Session 4S: April 28Th, 2021

Welcome Secondary Educators!

P‐12 Mathematics Professional Learning

Session 4 – Secondary 1:30 –2:30 Stacks of Cups (These items were given to all grades 6 ‐10)

What is the height of one cup? (in cm)

Explain how you figured out the height of one cup. From an equation solving perspective to a functions perspective.

The most important change in secondary grades. From the way we treat proportional relationships in grades 6 and 7, to the way we use mathematics to answer questions about linear, exponential, quadratic and periodic phenomena in algebra through Calculus and/or data science.

ENH are the responses only of the students in Enhanced courses. Today’s Diagram

Situation

> = Function Equation Iconic viewpoint

Situation

> = Function Equation

input  output expression = expression Iconic viewpoint

Situation

> = Function Equation input  output expression = expression

x  2x ‐ 12x ‐ 1 =3 CIconicviewpoint

Situation

> = Function Equation

input  expression = expression output 2x ‐ 1 =3 x  2x ‐ 1 y = 2x –1 Function Equation y = 2x –1 2x ‐ 1 =3

6 6 y y y = 2x-1

4 4 y = 2x-1 y = 3

2 2 x x

5 10 5 10 2

Solution: x =2 Stacks of Cups (These items were given to all grades 6 ‐10)

Define two functions that model this situation.

Breakout rooms. Summary Situation

> = Function Equation t  3t 3t = 40 input  expression = expression

output v 100 120

v 80 v = 96 100

80 60 v = 3 t 60 40 v = 3 t 40 20 20 t t 10 20 30 32 40 10 20 30 4 Difficulties often come from bugs in prior knowledge How to debug prior knowledge

• Repetition inefficient and ineffective; does not produce durable learning, improvements are fleeting • Debugging is efficient and durable • Recognize mistake in example work of other student • Explain mistake and how to fix it • Explain how to avoid that mistake in future • Try it • Connect to and reflect on underlying principle in the example work AlgebrabyExample

SERP Developed by HS math depatments from high end districts including Evanston, Ill.; Ann Arbor, Mich; Arlington, Va; Shaker Heights, OH; Madison, WI. Working with Carnegie Mellon learning scientists. Free https://www.serpinstitute.org/algebra‐by‐example Teaching Moves

Give each student a composition book or spiral notebook. • “Each student is their own first audience, partners are the second audience and the whole class is the third audience and I am not the audience, I am the teacher.” • “Each of your notebooks has your own thinking AND ideas you liked that you got from other students. Write their names next to their idea you liked.” • “ Revise your notebook when you revise your thinking…everyday, I hope! But don’t erase, line out” • Make math more like other subjects and less peculiar. Re‐use notebook routines from other subjects • The day before a quiz or test, direct students to prepare their notebooks so they can use them during the test. Have them lay out the most important ideas and examples on a few fresh pages. Encourage students to go find a good idea in the notebooks of others.

• 5 minutes a day is more effective than 10 or more minutes at a time • Like language learning, best learned in use • quick “etudes” better than pull outs • Stations, menus: games, tasks, etudes • Debugging is best in meta‐cognitive action e.g. number talks, student analysis of fictitious students errors, draft/revision

philip daro, all rights reserved. do not copy or distribute 7/29/18 18 without permission of the author. Explain the mathematics when students are ready • Toward the end of the lesson • Prepare the 3‐5 minute summary in advance, • Spend the period getting the students ready, • Get students talking about each other’s thinking, • Quote student work during summary at lesson’s end • Engage students in formulating summary • Students write summary in notebooks, use draft‐feedback‐revise for highest priorities

philip daro, all rights reserved. do not copy or distribute 7/29/18 19 without permission of the author. Core task: prepare explanations the other students can understand Need to pull opinions and intuitions into the open: make reasoning explicit Make reasoning public The more sophisticated your thinking, the more challenging it is to explain so others understand Students Explaining their reasoning develops academic language and their reasoning skills

Think‐pair‐share starts with each individual’s thinking, but then with each succeeding step, students are learning the thinking of others. All the students’ thinking is moving toward grade level thinking at each step. Think‐pair‐share starts with each individual’s thinking, but then with each succeeding step, students are learning the thinking of others. All the students’ thinking is moving toward grade level thinking at each step.

