015, working on Financial Literacy. primary students for 20 years Fridays to support teachers recently completed his Master’s and is currently teaching and district projects in the in Mathematics Education at Michael Pruner grade two in the Surrey School area of numeracy. She has Simon Fraser University. district. She is passionate about taken the lead on many book Michael is a high school creating learning experiences clubs and workshops some of Jeff Irvine mathematics teacher from that are rooted in best practices, which included mathematical North Vancouver, and the as well as helping students see Jeff has been a secondary mindsets, number talks and past president of the BCAMT. that mathematics is much more mathematics teacher, fractions. She is a strong Michael is also a PhD student than a worksheet. department head, vice principal advocator of promoting student at Simon Fraser University and Education Officer fo r th e engagement and enjoyment in in mathematics education. Province of Ontario, where mathematics. He has recently discovered

Vector • fall 2019 5 Alana Underwood

Alana has been teaching elementary-aged students for the past eight years. She has a passion for mathematics and focuses on making it engaging for students in her classroom. She recently completed a masters in Numeracy at SFU. Glen Van Brummelen

Glen teaches at Quest University. He has served as president of the Canadian Society for History and Philosophy of Mathematics. He is a 3M National Teaching Fellow and received the MAA’s Haimo Award for Distinguished Teaching. Among his books are ‘Heavenly Mathematics’ (Princeton, 2013) and : A Very Short Introduction (Oxford, to appear December 2019). David Wees

David works as a senior curriculum Designer for DreamBox Learning, working remotely from his island paradise. He has worked as a teacher in NYC, London, Bangkok, and Vancouver. He has six years of experience as a mathematics coach. He is the co-author of a textbook designed for the International Baccalaureate program. He has his Bachelor of Science in mathematics and master of Educational Technology.

Vector • fall 2019 6 President’s Message

Welcome to the fall issue of Vector. I Teacher Awards: In recognition of teacher excellence, we present am inspired by the relevant and three awards per year. Each award includes funding to attend the thought-provoking articles fall conference, and the Ivan L. Johnson award offers funds for the shared by BC mathematics winner to attend the NCTM Annual Conference. educators. I am grateful for each author’s leadership, BCAMT Listserv: There may be no better place to find assistance commitment and effort as and advice than our listserv. Our members are our best resource. they share their experiences Email Colin McLellan at [email protected] to join. with readers across the province. I know that there will be many more discussions and professional development opportunities for all of us to work together and In reading these articles we, as educators, have the chance to jump exemplify a caring and supportive mathematics curriculum. I am into other people’s classrooms, leading us down new pathways. My personally grateful for the support from my school and provincial appreciation also extends to our dedicated editors, Sean Chorney colleagues as we all move forward toward this goal. and Susan Robinson, for uniting these authors and ensuring our distinguished BCAMT Vector publication remains an influential Three teachers who are demonstrating exceptional teaching and part of the mathematics conversation here and abroad. innovation meeting student needs are our BCAMT 2019 award winners. Rebecca Kaukinen from St. Margaret’s School was Grade 11 and 12 teachers have now joined K-10 teachers in the full presented with the Outstanding Elementary Teacher award. Some implementation of the new BC curriculum. As most of us do not of Rebecca’s focuses are: STEM-based projects, parent outreach and solely teach mathematics, this is an extensive undertaking in which providing support to girls in mathematics. The BCAMT executive we need to be patient with ourselves during the learning process. also awarded Cody Hilton with the 2019 Outstanding Secondary We need to work together to ensure we are not only supporting our Teacher award. Cody incorporates thinking classroom methods students but our colleagues as well. Take advantage of professional and PBL authentic tasks in his teaching as well as helping students learning opportunities, create a book club focused on assessment, as a Numeracy Resource teacher. or hold discussions about how to make the competencies come alive in your classroom. The BCAMT executive was especially excited to present the Ivan L. Johnson Memorial award to Janice Novakowski for her dedication To support this journey, the BCAMT has many supports in place and effort of promoting and progressing mathematics education for our members. I urge you to explore these supports and join in BC, Canada and the USA. In addition to the work she does in the larger conversation in the province. Please visit our website for her district, Janice leads Reggio-inspired workshops around the information about what the BCAMT can offer. province and has significantly contributed to create the Balanced Numeracy Network website (bcnumeracynetwork.ca) designed to Chapter support: Consider welcoming our executive to your help mathematics educators on their journey. district or school to help with the curriculum transition or to revisit important mathematics education practices. Let us know I am very fortunate to lead a dynamic and committed executive about your needs on the application on our website (https://www. and am very proud to be able to say that we are the largest bcamt.ca/). provincial specialist association in BC. It is the BCAMT history of collaboration and leadership that has led us to where we are today. BCAMT Grants: Each year we approve and fund inquiry projects Our executive never shies away from supporting and advocating around the province. Last year we provided $8000 to teacher for our BC teachers. We strive to find ways to reach every teacher, groups. This year we hope to provide more! Don’t miss the no matter the location. The enduring goal underpins what our application deadline of November 30th. executive has always believed, that mathematics is for all: students and educators. Vector • fall 2019 7 Mathematics Catcher: Indigenizing Mathematics Through Culturally-Based Storytelling by Alana Underwood

Introduction In British Columbia, the redesigned curriculum encourages students see themselves and their practices” in the stories (V. Jungic teachers to “incorporate Aboriginal content into all subject personal communication, Feb 5, 2018). This personally connects areas from Kindergarten to Grade 12” (BC curriculum, 2015). Aboriginal students to the mathematics found in the film and the However, teachers tend to be “nervous to venture into the realm subsequent mathematics activities that align with these films. The of Aboriginal education because they have very little knowledge of Aboriginal content of these films do not only benefit Aboriginal Aboriginal culture and are afraid of making mistakes or offending” students. Research underscores the benefits to “identity, well-being (Hodge, 2016). As a teacher, I relate to this statement—trying to and achievement [of all students] when Indigenous knowledge incorporate Aboriginal education into mathematics has been systems are at the heart of learning” (BC curriculum, 2015). a personal struggle. I find myself periodically introducing one- off, superficial examples of First Nations’ cultures, and only About the Films doing so to fulfill curriculum requirements, not to enhance my The Math Catcher film series incorporates problem-solving and lessons or benefit my students’ learning. To move away from this Aboriginal traditions into three to four minute-long animated type of tokenism and towards teaching mathematics with more stories. In these films, students watch Small Number, “a bright and authentic Aboriginal perspectives, I experimented with different playful [Aboriginal boy] who recognizes patterns and calculates ways to incorporate stories and films from Math Catcher into my quickly,” engage in Aboriginal traditions, go on adventures and mathematics lessons that, from a student’s suggestion, evolved get into mischief. Jungic feels that storytelling, accompanied by into a literacy unit I feel that indigenizing mathematics through pictures and open-ended questions can help students experience culturally-based storytelling demonstrates a tangible way to enrich mathematics in action, thus encouraging "young people to enjoy students’ learning, indigenize the curriculum and give students a mathematics and dispel myths that mathematics is boring and different outlet to explore mathematics with literacy. To find more abstract”. information about this resource, please visit the Match Catcher website (https://www.sfu.ca/mathcatcher.html). The stories in these films are not traditional First Nations’ stories but antidotes inspired by First Nations’ traditions. Based on Jungic’s The Creation of Math Catcher personal experiences with the Aboriginal community, every aspect The Math Catcher program was created in 2011 in response to of these films have been “researched, checked and double- checked” the low mathematics completion rate of Aboriginal high school by Jungic, his team of researchers and Aboriginal colleagues to students. “Presently only 2% of BC’s Aboriginal population ensure Aboriginal culture is respected and historically accurate. completed Principles of Mathematics 12 compared to a completion Each of the films ends with a broad open-ended question inviting rate of 25% for the whole BC population” (Math Catcher website). students to think critically about a topic related to the film. Dr. Veselin Jungic, the creator of Math Catcher and a professor in the department of mathematics at Simon Fraser University feels The First Attempt that the cultural component in the Math Catcher films makes I decided to introduce the film Small Number Counts to 100 to mathematics more relatable to Aboriginal students “because my grade 4/5 students. In this film, Small Number chases a black

Vector • fall 2019 8 cat with a white stripe into the forest and finds out too late that the cat is actually a skunk. The skunk sprays Small Number, and due to his smell, his grandmother wants to keep him out of her tipi. To occupy him and keep him outside, she instructs him to figure out which is the 100th tipi in the circle of seven tipis (if he first starts with his own). Without counting the tipis, Small Number quickly figures out that the 100th tipi is his Auntie Rena’s which is the tipi just south of his grandparents’ tipi.

To begin the lesson, I asked my students to watch for examples of First Nations’ culture while watching the film, then without further explanation, I played Small Number Counts to 100. As soon as the film ended, students slowly turned their attention to me and I immediately asked them what examples of the First Nations’ culture Image 1: This is an example of a student's solution. they saw in the film. After about ten seconds of wait time, a few She determined that Small Number’s tipi was “C” and hands went up and some students gave some examples of cultural his Auntie Rena’s (the 100th tipi) is “D”. To do this, she components: explained that she noticed that every 15th tipi lands on the subsequent tipi. 15th= C 30th= D 45th= E 60th= F 75th= G • Small Number lives in a tipi 90th= A. She explained that the 15th tipi after 90 would be • He wears clothing made from animal skin B but that would be the 105th tipi (crossed out). Since that • He bathes in the river tipi is about 100, she went back to A (the 90th tipi), counted 10 from there and landed on D. After students shared these examples with the class, I put my students into pairs and they set out to figure out how Small Number While my students actively participated in the mathematics found the 100th tipi accompanied with various extension questions problems, I walked around the class listening to their conversations (Table 1). During this mathematics activity, I did not have to ask any and did not hear any of my students mention anything about the group to stay on task or encourage students to challenge themselves First Nations’ culture or the film. With this lack of independent to find different ways of solving the problem. All of my students discussion about the Aboriginal culture, I realized that throughout were intently trying to find creative and unique ways to get to the the entire lesson, my students did not have the opportunity to 100th tipi (Image 1). make any personal connections with the content of the film. Why would they discuss a topic that has nothing to do with them? Accompanying Math Questions: Without engaging my students in the lesson or with the film (with an Aboriginal focus) I asked myself if I was indigenizing 1. There are seven tipis in a circle. If Small Number the curriculum? starts on tent A (his grandmother’s tent) and the rest of the tipis are labeled B, C, D, E, F and G which What Went Wrong? letter is the 100th tipi (his Auntie Rena’s tipi)? In my opinion, my students did not fully engage with the First 2. Small Number’s Grandmother’s tipi is letter B. What Nations’ culture in my first lesson due to my initial question— are some examples of patterns she has to ask him to “look for examples of the First Nations’ culture.” When I gave my walk in to keep him out of her tipi for the longest students this task, I was feeding into the tokenism that I was trying amount of time? to avoid, and I was setting my students up as observers instead of participants. I put too much emphasis on the content of the Table 1 film and focused on having students find information about First Nations’ culture, which did not allow them to experience the film or engage with the culture. This first attempt highlighted that to indigenize the mathematics curriculum in a beneficial way,

Vector • fall 2019 9 using the Math Catcher resource, I needed to give my students opportunities to interact and engage with the First Nations’ culture and the films in a genuine way.

Incorporating the First Peoples’ Principles of Learning To encourage students to engage with the film and its cultural content in a meaningful way, I tried to incorporate aspects of the First Peoples’ principles of learning while using the Math Catcher films in my mathematics lessons.

• Learning requires exploration of one’s identity, • Learning is embedded in memory, history, and story, Image 2: In this film, Small Number lives in a small • Learning requires time and patience, village by the water with his mother and father. It is a • Learning is holistic, reflexive, reflective, experiential, crisp autumn day and Small Number is helping his father and relational (focused on connectedness, on reciprocal to prepare the nets for tomorrow’s salmon harvest. “There relationships, and a sense of place). is a school of salmon by Straight Line Beach. We need to set our net in the morning while the tide is still high,” says These principles of learning “contribute to a more holistic Small Number’s father. and experiential experience of mathematics and benefits all learners” (First Peoples’ Principles of Learning). I feel having students experience mathematics with these Learning Requires Exploration of One’s Identity principles while using the Math Catcher films, is a more In the first few minutes of the lesson, I told students how interesting natural way to structure a lesson compared to my first lesson. and engaging the lesson was going to be for them. Thinking back on my attempt using the Math Catcher film, I now realize the first thing I did was remove my students from the activity because I Small Number and the Salmon Harvest: An account of did not make this activity about them. To connect students to the using the Math Catcher resource film, frame their thinking and guide their understanding, I have since started each Math Catcher lesson with a personal question I have incorporated significant parts of the mathematics that encouraged students to explore their identity, so they could lesson using the Math Catcher film Small Number better connect to the film. I found that starting lessons with a and the Salmon Harvest as a resource to indigenize connection question reinforced to students that their opinions and the curriculum. I feel that this lesson highlights the ideas were valued, and that their learning experience was going to benefits of using Math Catcher films as a resource and be personal. Furthermore, while watching and interacting with incorporating the First People’s principles of learning into the Match Catcher films on a personal level, many non-Aboriginal mathematics. My students were particularly interested pupils see components of the traditional Aboriginal culture in their in the plot of this film and I found the discussion at the personal identity. end of the film demonstrated students’ understanding of First Nations culture and their mathematical thinking: Account of students’ responses to the personal reasoning, making connections between ideas, strategies question: Have you ever gone fishing? and justifying their answers. Students are in a circle sitting on desks, on chairs or on the carpet learning forward and waiting for the personal question that will connect them to the film. This is the fourth Small Number film they have watched and most of them are getting to know this routine.

Vector • fall 2019 10 Learning Takes Patience and Time Me: Have you ever gone fishing? [Hands fly up and I noticed that while using the Math Catcher films as a resource, students lean forward. Some students immediately turn there was an increased amount of discussion at the beginning to the person next to them, eyes wide and hands moving, of mathematics lessons. On average, students talked about the talking about their experiences. It takes over 30 seconds film, the open-ended question at the end of the film, and the for me to get everyone settled.] Cole? mathematics activity connected to the film for about 20-30 minutes before they started solving the mathematical problem. At first, I Cole: I go fishing all the time with my dad. I caught a was worried that this amount of discussion time was taking away huge [demonstrates with his hands] salmon this summer from students engaging to the mathematics activities but, student at the lake. Islay (one of his friends in the class), and her discussion after Math Catcher films demonstrated mathematical family comes with us (to the lake) sometimes. thinking and in depth understanding of First Nations culture. Allowing students to discuss for (up to) half of the mathematics Islay: [cuts in smiling] We ate Cole’s fish! It was so good. period honored the fact that learning takes patience and time and is not all about numbers—two parts of mathematics that are not Reflection frequently valued in mathematics class. Cole was excited to be able to share his fishing story with the class. This is a big part of his life and through the Account of student to class discussion enthusiasm in which he told his story; I could tell that he feels this is a big part of his identity. Cole is not usually When asked to share some ideas with the class, most someone who readily shares ideas and participates during students were shifting in their seats, arms stretched mathematics lessons but was one of the most engaged up high and hands waving. Although given the choice students during this discussion and his participation to share their observations, connections or questions continued throughout the mathematics activity. students who participated in the class discussion were the most interested in sharing their solutions to the open- ended question: Why did Small Number think that during Learning Is embedded in Story a low tide the catch would be much smaller? While watching the Math Catcher films, I wanted to connect students to the storyline of the film and its characters, Me: Does anyone have anything to share? Andrew? demonstrating that stories can transmit knowledge. To do this, as previously stated, while students watched the films, they were Andrew: I think they would catch more salmon because expected to make observations, identify questions and make during a high tide, the current pushes the salmon into the connections. In my opinion, having students interact so deeply net and makes it harder for them to swim out. In a low with the context of a question is rare in mathematics. Students tide, the current would be going the wrong direction. I are often taught various strategies to help them efficiently solve don’t think they would catch any fish in low tide, and they “story-problems” (mathematical problems that are put into words), might lose their net. and most of these strategies include stripping away the context of the story-question to highlight the mathematical components. For Elaina: I agree with Andrew. It would also be better in a students, this reinforces the fact that mathematics is an abstract high tide because they wouldn’t have to carry their canoes subject that needs to be taken out of context to be understood. I as far to get from their tipi to the water. feel that encouraging students to value the context of the story in Math Catcher films makes the ensuing mathematics activity more Nicole: I don’t think it matters if Small Number fishes relatable to students and proves that learning can be embedded during high tide or low tide. The only thing that matters in story. is that they went in the morning because that’s when the salmon wake up and need breakfast. I learned at the Salmon Festival that they swim to the surface to find food

Vector • fall 2019 11 curriculum, that encouraged teachers to incorporate “First Peoples and that’s where Small Number’s net was, so that’s why cultures” into their literacy program, I started planning a short they caught so many fish. story unit based on this resource.

