Math 97 Pedagogy & Assessment Plans

MATH 180 PEDAGOGY & ASSESSMENT PLANS Rev: Winter 2007

Math Knowledge Weekly Topic (& relevant sections of the text) Outcomes 1. The role of the educator; the concept of “numbers” (2.1, 2.2, 2.4) ~1, ~3, 9 2. (Short) Concept of numbers and features of number systems (2.2, 2.3) 1, ~3 3. Types of numbers – purpose and representations (2.2, 3.3, 6.1, 7.1, 8.1, 9.1, 9.2) ~2, 3 4. Structure of numbers; Pattern recognition (5.1, 5.2; 1.1, 1.2) 2, ~5, ~7, ~8 5. Concept of the operations (+ and – with wholes, dec, frac; 3.1, 6.2, 7.2) ~4 6. Concept of the operations ( and  with wholes, dec, frac; 3.2, 6.3, 7.2) 4 7. (Short) Algorithms (estimation, + and – with wholes; 4.1, 4.2, 4.3) PPTL 8. Algorithms (+ and – with dec, frac;  and  with wholes; 4.2, 6.2, 7.2) PPTL 9. Algorithms ( and  with dec, frac); Ratios, Percents, Proportions (6.3, 7.2, 7.3, 7.4) PPTL, ~6 10. Ratios, Percents, Proportions and Problem Solving (7.3, 7.4, 1.1, 1.2) 5, 6, 7, 8 11. (1 class day – student evals?; Finals)

Typical Weekly Routine Day Class Activity Outside of Class  Finish leftover tasks from previous Friday and weekend. (See those Homework presentations days below for details.) Present  Reread your class notes to identify the key ideas for the week. “Question(s) of the Week,” Monday  Thoroughly read the relevant parts of the assigned readings. Make “Puzzle of the Week,” your notes more complete by writing down additional details from “Standard(s) of the Week” the reading about the key ideas. Lecture on key ideas  Write down one or more specific questions to ask before, during, or after class tomorrow. Activity on major ideas  Try all problems on the homework assignment. Limit yourself to Discussion and summary of 15 minutes on each single problem/task UNTIL you have tried Tuesday discoveries everything. Self-reflective writing  Practice the basic skills for the next proficiency test.  Spend 15 minutes creating a rough draft of a concept map for this Homework presentations week’s ideas. Supplemental  Try to answer the “Question(s) of the Week.” lecture/discussion Wednesday  Finish your homework. Initial attempt to articulate  Review your notes and give yourself a practice test (write major elements of topic as vocabulary words or the name of a technique and 2 - 3 practice well as links between them problems on a blank page, close your book and notes, and try to write descriptions or answers in 15 mins). Homework due at start of class  Write 2 – 3 sentences in your notes describing what you know you New homework assigned did well on the test, and what two specific ideas or techniques you Thursday Quiz/Test still need to memorize/practice. Discuss Questions, Puzzle,  Start on Friday’s tasks or take a break. and Standards for the week Friday –  Skim the readings you are asked to do as part of your new assignment.  In your notes, write down descriptions of the 3 – 5 most important ideas from the reading.

of 23 MATH 180 PEDAGOGY & ASSESSMENT PLANS Rev: Winter 2007

 In either a separate section, or after your descriptions of ideas, write up to 10 key words and definitions that appeared in the reading. Circle or star definitions you don’t understand.  Practice the basic skills for the next proficiency test.  Take a break, then  try at least three problems from the homework to be ready for presenting on Monday. Weekend –  In your notes, write down one or more specific questions you still have about the words, skills, or concepts you’re studying.  Do any of the tasks you did not do earlier in the week.

Learning Objectives Procedural Proficiency at the Elementary Student Level  Able to accurately and efficiently perform computations and tasks expected of students in grades K-8 without the use of a calculator or notes. Procedural Proficiency at the Elementary Teacher Level  Able to accurately explain and use alternative strategies to perform computations or other tasks expected of students in grades K-8. (Some strategies will make use of a calculator.) Robust Understanding of Core Concepts Depth of understanding at highest levels of Bloom’s Taxonomy:  Able to analyze mathematical techniques or student work to describe core assumptions and ideas being used.  Able to synthesize a variety of ideas into concept maps showing the relationships between mathematical concepts and techniques.  Able to evaluate and articulate one’s own understanding and uncertainty of mathematical ideas. Increased Awareness of Own Attitudes and Behaviors About Mathematics and Learning  Able to describe one’s own attitudes and behaviors, along with how they change and their consequences for one’s future students.

Thorough Knowledge of Specific Mathematical Content At the end of this course, you should be able to: 1. explain the concept of “number” along with the key features of enumeration using systems other than base-10, 2. describe the structure of and types of numbers in the real number system, 3. represent numbers in a variety of ways, 4. describe the central idea(s) underlying each of the basic operations, 5. use the understanding of the role of each operation and the different types of numbers to correctly set up and solve problems presented in words, 6. explain how ratios, proportions, and percentages describe relationships between quantities, 7. describe differences between genuine problem solving situations and routine exercises, 8. describe problem solving strategies (like Polya’s 4-step process) and apply them to a wide variety of problems, and 9. name the primary national and state standards governing K-8 mathematics teaching, and summarize several major learning outcomes.

Details on Class Activities

 Lecture – This is an opportunity for me to highlight major concepts and provide additional explanation of ideas presented in your book or other materials.

 In-Class Exploration – Each week you will spend class time working on activities that will help you deepen your understanding of the ideas. As a potential teacher, it is not enough for you to simply mimic what someone else tells you to do; you MUST be in the habit of actively developing your own understanding of the “whys” and “hows” of new ideas and techniques. Working with others in groups can assist you with this, but be careful not to rely on others to get you unstuck.