• Think. Students work solo and silently, thinking about the problem or question. Stop ‘think’ step before students get the answer. About 1 minute. After the answer, no interest in thinking. • Pair. Each student expresses their initial thinking to the partner, then listens to the partner express their own initial thinking. Then students work in collaborative pairs to prepare a presentation to “share” with the class. 3‐ 20 minutes depending on its role in lesson. Remind the assignment is aprepared presentation, not just improv. • Share. 3 or 5 student‐pairs are selected to present their prepared presentations to the class. The presentations are selected to capture as much of the variety of thinking and representing that emerged during pair work. 3‐20 minutes depending on role in lesson.

Pose a task for the class without fuss: Just in time, not just in case. If some can’t start: • Say, “anyone who wants to talk some more about this problem, join me on the rug.” • Tell students whenever they are ready, return to their seats and go to work. • Call on one student to think aloud about the task. others help find what’s confusing. • Guide with questions, given where the student starts • Call on another student; repeat. • When only a few students are left, guide more directly and explicitly to productive approach; • re‐engage math from earlier grades as needed for this specific task; don’t get bogged down in reteaching a whole topic.

When partners appear stuck and not working productively or have finished too quickly. • Direct the partners to get up and walkabout the room to find a good idea in the written work (notebooks) of other students. • Encourage asking other students to explain their work or ask others to give feedback on their explanation • Direct them to bring a good idea back and write it in their own notebooks, citing from whom they got the idea. • Tell them to use the idea to continue working on the problem. • Return to the pair and check in: what’s the idea you liked? How have you used it in your own work?

Anytime Many purposes Especially when any student says something that points to target of unit Or when teacher needs a breather Or when students aren’t talking when you want a discussion Or an interesting disagreement

For a few minutes… Language development Draft feedback and revise cycle Embedded tutoring

The focus is on listening: understanding each student’s thinking De‐mystifies thinking Teacher writing student thinking Celebrates difference, agency and belonging Teaches Expressing thinking in writing using symbolic expressions Culture of risk‐free BUT: have to guide volunteering so all get attention over time, can’t let ew ew ew dictate opportunity to learn

Turn and talk Think‐pair‐share Whole class discussion Partners Groups of 4 or more Revision Get ideas from others to use in your own work Give responses to others’ work Conferencing Lesson Design

Don’t Start a lesson with an explanation that assumes the students listen with just the right prior knowledge. Gap in opportunity to comprehend Do Start the lesson by using today’s problem as a prompt to reveal the variety of ways of thinking about the problem. Normalize Variety of initial thinking Extend varieties of student thinking toward priority goals of the unit, toward grade level way of thinking about the problem. Pull from several directions toward “standard” goal Direct discussion summarizing mathematics followed by practice (goldilocks!) using newly learned mathematics Develop language of mathematical representations to enable communication and comparison of student thinking about the problem Language development is an everyday experience, not a topic

• Re‐engage earlier content inside grade level problems • Connect grade level work to earlier grade level content by asking students to explain or illustrate the connection; • Deepens insights into mathematical coherence • Opportunity to finish unfinished learning • Embed earlier content in efficient routines • Number talks • Student presentations (compare 3‐4 ways of thinking) • Stations, menus

• ….explain well enough so others can understand • NOT answer so the teacher thinks you know • Listening to other students and explaining to other students

Most good discussion questions are applications of 3 basic math questions: 1. How does that make sense to you? 1. Where does her idea here (pointing) correspond to your ideas? 2. Why do you think that is true 1. Do you agree? 2. Convince yourself, convince a friend, convince a skeptic. 3. Is that always, sometimes, never true? 3. How did you do it? 1. Strategy, purpose of a move 2. Validity of a move

• Prepare an explanation that others will understand • Understand others’ ways of thinking

philip daro, all rights reserved. do not copy or distribute 7/29/18 33 without permission of the author. Assignments like writing can reverse Mathew effect Collaboration, but each student produces an individual product…just like writing assignment Get ideas from other students Give suggestions to other students Notebooks: get ideas from your notes “whoever wants to talk some more, come to the rug” Students become assets for each others learning Good teammates Productive struggle