Reflections Students’ Short Stories In this part of my students’ discussion, I feel that their In this unit, students created their own Small Number stories using mathematical responses showed advanced mathematical a similar style as the stories read in class. To highlight the style of thinking: they were justifying their answers by using the Small Number stories, students were asked to create a three- prior knowledge, making connections to each other’s circle Venn diagram and compare three Small Number stories ideas and thinking of reasonable solutions to the of their choice. From this exercise, students noticed that all the questions. Furthermore, the dialogue amongst students stories had dialogue, that Small Number was the main character demonstrates a good understanding of First Nations’ and that the concluding questions can all be solved by using an culture. understanding of the environment or with mathematics. Through this exercise, students were also able to notice the variations in the texts such as who Small Number lives with, where he lives and the Learning Is Experimental time period in which the stories take place. I feel having students The open-ended questions at the end of the film encouraged analyze the stories after completing the mathematical activities my students to thinking mathematically and share their ideas highlighted many details that students had not previously without the fear of being wrong. I noticed that students who rarely expressed or noticed when they previously watched the videos. participated in mathematical discussions opened up and offered That being said, while students wrote their short stories, they ideas to their peers. The openness of the question removed the had the freedom to make their stories quite unique, due to the pressure of giving the “correct” answer because there simply was differences they had found in the Small Number stories, but at the not one correct answer. Instead, the openness gave students with same time, they were challenged to incorporating the similarities different experiences and strengths an opportunity to contribute between Small Number stories into their texts. confidently to the discussion. I found that this confidence and engagement with the context of the film continued into the subsequent math activities.

What Students Think My students enjoyed mathematics lessons using the Math Catcher resource. Whenever I mentioned Small Number, students would excitedly position themselves around the projector and start chatting excitedly. Furthermore, during discussions, students expressed how much they liked the quality of the animations and the story lines of the films.

After playing the nine Math Catcher films for my class, I asked my students to write what they thought about using this resource in mathematics. Most of the responses students wrote were about enjoying the mathematics challenges accompanying the films, the interesting plots of the films and that using this resource gave them a unique and engaging mathematical experience. One of my students even asked if the class could write their own Small In this story, Small Number, his sister, his cousin, his mother and Number stories. I was excited when I read this suggestion as I his father go fishing. They catch seven fish. How do they divide the feel like it showed that my students wanted to continue working last fish fairly? with the Math Catcher resources. After looking at the grade 4/5

Vector • fall 2019 12 After each writing period, students were given the opportunity Hodge, M. (2016). Aboriginal ways of knowing and learning, 21st to share parts of their stories. When sharing, most students century learners, and STEM success. presented their concluding questions. When other students heard their peers’ questions, they would raise their hands eagerly to give Math Catcher site. (n.d.). Retrieved March 23, 2018, from the most cleaver suggestion to the problems. https://www.sfu.ca/mathcatcher/about.html

Some examples of questions students concluded their Nolan, K., & Weston, J. H. (2015). Aboriginal perspectives and/ stories with: in mathematics: A case study of three grade 6 teachers. Retrieved December 01, 2017, from http://ineducation.ca/ineducation/ • How can Small Number figure out the depth of the article/view/195/78 lake? • What regular shapes can figure-skates create on the Archibald, T., Jungic, V. (2016). Mathematics and first nations in ice? western Canada: From cultural destruction to a re-awakening of • How many different combinations of beads can Small mathematical reflections. In B. Larvor (Ed.) Mathematical cultures: Number’s grandmother create on her bracelet with The London meetings 2012-2014, pp. 305-328. Springer. one blue, one yellow and two red beads? • How do you think Small Number’s family should Jungić, V., and MacLean, M. (2016). Small Number: Breaking the divide the last fish fairly? pattern. CMS Notes, Volume 43 No.6, 1013.

Once the stories were finished, during the following math period, students were given the opportunity to read each other’s work and try to solve each other’s problems. Their stories and answers incorporated aspects of the First Nations culture as many stories highlighted Small Number’s interconnectedness with the environment and reliance on fishing and hunting.

Conclusion In conclusion, I feel that incorporating Aboriginal perspectives in the curriculum benefits both indigenous and non-indigenous students, but I do not think that there is a simple “fix all” method to do this, or that incorporating culturally based stories is the best solution. Experimenting with how to use the Math Catcher films as a resource in my class, I feel that indigenizing the curriculum is not about teaching students facts about First Nations’ people but instead it pertains to encouraging teachers to immerse their students into Aboriginal pedagogical practices, thus, allowing students to experience First Nations’ culture.

References

Building Student Success - BC's New Curriculum. (n.d.). Retrieved December 01, 2017, from https://curriculum.gov.bc.ca/

First Peoples Principles of Learning. (n.d.). Retrieved March 23, 2018, from https://firstpeoplesprinciplesoflearning.wordpress.com/

Vector • fall 2019 13 14 Vector • Fall 2018 Rekenreks: A Math Tool that Builds Number Sense by Stephanie Turner

Bring Out the Manipulatives! would end up rushing through the concrete part when it came to Primary teachers are great at putting manipulatives into the addition and subtraction, and move quickly to the representational hands of their students. We know that our youngest learners need stage. It was always easier to just have students draw dots or tally concrete materials to help them understand the abstract concepts marks to solve the equations. I would also try number lines, as they that are the building blocks for their mathematical journey, from seemed like an easy-to-manage tool for addition and subtraction. Kindergarten to graduation. Walk into any primary classroom So why did they always lead to such confusion in my grade-one and you are bound to find tubs of pattern blocks, Unifix cubes, students? Where do I start counting? How can I keep track of when Cuisenaire rods, counters, and collections of objects gathered over to stop? Do I go backwards for subtraction? were common questions the years by the teacher. I too had all of these manipulatives in my my first graders grappled with. classroom, as I began my teaching career. I can now see that by moving too quickly to the representational and abstract stages, my students were not being given the opportunity to develop important understandings around the meaning of numbers and the relationships between numbers. These crucial number concepts can only come from the use of visual models and manipulatives specifically designed to allow students to construct this knowledge for themselves. I was learning the hard way that Manipulatives would appear when it was time to practice counting I could not simply put objects into the hands of my students and to ten with my Kindergarten students, to describe attributes of expect that they would learn what I wanted them to. The following geometric solids, or to work on patterning. Somehow, no matter quote has had a profound effect on how I view math manipulatives what grade I was teaching, when it came time to work on addition and their role in developing young students’ number sense: “… and subtraction skills with my primary students, I found myself materials cannot transmit knowledge: the learner must construct falling back to worksheets with pictures of little birds or insects the relationships,” (Blanke, 2008). for the students to compute. The concrete materials would quickly disappear because of my own frustrations around managing the Knowing is Half the Battle objects and students at the same time. When I look back and reflect on the early years of my teaching I always had good intentions of following the CRA (concrete to career, I see that I was passing on to students my lack of conceptual representational to abstract) sequence of instruction. I would understanding of mathematics. The famous quote from Maya start by bringing out tubs of counters or cubes, for students to use Angelou, “When you know better, you do better,” has become when solving their addition and subtraction problems. However, very significant to me along my journey in changing how I teach I seemed to give up after just a lesson or two, as the manipulatives mathematics. I remind myself that with minimal training in I was providing seemed to be getting in the way. Some students mathematics instruction, teachers often teach mathematics the would play with the cubes, others would not use them, and others way we were taught as students. My own mathematical journey had to make sure all their cubes were of the same colour and that as a student was an anxiety-inducing cycle of textbook exercises, they could connect! In addition, many of my students seemed to get bewildering algorithms, worksheets, and tests. I want to do better stuck at counting one, by one, by one (the direct modeling stage.) I for my students.

Vector • fall 2019 15 When I started incorporating ten-frames into my mathematics A Model and a Tool program many years ago, I felt much better about what I was doing. The Rekenrek, or arithmetic rack, was developed by Adrian Students were now working with a visual model that would help Treffers, a mathematics curriculum researcher at the Freudenthal them to understand our base-ten number system. They could fill Institute in Holland. It consists of two rows of ten beads, divided in empty ten-frames with my baskets of counters when adding into sets of five by using two colours, red and white. The beads can numbers up to 20. Students began subitizing numbers to ten, and slide across the rods, like an abacus. I was intrigued by this new visually manipulating models of numbers in their head. The ten- manipulative, and started to look into how it is used in primary frame became an important mathematical model, and a useful classrooms to assist students with their addition and subtraction mathematical tool, in my classroom. It was a good start on my facts. I quickly realized that the Rekenrek could be the answer path to effective numeracy instruction. to my management challenges when it came to using concrete manipulatives for addition and subtraction work. It is portable, contained, and easily accessible for students. I was also very excited about my discovery because the Rekenrek was being touted as a tool that would not only assist students in building their automaticity with their basic facts, but would build their number sense too! It was not just a tool to help students with computation, in fact, it was a visual mathematical model that would support young learners in the construction of crucial mathematical understandings.

Frykholm (2008) describes the Rekenrek as a powerful model that combines the various strengths inherent in the most commonly used models in the classroom: counters, number lines, and base- ten blocks. He essentially overrides the limitations of each. He goes on to provide a rationale for using Rekenreks in the primary In 2015, BC’s redesigned mathematics curriculum pushed me classroom, stating that the Rekenrek will provide students with an to continue my mathematical journey. I wanted to provide opportunity to develop their understandings around cardinality, mathematics instruction that was rooted in best practices and subitizing, decomposition, anchoring in groups of five and ten, hopefully avoided the “I hate math” scenario all too common in the concept of one/two more and one/two less, and strategies classrooms. This negative attitude towards mathematics can arise such as halving, and doubling. If the Rekenrek was going to help in part from frustration, when students lack an understanding of my students develop strategies for addition and subtraction, and the mathematics we are asking them to do. I had to deepen my own also advance their number sense, I was in! I was eager to give the understanding of how primary students develop number sense in Rekenrek a try. order to help my students think flexibly about numbers. I also needed to solve my manipulative problem when teaching students For the last three years, I have been exploring how to use the addition and subtraction. As part of my professional development Rekenrek with grade one and two students, adding to my repertoire I spent some time researching manipulatives and came across a of activities each year. There are numerous resources available to tool for computation that I had never seen or heard of in 15 years assist teachers in how to use the Rekenrek, (also known as a Math of teaching primary students: the Rekenrek. Rack), with students. I use the Rekenrek as both a visual model for instruction and as a concrete tool for students to work through addition and subtraction problems.

Vector • fall 2019 16 This is an example of an image that is flashed to students, who them forward. If I see a student that has an incorrect answer, I will are then asked “How many did you see?” and “How did you see point to the equation and their solution, and ask them to “prove it it?”. The power of the Rekenrek exists in its structure. Students to me.” Upon computing the equation again, students usually find are able to relate 14 to 15, and to 20, and are able to easily see how their mistake. At the end, we gather together and I ask students to to decompose 14 into 10 + 4, and into 5 + 5 + 4. The relationship share strategies I want to highlight to the group. between addition and subtraction can also be highlighted with the Rekenrek. How does 20 – 6 relate to 14 + 6?

Figure: A 100-bead Rekenrek for double-digit work

Growing Math Minds For me, the capacity of the Rekenrek to grow students’ number sense, alongside their strategies for addition and subtraction, is what makes it so powerful. I have been asked by other teachers if I use the Rekenrek instead of ten-frames or other mathematical models in my classroom. The thinking behind this question may Here my grade two students are practicing their basic math facts be “if it so effective, why use anything else?” I always respond with to-and-from 20. This group of learners understand the operation an emphatic “no”. In order to develop the flexible thinking we of addition and are ready to work towards automaticity of some desire in our students, we need to expose them to multiple visual basic facts. They have all written an equation on a sticky note. representations and allow them to build their understandings Students move to each other’s desks with a whiteboard, solving and over time. The Rekenrek has simply been added to my “visual recording their answers. Students who are able to use mental-math repertoire” that I can access as I curate mathematical opportunities strategies do so, and students who require a tool use the Rekenrek that will allow children to grow their number sense during our placed on each desk. This activity gets students up and moving, year together. I continue to have my students work with ten- and they are allowed to work towards developing automaticity for frames, number paths or number lines, and base-ten blocks as well, the basic facts at their own pace. This activity is differentiated on depending on the age group. many levels. There is no set amount of equations to solve as with a worksheet, students work as quickly or as slowly as they need The Rekenrek has helped me to see the importance of evaluating to, and they use strategies they understand. While students are all of the manipulatives I place in front of my students. What is working, I travel around and observe and ask questions to gain their purpose? Which manipulatives will best support students in insight into where students are, in their progression of addition and the creation of a link between the concrete objects and the abstract subtraction. I also give “nudges” to students in order to help move symbols, and also with mathematical ideas. It would be my hope

Vector • fall 2019 17 that all teachers, not just primary teachers, are asking themselves References these questions. As students are introduced to multiplication, division, fractions, and decimals, the manipulatives remain just as Blanke, B. (2008). Using the Rekenrek as a Visual Model for Strategic important and should be carefully chosen. As with everything we Reasoning in Mathematics. The Math Learning Center. Salem, OR. do in the classroom, teachers must look closely and evaluate for efficacy. Time is precious in a classroom and we must ask if the Frykhom, J. (2008). Learning to Think Mathematically with the instructional strategies and tools we are using with students are Rekenrek. The Math Learning Center. Salem, OR. yielding the results we want. The Rekenrek has proven its worth to me, and I would love to see it become more widely used by primary teachers in our province, as we all help to grow the math minds of our students.