 Weekly Assignments and Presentations – You will have a formal assignment including problems from the book on a weekly basis. This will help you practice techniques and answer questions that highlight some of the main ideas. Members of the class will be asked to present their reasoning and methods on some of these problems in class each week.

 Concept Maps – These are a standard type of diagram used to show relationships between major ideas. It will be one of the ways I help you develop, and also measure your ability to synthesize the concepts and techniques.

 Self-Reflective Writing – The best teachers and students engage in metacognition, meaning “thinking about how they think.” It provides us with an opportunity to understand why we do or don’t grasp an idea, which offers suggestions for how to adjust our learning. I will ask you to do this in small amounts on a weekly basis, and use it as a window into your understanding of the course material during your midterm and final self-evaluations.

 Student Proficiency Tests – You will have four, 10-minute proficiency tests on basic knowledge expected of the students you may teach. They will cover decimal arithmetic, fraction arithmetic, ratios/percents/proportions, and equation solving/graphing – topics included in the WEST-B screening exam that must be passed by all preservice K-8 teachers. You MUST pass each of these tests with an 80% or higher average in order to pass this class REGARDLESS of your other work in this course. You will have an opportunity to retake each of these tests at least once during the quarter.

 Teacher Proficiency Tests – These will be regular quizzes on the new ideas presented in this class. Unlike the student proficiency tests, these are NOT “high stakes” tests that by themselves can determine whether you pass the class. As a result, they also cannot be retaken to improve your score.

Topics, Outcomes, Week Daily Plan & Sections of Text 1 Introductions Monday 1/8  Have them read cover page of syllabus, write into info as I Establishing process take roll. Summ  Write introductory info to share with groups (& hand in?): The Role of an Educator o Name, where you are from. The Concept of “Number” o [Why do you want to teach?] (2.1, 2.2, 2.4) o What level do you want to teach, and why? o What’s your experience with children (age, setting)? Math Knowledge Outcomes: o What’s your definition of a very smart (or “gifted”) [MK~1] explain the concept of o child? “number” along with the key  What are the core differences in the mission of elementary features of enumeration using teachers compared to day care providers? systems other than base-10 Assign: Write about the following for in-class discussion.  Teaching – What is the purpose of a “standard” like the [MK~3] represent numbers in a o Essential Academic Learning Requirements (EALRs) or variety of ways Grade Level Expectations (GLEs)? (Compare w/ syllabus learning objectives & course components) [MK 9] name the primary o  Math – What is a “number”? For example, “eight” is more national and state standards than just a word or symbol. governing K-8 mathematics  Write two questions you have about the learning objectives teaching, and summarize and course components. several major learning outcomes Tuesday [Standards, 2.1, 2.2]  Group creation. Procedural Proficiency (Student) Student Proficiency Test #1  Hand out HW assignment with QotW, PotW, SotW on same page.  Add/Subtr decimals  [25 min] Discuss purpose of a standard Features: Essential knowledge/skill; Achievable  Mult/Div decimals o (Determined how? What fraction of people? –  Round Military analogy); Consequences if not met; Implications? Robust Understanding: o Discuss my syllabus in light of the above.  [MK1] Writing task on analyzing the concepts of  [25 min] Discuss concept of a number. “standard” and “number.” o What’s challenging about answering this question? o Not just a symbol, sound, or a specific collection – it’s Question(s) of the Week: an essential feature that links many different things.  What is a “number”? [Read first part of Ch.2 of Russell’s Introduction to Mathematical Philosophy] Puzzle of the Week: o Concept of a set and elements. Subset relationship  P. 35 2(a) (Sec. 1.2 B) gives us order relations.

Standard(s) of the Week: Wednesday [2.1, 2.2, 2.4]  GLE 1.1.1, Grade 2:  List following key terms related to “number” “Represent a number to at least o Set, subset, element 1000 in different ways, o Relation, function, one-to-one correspondence including numerals, words, o Ordinal, cardinal numbers pictures, and physical models;  Activity: Distribute chips, place 4 dots on board, ask translate between students to show me how many using their chips.

representations.” o How did they know? o Associating “set” of dots with set of objects. (1-1 corresp.)  If done by “counting,” they associated with what set? o What other sets are relevant? [Number the list] (Arabic #s – they must describe symbol to draw; other lang; other numerals; other collections of objects!)  Complexity of what’s being asked of students – show number of links as number of different sets grows.  Step back and highlight another layer of complexity: using chips in front of you, show me “two.” o Does “two” mean “two chips” or does it mean “chip two?” o Concept of showing an amount versus showing placement. (Cardinal vs. ordinal)  Formal definition of terms: set, subset, element, cardinal #, ordinal #.  Highlight signaling of cardinal number through use of number word as adjective, but we eventually drop the noun it modifies as part of the process of abstraction.  Diagram ways of enumerating the set of four dots to arrive at conclusion it shows “four.” o Correct way: Map each dot to one word in counting chant (order of chant matters, assignment doesn’t!) o They propose incorrect ways (saying two names for one dot – double-counting dot – relation only; saying one name for two different dots – repeating oneself – function)  Define relation, function, 1-1 correspondence  Articulate benefits of realizing these elements of learning – constructivist approach of honoring logic of what individual is doing while honing in on erroneous assumptions.  Finish with intro to different number systems.

