Effective Math Instruction for Special and Struggling K-6 Students

1. Making Math Work Far More Effectively for our Special and Struggling K-6 StudentsColorado Council for Learning DisabilitiesFebruary 22-23, 2019Steve LeinwandSLeinwand@air.orgwww.steveleinwand.com

2. And what message do far too many of our students get? (even those in Namibia where I found this!)

4. Overview This workshop will be predicated on the understanding that special and struggling students can learn mathematics so long as teachers focus on the two core aspects of differentiation: alternative approaches and multiple representations – both of which maximize access for all students. We will also discuss effective instructional practices such as ongoing cumulative review, effective questioning, gradual reveal, ensuring high expectations for all, using low-floor/high ceiling tasks, and the use of contexts in which to embed the mathematics being taught – all of which foster engagement.

5. Agenda • Introductory context setting • Let’s do some math and reflecting • So what are the take-aways? • Special and struggling students • Specific Strategies • Putting in all together into powerful lessons that support differentiation • Nurturing a culture of collaboration

6. Me and Us • 45 years • 3 jobs • About 200,000 air mile per year (about 400 hours or 50 8-hour work days for reading, sleeping and catching up) • Representations because the members of every class learn in different ways • Most PD… • Validate outliers and sprinkle empowerment dust • My plea for breaking the cycle by reducing our professional isolation.

7. OKAY? Ready to dig in?

8. Introductory Gauntlet “Gauntlet”????

9. Thank you Wikipedia To "throw down the gauntlet" is to issue a challenge. A gauntlet-wearing knight would challenge a fellow knight or enemy to a duel by throwing one of his gauntlets or gloves on the ground. The opponent would pick up the gauntlet to accept the challenge. Consider my gauntlet throw my friends.

10. The Gauntlet: We have a problem: • what is typical ignores research and student needs and underserves; • what is needed must be envisioned, supported and nurtured if we are to truly make mathematics work at HIGH levels for ALL. And it fall to us to provide the leadership – in our classrooms and our schools – to make this happen.

11. And one more introductory set

12. So…the problem is: If we continue to do what we’ve always done…. We’ll continue to get what we’ve always gotten.

15. 7. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

16. Ready, set….. 10.00 - 4.59

17. Find the difference: Who did it the right way?? 910.91010 - 4. 5 9 How did you get 5.41 if you didn’t do it this way?

18. So what have we gotten? • Mountains of math anxiety • Tons of mathematical illiteracy • An untenable discrepancy between the haves and have nots of mathematics • HS programs that barely work for more than half of the kids • A society that fails to live up to the Jeffersonian ideal of an informed and educated citizenry • Gobs of remediation and intervention • A slew of criticism Not a pretty picture!

19. If however….. What we’ve always done is no longer acceptable, then… We have no choice but to change some of what we do and some of how we do it.

20. But what does change mean? And what is relevant, rigorous math that works for all?

21. So let’s do some math!

22. Grade 3 Good morning third graders! We’ve been studying fractions for a while. Let’s review with a focus on common misunderstandings.

23. In your groups, write down 5 things you see

24. Tim and Sari Tim eats 1/2 of a pizza. Sari eats 2/4 of a pizza that is the same size. Who eats more pizza? How do you know? Can you use a drawing to convince us?

25. Tim and Sari 2 Tim eats 1/4 of a pizza. Sari eats 1/5 of a pizza that is the same size. Who eats more pizza? How do you know? Can you use a drawing to convince us?

26. Tim and Sari 3 Tim and Sari each eat ½ of a pizza. Tim says they both ate the same amount. Sari says she ate more than Tim. Show with pizza drawings for Tim and Sari that both can be right!!!

27. Tim and Sari 4 Tim eats 3/4 of a pizza. Show this. Sari’s pizza has 8 slices and is the same size of Tim’s. Show this. How many slices does Sari need to eat so that she and Tim eat the same amount? Show with pizza drawings how you know.

28. Your turn How does that 5-slide mini-lesson provide mainstream classroom access for special and struggling students?

29. Straight from the textbook Sarah has picked 2605 apples. She has 91 boxes. How many apples will Sarah put in each box if each box holds the same number of apples? UGH! Brain-numbing! But so typical. Pluck out the numbers. Convert “in each” to divide. Divide and done. Now do 7 more just like it with little additional learning.

30. Adapting what the text bestows by turning practice into opportunities for learning with a focus on our questions. Sarah has 91 empty boxes. • What can you infer about Sarah? She had 2605 apples to pack into the boxes. • Now what can you infer about Sarah? • What do you think the question is? • So what does the 91 tell us, what about the 2605? • About how many apples do you think would be in each box? More than 100? Less than 100? Convince us • Can you draw a picture? • Can you create a number sentence? • Do you multiply or divide? Why? • So about how many apples would be in each box if…

31. Let’s Try It Grade 2: Robert has 85 stickers. Jen has 57 stickers. How many more stickers does Robert have than Jen? UGH! The absurdity and terrible waste of one problem, one skill, one answer Versus: Instead, turn and tell your partner what we can do with this.

32. Your turn Robert had 85 stickers. What do you notice? Or tell me 3 things about Robert. Jen has 57 stickers. Now what do you notice? What’s the question? What’s the answer? And with partners and white boards we’re off to the races!!!

33. Alternatively: • Good morning my 2nd grade mathematicians! • Today we are going to solve problems involving subtraction of two-digit numbers. • Ready: I have $95. You have $47. • Turn and tell your partner two things you notice? • Turn and tell your partner what the answer to the hidden problem might be. • $142 $48 $71 $150 $50 You Me Other? • Let’s focus on how much richer I am than you. How do you know? • Let’s try another problem.