Vector • fall 2019 18 Why History of Mathematics? by Glen Van Brummelen

Among the several new courses added to the BC high school The history of mathematics can reach these students in several curriculum, the History of Mathematics course may raise the ways: most eyebrows. Although topics in geometry and statistics are often offered in other jurisdictions, the history of mathematics Motivation: All mathematical subjects arose due to some need, seldom exists as its own course except for teacher training in some sometimes within mathematics but often from outside of it. American colleges. Even then, the goal is to provide background Students who participate in an environment where the need comes to help prospective teachers design their mathematics classes, not first will recognize that what they are doing means something, and to prepare to teach the subject on its own. Why does the subject will be inspired to pursue solutions. In such a setting, it is natural deserve sustained attention in its own course? to portray mathematics properly as inquiry rather than as edifice. Now, the demand for mathematics today is often portrayed to come Among many possible answers to this question, I find mine in the from science and technology, and certainly such motives are more phrase “humanizing mathematics”. By definition, mathematics is than appropriate for the classroom. But let’s broaden the possible an activity performed by humans, intended for a human audience. options with an example from well outside of science—local The fact that the phrase exists at all is as clear a sign as can be that practices of art. From basket weavers in Mozambique to modern many students are missing the point of mathematics. To them it musicians using 20th century mathematical objects like fractals in is a noun—a collection of abstractions—rather than a verb, a kind their compositions, the creative artistic process has often provoked of activity. Laudably, curricula are growing in the ability to pass mathematical questions. along problem solving strategies to our students. But this is not enough. As the new BC math curriculum emphasizes, we need to One such episode is records of meetings between artisans and raise with our students the bigger questions. Why is mathematics geometers in late 10th-century Iraq, some written by the great so successful in understanding the physical world? Why did mathematician Abū’l-Wafā’. Decorations of walls in palaces were we choose to represent unknown quantities with letters? What not to include images of people or animals, for religious reasons. is different about how we use mathematics to affect our world, Instead, the local artisans designed elaborate geometric patterns, compared to those who went before? many of which still grace the walls of historic Muslim buildings. The most famous example is the spectacular Alhambra, a palace After four years of intense study of undergraduate mathematics, in Granada, Spain (see Figure 1). These patterns are born from I had no idea why the questions my professors were answering in simple geometric constructions that are repeated to tile the entire their lectures were important. In a sense, then, I chose my career in plane. Then, each line segment or arc in the diagram is elaborated the history of mathematics as an escape from irrelevance. Many of in some way. The results can be stunning. For instance, Abū’l- our students, especially those not immediately attracted to STEM, Wafā’ describes how to embed an equilateral triangle in a square, need a big picture to feel part of the conversation. The questions as follows (Figure 2): extend the base GD by an equal distance to above were not written by me. They are taken from a list of “why” E. Draw a quarter circle with centre G and radius GB; draw a half questions I solicited from my humanities-oriented history of circle with centre D and radius DE. The two arcs cross at Z. Then mathematics students at the beginning of my course. They shied draw an arc with centre E and radius EZ downward, to H. If you away from math not just because they don’t feel competent, but draw AT = GH and connect B, H, and T, you will have formed the also because they have not heard answers that are meaningful to equilateral triangle. (If you want to prove it, connect GZ and ZD. the way they think. Consider first the angles in triangle GZD; next, consider the angles in triangle BGZ. After that, you’re on your own!)

Vector • fall 2019 19 and x. Combine them (Figure 3a), and the resulting rectangle has 7 area 12 . Cut off half the original rectangle and move it below the square. The result is a shape that is almost a square, but with a small 1 # 1 1 shaded square left out. Its area is 3 3 = 9 . If we add the shaded square, we know that the larger square of Figure 3b has area 7 1 25 5 1 12 +=9 36 , so its side length is 6 . But the side length is also x + 3 ,

5 1 1 so x =-= 6 3 2 .

Figure 1: A wall at the Alhambra (Granada, Spain)

B A

Z

T

Figure 3a: A Babylonian

G H D E Figure 2: Abū’l-Wafā’’s construction of an equilateral triangle in a square

Research: Having posed bigger problems, we are responsible to provide resources for tackling them. History is an obvious tool here as well; after all, , Newton, Euler and others became who they are because of their ability to solve the big problems. We learn how to approach problems from those who went before us. Often their major insights have involved some sort of leap across the gap between the divide in mathematics between arithmetic and geometry. For instance, the recognition that the function fx()= ex is its own derivative allowed Euler and his successors to solve many differential equations — and, with the introduction of i =-1 , he Figure 3b was able to combine the inherently geometric discipline of trigonometry with the algebraic analysis of exponential growth This process does a lot more than solve a quadratic equation; it into an amazing unity. proves the quadratic formula. Besides gaining the quadratic formula as a tool for future use, students learn several significant lessons Closer to the 11th grade classroom, we turn to ancient Babylonian about problem solving. Observing the successes of our predecessors schoolchildren’s solutions to the quadratic equation, which were can lead to ideas for our own successes as well. Changing the way a based on “cut-and-paste” geometry. Consider one tablet where a problem is represented (in this case, from to geometry) can 2 2 7 child solves xx+=3 12 . Although his geometry operates below the lead to surprising insights and novel solutions. Most importantly, surface of the text, it is clear that the reasoning is as follows: x2 is they see that mathematics moves from the “mess” of the struggle 2 a square whose side is the unknown x; 3 x is a rectangle with sides to the discovery of paths to resolution.

Vector • fall 2019 20 Critical Thinking: Every mathematical community makes shared students to a deeper appreciation of mathematics as much more decisions about the validity and power of various competing than a calculating machine, but rather a living discipline and driver approaches. For instance, medieval Indian mathematical of social change. astronomers were comfortable using iterative solutions to equations—a series of guesses or approximations, each one The biggest driver of social change these days in fact a calculating improved over the previous one using some procedure. However, machine, the computer. In both the 19th and 20th centuries, ancient Greek and medieval Islamic astronomers preferred direct mathematicians studied algorithms themselves, recognizing both arguments and calculations. This may be due to a commitment to their power and their limitations. The person usually considered to pure mathematics as the only path to true knowledge, propounded be the first to compose an algorithm implementable on a computer by the great astronomer Claudius Ptolemy. It is difficult for modern was Ada Lovelace (1815-1852), daughter of poet Lord Byron. She students to understand that such commitments are also present maintained a working relationship with Charles Babbage, whose today. For instance, we represent geometric magnitudes like Analytical Engine would have been easily the first real computer, if lengths and areas using numbers—a choice that practitioners of it had ever been built. Lovelace gave a prescription for computing the Euclidean tradition rejected. As a result, our mathematics leans Bernoulli numbers (fundamental in ), and was one heavily toward arithmetic and algebra, exploiting the power that of the few at this time to consider the possibility that the Analytical these algorithmic methods provide. In order to think creatively, one Engine or a machine like it could do much more than just needs to make informed judgments about such alternate avenues calculation—although she rejected the possibility of true artificial of attack; one must know what the community’s rules are before intelligence. This remarkably early analysis of algorithms reached one decides to bend or break them. an extraordinary conclusion with the work of Julia Robinson (1919-1985), an important figure in the resolution of Hilbert’s Consider, for example, the curious case of 12th-century Iranian tenth problem. This problem asked whether a computer algorithm scholar Ibn Yahyā al-Samaw’al al-Maghribī, who late in his life can be written to determine whether a Diophantine equation has composed Exposure of the Errors of the Astronomers in which he any integer solutions. The answer, “no”, implies that as powerful pointed out dozens of what he perceived to be mistakes in the as computers are, there are some things—meaningful things—that works of his colleagues and ancestors. One of these episodes they simply cannot do. involved finding a value for sin1o from a given value of sin3o. To find a precise solution turns out to require solving a cubic equation, Many of the changes brought about by mathematics were not in and this cannot be done with geometric methods—it is equivalent the realm of science and technology. Consider the birth of non- to the problem of trisecting the angle with ruler and compass, now Euclidean geometry. In the early 19th century, after dozens of failed known to be impossible. Violating his own commitment to the attempts to prove statements equivalent to the assertion that the perfection of mathematics, Claudius Ptolemy had been forced into approximating (an equivalent of) sin1o. Al-Samaw’al’s creatively angles of a triangle sum to 180°, three separate mathematicians restored the role of pure mathematics in philosophy by redefining considered the unthinkable: what if it’s false? Carl Friedrich the number of degrees in a circle from 360 to 480. The sine of 3° Gauss, Nicolai Lobachevsky, and János Bolyai discovered that becomes the sine of four units, and he can apply the sine half-angle a new and consistent geometry may be born from denying this formula twice to find the sine of one unit. Sometimes, changing assumption. Through this bold step, they created “worlds out of the rules of one game allows one to conform better to the rules of nothing”: elliptical and hyperbolic geometry. These non-Euclidean another. geometries remained creations of the mind for decades, but starting in the early 20th century they have become candidate models for Implications: Students motivated by humanistic questions admit the geometry of the universe in which we live. This episode reveals readily that mathematics has had a major effect on our world, that seemingly obvious assumptions that have stood for millennia but usually they can speak only vaguely about its uses in modern are not necessarily on solid ground. science and technology. This lack of clarity contributes significantly to the widespread misperception that mathematics is only a toolbox Students aware of cosmic shifts like these participate in a rich of algorithms allowing manufacturers and tech firms to build new educational experience. They are able to orient themselves within devices. Witnessing the various breakthroughs that have changed the intellectual landscape, and are can act in their profession with how people live their lives and perceive their world can bring these reflectiveness and purpose.

Vector • fall 2019 21 Communication: The rich settings exemplified here provide ample opportunity for our students to expand their options for writing and otherwise presenting ideas in a mathematics class. History encompasses entire narratives from initial conception to final product and societal impact, providing rich opportunities for interpretive short answers and essays that can go far beyond learning how to write up an effective solution.

Diversity: Attempts to improve intercultural understanding, including with indigenous societies, have been spreading throughout the school curriculum. Mathematics poses a unique quandary for these efforts, since most popular accounts send Figure 4b: The in-out complementarity principle the message that mathematics is separate from culture. “Isn’t for everyone?” Curricular projects have been helping to bring to the Of course, who has the opportunity to do the thinking is just as classroom certain examples of indigenous mathematics realized in important as how the thinking is done. In this respect, the human daily life, extending the project of ethnomathematics that has been race does not have a very good track record in allowing equal working on such matters for a few decades. However, history also participation for everyone in all societal endeavours, including can help to deepen considerably our questioning of how culture mathematics. It is an uncomfortable but unavoidable fact, for affects mathematics. Why are western cultures the only ones to use instance, that that very few women have been in a position axiomatic-deductive systems to verify mathematical knowledge? to be able to contribute to pushing forward the boundaries of Are other ways of knowing mathematics possible or desirable? Can mathematical knowledge until quite recently (and even now, one think differently than we do now? equality has not yet close). A history of mathematics course can engage in these issues readily. One approach that historians are A simple example of a positive answer to this question comes from starting to take to help bring female stories to the forefront is pre-modern China. Figure 4a is a standard diagram representing to illuminate their experiences with mathematics in other ways, a c two similar triangles; we all recognize that b = d . Chinese especially in their education. This broadens what we mean by “the geometers and surveyors rarely used this ubiquitous tool. Rather, history of mathematics”, making it more inclusive of all aspects of they relied on the in-out complementarity principle. In Figure 4b, human experience. This approach is in its early stages, but see pp. the large rectangle is cut by a diagonal line. At any point on the 59-68 of Jacqueline Stedall’s The History of Mathematics: A Very diagonal, draw vertical and horizontal lines. The reader might Short Introduction for a good start. pause here to consider why the two shaded rectangles have the same area … Now that you have paused, we see that the two My colleague Clemency Montelle (University of Canterbury, NZ) a c rectangles’ areas are ad= bc , which of course is equivalent to b = d . and I are working on a book to bring episodes like these to the The two methods are able to accomplish the same geometric goals, classroom and to the general public. In the meantime, we conclude but using the in-out complementarity principle can lead to with a list of resources for teachers to help design interesting and diagrams and thought processes that look very different from those effective lesson plans. produced by the use of similar triangles. References

Parts of this article were modified from “Why use history in a mathematics classroom?”, Canadian Mathematical Society Notes 47 (1) (2015), 16-17, and a companion article in Models and Optimisation and Mathematics Journal 3 (1) (2015), 36-41.

Figure 4a: Similar triangles

Vector • fall 2019 22 Resources: Joseph, George Gheverghese. The Crest of the Peacock: Non- European Roots of Mathematics, 3rd edition, Princeton, NJ: This list is not intended to be comprehensive. There are a Princeton University Press, 2011. number of general textbooks in the history of mathematics; some are more appropriate than others for a high school course. »» The most comprehensive general source for the mathematics of non-western civilizations. Berlinghoff, William; and Gouvea, Fernando. Math Through the Ages: A Gentle History for Teachers and Others, 2nd edition, Shell-Gellasch, Amy; and Jardine, Dick, eds. The Courses of History: Farmington, ME: Oxton House, 2014. Ideas for Developing a History of Mathematics Course, Washington, DC: MAA Press, 2017. »» A compact and accessible, yet surprisingly deep text. Contains a nutshell history of the subject and 30 short episodes appropriate »» A varied collection of approaches to designing a history of for the high school classroom. An expanded edition published mathematics course. by the Mathematical Association of America contains 60 extra pages of questions and projects. MAA Convergence, https://www.maa.org/press/periodicals/ convergence Katz, Victor. A History of Mathematics, 3rd edition, Boston: Addison Wesley, 2009. »» A high quality free online journal designed “to help you teach mathematics using its history”. Geared to 8th grade through »» At a higher level, but as authoritative as history of mathematics undergraduate. textbooks get. The Story of Maths, BBC/, 2008. Barrow-Green, June; Gray, Jeremy; and Wilson, Robin. The History of Mathematics: A Source-Based Approach, 2 vols., Washington, »» This comprehensive four-part documentary series takes a DC: MAA Press, 2019 (due to appear in May). multi-cultural approach. Its last episode, uniquely, takes on the implications of mathematical developments from the 20th »» A comprehensive text that emphasizes working with original century to almost the present day. historical materials. Transformational Instruction in Undergraduate Mathematics via Shell-Gellasch, Amy; and Thoo, John B. Algebra in Context: Primary Historical Sources (TRIUMPHS), https://blogs.ursinus. Introductory Algebra from Origins to Applications. Baltimore: Johns edu/triumphs/ Hopkins University Press, 2015. »» A five-year National Science Foundation-funded project to »» Contains a wealth of historical episodes from number systems develop teaching modules for the undergraduate mathematics to number theory. curriculum based on original sources. Some of the units deal with high school mathematics as well. They are looking for site Stedall, Jacqueline. The History of Mathematics: A Very Short testers at the moment. Introduction. Oxford, UK: Oxford University Press, 2012.

»» Not a survey of historical developments in mathematics, but rather a short statement of recent approaches to the subject that are opening up the history of mathematics to new ways of thinking.

