1 Making Math Work for Special Education Students Phoenix, AZ February 7, 2014 Steve Leinwand

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

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A Simple Agenda for the Day The Math “Special” Instruction Access Culture of Collaboration 4

An introduction to the MATH 5

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

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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. 9

Ready, set…

Find the difference: Who did it the right way?? How did you get 5.41 if you didn’t do it this way? 11

So what have we gotten? Mountains of math anxiety Tons of mathematical illiteracy Mediocre test scores HS programs that barely work for more than half of the kids Gobs of remediation and intervention A slew of criticism Not a pretty picture! 12

13 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.

But what does change mean? And what is relevant, rigorous math for all? 14

Some data. What do you see?

Predict some additional data

How close were you?

All the numbers – so?

A lot more information (where are you?) Roller Coaster454 Ferris Wheel253 Bumper Cars152 Rocket Ride404 Merry-go-Round102 Water Slide304 Fun House203 19

Fill in the blanks Ride??? Roller Coaster454 Ferris Wheel253 Bumper Cars152 Rocket Ride404 Merry-go-Round102 Water Slide304 Fun House203

At this point, it’s almost anticlimactic! 21

The amusement park RideTimeTickets Roller Coaster454 Ferris Wheel253 Bumper Cars152 Rocket Ride404 Merry-go-Round102 Water Slide304 Fun House203 22

The Amusement Park The 4 th and 2 nd graders in your school are going on a trip to the Amusement Park. Each 4 th grader is going to be a buddy to a 2 nd grader. Your buddy for the trip has never been to an amusement park before. Your buddy want to go on as many different rides as possible. However, there may not be enough time to go on every ride and you may not have enough tickets to go on every ride. 23

The bus will drop you off at 10:00 a.m. and pick you up at 1:00 p.m. Each student will get 20 tickets for rides. Use the information in the chart to write a letter to your buddy and create a plan for a fun day at the amusement park for you and your buddy. 24

Why do you think I started with this task? -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!!! 25

26 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.

27 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.

An introduction to SPECIAL 28

SPECIAL EDUCATION Students with disabilities are a heterogeneous group with one common characteristic: the presence of disabling conditions that significantly hinder their abilities to benefit from general education (IDEA 34 §300.39, 2004). 29

More practically: SPECIAL: -Different -Better -More individualized, but still collaborative and socially mediated -Differentiated 30

But How? Mindless, individual worksheet, one-size-fits-all, in-one-ear-out- the-other practice is NOT Special! 31

For SwD to meet standards and demonstrate learning… High-quality, evidence-based instruction Accessible instructional materials Embedded supports –Universal Design for Learning –Appropriate accommodations –Assistive technology 32

SwD in general education curricula Instructional strategies –Universally designed units/lessons –Individualized accommodations/modifications –Positive behavior supports Service delivery options –Co-teaching approaches –Paraeducator supports 33

Learner variability is the norm!  Learners vary:  in the ways they take in information  in their abilities and approaches  across their development  Learning changes by situation and context 34

Two resource slides: g/files/updateguidelines2_0.pdf But compare Amusement Park (teaching by engaging) to Networks and UDL (teaching by showing and telling) and notice that these are summaries for nerds. 35

3 Networks = 3 UDL Principles 36

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So what is a more teacher-friendly way to say all of this? 38

Join me in Teachers’ Chat Room They forget They don’t see it my way They approach it differently They don’t follow directions They give ridiculous answers They don’t remember the vocabulary They keep asking why are we learning this THEY THEY THEY BLAME BLAME BLAME An achievement gap or an INSTRUCTION gap? 39

Well…..if….. They forget – so we need to more deliberately review; They see it differently – so we need to accommodate multiple representations; They approach it differently – so we need to elicit, value and celebrate alternative approaches; They give ridiculous answers – so we need to focus on number sense and estimation; They don’t understand the vocabulary – so we need to build language rich classrooms; They ask why do we need to know this – so we need to embed the math in contexts. 40