Thursday  [10 min] SPT #1 – Decimals  20 min HW, 10 on Presentations of 2.2 (p. 67) #2, 5, 9ac o #5: Circle subset, equiv to subset, position in counting chant  Distribute new assignment.  [15 min] General questions, “of the week” stuff  [18 min] Chip abacus and different bases

Topics, Outcomes, Week Daily Plan & Sections of Text 2 The Concept of “Numbers” Monday 1/15 Features of Number Systems (2.2, 2.3) Tuesday [2.2, 2.3] Short  Chip abacus concept – use two colors of chips. Math Knowledge Outcomes: o Provides tangible representation of number and visual [MK1] explain the concept of Summ o meaning for the different columns. “number” along with the key  Provide them with a collection of chips. features of enumeration using  Ask them to describe how many chips they have (pooled in systems other than base-10 pairs). o [Suppose a group says “thirty six”] Ask how to write o [MK~3] represent numbers in a that in symbols, and how I’m supposed to recognize variety of ways the “3” in what they’re telling me. [Get at bundling concept.] Procedural Proficiency (Teacher) o Have all groups reorganize their chips in this manner. Teacher Proficiency Test #1 o Ask if “36” is the same as “63” – they must prove it.  Discoveries:  Identifying and classifying o Order of symbols matters. [Positional system] error types in counting o Value of one symbol is multiplied by the “bundle”  Analyze a child’s number size of 10 [Multiplicative system] system for recognizable o Cumbersome because of having to keep all the chips features. around  Want to use fewer objects – have each person create 3- Robust Understanding: column abacus on a sheet of paper: columns are “units,”  [MK1] Next assignment “first grouping/bundle,” and “second grouping/bundle.”  [MK3] Construction/  Start with 25 chips in the “units” column. Go through articulation of own number exchange process. They try with 36. system (Synth)  Explore Babylonian system – describe thirty six chips, then 142, then 471. Compare abaci with base-10, and the Question(s) of the Week: symbols being used. (Can use “red” = 10 and “white” = 1  If a child memorizes the for the Babylonian numerals.) Repeat with base-3 -> alphabet in reverse order (z y x emphasize naming like “two-one-one” not “two hundred w …), does it have a significant eleven” effect on his/her ability with o “Base n” – number of numerals, size of grouping language? What if a child memorizes a list of numbers in  Brainstorm other bundling representations (pennies, dimes, reverse order (100, 99, 98, …)? dollars; poker chip color)  Discuss the features/cognitive issues children must grasp Puzzle of the Week: to handle our number system [Concept map details]  Page 19, prob. 18 in Sec. 1.1 B o Order matters o One symbol represents a bundle of a certain size Standard(s) of the Week: o Bundle size is implicit  GLE 1.1.1, Grade 3 evidence of o Can make bundles of bundles learning: “Represent and o Each symbol is a specific quantity discuss place values of digits of o Both adding and multiplying taking place whole numbers [up to 10,000] o Absence of 0 as placeholder allows ambiguity using words, pictures, or o Other? [Issues related to “number” or numeration] numbers.”

Wednesday [2.2, 2.3]  [15 min] Groups discuss HW/post solution attempts Preferred: 2.2A #11 cdeil; 2.2B #15 – explain pattern, 52,603 2.3A 4 – 9

 [15 min] Groups present their work, I grade, then comment  Definitions of classification terms: o Positional – the physical location of a symbol affects the value of the overall number o Place valued – each location in the number system changes the meaning of all symbols placed there in a consistent way o Additive – the overall value of a set of symbols comes from adding the values associated with each symbol o Multiplicative – the value of a symbol sometimes represents a fixed multiple of that symbol (can be determined by place value or accents) o Subtractive – the overall value of a set of symbols comes from subtracting the values associated with each symbol o Has zero – There is a symbol which represents “none” of a particular amount.  I present a number system: “Celeste” o Up to 4 ticks (s, st, sta, star) at compass points, then cap for crosshairs with concatenation of name o 8 is a ring (mun), add a new tick for each 8 (muna, muni, muno, munae) up to 40. Two of these is a double-ring, called “sol.” o Analyze this system, then ask students to present their own.

Thursday [2.1-2.4]  HW #2 turned in; #3 assigned  Begin discussion of concept map – refer back to GLE. o Brainstorm key terms, skills, concepts o Group similar ones o Create networks of ideas  Explain “Standard of the Week” in your own words  Question/Puzzle of the Week  [10 min] TPT #1 – Number systems/Numeration o ID errors in counting patterns o Compare/contrast a child’s number system to the Hindu-Arabic base-10 system, highlighting features the child’s system shares. (Constructivist POV)

Topics, Outcomes, Week Daily Plan & Sections of Text 3 Types of Numbers – Purpose and Monday [2.2] 1/22 Representations  Return/discuss TPT #1 (2.2, 3.3, 6.1, 7.1, 8.1, 9.1, 9.2)  [15 min] Groups discuss HW/post solution attempts Preferred: Summ Math Knowledge Outcomes: o [MK~2] describe the structure  [15 min] Groups present their work, I grade, then of and types of numbers in comment the real number system,  Guided discussion – begin by brainstorming on: o [MK3] represent numbers in a o What are numbers for? variety of ways o Why do we have numbers, and not just stop at describing quantity with “none,” “one,” “two,” Procedural Proficiency (Student) “some,” and “many?” Student Proficiency Test #2 o Examining above questions leads to understanding what motivates a need to learn the concept – at what  Each of the operations, some stage/what view of the world would cause children to mixed numbers; Reducing find numbers meet a need they have? . Precision, standardization, communicating with Robust Understanding: someone who is not present (can’t be shown)  [MK1] Written explanation of . Response to cultural values: individuality, concept of number and ownership, competitiveness, “more is better” (no features of numeration (Anal) end to this, unlike with “many”) Concept Map (Synth) Tuesday [2.2, 6.1, 7.1] Question(s) of the Week:  Guided discussion – begin by brainstorming on:  What are the advantages of o Why does it make sense to call 1, 2, 3, … “natural #s” fractions (especially when o How does adding 0 to this list make it “whole?” compared to decimals) that o What are the roles of zero – why is zero given make them worth the “number” status and not just left as the word “none?” frustration of trying to . Placeholder role; consistency in symbolic comprehend? representations of numbers and calculations; comparison with other sets (“showing” Puzzle of the Week: emptiness, the “0” symbol being an empty  p.246 #26 in Sec. 6.1B circle); indicating a reference point ($, temp) o Why do we need fractions or decimals, and is there Standard(s) of the Week: any benefit to having both?  GLE 1.1.1, Grade 6: . Representing subdivision/pieces; Dec – “Understand the concept and standardized, easy comparison, linear models; symbolic representations of Frac – compact symbols, highlights relationship integers as the set of natural between two quantities, set models numbers, their additive . Frac complexity – showing two quantities with inverses, and 0.” Grade 7: their own meaning, plus a new meaning together “Understand the concept and … (c.f. analogies) of fractions, decimals, and . Response to cultural values: are pieces integers.” considered of value (herders, Nat. Amer., children – 2 quarters is better than $1)