Vector • fall 2019 23 A Public Math Gathering in Chicago by Janice Novakowski

Having been an enthusiast of the work of the Public Math group for families to play and problem solve together in joyful ways. On (public-math.org) over the years, I was honoured to receive an my way to Chicago, I was curious as to how I would be inspired invitation to a Public Math Gathering in Chicago this past August. and learn from the experiences and those participating with me. Public Math is a collective of mathematics educators, researchers, designers, artists, parents and citizens. Their mission states: Public The group met together for the first time on the Friday evening at Math creates mathematical opportunities in the spaces that diverse a neighbourhood park in Chicago. We were part of a community children and families inhabit and interact with in their daily lives. I initiative that brings the arts into the community. Families were would be joining the founding group of the collective along with able to play and create together making Frankentoys (think Toy educators, museum folk and artists from Minnesota, Wisconsin Story), design their own buttons, make music and engage in and the Chicago area. mathematical play. Christopher Danielson brought a suitcase full of tiling turtles (Image 1) and pattern blocks as well as a new math I appreciated the diversity of perspectives brought together for this vending machine full of mini tiling turtles and triangle shape gathering. As a mathematics educator, I have my own beliefs about puzzles. One of the things that struck me was how all the signage why community engagement and public mathematics projects are was accessible to the community, in both English and Spanish. I important, but it is important to hear from a range of voices. Why wondered how that intentional move welcomes families into the public mathematics? For me, I have encountered some limited space and what else we can be doing in our own communities views of what mathematics is and can be from both children and to make mathematics and education in general more welcoming their parents, and I want to engage families in my community to families. Although both children and adults were drawn to to think about what mathematics is and how it can be a part of the tiling turtles, we all noticed the time spent over at our “math family life. I also want to nudge thinking about what counts as table” was not as sustained as with other areas. Was this because mathematics. Many elementary children and their families think the children (and some adults) were creating something they could of mathematics as arithmetic and calculation (and those are take home with them in the other areas? Was it because once the foundational important aspects of mathematics) but mathematics novelty of the mathematics materials wore off, children were not is so much broader, connected and playful. I think we need to make sure how to sustain their play and inquiry with the materials? This mathematics visible in our communities and provide opportunities was something we were left to consider.

Image 1: Friday night in the park: Tiling turtles is always a compelling material for visitors to public math events.

Vector • fall 2019 24 Saturday morning brought us to the “Twitter-famous” Mr. Bubble laundromat (Image 2) which has become known in the mathematics education world because of public math projects that have happened on-site. In small groups, we walked around the laundromat, noticing mathematical opportunities as well as becoming familiar with the two projects already in place: The washers and dryers were numbered using stickers based on Dan Finkel’s Prime Climb game (Image 3), and the Chelsea Clinton Foundation had materials, children’s furniture and multilingual posters in the laundromat supporting literacy and numeracy (Image 4). We began to consider possibilities for mathematical play, tasks and conversations that could happen in this space.

Image 4: Signs in Spanish were posted in the laundromat asking “What do you notice?” and “What do you wonder?”

We spent Saturday afternoon at a local elementary school engaging in the design process in teams of three or four. Each team chose a space in the laundromat for which to create mathematical experiences, and then we created prototypes (Image 5), got feedback from each other and worked on our final designs before walking back to the laundromat to install our creations.

Image 2: Mr Bubble has gained notoriety on social media in the mathematics world because the owner has been open to creating opportunities for the space to be used to encourage public math.

Image 3: Prime climb stickers–circles that are colour coded with Image 5: The design of a prototype for the children’s area in Mr prime factors. The stickers matched the numbers that were already Bubble. One group’s big idea was to encourage children to explore in place on each washing machine. shapes that could be found in the laundromat.

Vector • fall 2019 25

Image 6: Visitors to the laundromat were encouraged to pick up a shape finder and look for the shapes in the environment, both inside the laundromat and through the windows.

Images 8 and 9: Putting the plans into place: Using the existing furniture in the space, the working group considered ways to enhance the mathematics play opportunities by creating outlines of shapes.

Working with museum educators and artists and those closely connected to the Chicago community brought new insights into the importance of public math projects. Projects need to be Image 7: A second group created cards that people could use to move accessible to all. How do I think we helped this to happen? I think around and create different designs and patterns. a colourful, playful appearance of the visuals and materials was helpful as was our intention to make signage language accessible to On Sunday morning we made our way back to Mr. Bubble to see the local community. I also think that many of our installations did how our materials had been used by visitors to the laundromat not read purely as “math” as they were often more interdisciplinary overnight and to continue to tweak things and observe interactions or puzzle or game-like which I think nudges the conversation about (Images 8 and 9). what mathematics is. The laundromat and that context was also a good reminder of the importance of creating awareness that

Vector • fall 2019 26 mathematics is all around us (not just in the classroom) either in mathematics materials to play with and tasks think about. Being our environment or in our daily or routine practices like doing involved in this Public Math Gathering has nudged my thinking the laundry. about ways to make mathematics visible and engaging in public spaces across our community. Follow the hashtag #publicmath on I am involved in a community engagement project around Twitter to see what educators across North America are doing! mathematics in my school district called Math Play Space and we “pop-up” at community, district and school events with

Vector • fall 2019 27 NCTM Annual Meeting and Exposition, San Diego 2019 by Alex Sabell

As a result of being the 2018 recipient of the BCAMT Ivan L. thought long and hard about the areas in my practice that I felt Johnson Memorial Award, I received the experience of a life time: could be improved, and then used the Focus Strands to help narrow a trip to San Diego to attend the 2019 NCTM Annual Meeting my search. As well, I considered recommendations and advice from and Exposition. With over 600 sessions, this event is one of the other conference attendees around sessions and presenters who I largest annual meetings for mathematics education in the world. might enjoy. In the end, I was happy with my choices and left with It is unique in that it brings together teachers, administrators, a list of presenters I wanted to see or read about in the future. researchers and educational leaders from around the globe, creating an energetic community of like-minded people who are The conference provided me with specific ideas to incorporate passionate about mathematics and learning. My attempt in this in my mathematics program and the inspiration to grow, but I brief meetup will be to convey some of my takeaways and advice knew that there would still be the challenge of applying what I had around making the most of the conference. learned to my teaching. It is normal to attend conferences with every intention of putting into practice what we have learned, but There was something really special about the “vibe” at the time passes and our teaching days get filled with other things. Or, conference from the moment of arrival. Despite not knowing we get overly enthusiastic and try to change too many things at anyone, it felt as if we were all connected through feelings of once, become overextended and end up reverting to our old ways excited anticipation and an eagerness to absorb everything we of doing things. To counteract some of these issues I devised a plan could from the experience. The positivity was infectious and casual which included documenting my experience, reviewing session conversations with other attendees often turned into sharing and slides after sessions, synthesizing information to create a resource, learning opportunities. There were many occasions to chat and and making a plan of implementation. After attending 14 sessions make connections with presenters and educational leaders. This in three days, I could scarcely recall all of what I had seen and heard was made possible through conference structures such as organized without my notes. Having a written record was helpful in reliving socials, access to a networking lounge, and a service called Bar where you could sign up for meetings with various conference speakers. This feeling of connectedness and positivity enhanced the whole experience.

Another significant attribute of the conference was the selection of sessions offered. The sessions were organized into “Grade Bands” and “Focus Strands.” Focus Strands included such topics as: assessment; building on students’ strengths; professional advocacy; beyond the classroom walls: empowerment, access and equity; creating inclusive classrooms; building mathematical knowledge; Enhancing Mathematical Thinking Through Reading, Writing, Speaking and Listening; For the Joy of Mathematics; New Teachers. There was an online program and app that allowed you to build a schedule. Sessions were first-come-first-served. The most challenging part was deciding what to attend. To help choose, I

Vector • fall 2019 28 the experience and remembering the impact that each session had The Bingman/Licari session really resonated with me because on me, and the session slides were a great way of reviewing the it tackled some of my own inner conflicts around summative content. That said, if I found that a presentation was not beneficial assessment. As educators, we work hard to foster ideas around to where I was in my journey, I made a note of that and didn’t having a growth mindset, but often our practices contradict this, take as much time writing things down. When I returned home I especially when it comes to summative assessment. The presenters went through my notes and the slides, chose the presentations that offered a structure wherein students were thoughtfully assessed offered me something relevant and valuable, created new notes on the same task multiple times. More specifically, students were and came up with a plan to incorporate what I had learned. This given the time, space and feedback in-order to improve and process really helped me get the most out of the conference and make corrections, so that when they undertook the summative allowed for positive change to happen in my classroom. task it truly reflected their understanding and ability, as well as their progression of growth. This session sent a powerful message While every session offered something of value, two sessions around the importance of assessing what we value and provided that have impacted my classroom practice more than others were me with a way to assess that fits with my beliefs. John SanGiovanni’s “Number Sense and Reasoning Routines to Jumpstart Math Class” and “Destroying the Hypocrisy of The inherent problem with writing an article about one’s personal Summative Assessment and Growth Mindset” presented by experience during an event like the NCTM Annual Meeting and Kathryn Bingman and Byron Licari. Exposition is the challenge of fully conveying the atmosphere and the feelings associated with the learning, interaction and John SanGiovanni was a presenter strongly recommended to subsequent inspiration that takes place. Hopefully I have been me; however, his session conflicted with another that I wanted able to relate some sense of that here, but my ultimate intention is to attend, so I waited to see him present at the Ignite session (an to motivate you to attend the conference yourself and experience intense series of 5-minute presentations where each presenter firsthand what I feel I can only inadequately describe. conveys their passion for a topic). I was so inspired that I went home that night, watched a 50-minute video of him presenting a session on routines to jump start mathematics class, reviewed his slides, and then went out to buy two of his books. What I took away from these resources were a number of new routines designed to engage, build confidence, develop students’ reasoning, critical thinking skills and sense making.

Vector • fall 2019 29 A Culturally-Relevant Financial Literacy Curriculum by Alisha Parashar-Choukulkar

This article examines the role that cultural background plays for I developed looks at how financial behaviour and attitudes such as youth in the acquisition of financial knowledge, behaviour and setting goals and planning can impact the future. My course looks attitudes. The findings suggest that an effective culturally-relevant at how our personal experiences, social media, popular culture financial literacy program would include experiential knowledge values and families can shape how we earn and spend money. and use finances to not only educate but also to empower youth I acknowledge the role that families play in the acquisition of and families. financial knowledge, behaviour and eventually the development of financial attitudes. Introduction/Background Turner Secondary (a pseudonym) is a school located in Southeast Each year, my class is comprised of students with wide-ranging Vancouver. The majority of its students are of Filipino and South backgrounds: some are first generation immigrants; a few are Asian descent. Enrolment continues to grow, as many newly international students; others are born and raised in Canada. arrived Canadians are taking advantage of the many settlement Although financial literacy is a complicated topic, the students services Turner has to offer. As a business education teacher, enjoy the lessons because it is equipping them with knowledge I introduce students to the fundamentals of marketing and they will need to be successful in the community. Many youth, entrepreneurship and educate them on how business and financial by owning debit cards, assuming responsibility for a phone bill, skills can help them navigate successfully in the community. or taking part-time jobs, are already engaging with finances. However, in accordance with our school’s mission, I also feel a Some of my students report dealing with money directly, through responsibility to develop my students’ financial literacy. allowances, bill payments, employment, or corresponding with family and friends. Some students, through avenues such as Last year, I created a pilot financial literacy course for grade 8 witnessing conversations at home, have had indirect experience students. This is a preview course offered only during the summer. with finances. The definition of financial literacy I use in the course is as follows: the knowledge and understanding of financial concepts, and the Background skills, motivation, and confidence to apply this knowledge and When I started developing this course I was worried given my understanding in making effective and accurate decisions across a limited training in financial education, that I would not be able range of financial contexts (PISA 2012 Framework). The aim of to successfully teach financial literacy. I looked for resources the course is to equip students with the essential skills they need online and guest speakers to come in and share their knowledge to make sound financial decisions in the community. The course regarding finances. Junior Achievement and Van City were two of covers a variety of topics, including budgeting, wages, banking, the organizations I invited into my classroom to teach my students and real estate. Although financial literacy is not currently a stand- about finances. Although the sessions with these organizations alone subject during the school year, financial literacy is included were informative and useful, I often wondered what my role was in the Kindergarten–Grade 11 new math curriculum. Students in the classroom and how I could make the content more relatable are required to learn how to count money, calculate percentages and engaging for each student in my class. Some of the students and budgeting as part of the new mathematics curriculum in were being exposed to concepts such as banking for the first time, British Columbia. While the mathematics curriculum teaches and I wanted to make sure the knowledge they were receiving was students how to do certain financial applications, the pilot course adequately preparing them to become financially capable citizen.

Vector • fall 2019 30 My goal was to create a culturally-relevant financial literacy According to the results, the European respondents seemed to have curriculum catered to Turner Secondary students preparing for more experiential knowledge compared to the Filipino respondents, their futures while acknowledging their diverse backgrounds. who displayed more skill-based and declarative knowledge. For I investigated what students already knew about financial example, more European respondents reported having bank literacy. I then examined the extent to which their cultural accounts and paying jobs compared to Filipino respondents. background informed their knowledge, behaviour and attitudes. I Although more European respondents reported engaging in ultimately sought to understand how some students were already financial activities compared to the Filipino respondents, the latter knowledgeable about the financial landscape of Vancouver and group performed better than the average respondent for all three what bearing their family’s culture has on this knowledge. By family financial knowledge questions in the financial knowledge section. culture, I refer to the beliefs, attitudes and practices families have The Filipino respondents had basic financial skills and generally regarding finances. performed above average but either did not engage in or were unsure if they engaged in certain financial practices. Financial knowledge and skills are taught at school as well as at home. Financial literacy education, therefore, can be more Further research could examine why European respondents comprehensive and effective if it considers the knowledge, skills had more experiential knowledge than the Filipino group. I and attitudes students inherit from their families. Current financial focused only on how the cultural background of the respondents literacy education programs tend to assume that all individuals impacted their financial knowledge; however, it is likely that the have an equal understanding of finances, ignoring the diverse socio-economic backgrounds of the family impacted the financial circumstances which cause people to interact with money in knowledge, behaviour and attitudes of the respondents. very different ways. Financial education programs that do not acknowledge the diverse backgrounds of participants run the Theme 2: Role of the Family in Financial Literacy risk of replicating current financial inequities in the community. A culturally relevant financial literacy curriculum has two Families play a critical role in shaping the financial attitudes and components: (a) it is based on the knowledge and experience of its behaviours of the respondents. By having direct and indirect participants; and (b) it reflects financial trends in the community. discussions, giving advice, modelling behaviour and giving In order to enhance my curriculum so that it was culturally money, family, especially parents, influence the spending and relevant, there were three questions I wanted to answer: saving habits of children and how they view their financial futures. When asked who they talked to about finances, a majority of What do grade 8 students at Turner Secondary understand about respondents (95.9%) reported parents. When creating a financial financial matters? literacy curriculum, the impact of family needs to be addressed. How do Grade 8 students engage with, and acquire knowledge, Respondents were asked about the types of financial conversations about finances? they hear at home. It is possible that cultural considerations inform What role do culture and family play when students are learning parents’ decisions as to which topics are worth discussing with their about finances? children and which are less important for their child’s future or even inappropriate to discuss. Retirement and pensions are sophisticated Findings topics, so I am not surprised that less than one-quarter of the total Theme 1: Cultural Differences in Declarative Knowledge vs number of respondents (24%) reported hearing about these topics Experiential Knowledge at home. On the other hand, a major cultural outlier for retirement and pension was the European respondents (56%). Retirement and Overall, the respondents performed well in the declarative pension are often associated with economic stability and a safer knowledge portion of the questionnaire and were able to do future. It is possible the families of the European respondents in financial calculations and understand concepts. As illustrated by the study have the means and resources to plan for retirement and the data (see table #1), there were no serious obstacles to calculating that is why this subject comes up more in conversations among gross pay or understanding financial concepts such as budgets and this cultural group than others. bankruptcy. There were also no significant differences across the cultural groups in understanding and calculating gross pay as well A majority of the respondents reported speaking to their parents as familiarity with financial concepts. about finances as well as receiving advice from parents regarding