34. Your turn again What are some of the elements of these approaches that enable you to provide mainstream classroom access for special and struggling students?

35. What started this thinking and designing?

36. Grade 6 SA Harlem Central Tues Dec 8, 2015 • Lesson 6 in the Expressions unit (6.EE standards) • Ally and Mabubar co-teaching • 19 Scholars • Driven by a number strings mini-lesson, a Math Workshop task and an exit ticket • “Our goal for today is to “identify, create and understand equivalent expressions.” • “Zayasia, can you please repeat our learning goal?” • “Let’s begin with out number strings.”

37. Number strings for today’s Mini Lesson Are they equivalent? How do you know? • 4(8) = 4(3 + 5) • 4(8) = 4(a + 5) • 4(8) = 4(3 + b) • 3x + 3y = 3(x + y) Let’s summarize: For each: Always, sometimes, never equivalent?

38. Math Workshop Task Jan normally rides her bike to and from work. Her normal route is 18 miles from home to work. One day she goes to a coffee shop on her way to work and on her way home. This adds x miles to her trip each way. (“What do you notice?” “What’s the question?”) Great: Write and show the distance Jan travels using a diagram or picture and two different, but equivalent, expressions.

39. Resulting in: x 18 2 2 18 x • 18 + x + 18 + x 18 x • 2(18 + x) • 2(x + 18) • x + x + 18 + 18 • 36 + 2x “Is everyone correct? Turn and tell your partner why?” “What do the numbers and variables represent?” “Which expression is simplest or easiest to use? Why?”

40. Lesson 6 Exit Ticket Which of the following represent equivalent expressions? Explain or show your process of determining which expressions ARE equivalent. Select all that apply: a. x + x + x + x = 4x b. 15 y + 5x = 3(5y + x) c. 6(2 + x) = 12 + 6 d. 3(x + y) = 3x + y

41. Why do you think I started with these tasks? • Standards don’t teach, teachers teach • It’s the translation of the words into tasks and instruction and assessments that really matter • Processes are as important as content • We need to give kids (and ourselves) a reason to care • Difficult, unlikely, to do alone!!!

42. Let’s be clear: We’re being asked to do what has never been done before: Make math work for nearly ALL kids and get nearly ALL kids ready for college. There is no existence proof, no road map, and it’s not widely believed to be possible.

43. Let’s be even clearer: Ergo, because there is no other way to serve a much broader proportion of students: We’re therefore being asked to teach in distinctly different ways. Again, there is no existence proof, we don’t agree on what “different” mean, nor how we bring it to scale.

44. So let’s step back and generalize. Turn and tell your partner what made those tasks effective?

45. Elements of Quality • Clarity of goals (not Lesson 4.5 or pages 214-217) • Context (not naked) • Rich tasks (not exercises) • Focused intentional questions (not punting) • Opportunities for discourse (not just telling) • Gradual release (not just a dumping) • Multiple representations (not one way) • Alternative approaches (not one way) • Explanations and justifications (not just answers) • Common errors and misconceptions (not just right correct approaches) • Sense-making by students (not lecture) • Evidence (not I taught it and let the chips….)

46. One by one, just consider how each of these 8 MTPs live in this lesson? • Establish mathematics goals to focus learning. • Implement tasks that promote reasoning and problem solving. • Use and connect mathematical representations. • Facilitate meaningful mathematical discourse. • Pose purposeful questions. • Build procedural fluency from conceptual understanding. • Support productive struggle in learning mathematics. • Elicit and use evidence of student thinking. * MTPs = Mathematical Teaching Practices from NCTM’s Principles to Actions

47. Six Key Processes Formal Cognitive Process Student-friendly terminology Storytelling Building Drawing and using symbols Seeing Describing what and how Discussing why • Contextualizing • Physically constructing • Representing graphically and symbolically • Visualizing • Verbalizing • Justifying

48. Contextualizing or story telling: Understanding that life can be described mathematically is at the foundation of fluency. Equations exist because they are a shortcut to explain situations, look at reality and make predictions. Too often we present equations without giving them context, leaving children without understanding and causing misconceptions. • Physically constructing or building: Children need to manipulate materials to develop an understanding of theaction of operations, which is then extended to the visual or pictorial level and then to abstraction. Fluency demands multiple models and making connections between and among models. • Representing graphically and symbolically or drawing and using symbols: Seeing and using models and relationships between models supports visual memory, building relationships, and mental fluency and enhances long term memory. For many students, fluency depends on being able to visualize concepts in different ways and understand the relationships between these different representations.

49. Visualizing or seeing: Children learn to visualize quantities and the relationships between them. As children accumulate a visual repertoire, their numerical fluency grows because they are able to “see” the mathematics in which they are engaged. • Verbalizing or describing what and how: Understanding of operations is achieved when students describe and explain what they did with the materials that they manipulate and the pictures they draw. Students need to describe the representations they create and how different representations are similar and different. Students should always be expected to describe the “what” and “how” of the mathematics they are learning using informal and, gradually, formal mathematical language. • Justifying or discussing why: Discussion about relationships and justifying solutions to problems is fundamental to developing metacognition and is crucial to long-term fluency. Justifying answers the question “why?” and it is one of the best ways to monitor the development of numerical fluency in students.

50. Self-reflection:Which of these do we see and don’t we see in your classroom? Formal Cognitive Process Student-friendly terminology Storytelling Building Drawing and using symbols Seeing Describing what and how Discussing why • Contextualizing • Physically constructing • Representing graphically and symbolically • Visualizing • Verbalizing • Justifying