Wednesday [3.3, 8.1, 9.1?]  SPT #2 tomorrow – fraction arithmetic (includes mixed numbers), reducing  [15 min] Groups discuss HW/post solution attempts Preferred: 6.1A #2c, 3, 4d, 18ab (&cd?); 7.1A#2b, 7c

 [15 min] Groups present their work, I grade, then comment  Discussion of concept map  Guided discussion – begin by brainstorming on: o What is the purpose of exponents – why do they exist? How does multiplication itself serve a related purpose? [Increasing shorthand/compression of +] o What is a negative number? Can you touch one? Where/why do they come up? . “Does color have a negative?” Same meaning? . Take away too much and want to record that amount – words (debt, loss) vs. symbol; Notion of opposites; only exist in opposition to something else – need a reference direction o Brainstorm quantities, determine if there is a “negative” of it and what conveys direction . Ground level; possession/debtedness o Annihilation concept – Papy minicomputer & chips . Significant understanding of world – application to electricity; weather (pressure); “nature abhors a vacuum”

Thursday [8.1, 9.1, 9.2]  HW #3 turned in; #4 assigned [concept map for number systems/preliminary self-eval?]  More on annihilation w/ minicomputer  Guided discussion – begin by brainstorming on: o What is a rational number? What numbers do you personally use that are NOT rational numbers? o Level of completeness of number system – closure of operations. o Other types of numbers and their reason for existence/purpose.  Explain “Standard of the Week” in your own words  Question/Puzzle of the Week  [10 min] SPT #2 – Fraction arithmetic  [10 min] SPT #1 (for those who hadn’t yet)

Topics, Outcomes, Week Daily Plan & Sections of Text 4 Pattern Recognition; Monday [1.1, 5.1] 1/29 Structure of Numbers  [10 min] Groups discuss HW/post solution attempts (1.1, 1.2; 5.1, 5.2) Preferred: 1.1 A #4, 6/9, 10, 13/19

Summ Math Knowledge Outcomes:  [15 min] Groups present their work, I grade, then o [MK2] describe the structure comment of and types of numbers in the  [15 min] “Problem Solving” real number system o Is being a “good problem solver” an asset? Why or o [MK~5] use understanding of why not? the role of each operation and . Does the answer depend on the problems being the different types of numbers “math” or “real life?” How are these different? to correctly set up and solve [Definition of “classroom math” vs. data analy.] problems presented in words o What are the attributes of a “good problem solver?” o [MK~7] describe differences . Persistent, patient, creative, resourceful, willing between genuine problem to take risks/try, observant of patterns, adaptable, solving situations and routine self-reflective, organized(?) exercises o Where are we taught this – both informally and o [MK~8] describe problem formally? solving strategies (like Polya’s . Why in math class? [Logos training – context- 4-step process) and apply them independent strategies for data-gathering and to a wide variety of problems reasoning] . Why use seemingly irrelevant number and shape Procedural Proficiency (Teacher) puzzles rather than real-world stuff like reducing Teacher Proficiency Test #2 littering/the amount of garbage we produce; overpopulation? [Can experiment, manipulate,  test, see results, control variables, limit emotional distraction – pathos] Robust Understanding: o What are ways people approach a problem to which  [MK8] Written descriptions there is no answer already known? of problem solving strategies (Analysis of ideas being used) Tuesday [1.1, 5.1]  [30 min] Problem-solving strategies applied to HW, Question(s) of the Week: WASL, and Sudoku (highlight Tetris tetrominos on p.10)  Why are prime numbers so o “Systematic ‘Guess & Check’” – Empirical/deductive important? (scientific) approach [generates patterns to observe] o Model the situation – draw a picture, use Puzzle of the Week: symbols/variables in place of quantities  Focus of entire assignment o Use cause-and-effect (deductive) reasoning

Standard(s) of the Week:  [30 min] Patterns in numbers – divisibility  GLE 2.1.1, Grade 4 evidence of o What does it mean for something to be divisible by learning: “Generate questions something else? (Can’t everything be divided?) that would need to be answered o Draw a picture strategy: Arrays to visualize numbers in order to solve the problem.” and ID groupings (both sums and products)  GLE 2.2.3, most grades: o Model situation with symbols: From dots to factor “Apply a variety of strategies to trees construct solutions.” o Systematic exploration/recognizing patterns: Sieve of Erathostenes; Uniqueness of prime factorization

[F.Thm. of Arith.]