Not doing lots of problems they already know how to do. Not doing lots of problems they don’t know how to do. Doing a few problems they can learn from by struggling through them. What are they learning? ‐ not how to get the answer ‐ but the mathematics: developing expertise, comprehending concepts, learning to use mathematical expressions and representations to communicate and as thinking tools. Beyond “whatja get?” and “howdja do it?” Cathy Humphreys discusses “the investigative process” with her students (Movie 4.1 on Insidemathematics.org). This process maps thinking, but also discussion when students when students investigate collaboratively. Whole class discussion can teach, model and ‘authorize’ this process and the conversational agenda it implies. Less wide‐more deep

People are realizing that Answer getting, as important as it truly is, is not the the goal. Making sense and making explanations of mathematics that make sense are the real goals. Learning tricks is superficial; understanding is deep. Differences among students

• The first response, in the classroom: make different ways of thinking students’ bring to the lesson visible to all • Use 3 or 4 different ways of thinking that students bring as starting points for paths to grade level mathematics target • All students travel all paths: robust, clarifying Reengage earlier knowledge

• Inside grade level problems • Unfinished learning • Logical coherence Closing in on the mathematics

• First 2/3 of lesson driven by variety of student thinking • Last 1/3 of lesson driven by grade level mathematics: mathematical target of the unit

• 3/3 of the lesson driven by mathematical practices 5 – minute Transition Time

Welcome Elementary Educators!

We will begin at 2:30 Goldilocks

TEACHING!

strong

pure telling pure discovery weak weak Direct Instruction

• Yes…at end of lesson when students are ready • Not at the beginning when readiness has maximum differences 2 + 2 = 4

Fluency still important • Single digit addition facts and related subtractions • Single digit multiplication facts and related divisions • Standard calculation methods for 4 operations on multi‐digit numbers • Equivalent fractions • Number line and fractions • 4 operations with expressions with letters, Practice

Not whether practice is good or bad, but… WHAT is practiced How much practice is beneficial for what Why students get problems wrong: some causes practice benefits, some causes practice does not help Cost of practice, invest wisely: can make math boring for many and discouraging for others Goldilocks and balance Balance

Conceptual understanding AND Skills AND Applications

These are three dimensions of rigor that depend on each other Every problem involves all three in different balances Rebalance everyday

Mathematical Practices cut across all three dimensions of balance and rigor The reward for living the practices: Expertise

A. Using mathematics to make sense of problem situations, formulate models of the situations and solve unrehearsed problems. B. Facility with reading, writing and revising mathematical representations: symbolic expressions, graphs, diagrams, tables and sentences. C. Recognition and use of the correspondences between representations, between the concrete and the abstract. Communicate about the correspondences across different students’ thinking. D. Understanding, analyzing and formulating viable mathematical arguments. E. Understanding the thinking of others, communicating one’s own thinking in mathematical contexts, the ability to collaborate in a team working with mathematics and the ability to change one’s mind for good reasons. Mathematical Practices Develop

Develop expertise as we do in reading and writing expertise; The Practices define mathematical expertise for P‐12 Develop character and identity as a mathematics learner The Practices define good mathematical character Develop classroom culture The Practices define the mathematical values of classroom culture Mathew Effect Example: soccer

Two little girls join the soccer team. They have nearly identical athletic skills. Both are good runners, but Maria runs a half step faster than Gail. When the first soccer lesson begins, there’s one ball. They both run forward, and Maria gets to the ball first and kicks it. After they’ve done this 20 times, Maria has practiced kicking the ball 19 times, and Gail has had only one opportunity. A week later, Maria has learned a lot more about kicking the ball than Gail.

1/21/20 copyright Philip Daro, all rights reserved The differences in opportunity multiply over time In a nutshell, a small difference in prior learning causes the system to generate a difference in opportunity to learn where those with more prior learning get more opportunity to learn and those with less get less. The differences in opportunity multiply over time, and the differences in opportunity generate greater differences in learning. It’s a vicious circle. Math achievement is especially affected by Matthew effects.