Vector • fall 2019 31 financial prudency. Despite receiving financial advice from and Curriculum. I have started to include financial literacy in my ELL having conversations with parents, siblings, aunts and uncles Social Studies curriculum so that students who are new to Canada and grandparents, Filipino, South Asian, and Southeast Asian have opportunities to apply financial skills in an environment respondents reported having negative and or ambivalent feelings which acknowledges the diverse backgrounds of the students so towards their financial futures. More than half of the respondents that they are empowered. My culturally-relevant financial literacy (72%) felt positive about their financial futures and the response program addresses these two concerns by educating youth and frequency jumps to 100% for European and Chinese respondents. their families on basic financial skills and using the skills and While only 6% of all respondents feel negatively about the future, knowledge to empower youth who have negative or ambivalent 16% of Filipino students have negative feelings regarding their feelings towards their financial futures. I use financial literacy as a futures. Some of the fears are due to expenses they will incur as tool to motivate youth, help them access financial resources and to adults and self-doubt. The respondents especially doubted their become confident financial citizens. I try to work with families in ability to find a job. Some examples from Filipino respondents the community to close the gaps of financial practices and financial exhibiting a negative feeling is as follows: “I feel like in the future efficacy between cultural groups identified in this study. having a job gives me more difficult and give a more stress, because of some finances are getting higher every year.” This respondent Practical Financial Education is talking about the high cost of living. Regardless of whether the The first part of the course acknowledges what students know respondent finds work, it is difficult to save for the future when the about finances and where this knowledge comes from. Students cost of living is so high. Even though most respondents had a good discuss what they know about finances and interview their family understanding of financial knowledge, talked to family members members and friends regarding their experiences with finances. By about finances either sometimes or every day and displayed starting with real experiences, I find students are more engaged in financially prudent behaviour, some respondents from the Filipino, the class because they are able to make connections and understand South Asian and Southeast Asian cultural groups felt negative or the value behind each of the lessons. I then take what we learn uncertain about their financial futures. in the classroom and give students the opportunity to apply it in the community. An example of how I teach practical financial Before undergoing this study, I thought that a culturally-relevant knowledge which I recently incorporated into my class involves a curriculum would be most effective as a stand-alone subject in partnership I created with a local credit union to help my students the BC Curriculum. The data shows that financial literacy is not open up a junior chequing and savings account. The students lacking in the curriculum as respondents possess declarative independently opened and monitored the account while learning knowledge and financial skills are part of the new curriculum. about important financial concepts and skills which will help them Although offering a culturally-relevant financial literacy course reach the financial goals in the future. Opening the account has would continue to benefit students, making this a community- helped my students in developing practical financial skills and has based program which can be accessed by families would make given them an opportunity to further their knowledge in basic it even more effective. A culturally-relevant financial literacy banking. In order to be accessible to families, I invite parents and program should reflect the learning going on inside the classroom guardian to join us on this trip to help families engage with their and the home and attempt to close the cultural gaps in financial children in topics that may not be discussed at home. practices and financial efficacy previously identified. The culturally relevant financial literacy course is new and there Conclusions: Developing My Curriculum have been many advances since its inception. The students enjoy Looking at the data, what the BC curriculum is currently lacking is being given an opportunity to discuss what they already know practical financial education for those students who are engaging about finances and comparing Canadian currency to currency in or who are getting ready to engage in financial transactions. from their home countries. Students are engaged because they get The curriculum is also not developing the financial efficacy and to reflect on financial practices they see at home and understand confidence of students to engage in financial transactions, as how that impacts their decisions. Students also enjoy the practical students are doubting their financial futures. In addition to teaching aspect of the course being able to open up a savings and chequing financial literacy in the summers as a stand-alone subject, I also account. Given this is a new program, there have been some thought it was important to incorporate financial literacy in the setbacks, making it difficult to implement my vision. Some of the English Language Learning (ELL) and Grade 8 Business Education assignments require students to interview family members about

Vector • fall 2019 32 financial practices. Family members are busy and although they conversations at home while others reported feeling negative or appreciate the curriculum their children are learning, they do not ambivalent. Further research could look into why the respondents have time to participate in the projects or help their children at felt calm and more responsible after listening to these conversations home. Another challenge is the role of financial institutions and and how financial literacy can be used to empower everyone, their potential influence on the students. Some staff members have regardless of their age, socio-economic status and cultural voiced concerns about my partnerships with credit unions and do background. The youth who reported feeling empowered after not think it is in the best interest of the students to expose them listening to financial conversations discussed reaching vocational to institutions at such a young age. Balancing practical education and financial goals through planning, saving and delaying along with consumer education is something I will continue gratification. Understanding the nature of the conversations working on so that students are making real life connections to that respondents had at home could help in developing financial the content while being able to assess institutions critically. confidence among all youth. These skills and concepts would be integral in a culturally-relevant financial literacy program. Empowering Youth The results of my study show that learning financial skills at By understanding their experiences with finances and giving them school and adopting financially prudent behaviour by listening to opportunities to practice financial skills, I hope to empower my conversations at home do not necessarily result in youth feeling students through this curriculum and give them the confidence positive about their financial futures. Some respondents reported they need to be financially capable citizens. feeling empowered, calm and responsible after listening to financial

Vector • fall 2019 33 Appendix:

Financial Literacy Curriculum Part 1: Part 2: Part 3: Our environment (peers, school, family, Money is a medium of exchange which There are different ways to accumulate, social media,) impacts our values, which helps us achieve our financial goals and save and grow wealth depends on their impact our financial goals, behaviour and make the correct decisions which align with values, goals and risk tolerance. eventually the financial decisions we make. our values.

Activities Activities Activities • Interview a family member about their • Cashless society: Trade goods in the • Students use what they learned about personal views on finances classroom and compare bartering their values and personal choices in • Reflect on interaction with friends to present day shopping and the Part 1 to assess their risk tolerance regarding finances importance of currency in a society • Learn about investment products • Review popular culture and its • Canadian Currency: Examine through front loading: RRSPs, GICs, potential impact on youth’s financial Canadian currency and how it TFSAs, mutual funds attitudes and behaviour. compares to currency around the • Stock Market: students learn how to • Define personal values world in terms of exchange rates invest in and monitor the stock market • Create Short, medium long-term • How to find part time work: Search through online games financial goals based on values online for work, create a resume and • Home Ownership/Renting: Students • Decision making model: Make prepare for a job interview compare the cost of owning a home in decisions according to our values • Wages: Learn about gross and net pay, Vancouver to renting as well as how federal and provincial taxes, Employment Insurance, and CPP are deducted from earnings • Banking (a local credit union gives $5.00 to students to open up a junior chequing/savings account): Research online banking and compare credit unions and banks as well as chequing and savings accounts

Application Application Application Students create a “Decision Making Students look for a part time job, create Students are given a certain amount of Model” to make financial decisions based a paystub and start to put their money in wealth to invest in different portfolios on their individual values and goals. a bank account. Students practice how in accordance to their personal risk to deposit, withdraw and monitor their tolerance and financial goals. finances in person, online.

Vector • fall 2019 34 Table 1: Understanding how to compute financial equations and concepts

South east Topic All Responses Filipino South Asian European Chinese Asian

Calculating gross pay 91.9% 93% 80.3% 96.8% 93.3% 100%

Understanding the concept 88% 87.5% 79% 100% 90% 91.5% of budget and debt

Computing interest 71% 80% 63% 63% 60% 33%

Table 1 identifies the financial knowledge of the respondents.

Table 2: Experiential Knowledge

Response All South east Question Filipino South Asian European Chinese Categories Responses Asian

In the bank 48% 25% 67% 77% 40% 33%

In my room 62% 80% 75% 33% 80% 33%

Where do you With family 20% 15% 16% 22% 40% 17% keep your members money? Take it with me 22% 35% 0.08% 11% 40% 0% everywhere

Other 20% 20% 17% 11% 20% 33%

Paying job 14% 0% 25% 33% 0% 17%

B-day/gift 80% 75% 92% 89% 100% 67% money

Do you receive Allowance 42% 40% 42% 56% 40% 67% money? If so, in what ways? I don’t work 50% 65% 42% 33% 40% 50% for it

I don’t receive 0.06% 0% 0% 11% 0% 17% any money

Other 0.02% 0.05% 0% 0% 0% 0%

Table 2 illustrates the financial behaviour and habits of respondents.

Vector • fall 2019 35 Table 3: Financial Literacy and Families

South All South Questions Response Categories Filipino European Chinese East Responses Asian Asian

Housing 66% 55% 54% 89% 80% 67%

Saving 70% 70% 54% 78% 80% 67%

Budgeting 34% 60% 8% 11% 20% 33%

Job loss 14% 25% 8% 0% 0% 0%

Retirement/Pension 24% 10% 15% 56% 0% 0%

What are some Taxes 72% 60% 69% 89% 100% 67% financial conversations Loans/mortgages/debt 42% 35% 38% 56% 60% 33% you hear at home? Bills 78% 75% 77% 78% 80% 67%

Banking 62% 55% 46% 100% 60% 67% Household Expenses/ 76% 70% 70% 78% 80% 67% Groceries Scams 24% 0% 31% 78% 60% 50%

Other 10% 5% 0% 0% 40% 17% Only when there is a 22% 45% 0% 22% 20% 17% problem How often do you Sometimes 36% 35% 38% 44% 80% 17% discuss finances with your family? Often 38% 35% 54% 33% 0% 50%

Everyday 4% 0% 8% 0% 0% 17%

Parents 95.9% 89% 100% 100% 100% 100%

Siblings 18.4% 16% 31% 0% 0% 5%

Who do you discuss/ Aunts/uncles 6.1% 0% 8% 11% 0% 33% learn about finances from in your family? Grandparents 8.2% 5% 15% 0% 0% 17% Cousins 6.1% 0% 15% 0% 0% 17%

Other 0% 0% 0% 0% 6.1% 17%

Table 3 examines who respondents speak to about finances at home and what topics are discussed.

Vector • fall 2019 36 Table 4: Feelings Regarding Financial Future

Response All South East Questions Filipino South Asian European Chinese Categories Responses Asian

Positive 72% 58% 63% 100% 100% 60%

How do you feel Negative 6% 16% 0% 0% 0% 0% about the future?

Ambivalent 19% 21% 31% 0% 0% 40%

Table 4 discusses how respondents feel about their financial futures. The question asked for open-ended responses that have been thematically coded into the following: Positive, Negative and Ambivalent.

Vector • fall 2019 37 Connecting Our Colleagues: A Book Club Study Using Open Questions by Tamara DeFord and Marie Fanshaw

In this article we will highlight the discussions and the benefits of This is the fifth mathematics book club that we have offered and bringing teachers together for a book club using the book Exploring each time we refine the way we run it. Teachers from past book Open Questions For Rich Math Lessons by Marian Small. The book clubs indicated that they value the time with their colleagues to club was led by Marie Fanshaw and Tamara DeFord who are both have collaborative, supportive conversation about classroom numeracy resource teachers in School District 57 (Prince George). practice. Because of this, we decided to prioritize time and space This article is a summary of a book club that was supported by a for this conversation to happen and to provide information and grant from the BCAMT. structures to help focus the conversation around open questions.

Finding a Good Fit Unpacking the Book In the fall of 2018, a small team from our district was fortunate For this year’s book club, we opted to hand out the books at the enough to participate in a two-day workshop series with Marian first scheduled meeting. This meant that our participants would Small with a focus on open questions. These sessions inspired us need time during our first meeting to read and unpack the book. to take what we had learned and share our learning with other Teachers were given time to explore the introduction of the teachers. Many of our district’s teachers are already familiar with book and think about how open questions might help them to Dr. Small’s work, so we decided that her open questions book series address some of the challenges that they faced with mathematics would be a perfect fit for this year’s elementary book club. instruction in their classrooms.

In our district we are trying to find creative ways to bring teachers We provided teachers with guiding questions to frame the together for after-school workshops and professional development. discussion at their tables: We have found that offering a book club attracts a small group of teachers who are interested in enhancing math instruction in their • When you think about your classroom and the various classrooms and are also motivated to connect with colleagues and applications for this book, how would it “fit”? share their classroom experiences. • How could having multiple grade levels within this resource be helpful for us as classroom teachers? Starting Points • Is there a culture or a structure in your classroom that would In November we sent out an expression of interest through our need to be in place for you to move forward with open DTA (District Teachers Association) PD reps. The expression of questions in your classroom? interest gave a brief introduction to Marian Small’s book and asked teachers to let us know what they were hoping to take away from From the table-talk discussions some common themes began to our book club. We were hoping for 15–20 teachers as we have found surface. Teachers immediately recognized the teacher-friendly that this is a workable number for us to facilitate and accommodate layout of the book and appreciated how the questions were in our learning commons work space. We decided to break the scaffolded from beginning stages to the consolidation stage. book into three sections and hold three one-hour meetings to discuss the contents of the book. We were able to attract a diverse With respect to classroom culture, we talked about the value of group of teachers because the book was available for grades K-3, working with a partner and how these questions fit nicely with the 4-6 and also in a French edition. curricular competencies. We also talked about what we value in

Vector • fall 2019 38 our mathematics classes. The importance of feeling safe and being guidelines for creating open questions. We shared some resources willing to take a risk is important for both students and the teacher. from Marian Small’s website and talked about alternate ways that students could solve open questions such as using whiteboards or At the end of this meeting, teachers were encouraged to try out a placemats. We really tried to plant some seeds and wanted to see couple of open questions from the book in their classrooms and to where our teachers would go with it. be prepared to discuss their experiences at the following meeting. Teachers Taking the Lead Carrying On We believe when you bring a group of teachers together for a In the subsequent two meetings, we prioritized collegial talk about common purpose they will take it and run with it. The only ask of lived classroom experience. We encouraged teachers to share teachers in our book club was for them to try at least one or two something they tried, something they noticed about their students’ questions from the book and to share their experience with their thinking and their next steps in implementation. Teachers shared colleagues in our group. Our group (of course) did not disappoint. successes and challenges and were able to offer each other ideas and suggestions based on their varied classroom experiences. The A couple of teachers had recently attended a workshop on first time our teachers met we did not group or sort the teachers; loose parts and they decided to take a question on representing however, for the last two meetings we had teachers sit in groups numbers from the Open Questions K-3 book outdoors. The based on which open questions book they used. teachers described using loose parts as a move away from formal instruction, and using counting collections to discover Our Job as Facilitators mathematical concepts through play. Students are encouraged to One of the great things about our book club is that we have keen experiment and collaborate, while discovering patterns and shapes teachers who truly embrace a positive mindset when it comes to and counting, sorting and building. Loose parts should be student- mathematics. That being said, mathematics is only one of many led and allow for students to discover on their own, with some subjects they teach so they don’t always have the time to search out adult direction/input to guide them on the way. This is where an “what’s new” in mathematics. This is where we come in. open question is useful, such as “Can you tell me what number they were trying to represent?” Loose parts leads to experimentation At each book club we try and provide some tools that will help and collaboration and mathematical talk between students and move teachers forward or that might be related to our book club student-to-adult that would not be possible in regular interactions topic. We try to remember that teachers “don’t know what they with regular classroom objects. don’t know” and sometimes just need a reminder about something that has gotten lost in the shuffle.

To start off the first meeting we chose some open questions from the nrich.maths.org website and then moved forward by introducing tasks from Jo Boaler’s youcubed.org site during meeting number two.

It was interesting to see the different approaches to solving the questions we chose (some people used manipulatives, while others made tables and charts). It opened up the door for us to talk about how our students need different tools and opportunities, and the importance of having concrete materials available to our learners.