Wednesday [1.2, 5.1, 5.2]  TPT #2 tomorrow – WASL problems + analysis  [15 min] Groups discuss HW/post solution attempts Preferred:

 [15 min] Groups present their work, I grade, then comment  Discussion o Inductive vs. deductive reasoning – pros and cons . Inductive: Child needs to touch burner to know it’s painful . Deductive: Child sees burner make hot food, knows hot food sometimes burns, concludes burner is unpleasantly hot o Power of deductive reasoning: demonstrating certainty of that which cannot be shown through experiment . p. 221’s proof that the number of primes is infinite o Demonstrating deductive nature of argument proving divisibility rules

Thursday [5.2]  HW #4 turned in; #5 assigned [self-eval]  Discussion: o What is the value of factoring? . Uniqueness of representation o Greatest common factor/Least common multiple – which is larger than the numbers you start from? (Linguistic confusion) o Algorithms for GCF, LCM

 Explain “Standard of the Week” in your own words  Question/Puzzle of the Week  [20 min] TPT #2 – WASL problems + analysis

Topics, Outcomes, Week Daily Plan & Sections of Text 5 Concept of the operations of + Monday [3.1] 2/5 and – with whole numbers,  [10 min] Groups discuss HW/post solution attempts decimals, and fractions Preferred: (3.1, 6.2, 7.2) Summ  [15 min] Groups present their work, I grade, then Math Knowledge Outcomes: comment o [MK~4] describe the central  Collect HW idea(s) underlying each of the  [20 min] TPT #2 – WASL problems + analysis basic operations  [15 min] Lecture/Discussion: o [MK~5] use understanding of o What is the essential idea behind addition? the role of each operation and . Put together, join, “and” the different types of numbers o How does this help us recognize English words or to correctly set up and solve phrases that reflect addition? problems presented in words o How does addition relate to counting? . Put together approach – count each set, smash Procedural Proficiency (Student) together, count result; only physical relationship Student Proficiency Test Retake between addends and sum . “Counting on” approach – shows continuity of Robust Understanding: sum through counting; challenge is duality of  Evaluate and articulate one’s tracking “four more places” in the number chant own understanding and uncertainty of mathematical Tuesday [3.1, 6.2, 7.2] Lecture/Discussion: ideas (Eval) o How do the meanings of addition reveal themselves in  Able to describe one’s own different types of representations? attitudes and behaviors, along . Set model – smash idea with how they change and . Measurement model – counting on their consequences for one’s . Fractions – not every single object is a “one”: future students. (Anal, Eval) challenge of changing unit size . Integers – again, objects have different attributes Question(s) of the Week: o What is the essential idea behind subtraction?  What is the difference between . Take away; (directed) difference; “undo” the concept of an operation and addition (inverse, missing addend) the skill for using it? . Forwards questions & backwards questions o How does this help us recognize English words or Puzzle of the Week: phrases that reflect subtraction (rather than division)?  p. 257 #26 in Sec. 6.2 A o How does the meaning of subtraction reveal itself in different types of representations? Standard(s) of the Week: . Sets – take-away; comparison; missing-addend  GLE 1.1.5, Grade 2: . How does the significance of the minuend and “Understand the meaning of subtrahend change across approaches? [Prelude addition and subtraction and to noncommutativity] how they relate to one another” . Which of these approaches facilitates a transition  GLE 1.1.5, Grade 2 evidence of to subtraction of negative numbers? [Flaw is learning: “Use joining, different attribute – cannot take away/compare separating, part-part-whole, and red with white; introduce “adding zero” concept] comparison situations to add . Measurement model – any better? Start with 2-5 and subtract.” “Illustrate and explore what model suggests [“put together” addition and subtraction using with new objects] words, pictures, and/or . Directed difference – How does subtraction need numbers.” Page 12 of 23 MATH 180 PEDAGOGY & ASSESSMENT PLANS Rev: Winter 2007

to be redefined to understand 2 - 5?

Wednesday [3.1, 6.2, 7.2]  SPT Retake tomorrow  [15 min] Groups discuss HW/post solution attempts Preferred:

 [15 min] Groups present their work, I grade, then comment  Lecture/Discussion: o What is the meaning/significance of the properties of addition and subtraction? . Associativity/Commutativity – changing order of operations vs. changing order of symbols . Prove these with chips – how? . Identity (root “identical”) . Closure property & children’s error with computing 2-5. (Absence of such numbers) . How do we know 2-5 is not 3? . How to guide a child toward discovery of negative numbers

Thursday [3.2]  HW #5 book work turned in; #6 assigned  Leftover discussion of previous days’ topics o Addition facts & strategies o Operations in other bases  Explain “Standard of the Week” in your own words  Question/Puzzle of the Week  [10 min] SPT Retake