• “I do – We do– You do” design breaks down for many students • Because it ignores prior knowledge

• I – we – you designs are well suited for content that does not depend much on prior knowledge…

• You do- we do- I do- you do Minimum Variety of prior knowledge in every classroom; I ‐ WE ‐ YOU

Student A Student B Student C Student D Student E

Lesson START CCSS Target Level Level Variety of prior knowledge in every classroom; I ‐ WE ‐ YOU Planned time Student A Needed time Student B Student C Student D Student E Lesson START CCSS Target Level Level Variety of prior knowledge in every classroom; I ‐ WE ‐ YOU

Student A Student B Student C Student D Student E

Lesson START CCSS Target Level Level Variety of prior knowledge in every classroom; I ‐ WE ‐ YOU CCSS Target Student A Student B Student C Student D Student E

Answer‐Getting Lesson START Level You ‐ we –I designs better for content that depends on prior knowledge

Student A Student B Student C Student D Student E

Lesson START Day 1 Day 2 Level Attainment Target Progressions in the variety

Mathematics in the progressions He posed the following problem: 20 +60 =

I could observe directly about 2/3 of the students. These were the different ways I observed students solving the problem: A. (3 students) Went to the 100s chart on the wall, started at 20 and counted from 1 by 1s all the way up to 60, read the number “80” where counting stopped. B. (5 students) Went to 100s chart, started at 20. Counted down the column 6 rows (30,40,50,…). B recognized 60 as 6 of 10 and counted six rows of 10. C. (3 students) Wrote two tens (10,10) then next to that wrote six 10s. then counted how many tens; there were 8 tens. Said “8 tens is eighty”. D. (1 student) Wrote; 20 +60 800 A 100s chart has the numbers from 1 to 100 in 10 rows of 10. The first row goes from 1 to 10. Thus the columns increase by 10. Used mostly in grades k‐1.

1/21/20 Copyright Philip Daro, all rights reserved Show why 4/5 is closer to 1 than 5/4 Use pictures and/or words on paper or a white board to create a visual diagram Click to a text 1/21/20 copyright, Philip Daro. all rights reserved 1/21/20 copyright, Philip Daro. all rights reserved 1/21/20 copyright, Philip Daro. all rights reserved “How do you know ¼ is bigger than 1/5?”

• Pose a question that prompts students to formulate a function

Jason ran 40 meters in 5 seconds • How far in a given time • How long to go a given distance • How fast is he going • A single relationship between time and distance, three questions • Understanding how these three questions are related mathematically is central to the understanding of proportionality called for by CCSS in 6th and 7th grade, and to prepare for the start of algebra in 8th

Skip counting: 40, 80, 120,… Table of seconds and meters Equation translation of word problem and solve Find unit rate and use unit rate to find answer

All approaches get the same answer. Comparing approaches draws attention to the progression from third grade skip counting to 6th grade ‘unit rates’. Reveals the coherence of mathematics. Opportunity to kick the ball of earlier content within grade level work…finish unfinished learning. Priorities: What is most important about each topic? 1. How today’s content extends prior content 2. How today’s content supports future content 3. Using the content to solve unrehearsed problems 4. Development of mathematical representations and language

FOCUS: Getting all students on track to be college and career ready The 3 Math Shifts

1. Focus: Focus strongly where the Standards focus.

2. Coherence: Think across grades, and link to major topics within grades.

3. Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application. Draft Your Vision of High Quality Math Instruction

What does it What does it What does it look like? sound like? feel like?

Students

Teachers

Leaders Developing a Vision - Instructions

1. Independently, imagine an ideal future state of math instruction. 2. Name what students, teachers, and leaders would be P-2 3-5 doing, saying and feeling in this ideal future. 3. When prompted, capture those ideas on your assigned post-its within the Padlet.

6-8 9-12 Pause and Reflect

● What appears to be trends across all grade bands? ● Where does the vision diverge? Closing Remarks

● Submit 3 words that express your hopes for math in the Bellevue School District 5 – minute Transition Time

• Change prior answers into new questions. • The new knowledge answers these questions. • Teaching begins by turning students’ prior knowledge into questions and then managing the productive struggle to find the answers • Direct instruction comes after this struggle to clarify and refine the new knowledge. 2nd Grade 3rd Grade Upgrade by comparing without decimal corrects to decimal incorrects. What’s different about these? Upgrading deepens place value understanding Upgrade Prior Knowledge

•Instead of teach from scratch •More efficient •Deeper understanding •Coherent mathematics Many Difficulties often come from bugs in prior knowledge Debugging is very efficient Reteaching is very inefficient debug