We also brought in some intentional focus points from Marian Small’s fall 2018 workshops. Two areas that we felt would be helpful in moving our teachers forward was to take a closer look at Figure 1: Loose parts open question: “Choose a number less than parallel tasks versus open questions and unpacking strategies and 10. Show that number in 3 ways.”

Vector • fall 2019 39 Success and Wonderings They wondered if the language might be difficult for beginning The feedback that we received was once again very positive. kindergarten immersion students who come in with little or no Teachers enjoyed the one-hour format and see Figures 2 and 4 French. appreciated the time to network with their colleagues. We had five teachers from our French Immersion schools and they expressed Our last wondering is . . . what book should we do next year? their appreciation for actually having questions available in French.

Figure 2: Teacher reflection Figure 3: Reflective practice

Figure 4: Teacher feedback

Vector • fall 2019 40 Lego Robotics CoLab in SD73 by Elizabeth deVries

The LEGO CoLab Vision take on various programming challenges Elizabeth deVries is the district Coordinator with the laptops, where they have to write of Technology in SD73, Kamloops- the code to move their LEGO Mindstorms Thompson. Through collaboration with EV3 robot in specific ways and complete other teachers she has begun to learn about particular tasks. As a final celebration of the potential that lies with having students their learning, and as a presentation to an engage with various platforms of robotics. audience, the programmers invite a class LEGO Mindstorms is a set of interlocking from another school to come and work with and articulating parts with various sensors them. They spend time demonstrating the and motors that connect to an electronic and skills they have developed and the models programmable component called the brick. that they have built. Following the sharing Students can explore through play and dig opportunity, the programmers guide their into many different curricular competencies and outcomes by partners from the other class in taking on some programming being given different challenges to complete with LEGO robotics. challenges together; they teach the new students how to use the software and how to manipulate the variables within the block- With much gratitude to the BCAMT and the Future Launch programming code in order to manoeuvre the LEGO robot Program at RBC, SD73 received funds to be able to purchase fifteen through some kind of physical challenge. Together, the students sets of LEGO Mindstorms EV3 Core Sets and Expansion Kits and work collaboratively to build skills, solve problems and apply their Lenovo laptops to support the software for programming the understanding of measurement. robotics kits. The class set is owned by SD73, and classes wishing to access it must agree to have their class learn from a group of After the learning opportunity where one class has visited another students from another school and, in turn, share their learning school to learn from the students that have experience with LEGO by teaching another group of students what they have learned. robotics, the visiting class returns to their own school with the Through this hands-on, collaborative learning model, nicknamed CoLab kit of LEGO robotics and laptops to take their turn to the “CoLab” (because students are collaborating and working continue to deepen their learning. That class continues to work directly with the devices as one might in a laboratory), students and build their expertise and then they, in turn, invite another build connections through shared experience, develop leadership, class to come and learn from/with them. The pattern of developing practice communication skills, and connect mathematics to student capacity in leadership and programming skills carries on their understanding of programming. It across classrooms and across the district. is an exciting opportunity for students to interact with peers and build skills in The LEGO CoLab Project collaboration, programming and problem In February 2019, teachers came together solving. in SD73 to plan their approach to integrating LEGO robotics within the In the CoLab model, teachers engage curriculum. The half-day provided an students through a variety of challenges. opportunity for teachers to learn how The model starts with construction to build their own robots using LEGO challenges whereby students apply their Mindstorms and try their hand at creative ideas to build a robot using a programming the robots to move and LEGO Mindstorms EV3 kit. They then manoeuvre in different ways. They spent

Vector • fall 2019 41 time programming the sensors of the robots, and considered on in order to support their students in continuing the learning: the the curricular connections they would like to focus on with their second goal of the project is currently in progress! The teachers classes. The teachers left with a schedule of their time with the kits from SD73 who are preparing to connect with the LEGO Robotics and a partner class with whom they would work. The collection CoLab have identified that students will be deeply engaged in of resources are shared online between all parties, with a list of the curricular competencies in mathematics through this work. curriculum connections and challenges. Students will be challenged to engage in:

LEGO CoLab Project Goals • reasoning and analysis: using logic and patterns to be able to Increase student access to a real-world application of solve puzzles and play with the robots, estimating reasonably mathematics and programming; in order to modify variables in their code and complete their Increase teacher confidence in integrating this aspect of challenges; technology within their mathematics curriculum; • understanding and solving: developing, demonstrating and Develop a set of key challenges and activities for students applying mathematical understanding of measurement, to work through, engaging in reflection and self-assessment geometry, ratio and more as they challenge themselves to play of problem-solving, critical thinking and communication through specific tasks; throughout. • communicating and representing: explaining and justifying their decisions in their code and then communicating their With a kit of fifteen LEGO Mindstorms sets to be shared within mathematical thinking in order to change the program they SD73, we have the opportunity to meet our first goal of the project. have created according to their task; Groups that are working with the kits already have shown a high • connecting and reflecting: using mathematical arguments in level of excitement and shared that they understand that learning order to support their personal choices as they change their how to program a robot and building their skills with coding has code to flex their thinking to accomplish their task. opened their eyes to the skills they would us in some careers in (BC new curriculum: https://curriculum.gov.bc.ca/) computer science or programming. While watching students as they work with the platform and the programming software, I am In order to support our third project goal, the teachers brainstormed always impressed at the speed with which they recognize what potential curriculum links with the LEGO robotics work. They variables need to change in their program in order to accomplish identified that this learning embeds a rich opportunity to connect their goal. I am constantly marveling at the abilities of young the Core Competencies in an intentional way. Throughout the programmers to take a challenge, program it watch it run, and CoLab, students are encouraged to reflect on the following Core then go back and debug (if necessary) in order to get it working. Competency targets:

After the work in February, teachers expressed that they were • I can design and continue to develop my code as I design the feeling confident and ready to bring the work into their classrooms algorithm for LEGO EV3 to complete a challenge;

Vector • fall 2019 42 • I can question and investigate with LEGO EV3 in order to Challenge One: Shape Master create the algorithm for LEGO EV3 to complete a challenge; Program the EV3 to move and create a series of shapes in its • I can connect and engage with others as I am working with wheel movements understanding perimeter of complex shape a small group to create the algorithm for LEGO EV3 to and angle measurement. complete a challenge; • I can present information in a way that makes sense when Challenge Two: Map Master I am working to create the algorithm for LEGO EV3 to Program the EV3 to navigate on a map to be able to apply complete a challenge; measurement, angles and movement within specific locations • I can make my ideas work and change what I am doing as I and routes. work to complete challenges with LEGO EV3; • I use my experiences to direct my future work as I complete Challenge Three: Race EV3 challenges with LEGO EV3. Pre-set a route and have two or more groups race their robot to get to the same spot, through a series of movements. Extension Using the above target statements as reflective prompts challenge to see how different builds might move at different embedded in the work, students engage in reflection throughout speeds, and to see how different surfaces influence speed. their challenges. Through this process, students become more comfortable detailing how they are engaging their specific skills Challenge Four: How fast is EV3? and will be able to indicate success in the target areas. See the Using the manipulation of variables within the programming BINGO collection of reflection prompts below. platform to have students measure displacement over time and calculate the speed. Extension challenge to see how different In February, the teachers also established some main challenges to builds might move at different speeds, and to see how different engage their students in wrestling with the mathematical concepts surfaces influence speed. found in this work as well: Challenge Five: Sensoriffic! Program EV3 to use the Ultrasonic sensor in measure mode to read the distance output and create the drawing of the line or surfaces that yield from that specific series of data points.

Challenge Six: A-maze-ing Use the Ultrasonic sensor in measurement mode to navigate a series of obstacles without collision.

I am so excited to see how the students perform and improve with this work in the coming years. On behalf of all the learners in SD73, thank you to the BCAMT for this opportunity.

Vector • fall 2019 43 Whenever I do cartoons for Vector, I always aim to make the worldometers.info/)—real time world statistics in REAL TIME, mathematics as accurate as possible. One thing I really struggle and it sparked a lot of conversation and debate about how we with is innumeracy in the media, particularly in today's era. In this interact in the world. I chose to only show specific parts of the cartoon, I located some data regarding climate change, knowing statistics so we could focus on areas that I wanted the students to that there are a myriad of statistics out there. To this end, I'm explore. Watching and listening to students “using” mathematics to fully aware that statistics and data can be presented in a variety push a point about the world we live in is so encouraging. Guiding of ways for a variety of purposes—a great book on this matter students to be critical thinkers in the midst of rampant innumeracy is Innumeracy by John Allen Paulos. The cartoon asks what is is more important than ever, in an era of “fake news,” conspiracy important when it comes to data and statistics. Who is asking the theories, big business and political statistics and data. questions? What kind of questions? What do we mean by answers? Who is doing the answering? Ray Appel

When I taught grade 8 mathematics a few years ago, I loved the debates we got into. We examined WorldoMeters (https://www.

Vector • fall 2019 44 BCAMT Award Winners

Cody Hilton Rebecca Kaukinen Outstanding Secondary Teacher Outstanding Elementary Teacher

Cody Hilton has been a mathematics Rebecca Kaukinen has shared teacher in the Kamloops-Thompson her love of mathematics with School District for thirteen years. students for over 20 years. She is He has an MEd in Curriculum an elementary school teacher at and Instruction with a focus on St Margaret’s, an all-girls school, distance learning. During his where she currently works with studies, Cody developed a curiosity Kindergarteners to develop a solid for mathematical instruction foundation of number sense. She and assessment that has changed is a highly respected teacher at her his current practice. Cody has school and within the Independent implemented assessment techniques that are modelled after the Schools Association of British Columbia. Rebecca is a lifelong new curriculum and the new Graduation Numeracy Assessment, learner taking advantage of professional learning opportunities including the use of a four-point rubric for all of his assessments. when they arise and sharing her experience and enthusiasm with Cody is a big proponent of the use of vertical non-permanent others at the school. This is especially evident when she talks of the surfaces in his classroom to help facilitate his lessons. Cody mathematics learning happening in her classroom such as open- has taught in the International Baccalaureate program at his ended problem solving, number talks, esti-mysteries, WODB, a current school, NorKam Senior Secondary, since 2010. Having variety of math games, digitally recording student learning, and opportunities to participate in professional development with experimenting with an alternative to traditional “calendar time.” the IB program has fostered his teaching practice to focus on conceptual understanding in the mathematics classroom. One of Rebecca’s greatest accomplishments at St. Margaret’s School is her ability to champion mathematics education and Professional development has been a key factor in Cody’s truly influence girls’ attitudes in a positive way in the way she development as a teacher. One of his major influences has been talks about and encourages mathematical thinking, from the the work that the Kamloops-Thompson District has done with Dr. early years through childhood and middle school. She strives to Peter Liljedahl over the past several years. Cody is very grateful for encourage risk taking and helps learners embrace it as an important the many excellent professional development opportunities he has part of learning. She promotes the achievements of women in had throughout his career. He is convinced that his passion for the mathematics, assesses her students using methods beyond fluency profession is a direct result of the collaboration he has had with learning, and ensures that the contexts of mathematical concepts many other outstanding teachers and administrators throughout cater to all learners. Above all, Rebecca teaches her learners that the province. Cody is very humbled by, and appreciative of, this they can be capable and confident in the study of mathematics. award and looks forward to continuing to work collaboratively with passionate educators. When working and learning with parents, Rebecca encourages them to play games with their children from an early age, and to promote a positive attitude towards mathematics (such as, avoiding statements such as “I was never good at mathematics”). Parents participate in weekly math games and collaborative activities where the focus is on team work rather than speed. She also models verbal prompts for parents to use when discussing math problems and strategies. Rebecca is excited to share learning from the BCAMT Fall Conference with her students and parents. Vector • fall 2019 45 Janice Novakowski Janice served as an executive member of the BCAMT for several Ivan L. Johnson Memorial Award years and continues to work as a committee member for the Northwest Mathematics Conferences in British Columbia. She is Janice Novakowski has been a member of the BC Numeracy Network, a group of mathematics inspiring students and teachers educators working on sharing a balanced numeracy framework in the Richmond School District with teachers from across the province. and across the province for 28 years. From the onset of Janice’s Janice’s dedication to improving the teaching and learning of career as a primary teacher at mathematics is truly inspirational! Homma Elementary School, it was evident that she had a passion for mathematics. Janice created unique lessons that demonstrated her knowledge of what her students knew and where they needed to go, as well as deep understanding of the big ideas in mathematics. She was one of the early educators who began using children’s literature books in her lessons and her pedagogical approaches were highly engaging and multi-modal. Because of this, Janice quickly became a leader within the Richmond District, hosting sessions for other teachers after school, as well as teacher candidates. In 1993 Janice returned to UBC to complete her Masters in Mathematics Curriculum and Pedagogy and from 2002–2005 she worked as a Faculty Advisor and as a Mathematics Instructor. Many of the new teachers who were mentored by Janice share that she positively influenced their beliefs about themselves as mathematicians and inspired them to teach in ways in which all students see themselves as capable.

Janice has been in both school-based and district teaching positions over her career. She has worked as a district curriculum coordinator and teacher consultant in K-12 mathematics and numeracy, working side-by-side with teachers in their classrooms, co-teaching and co-planning and facilitating professional learning expereinces. She also initiates and facilitates collaborative professional learning communities in which teachers shared their learning journeys through published compilations which serve to further inspire other educators.

Most recently, Janice has been exploring Reggio-inspired principles and practices in mathematics. To support this work, Janice co- created a studio space at Grauer Elementary that has hosted both students and educators from Richmond, as well as around BC, Canada, North America and the world. She also published Reggio- Inspired Mathematics and co-authored another book called Pop Up Studio.

Vector • fall 2019 46 Using Self-Determination Theory to Engage Learners in Senior Mathematics Courses and Build Motivation and Higher-Order Thinking

by Jeff Irvine

Over my 45 years in education, I have been very interested in An added bonus is that by students taking an active role they are using instructional strategies that improve student motivation and more engaged and interested (Fosnot & Dolk, 2005). There is ample engagement in mathematics. Some of these strategies have been evidence that increased engagement is correlated with increased structured, like four corners, RAFT, menus, tiered assignments achievement in mathematics (e.g., Connor & Pope, 2013). I have and choice boards (for anyone unfamiliar with these strategies, found that teaching strategies that involve SDT, in which students there is a wealth of information available on the internet). All of have more agency in their learning, are particularly engaging and these strategies promote student choice and engender engagement. motivating for some topics and some situations. However, like all Some strategies have been very informal, like “Choose any five instructional strategies, there is no one size fits all and no strategy questions for homework.” All of the strategies have positively that is perfect for every situation. In my experience, SDT activities influenced student engagement and motivation. Several years ago, effectively engage students when I learned about self-determination theory, which integrates student choice with two other dimensions. • Introducing a concept; • Activating prior knowledge; Self-determination theory (SDT) has a great deal of endorsement • Encouraging higher order thinking skills; in educational research; for example, supporting student learning • Generating student interest and engagement during learning through SDT has been shown to enhance student engagement activities; and motivation, as well as promoting deeper and longer-lasting • Promoting student ownership of learning and increasing understanding of concepts (Deci & Ryan, 2008). There are three persistence. dimensions comprising SDT: student autonomy, usually involving choice; competence, the student’s need to have feelings of mastery Strategies based on SDT do not work particularly well for of some area of study; and relatedness, the need for social generating theory or for proving theorems. I found that introducing interaction and support, which can be provided through the use of a concept using student agency activities is a good start, which I student groups or pairs during instruction. The dimension of SDT then follow with cooperative learning or whole group discussion. that has received the most attention in the research community is to flesh out and provide detail. student autonomy, since it has been shown to have a significant impact on student motivation (Deci & Ryan, 2008; Irvine, 2018). Utilizing SDT in the classroom often involves structuring a student Student autonomy is often supported by providing some elements activity in which elements of SDT arise naturally in the course of of student choice during a mathematics class, although it should the activity. Here are classroom-tested activities that I have created be noted that this does not mean unlimited student choice. By over the years, together with comments and suggestions regarding providing students with some elements of choice, some of the time, where and when to use them. Usually, all the activities would be research has shown that this can promote student engagement, carried out in student groups, satisfying the relatedness dimension motivation, and persistence (Ryan & Deci, 2000). of SDT. Also, the confidence developed during these activities

Vector • fall 2019 47 contributes to the competence dimension of SDT. The next activity introduces ideas about arithmetic and geometric sequences. The total time required is approximately 35 minutes. The first activity is an interesting way to introduce limits in a Students find this activity engaging because they have many calculus course. It can also be used across multiple grades as a opportunities for choice. When done in small groups, this activity fractions activity. The activity takes very little time; it generates also satisfies the relatedness dimension of SDT. Competence will interest; it has a “low floor, high ceiling” (Gadanidis, Hughes, & be achieved during the more formal classroom activities that would Scucuglia, 2009) so that every student can become engaged; it follow from the sequence sorting activity. This activity requires encourages higher order thinking and conjectures; it addresses the students to recognize and verbalize patterns and reinforces the autonomy dimension of SDT through student choice. This activity key differences between arithmetic and geometric sequences in a typically takes less than 20 minutes to complete. fun way. Theactivity would be followed by formal lessons on the various types of sequences, and the activity could be revisited at Fraction Follies the end of the unit for review of key concepts.