Topics, Outcomes, Week Daily Plan & Sections of Text 6 Concept of the operations of  Monday [3.2] 2/12 and ÷ with whole numbers,  Collect Self-Evals decimals, and fractions  [10 min] Groups discuss HW/post solution attempts (3.2, 6.3, 7.2) Preferred: 3.2 #4, 5dg, 6cd, 9bc, 10ab, 14, 15; 3.2 B #13ab Summ 3.3 #8, 9, 11 Math Knowledge Outcomes:  [15 min] Groups present their work, I grade, then o [MK4] describe the central comment idea(s) underlying each of the  Lecture/Discussion: basic operations o What is the essential idea behind multiplication? o [MK~5] use understanding of . Repeated addition the role of each operation and o What English words or phrases reflect multiplication, the different types of numbers and how do they express the repeated addition idea? to correctly set up and solve o How do you show 3 x 2 problems presented in words . With a set model? [Roles of each factor?] . With a measurement model? [Roles of each Procedural Proficiency (Teacher) factor?] Teacher Proficiency Test #3 o New approaches arising from multiplication . Why do we connect areas/dimensions (“2 by 4”)  Identifying appropriate models with multiplication? [Array approach] and approaches to explain . Probabilities of events – counting number of situations given in words. ways two dice can be rolled [Cartesian product] . Non-uniform events/conditional probability – 3 Robust Understanding: cone options, two scoops, 2 syrups if in bowl  [MK4] Analysis of student [Tree diagram approach] work showing idea behind an o Which of these methods extends to decimals and operation (Anal); Concept fractions? Map (Synth) Tuesday [3.2, 8.2] Lecture/Discussion: Question(s) of the Week: o How do you show 3 x -2, -3 x 2, or – (3 x 2)?  How are the basic operations . If 3 x 2 means “put together” three sets of related to each other? What are “positive” objects twice, how about using “take two different ways to pair up away” and “negative”? the operations, and what is the . Use pattern of 3 x 2 = 6, 3 x 1 = 3, 3 x 0 = 0, etc. logical basis for those like with exponent law (can use commutativity groupings? of mult with natural numbers to decrease first factor similarly) – Sec. 8.2 Puzzle of the Week: o What properties of multiplication can be deduced  p. 233 #25 in Sec. 3.2 A from array models? . Commutativity; Associativity; Distributive Standard(s) of the Week: property  GLE 1.1.5, Grade 3: . How does this reduce the amount of “Understand the meaning of memorization required for “multiplication multiplication and division of facts”? whole numbers” o What is the essential idea behind division?  GLE 1.1.5, Grade 3 evidence of . Partitioning/Grouping; Repeated subtraction; learning: “Show and explain “undo” multiplication (inverse, missing factor) the relationship between o How do we recognize situations requiring division? multiplication (division) and o How does the meaning of division reveal itself in repeated addition (subtraction)” different types of representations? “Illustrate multiplication and Page 14 of 23 MATH 180 PEDAGOGY & ASSESSMENT PLANS Rev: Winter 2007

division using words, pictures, . Set models: Partitioning – Partitive division is models, and/or numbers” when the number of partitions (groups) is known, but size of each is not; Measurement is Puzzles for later when the amount to be placed in each group is  p.257 #26; Fibonacci; Pasc Tr. known, but the number of groups is not . Measurement models – Repeated subtraction & the division algorithm . Missing factor & the rules for zero o Note that addition and multiplication have one name for the components – addends or factors – while subtraction and division have two [reflects (non)commutativity]

Wednesday [3.2, 6.2, 7.2]  TPT #3 tomorrow – meanings of operations  [15 min] Groups discuss HW/post solution attempts Preferred:

 [15 min] Groups present their work, I grade, then comment  Lecture/Discussion: o Which of the whole number perspectives carry over to multiplication and division of decimals and fractions? . How can we represent it physically or visually? . What modifications are required? o Why is division by fractions so cognitively challenging? [Changing reference units] o Additional discussion of properties or representations.

Thursday [3.2, 6.2, 7.2, 8.2]  HW #6 turned in; #7 assigned  Leftover discussion of previous days’ topics (division by negatives?)  Explain “Standard of the Week” in your own words  Question/Puzzle of the Week  [10 min] SPT Retake

Topics, Outcomes, Week Daily Plan & Sections of Text 7 Algorithms for Computation: Monday [] – No class 2/20 Estimation, + and – with wholes (4.1, 4.2, 4.3) Tuesday [4.1] Lecture/Discussion: Short o Mental math techniques – clever uses of the Math Knowledge Outcomes: associative, commutative, and distributive properties [PPTL] … use alternative Summ o . Lists of summands & benefits of assoc/commut strategies to perform . Compensation (equal additions) for doing computations … subtraction . Multiplicative compensation and powers of 10 Procedural Proficiency (Student) o What is estimation, and why do we do it? Student Proficiency Test #3 . Is it important? When/where? . Is it more important than exact calculations?  Graph linear and nonlinear . When is it okay to do? equations by point-plotting . What consequences does estimation have for the  Solve a linear equation learning of number facts? involving parentheses o Why are there several ways to estimate? . What are the tradeoffs? Robust Understanding: o Comparison of strategies:  [MK4] Classroom presenta’ns . Range estimation – gives guaranteed low and on algorithms for +, – (Anal) high (high requires increasing even shorter numbers by 1 in the largest column in problem) Question(s) of the Week: . Front-end: one-column (w, w/o adjustment),  What are the advantages and two-column & tradeoffs disadvantages of the standard o Rounding vs. truncating algorithm for the addition of whole numbers? Wednesday [4.2]  Lecture/Discussion: Puzzle of the Week: o Algorithm for adding whole numbers  p. 185 #28 in Sec. 4.2 B . What’s confusing about it? . How can we show it physically? [Chip abacus] Standard(s) of the Week: . What are some bridges between the objects and  GLE 1.1.6, Grade 3: “Use symbols? computational procedures for  Expanded form of numbers addition and subtraction of  Intermediate algorithm 1 [What confusions whole numbers” does this address? Other advantages?]  GLE 1.1.6, Grade 3 evidence of . Pros & cons of Int. Algor 2, Std. Algor, Lattice learning: “Explain and apply method [include motor skills, visual perception] strategies or use procedures to o Algorithm for subtracting whole numbers add three 2-digit or two 3-digit . What’s confusing about it? numbers, and/or subtract . How can we show it physically? [Chip abacus] numbers with 1, 2, or 3 digits.” . What are some bridges between the objects and symbols? [Expanded form again] . Pros & cons of “subtract-from-the-base” o Connections with polynomials  SPT #3 tomorrow – graphing and solving equations

Thursday [4.3]  [15 min] Groups discuss HW/post solution attempts Preferred: 4.1A#6c (iii, iv), 28; 4.2A#2, 4a, 5a; 4.3A#1d,2c,4b