Show students worked examples of fictitious students Some have mistakes (caused by the bugs) Some are correct Ask students to turn and talk about the mistakes Ask them to WRITE a note to the fictitious student that explains the mistake and how to avoid the mistake Whole class discuss 5‐10 minutes, once or twice a week. Show 15 ÷3 = ☐

1. As a multiplication problem (3 x ☐ = 15 ) 2. Equal groups of things: 3 groups of how many make 15? 3. An array (3 rows, ☐ columns make 15?) 4. Area model: a rectangle has one side = 3 and an area of 15, what is the length of the other side? 5. In the multiplication table: find 15 in the 3 row 6. Make up a word problem Show 16 ÷3 = ☐ Use each of the 6 representations below What happens? 1. As a multiplication problem 2. Equal groups of things 3. An array (rows and columns of dots) 4. Area model 5. In the multiplication table 6. Make up a word problem Use paper, ready to hold up to camera to share. Probably need several sheets Readiness for grade level work

• Always some students will need intervention…no magic pills • Minimize the number who need intervention beyond regular instruction • Today is about regular instruction Revisit earlier knowledge

• Inside grade level problems • Unfinished learning • Logical coherence Depth in mathematics: Compare different ways of thinking

1. Reveal the logical coherence of mathematics: how today’s lesson builds logically on yesterday’s, last week’s, last year’s…go back into prior knowledge and make the logical connections 2. Coherence across representations: Probe correspondences between mathematical expressions, equations, diagrams, graphs, tables and the real‐world situation. 3. Demand making sense Start apart, bring together to target

• Diagnostic: make differences visible; what are the differences in mathematics that different students bring to the problem • All understand the thinking of each: from least to most mathematically mature • Converge on grade ‐level mathematics: pull students together through the differences in their thinking Next lesson

• Start all over again • Each day brings its differences, they never go away Teaching Moves

philip daro, all rights reserved. do not copy or distribute 7/29/18 99 without permission of the author. Explain the mathematics when students are ready • Toward the end of the lesson • Prepare the 3‐5 minute summary in advance, • Spend the period getting the students ready, • Get students talking about each other’s thinking, • Quote student work during summary at lesson’s end • Engage students in formulating summary • Students write summary in notebooks, use draft‐feedback‐revise for highest priorities

philip daro, all rights reserved. do not copy or distribute 7/29/18 100 without permission of the author. Core task: prepare explanations the other students can understand Need to pull opinions and intuitions into the open: make reasoning explicit Make reasoning public The more sophisticated your thinking, the more challenging it is to explain so others understand Students Explaining their reasoning develops academic language and their reasoning skills

For a few minutes… Language development Draft feedback and revise cycle Embedded tutoring

Turn and talk Think‐pair‐share Whole class discussion Partners and triads Groups of 4 or more Revision Get ideas from others to use in your own work Give responses to others’ work Conferencing Lesson Design

Don’t Start a lesson with an explanation that assumes the students listen with just the right prior knowledge. Gap in opportunity to comprehend Do Start the lesson by using today’s problem as a prompt to reveal the variety of ways of thinking about the problem. Normalize Variety of initial thinking Extend varieties of student thinking toward priority goals of the unit, toward grade level way of thinking about the problem. Pull from several directions toward “standard” goal Develop language of mathematical representations to enable communication and comparison of student thinking about the problem Language development is an everyday experience, not a topic

• ….explain well enough so others can understand • NOT answer so the teacher thinks you know • Listening to other students and explaining to other students

• Prepare an explanation that others will understand • Understand others’ ways of thinking

philip daro, all rights reserved. do not copy or distribute 7/29/18 114 without permission of the author. Assignments like writing can reverse Mathew effect Collaboration, but each student produces an individual product…just like writing assignment Get ideas from other students Give suggestions to other students Notebooks: get ideas from your notes “whoever wants to talk some more, come to the rug” Students become assets for each others learning Good teammates Productive struggle

• Inside grade level problems • Unfinished learning • Logical coherence Closing in on the mathematics

• First 2/3 of lesson driven by variety of student thinking • Last 1/3 of lesson driven by grade level mathematics: mathematical target of the unit

• 3/3 of the lesson driven by mathematical practices Congratulations to today’s winners: Maria Vazguez – Puesta del Sol

Melissa Willey –Stevenson

Your prizes will be mailed to your building tomorrow! Thank you for your participation!