(These are the instructions I give my students.) Sequence Sorting

1. Write down your favourite fraction Design a sorting rule and sort the sequences below into two groups. 2. Add 1 to the numerator and 1 to the denominator A: 3,5,7,9,11,13 B: 1,3,9,27,81,243 C: 2,4,8,16,32,64 3. Is the new fraction larger or smaller than the original fraction? D: -9,-2,5,12,19,26 E: 5,-10,20,-40,80,-160 F: 9,7,5,3,1,-1 How do you know? G: 7,8,11,16,23,32 H: 12,6,3,1.5,0.75,0.375 I: 1,4,7,10,13,16 4. Repeat steps 2 and 3 a bunch of times. What happens? J: 1,4,9,16,25,36 K: 2,7,9,16,25,41 L: 7,11,15,19,23,27 5. Explain your results. 6. If you repeated these steps 1000 times, what do you think My sorting rule: would happen? 7. What if you started with a smaller fraction? 8. What if you started with a larger fraction? 9. What if you started with a mixed number? 10. Prove your conjectures using limits. (This step may be withheld until students have additional knowledge of limits.) Group One Group Two

The graph in Figure 1 illustrates the outcomes starting with 3/4 Navneet used a sorting rule based on operations. Sort the sequences (lower curve) or starting with 5/3 (upper curve) as the number of above using Navneet’s rule: iterations gets very large.

Formed using + Formed using X

In which group should Navneet put the sequence 1,2,2,4,8,32?

Why?

Now design a sorting rule to help Navneet subdivide each of her two groups into two subgroups.

Figure 1: Graphs of y = (3+x)/(4+x) (lower curve) and y = (5+x)/ For Navneet’s sequences formed using + my subsorting rule is: (3+x) (upper curve); as x becomes large, illustrating that in both cases the curves approach the value 1.

Vector • fall 2019 48 Here are Navneet’s X sequences sorted using my rule: Part B: Using the“ !” function

In your journal, explain how you can use the “ !” notation (called the factorial notation) to solve the problems below.

1. How many different arrangements are there using all the letters of GARDEN? 2. How many six letter arrangements are there of letters chosen from FUNCTION? Where will you put the sequence 3,4,6,9,13,18? Why? 3. How many six letter arrangements of letters chosen from FUNCTION start with an N? When using this activity, it is important to accept very simple 4. How many eight letter arrangements of letters chosen from sorting rules, such as “these sequences contain a 4” and “these FUNCTION have all the vowels together? sequences do not contain a 4.” Such very basic sorting rules are a 5. How many five letter arrangements are there of letters chosen starting point. However, when sharing group sorting rules across from SEVEN? the class, students will recognize that more sophisticated sorting 6. Make up an arrangement question similar to the ones above rules generate more differentiation among the sequences. Whole- using your own starting word. Share with a partner. class discussion leads to student growth in understanding. 7. Investigate the relationship between 1! and 0!. Justify your response by referring to your work on arrangements. Notations such as the factorial (!) notation are not really worth a lot of class time, unless they can be incorporated into activities Part C: Investigating the function nPr requiring higher order thinking skills. The activity below, which may take 60 to 75 minutes, introduces the notation and relates it Determine the function of the function nPr. Syntax is number nPr to work on permutations and combinations. number (e.g. 5 nPr 3 ). In your journal, respond to this question: How can the function “!” be used to answer permutation Part A: Determining the meaning of “!” questions?

Your mission is to determine the meaning of the notation “!” Part D: Investigating the function nCr

1. On your calculator, enter the number 5, then “ !” and = or Determine the function of the function nCr. Syntax is number nCr 2. On your graphing calculator, enter the number 5, then press number (e.g. 5 nCr 3 ). In your journal, respond to this MATH, then PRB, then 4 question: How can the function “!” be used to answer questions 3. Examine the answer shown on the screen. Conjecture how about arrangements? the value was computed. 4. Try several other numbers instead of 5. Make a table of your results. 5. Based on the patterns of your table, conjecture what the function “ !” does. 6. Check your conjecture on several other input numbers. 7. Conjecture a value for 0! ? 8. “Break your calculator”: Is there a largest number n for which your calculator cannot compute n!? Try to find it.

Vector • fall 2019 49 Part E: Make up a question involving permutations or combinations and give it to a partner to solve. y = −4x2 + 7x + 1 y = 5x2 + 2x Part F: (Challenge Question) y = x2 − 49 y = −4x2 + 9x − 11 How many 5 letter arrangements are possible if the letters are y = 7x2 − 3x + 6 chosen from MATHEMATICS? y = −x2 + 4x − 5 y = 11x2 + 9x + 10 All of these can be used to introduce key concepts in an interactive y = 50x2 + 25x − 89 and engaging way. An example of this introduction to a big idea from Grade 12 Calculus is shown below. 7. Discuss with your group how you could prove or disprove your conjectures. The concept of tangency and its relation to the derivative is a key 8. Write an equation for your favourite polynomial. Then carry idea in calculus. This 40-minute activity sets the stage for more out steps 1 to 5 above and examine the relationship between formal work on the derivative as tangent to a curve, and begins your original equation and the equation of the slopes of the to generate student thought about the power rule for derivatives. tangents. By actually constructing tangents, measuring slopes, and creating the derivative function from those slopes, student understanding By using activities that support the dimensions of SDT, we allow is deepened in an active way without sacrificing a great deal of students to build their own understanding of concepts and engage class time. This activity allows students to get an intuitive feel for in higher order learning as well as dramatically increasing student the topic, which can then be made more rigorous with whole class engagement. The teacher’s role is to consolidate and extend that discussion in the consolidation phase of the lesson. The active understanding. The result is more motivated students with a deeper stance by students also generates increased engagement and understanding of concepts. Supporting student agency does not motivation. mean that students no longer need to have important facts and procedures at their fingertips. Consolidation of ideas will always Tangents to Curves be necessary, typically in more formal class structures. However, 1. Open Desmos, and graph y = x2 − 3x + 5 the deeper understanding that is a consequence of supporting 2. Draw tangents to the curve at x=-3,-2,-1,0,1,2,3 student autonomy and an active student stance in mathematics 3. Find the slopes of the tangents and complete the table below classrooms provides students with a better fundamental set of tools and strategies to enable them to transfer their learning to x Slope of tangent new problems and new situations (Irvine, 2015, 2017) and make -3 them more able mathematics learners, mathematics users, and -2 mathematics producers, as well as more interested and engaged -1 learners. 0 1 References 2 Conner, J. O., & Pope, D. (2013). Not just robo-students: Why full 3 engagement matters and how schools can promote it. Journal of 4. Graph the relation given by the slopes of the tangents and Youth and Adolescence, 42(9), 1426-1442. http://dx/doi:10.1007/ find its equation. s10964-013-9948-0 5. Conjecture the relationship between this equation and the equation of the original parabola. Deci, E., & Ryan, R. (2008). Self-determination theory: A 6. Conjecture the equation for the line defined by the slopes of macrotheory of human motivation, development, and health. tangents to each of the following curves. Canadian Psychology, 49(3), 182-185. http://dx.doi:10.1037/ a0012801

Vector • fall 2019 50 Deci, E., Vallerand, R., Pelletier, L., & Ryan, R. (1991). Motivation Irvine, J. (2018). Self-determination theory as a framework for an and education: the self- determination perspective. Educational Intermediate/Senior Mathematics preservice course. Journal of Psychologist, 26(3&4), 325-346. Instructional Pedagogies, 22, Article 4, 1-27.

Fosnot, C., & Dolk, M. (1995). “Mathematics” or “Mathematizing? Ryan, R., & Deci, E. (2000). Self-determination theory and the “In L. Steffe & J. Gale (Eds.), Constructivism in Education (pp. 175- facilitation of intrinsic motivation, social development, and well- 192). Hillsdale, NJ: Lawrence Erlbaum Associates. being. American Psychologist, 55(1), 68-78.

Gadanidis, G., Hughes, J., Scucuglia, R., & Tolley, S. (2009). Low Sha, L., Schunn, C., & Bathgate, M. (2015). Measuring choice to floor, high ceiling: Performing mathematics across grades 2-8. participate in optional science learning experiences during early Conference Papers—Psychology of Mathematics and Education adolescence. Journal of Research in Science Teaching, 52 (5), 686- of North America, 2009 Annual Meeting, 1-8. 709.

Irvine, J. (2015). Problem solving as motivation in mathematics: Just in time teaching. Journal of Mathematical Sciences, 2, 106-117.

Irvine, J. (2017) Problem posing in consumer mathematics classes: Not just for future mathematicians. The Mathematics Enthusiast, 14(1), Article 22, 387-412.

Vector • fall 2019 51 Math Challengers by Matt Branch, Chairperson of Math Challengers BC

For over 30 years, students have gathered at campuses across BC to In our second year running a Grade 10 competition, Karen Situ spend a day away from school in competition against other students from University Hill finished 3rd, Kaixin Wang from Magee from across the province. They are brought together by one thing: Secondary School came in 2nd and Munki Hahn from University their love of mathematics. This year was no exception when Hill was 1st. Munki Hahn defended his top-ranked seating and over 300 students competed at the provincial Math Challengers finished first in the Face-off Round. tournament on April 6 at UBC Point Grey Campus. The committee would like to thank one and all for participating It was a busy day all in all, and we would like to congratulate the in this year’s Math Challengers competitions and we hope to see following school teams who placed in the top three among all everyone back next year. Congratulations to all participants! A school teams who competed: special thanks goes out to the school coaches of these teams and individuals, who not only worked hard throughout the year to In Grade 8, St. George’s School placed 3rd, Semiahmoo Secondary prepare them, but also took the time to bring them to the regional School took 2nd place, and Norma Rose Point School came in 1st and provincial competitions and support them in their pursuit place (for the fourth year in a row). of mathematics. Without you, there would not be a competition: thank you! The rankings for Grade 9 are as follows: with University Hill Secondary School finishing 3rd, Moscrop Secondary School 2nd Thank you also to all the volunteers who took a day out of their and the University Transition team in 1st. week to help with the various tasks involved in running these competitions. For more information on how you can get involved For Grade 10, Lord Byng Secondary School came in 3rd, Fraser with Math Challengers, either as a participant, coach, or volunteer, Heights Secondary School placed 2nd and University Hill please check out our website at https://www.egbc.ca/Math- Secondary School took 1st place. Challengers/Math-Challengers-Home.

There were many successful individuals. Of these, we would especially like to recognize and congratulate the top three students at each grade level.

In Grade 8, Eric Shao from Semiahmoo Secondary came in 3rd place, Edward Lai from Sir John Franklin finished in 2nd place, and Kevin Liu from Norma Rose Point School came in 1st place. Special congratulations to Tim Zou, who placed 6th overall, but won the head-to-head title in the Face-off Round, successfully defeating each of the students who had ranked ahead of him in the individual competition portion.

Among the Grade 9 competitors, Andrew Kang from University Transition was 3rd, Jiangxu Wan from University Hill placed 2nd, and Er Bei from William Osler Elementary finished 1st. Rian Popat deserves particular mention as well, after winning the head-to- head title in the Face-off Round, despite placing 6th overall.

Vector • fall 2019 52 Social Justice and Mathematics

A current annotated reference list, compiled July 2019 by Janice Novakowski by Janice Novakowski

What role does social justice have to play in the teaching and Math That Matters 2: A Teacher Resource Linking Math learning of mathematics? If mathematics is one way to help us and Social Justice understand the world around us, it is imperative that students have Stocker, D. (2019). Math that matters 2: A teacher resource linking opportunities to study issues affecting them and their communities math and social justice. Ottawa, Ontario: The Canadian Centre for and to consider the ways mathematics can be used to solve Policy Alternatives. problems and analyze and communicate information. Another aspect of teaching mathematics with social justice in mind is to Teacher resource with fifty lessons linking social justice issues to consider the resources and contexts we use and whether they are mathematics, most with Canadian content. reflective of and responsive to the students we are teaching. A growing area of focus for the National Council of Teachers of Social Justice and Mathematics: A Time for Renewed Mathematics (NCTM) is access and equity in mathematics and Implementation? this is reflected in recent conferences, online discussions and Chorney, S. (2015). Social justice and mathematics: A time for publications. In our BC context, there is growing awareness of a renewed implementation? Vector, 56(2), 8-11. need for more inclusive curricular frameworks in mathematics and this is reflected in some of the K-12 mathematics curricular This article was written as the redesigned BC mathematics competencies as well as the range of senior mathematics courses curriculum was being released. The author discusses social justice now available. The First Nations Education Steering Committee and mathematics through two lenses: as content and as context. (FNESC) is currently updating their First Peoples Mathematics Also discussed is the issue of equity, not just in the world but in teaching resource to be aligned with the revised BC curriculum our classrooms. with goals of supporting all students in developing awareness of Indigenous knowledge and perspectives as well as supporting the Teaching Mathematics for Social Justice academic success of Indigenous students. With these areas of focus, Wright, P. (2016). Teaching math for social justice. Derby, UK: need and interest in mind, a current list of both print and online Association of Teachers of Mathematics. resources has been compiled. This ATM publication addresses inclusive practices in the Rethinking Mathematics: Teaching Social Justice by secondary mathematics classroom while integrating issues of social the Numbers justice into seven classroom-based projects/investigations. Gutstein, E. & Peterson, B. (Eds). (2013). Rethinking mathematics: Teaching social justice by the numbers (2nd ed.). Milwaukee, WI: Annual Perspectives in Mathematics Education: Rethinking Schools. Rehumanizing Mathematics for Black, Indigenous and Latinx Students Teacher resource with an introduction and overview of teaching Goffney, I. & Gutierrez, R. (Eds). (2018). Annual perspectives in social justice as an approach to support students understanding of mathematics education: Rehumanizing mathematics for Black, mathematics as a powerful tool in society. Several classroom-based Indigenous and Latinx students. Reston, VA: NCTM. investigations, written by contributing authors, directly connect the use of mathematics within social justice topics. NCTM’s annual publication focuses on current issues in mathematics education and is comprised of submitted chapters

Vector • fall 2019 53 that discuss the teaching and learning of mathematics. The chapter Mathematics for Social Justice: Resources for the topics address equity, access, privilege and systemic issues for Black, College Classroom Indigenous and Latinx students and share the efforts of educators Karaali, G. & Khadjavi, L.S. (Eds). (2019). Mathematics for social and researchers to provide mathematics learning experiences that justice: Resources for the college classroom. Washington, DC: MAA are humane, positive and powerful. Press.