 [15 min] Groups present their work, I grade, then comment  HW #7 turned in; #8 assigned  Lecture/Discussion: o Algorithms for + and – in base-5 . At what steps do we experience significant slowdown? What does this say about the building blocks for learning the base-10 algorith. o Tom Lehrer’s “New Math” song  [10 min] SPT #3 – graphing and solving equations  Explain “Standard of the Week” in your own words  Question/Puzzle of the Week  Alternative: Discuss the full spectrum of job duties of a faculty member o Teaching load o Official office hours + unofficial hours o Prep time + Grading o Phone + Email o Committees – hiring, textbook (& publishing cycle), curriculum devel, advising, letters of rec., student awards, grants/special projects, scheduling, tutor hiring/oversight, PT faculty hiring/oversight, student complaints o Amount of sleep, tradeoffs, enjoyment, compensation o Don’t expect much different in first couple of years of teaching (high attrition rate; student teachers taking TP with them)

Topics, Outcomes, Week Daily Plan & Sections of Text 8 Algorithms for Computation: +,– Monday [4.2, 6.2, 7.2, 8.1] Lecture/Discussion: 2/26 w/ Dec, Frac; × and ÷ with wholes o How do the algorithms for + and – change for (4.2, 6.2, 7.2) . Fractions  Why do we need a common denominator? Summ Math Knowledge Outcomes: [Counting same units as basis for addition] o [PPTL] … use alternative . Decimals – What changes in the algorithms? strategies to perform  Alignment – why? Quick fix for errors? computations … . Signed numbers – Failure of algorithm o Algorithm for multiplying whole numbers Procedural Proficiency (Teacher) . What’s confusing about it? Teacher Proficiency Test #4 . How can we show it physically? [Chips/Array] . What are some bridges between the objects and  Analyze a student’s attempt to symbols? use a common algorithm and  Intermediate algorithm 1 [What confusions identify the error pattern as does this address? Other advantages?] well as suggest an alternative Tuesday [4.2] Lecture/Discussion:  Effectively compute with a o Algorithm for multiplying whole numbers nontraditional algorithm . What are some bridges between the objects and symbols? Robust Understanding:  Expanded form of numbers – Connection  [MK4] Classroom presenta’ns with FOIL: do 27  14 & (2x+7)(1x+4); on algorithms for ×,÷ (Anal) show mult grid on p. 172 . Pros & cons of Int. Algor 2, Std. Algor, Lattice Question(s) of the Week: method [include motor skills, visual perception]  When computing in a different o Practice multiplication in base 4 base number system, which o Algorithm for dividing whole numbers steps of each algorithm are . What’s confusing about it? most difficult or confusing? . How can we show it physically? [Chips, bundle] How does your level of . What are some bridges between the objects and familiarity with the addition & symbols? multiplication facts tables  How does it reflect the “repeated affect your ability to compute? subtraction” concept?  Where’s the “grouping” idea? Puzzle of the Week: Wednesday [4.2, 4.3]  p. 185 #3 in Sec. 4.2 “Problems  TPT #4 tomorrow – Algorithms for Operations for Writing”  [15 min] Groups discuss HW/post solution attempts Standard(s) of the Week: Preferred: 4.2A#11, 31, B#20b, 23, 31; 4.3A#10c  GLE 1.1.7, Grade 4: “Apply strategies and use tools  [30 min] Groups present their work, I grade, then appropriate to tasks involving comment multiplication and division of whole numbers.”  Lecture/Discussion: Algorithm for dividing whole numbers  GLE 1.1.7, Grade 4 evidence of o learning: “Select and use . What are some bridges between the objects and appropriate tools from among symbols? mental computation,  Literal repetition (p. 181 #20) estimation, calculators,  Incorporating multiplication as shortcut – manipulatives, and paper and no concern about how close estimate is pencil to compute in a given [scaffold method] situation  Refining estimation Page 18 of 23 MATH 180 PEDAGOGY & ASSESSMENT PLANS Rev: Winter 2007

Thursday [4.3]  HW #8 turned in; #9 assigned  [20 min] TPT #4 – Algorithms for Operations  Leftover discussion of previous days’ topics, or multiplication and division in other bases  Explain “Standard of the Week” in your own words  Question/Puzzle of the Week

Topics, Outcomes, Week Daily Plan & Sections of Text 9 Algorithms for Computation: ×,÷ Monday [6.3, 7.2] Lecture/Discussion: 3/5 w/ Dec., Frac.; Ratios, Percents, o How do the algorithms for  and ÷ change for Proportions . Decimals – What changes in the algorithms? (6.3, 7.2, 7.3, 7.4)  Alignment – why not important? Effect of Summ adding zeros? Quick fix for errors? Math Knowledge Outcomes: . Fractions o [PPTL] … use alternative  Why don’t we need a common strategies to perform denominator? computations …  Straight-across division strategy o [MK~6] explain how ratios,  Developing trust in algorithm via patterns proportions, and percentages . Signed numbers – Failure of algorithm describe relationships between o Terminating/nonterminating decimals & the quantities relationship with fractions

Procedural Proficiency (Student) Tuesday [7.3] Lecture/Discussion: Student Proficiency Test #4 o What are ratios for? [Comparisons] . What are different ways of comparing two  Recognize RPP quantities?  Solve proportions  Inequalities (more/less than)  Convert percents  Difference (subtraction – amount more)  Solve word problems  Factor (multiplier – # of times more)  [MK4] Creating lesson on . Which situations are best served by each of these algorithms for × (Anal) approaches? (Precision; q’tity vs. relative size) o Ratio notation: three ways to represent – 6 to 5, 6:5, Question(s) of the Week: 6/5; highlight significance of order when it comes to  When computing with parts of interpretation – class ratio of F:M vs. M:F & effect numbers, which calculations o Connection between ratios and concept of fairness – are most easily done with why isn’t it enough for things to simply be required to decimals, and which are easiest be equal? [Ethnicity & teachers example] with fractions? o Concept of proportions – Puzzle of the Week: . Equivalent ratios/fractions (highlighting fairness  p. 255 #22 in Sec. 6.2A; p. 257 requirement by comparing preexisting ratios, #26 in Sec. 6.2B also bridges between old ideas and new ones) Standard(s) of the Week: . Scaling of data (applying old information to a  GLE 1.1.6, Grade 6: “Apply new setting – teacher:student ratio for school) strategies and use computat’nl . Analogies (same cognition & notation as procedures to add and subtract “Grass : Sky :: Green : ______”; Why so non-negative decimals and hard?) fractions.” o Solving proportions symbolically – first with whole  GLE 1.1.8, Grade 7: “Apply numbers to reinforce equivalent fractions idea and estimation strategies involving highlight “cross-multiply” shortcut; then with addition and subtraction of decimals integers and the four basic o Constructing proportions for a described situation operations on non-negative  Suppose your car travels approximately decimals and fractions to 320 miles on 14 gallons of gas. How much predict results or determine gas would be required to make a 581-mile reasonableness of answers.” trip?

. “Want, Know, Relationship” strategy . Role of finding pairwise relationships between the known & unknown information . Using pairs to create proportion – one pair dictates a single fraction, and a second pair dictates placement of #s in second fraction o Judging equivalence of proportions ("Have I set it up right?") – identifying when proportions give the same or different answers (equality of diagonals; pairs) o “Cross multiply” vs. “Multiply across” vs. setting up the problem horizontally & vertically Wednesday [7.3]  SPT #4 tomorrow – Ratios, Percents, Proportions  [15 min] Groups discuss HW/post solution attempts Preferred:

 [30 min] Groups present their work, I grade, then comment  Lecture/Discussion: o Practice setting up ratios (rates) & proportions . [Fairness] In the 2001/2 academic year, Highline had 2,006 Asian/Pacific Islander students and 1,550 students of African American descent. If a single class had the same mix as the overall campus, how many Asian/PI students would be expected in a class with 5 AfrAm students? . [Scaling] On a map, the distance between two towns is 2¾ inches. The legend shows a half- inch represents 30 miles. How far apart are they? . [Sampling] Researchers at the Department of Fish and Wildlife decide they need to measure the fish population in a particular lake. They first go to the lake and catch 41 fish, which they tag and release back into the lake. One month later, the researchers return and catch 58 fish, of which 16 are tagged. How many fish might the researchers estimate are in the lake? Thursday [7.3/4]  HW #9 turned in; #10 assigned o What is a percentage? How is it the same as, or different from a ratio? . Ratios – Comparison of two quantities measured in the same units; Compares part to part or part to whole; Clue word of “to”; Written as frac or : . Rates – Comparison of two quantities measured in different units (like distance, time, money, volume, etc.); Compares part to part; Clue word of “per”; Written as a frac or : . Percentages – Standardized comparison of two quantities in the same units; Compares part to whole; Clue word of “of”; Written frac/100 or %  Explain “Standard of the Week” in your own words  Question/Puzzle of the Week Page 21 of 23 MATH 180 PEDAGOGY & ASSESSMENT PLANS Rev: Winter 2007

 [10 min] SPT #4 – Ratios, Percents, Proportions

Topics, Outcomes, Week Daily Plan & Sections of Text 10 Ratios, Percents, Proportions Monday [7.4] Lecture/Discussion: 3/12 Problem Solving o Review (oral) – What is a percentage? How is it the (7.3, 7.4; 1.1, 1.2) same as, or different from a ratio? o Advantages of ratios? [Can reveal original raw Summ Math Knowledge Outcomes: numbers; highlights multiplier] o [MK5] use understanding of the o Advantages of percentages? [Standardized, so easier role of each operation and the to compare] different types of numbers to o Basic percents vs. percentage change vs. percentage- correctly set up and solve point change problems presented in words . Basic – I make $3000 a month and my rent is o [MK6] explain how ratios, $600. What percent of my income goes to rent? proportions, and percentages . % chng – My rent rose from $580 to $620. This describe relationships between is an increase of what percent? quantities . The president’s approval rating went from 51% o [MK7] describe differences to 32%, a drop of 19%. between genuine problem o Word-problem set-up stratgies: 15 is 83% of what? solving situations and routine . “Equation translation” – “is” becomes =, “of” exercises becomes  o [MK8] describe problem . “‘is’ over ‘of’” – translate to proportion solving strategies (like Polya’s Shortcoming of these methods – they’re tricks that 4-step process) and apply them obscure the reasoning; how do they translate to to a wide variety of problems solving “real world” word problems . “Part, Whole, %” strategy Procedural Proficiency (Student) o Terminating/nonterminating decimals & the Student Proficiency Test Retake relationship with fractions

Robust Understanding: Tuesday [7.4] Group project work time  [MK4] Creating lesson on o Check groups’ ideas algorithms for × (Anal) o Ask about components of project, division of labor o Their questions for me & hypothetical student Question+Puzzle of the Week: questions  (Resolving confusion over describing parts using improper Wednesday [7.4, 1.1, 1.2] fractions or ratios)  SPT #3 or 4 retake tomorrow Standard(s) of the Week:  Hand out final self-assessment assignment  GLE 1.1.4, Grade 6:  [15 min] Groups discuss HW/post solution attempts “Understand the concepts of Preferred: ratio and percent.”  GLE 1.1.4, Grade 6 evidence of  [30 min] Groups present their work, I grade, then learning: “Write or show and comment explain ratios in part/part and  Lecture/Discussion: part/whole relationships using o Practice with word problems words, objects, pictures, models, and/or symbols.” Thursday []  HW #10 turned in; reminder about project, self-eval  [10 min] SPT #3 or 4 Retake  Explain “Standard of the Week” in your own words  Question/Puzzle of the Week