A Qualitative Metasynthesis of Teaching Mathematics A collection of essays about politics and pedagogy in classrooms for Social Justice in Action: Pitfalls and Promises of followed by fourteen modules of lessons with social justice themes. Practice Harper, F. K. (2019). A qualitative metasynthesis of teaching Radical Equations: Civil Rights from Mississippi to the mathematics for social justice in action: Pitfalls and promises of Algebra Project practice. Journal for Research in Mathematics Education, 50(3), Moses, R.P. & Cobb, C.E. (2002). Radical equations: Civil rights 268-310. from Mississippi to the Algebra Project. Boston, MA: Beacon Press.

This paper discusses the results of a metasynthesis of thirty-five Sharing the power of community, this book shares stories from the qualitative reports of social justices learning experiences in diverse Algebra Project, enacted in twenty-five cities in the USA. classrooms contexts. The guiding framework used for analysis is Critical Race Theory. Possible pitfalls of teaching math through Living Proof: Stories of Resilience Along the social justice include: Mathematical Journey Henrich, A.K., Lawrence, E.D., Pons, M.A. & Taylor, E.G. (Eds). 1. the negative implications of not addressing issues of race (2019). Living proof: Stories of resilience along the mathematical directly; journey. Providence, RI: American Mathematical Society. 2. not critiquing liberal assumptions and thus perpetuating racist stereotypes; and A collection of contributed stories of struggle and persistence 3. time and curricular constraints can lead to superficial intended to inspire students of mathematics. Available as a free connections. pdf download at maa.org.

Promising practices to address racial inequities in mathematics Teaching Mathematics for Social Justice: Conversations classrooms include: with Educators Stinson, D.W. & Wager, A.A. (Eds). (2012). Teaching mathematics 1. normalizing the discussion of race so that social justice for social justice: Conversations with educators. Reston, VA: learning experiences can center voices of colour; NCTM. 2. encourage considerations of intersectionality (for example, consider not just race but also class, gender, sexism); Leading scholars in social justice mathematics share their research 3. create counterstories to critique mainstream claims; and perspectives through a collection of articles that connect both 4. needing to take a long term and interdisciplinary approach; theory/practice and education/social justice. and 5. plan time for students to take action toward change. Access and Equity: Promoting High-Quality Mathematics in Grades 9-12 Mathematics for Human Flourishing (available for pre- White, D.Y., Fernandes, A. & Civil, M. (Eds). (2018). Access and order) equity: Promoting high-quality mathematics in grades 9-12. Reston, Su, F. (2020). Mathematics for human flourishing. New Haven, CT: VA: NCTM. Yale University Press. Part of a grade-band series, this volume addresses issues of access, Building on Su’s other publications and presentations, this book equity and empowerment in secondary classrooms with chapters shares the beauty of mathematics and how it connects to our focused on the themes of engaging students in equity-based common humanity. classroom projects, developing students’ positive mathematical

Vector • fall 2019 54 identities and encouraging teachers to reflect upon their own Math and Social Justice: A Collaborative MTBOS Site equitable teaching practices. A collaboratively designed collection of resources for exploring social justice issues in the math classroom and striving for equity Also available in volumes for Grades 6–8, Grades 3–5 and Grades in math education. Pre-K–Grade 2 through nctm.org https://sites.google.com/site/mathandsocialjustice/home Critical Science and Mathematics Early Childhood Education: Theorizing Reggio, Play, and Critical Radical Math Pedagogy into an Actionable Cycle. A website to support educators in integrating issues of social and McCormick Smith, M. & Chao, T. (2018). Critical science and economic justice into their mathematics classes. mathematics early childhood education: Theorizing Reggio, play, and critical pedagogy into an actionable cycle. Education Science http://radicalmath.org 8, 162. #makemathjust This paper examines ways to support early childhood educators This hashtag is currently being used on twitter to explore creating in engaging in critical conversations with young children about social justice investigations and ways to help students use math identity, culture, diversity and other critical tensions and draws critically. upon frameworks such as the philosophy of early childhood education in Reggio Emilia, play-based pedagogy and critical With thanks to Twitter colleagues who contributed to this list. pedagogy.

Online Articles, Resources and Blogposts:

Special Issue of Teaching for Excellence and Equity in Mathematics Education: Through the Lens of Social Justice By TODOS, 2016 https://www.todos-math.org/assets/documents/TEEM/teem7_ final1.pdf

Elementary Mathematics and #BlackLivesMatter By Theodore Chao and Maya Marlowe https://www.bankstreet.edu/research-publications-policy/ occasional-paper- series/occasional-paper-series-41/elementary- mathematics-and- blacklivesmatter/

Re-Designing Mathematics Education for Social Justice: A Vision By Fahmil Shah https://www.bankstreet.edu/research-publications-policy/ occasional-paper- series/occasional-paper-series-41/re-designing- mathematics-education-for- social-justice-a-vision/

Vector • fall 2019 55 Book Review

Reviewed by Brian Taylor

David Acheson (born in 1946) is an English applied mathematician motion and the oscillations of a guitar string. The book concludes who now lectures widely, popularizing mathematics. His delightful with a chapter on chaos and Ed Lorenz. book, The Calculus Story, is divided into 28 short chapters, each telling part of the story of calculus. Along the way we are introduced There is never really enough material to properly learn any of the to Newton and Leibniz and some of the controversy surrounding concepts, if you don’t already know them, so there is a danger the “discovery” of calculus. Each chapter includes diagrams and that a student might get a little overwhelmed by not being able to equations, but only enough to aid understanding. It does not read follow the arguments being made. However, so long as the book like a textbook. There are some very interesting reproductions of is regarded as a bird’s eye view of the topic and not an exhaustive pages from original papers by Wallis, Leibniz, Newton, Berkeley, exploration it should spur interest rather than dampen it. and Euler, as well as some illustrations from the notebooks of Newton and Leibniz. The topic of infinite series comes up in a As a mathematics teacher I found The Calculus Story a thoroughly number of chapters and in one chapter the development of the engaging and entertaining read that contains quite a number of Madhava–Leibniz series is shown. interesting examples that I immediately wanted to include in my instruction. I think that this book would be an excellent suggestion The book loosely follows the development of calculus and its for any student of calculus. It might allow the students to better applications including a discussion of the differences in notation understand some of the reasons behind the techniques they are and the later efforts to put the subject on a more rigorous footing. learning, and give them a broader appreciation for the subject. There are some nice chapters on applications such as planetary

The Calculus Story—A Mathematical Adventure David Acheson, Emeritus Fellow of Jesus College, Oxford Oxford University Press, 2017 192 pages

image source: amazaon.ca

Vector • fall 2019 56 Problem Set

Problems curated by Adam Fox and Mike Pruner

Problem 1 Problem 4 How many diffferent cubes can be made such that each face has a A bicycle shop makes unicycles, bicycles and tricycles. If the shop single line joining the mid-points of a pair of opposite edges? Same owner receives a shipment of 8 wheels, how many different ways question for a dignoal stripe. can she build cycles using all of the wheels?

Source: John Mason, Thinking Mathematically Source: Teaching Children Mathematics, August 2016

Each face of a cube is painted either red or blue. How many ways Problem 5 are there to paint the cube? For this activity the students will need 1cm graph paper and a set of Cuisenaire Rods. Students can either be in pairs or two sets Problem 2: The Electric Circuit of pairs. Ask one child (or pair of children) to draw a design on There are 12 nodes spaced evenly near the edge of a circular the graph paper using no more than six Cuisenaire Rods. When circuit board. Pairs of nodes need to be connected according to they have drawn the design the other student (or pair of students) the following constraints: studies it carefully for 10-20 seconds. When the time has passed the design is turned upside down and the students who did not a. Opposite nodes need to be connected. make the design must recreate it using the Cuisenaire Rods. This b. Wires cannot cross can be a challenging task, so let them peek if they need help. The c. There is only room for 2 wires in the space between the graph paper is important because it ensures the design is orientated nodes and the edge of the circuit board. correctly. This activity forces the students to deal abstractly with different rod lengths, and also calls upon visual memory. The How can the nodes be connected? difficulty level can be adjusted by adding or removing rods.

Problem 3 Source: Idea Book for Cuisenaire Rods Find the minimum number of moves necessary to move the red dot to the opposite empty conrner on an n x n grid. All dots can Problem 6 only move horizontally or vertically into an empty space. Both regular hexagons have area 6. What is the total shaded area?

What if the board is 4x4? Or 5x5?

How do you know that you have the minimum number of moves? Source: Catriona Shearer on Twitter - @Cshearer41 Problems from Marian Small

Vector • fall 2019 57 Problem 7 4. You see the number 18 on the TV. What could the number Four equilateral triangles are arranged around a square which has mean? area 12. What’s the total shaded area? 5. You see the number 7 on the TV. What could the number mean?

6. Start at zero and skip count by a number until you reach 14. What were you counting by? What were you not counting by? How do you know?

7. With a friend, make a game that involves adding single-digit numbers. Teach another group of friends your game and play Source: Catriona Shearer on Twitter - @Cshearer41 it with them

Problems from Marian Small 8. Make an increasing pattern that contains more odd numbers 1. I add two numbers. I also subtract them. The “add answer” is than even numbers in the first nine terms. What is the pattern 10 more than the “subtract answer.” What could the numbers rule? be? 9. In a decreasing pattern the third term is 80, and the eigth term 2. Create a sentence that uses each of the four numbers and words is in the 60s or 70s. What is your pattern rule? What else do shown below. Other numbers and words can also be used. you know about the pattern? 3, more, 5, and

3. You see the number 4102 on the internet news. What could the number mean?

Vector • fall 2019 58 Math Links Fall 2019 Mathematics Websites

http://www.smartvic.com These free assessments are designed to reveal the mathematical thinking done by students. They are designed to be done online and so require students to have access to a computer. Although the assessments align to the Australian standards, the topic assessed by each test is made clear, making using these assessments possible for anyone.

https://variationtheory.com/ Instead of students practicing randomly selected questions, these questions are designed to deliberately vary, making it easier for students to notice mathematical relationships. The website includes advice for teachers and questions from a wide variety of topics.

http://goalfreeproblems.blogspot.com/ A goal-free problem is a mathematical task that asks students to determine everything they can from a given diagram or problem context, without specifying a specific question or prompt to solve. This website has a growing collection of problems designed to be goal-free.

https://www.resourceaholic.com/ This blog by Jo Morgan, a mathematics teacher from the United Kingdom, contains the many hundreds of resources she has collected from Twitter and resources she has created for use in her classroom.

https://mathsvenns.com/ Each of these problems contains clues or constraints around the outside and a Venn diagram in the middle that students fill in. The website contains advice for teachers on how to use the Venn diagram problems with their students.

Links selected and described by David Wees (http://davidwees.com). Previous websites can be viewed here: http://davidwees.com/m/ mathwebsites

Vector • fall 2019 59 Teachers Awards Information

The BCAMT sponsors awards in three categories (Outstanding Teacher, Ivan L. Johnson Memorial, and Service) to celebrate outstanding achievements of its members. Winners are honoured at a BCAMT conference and receive a commemorative plaque.

AWARDS & CRITERIA

Outstanding Teacher Awards (Elementary; Secondary; New Teacher with less than five years teaching experience) • shows evidence of significant positive impacts on students, staff and parents • has initiated innovative and effective programs in their classroom, school, district, or province (teacher research, technology, active learning, assessment, etc.) • has and continues to demonstrate excellence in teaching mathematics regularly in British Columbia (teaching style, knowledge of the curriculum, current curriculum trends, etc.) • has made contributions to mathematics education at the school, district or provincial levels (eg. workshops, seminars, conferences, community projects, curriculum development, publishing, etc.) • is not a current member of the BCAMT Executive

Service Award • has provided extraordinary service to mathematics education as an active member of the BCAMT for a significant period of time

Ivan L. Johnson Memorial Award

The Ivan L. Johnson Memorial Award is awarded in honour of long-time BCAMT executive member Ivan Johnson. Ivan donated money to the BCAMT for an award in which the recipient will receive significant funding to cover costs of attending the NCTM Annual Conference. • inspires teachers to try new ideas that improve the quality of mathematics education • consistently seeks ways to innovate practices in the mathematics classroom • actively engages in professional dialogue involving mathematics pedagogy • is not a current member of the BCAMT Executive, but is a member of the BCTF

Note: Nominees for the BCAMT Outstanding Teacher Awards will automatically be considered for this award. Previous winners of BCAMT Outstanding Teacher Awards may also be nominated. Recipients of this award are expected to contribute an article to Vector.

60 Vector • Fall 2019 SELECTION PROCESS • All nominations are reviewed by the BCAMT Awards committee (consisting of a minimum of five previous award recipients) who recommend the recipients to the BCAMT Executive for ratification. • Each nomination is considered for two years, after which time the application can be re-submitted with updated information.

HOW TO NOMINATE

Required documentation: • a completed nomination form (one person per form); • nominee’s curriculum vitae which demonstrates evidence of teaching, contribution, innovation, professional involvement and impact; • nominator’s summary (one page only) explaining concisely the reasons for the nomination; • two letters of support (one page each) with concise information about how the nominee fulfills the criteria.

Send all required documents listed below in an envelope to:

BCAMT Awards c/o Michael Pruner 2680 Standish Drive North Vancouver, BC V7H 1N1

Deadline: May 31, 2020

Vector • Fall 2019 61 Grant Application 2018-2019

Each year the BCAMT offers up to $8000 in grant funds to its membership. The funds must be used to further mathematics education in BC. These initiatives must meet the Goals and Objectives of the BCAMT. These funds are not meant for individual professional development.

The BCAMT values the sharing of ideas and expects that successful applicants will: • Write a 150-word summary explaining the initiative and results and also include photo(s) suitable for publication.

Fund distribution:: • If the initiative requests for $1000 or less, 100% of the grant is provided upon approval by the BCAMT executive. • If the initiative requests for more than $1000, 75% of the funds will be provided upon approval by the BCAMT executive, the remaining funds will be released upon completion of the grant requirements (see above).

All applications must be received no later than November 29, 2019. Applicants will be informed about funding after approval by the BCAMT Executive. Successful applicants may wish to re-apply for funding each year but are not guaranteed continued support.

This grant is not meant for individual professional development.

Application steps:

1. Fill out the Grant Application Form found at www.bcamt.ca/grants 2. Upload or email a one-page rationale for the funding request that outlines: a. the goals of your initiative (SMART goals: specific, measurable, attainable, realistic, timely) b. details of your initiative including estimated number of teachers or students affected (breadth of impact) as well as likelihood of success (depth or duration of impact) 3. Upload or email a detailed budget. (List expenses, other funding, etc.)

The above documentation needs to be received before November 29, 2019.

62 Vector • Fall 2019 Vector • fall 2019 63 WWW.BCAMT.CA