Pause….. Questions??? -Most intriguing/Aha point? -Most confusing/Hmmm point? 41

So……an introduction to Instruction 42

43 My message today is simple: We know what works. We know how to make math more accessible to our students It’s instruction silly! K-1 Reading Gifted Active classes Questioning classes Thinking classes

9 Research-affirmed Practices 1.Effective teachers of mathematics respond to most student answers with “why?”, “how do you know that?”, or “can you explain your thinking?” 2.Effective teachers of mathematics conduct daily cumulative review of critical and prerequisite skills and concepts at the beginning of every lesson. 3.Effective teachers of mathematics elicit, value, and celebrate alternative approaches to solving mathematics problems so that students are taught that mathematics is a sense-making process for understanding why and not memorizing the right procedure to get the one right answer.

4.Effective teachers of mathematics provide multiple representations – for example, models, diagrams, number lines, tables and graphs, as well as symbols – of all mathematical work to support the visualization of skills and concepts. 5.Effective teachers of mathematics create language- rich classrooms that emphasize terminology, vocabulary, explanations and solutions. 6.Effective teachers of mathematics take every opportunity to develop number sense by asking for, and justifying, estimates, mental calculations and equivalent forms of numbers. 45

7. Effective teachers of mathematics embed the mathematical content they are teaching in contexts to connect the mathematics to the real world. 8. Effective teachers of mathematics devote the last five minutes of every lesson to some form of formative assessments, for example, an exit slip, to assess the degree to which the lesson’s objective was accomplished. 9. Effective teachers of mathematics demonstrate through the coherence of their instruction that their lessons – the tasks, the activities, the questions and the assessments – were carefully planned. 46

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Yes But how? OR: Making Math Work for ALL (including SwD) 48

49 Number from 1 to 6 1. What is 6 x 7? 2. What number is 1000 less than 18,294? 3. About how much is 32¢ and 29¢? 4. What is 1/10 of 450? 5. Draw a picture of 1 2/3 6. About how much do I weight in kg?

50 Strategy #1 Incorporate on-going cumulative review into instruction every day.

51 Implementing Strategy #1 Almost no one masters something new after one or two lessons and one or two homework assignments. That is why one of the most effective strategies for fostering mastery and retention of critical skills is daily, cumulative review at the beginning of every lesson.

52 On the way to school: A fact of the day A term of the day A picture of the day An estimate of the day A skill of the day A measurement of the day A word problem of the day

Or in 2 nd grade: How much bigger is 9 than 5? What number is the same as 5 tens and 7 ones? What number is 10 less than 83? Draw a four-sided figure and all of its diagonals. About how long is this pen in centimeters? 53

54 Consider how we teach reading: JANE WENT TO THE STORE. -Who went to the store? -Where did Jane go? -Why do you think Jane went to the store? -Do you think it made sense for Jane to go to the store?

55 Now consider mathematics: TAKE OUT YOUR HOMEWORK. #1 19 # #3 185 (No why? No how do you know? No who has a different answer?)

56 Strategy #2 Adapt from what we know about reading (incorporate literal, inferential, and evaluative comprehension to develop stronger neural connections)

Tell me what you see. 57

58 Tell me what you see

59 Tell me what you see. 2 1/4

60 Strategy #3 Create a language rich classroom. (Vocabulary, terms, answers, explanations)

61 Implementing Strategy #3 Like all languages, mathematics must be encountered orally and in writing. Like all vocabulary, mathematical terms must be used again and again in context and linked to more familiar words until they become internalized. Area = covering Quotient = sharing Perimeter = border Mg = grain of sand

62 Ready, set, picture….. “three quarters” Picture it a different way.

63 Why does this make a difference? Consider the different ways of thinking about the same mathematics: 2 ½ + 1 ¾ $ $ ½” + 1 ¾”

64 Ready, set, show me…. “about 20 cms” How do you know?

65 Strategy #4 Draw pictures/ Create mental images/ Foster visualization

66 The power of models and representations Siti packs her clothes into a suitcase and it weighs 29 kg. Rahim packs his clothes into an identical suitcase and it weighs 11 kg. Siti’s clothes are three times as heavy as Rahims. What is the mass of Rahim’s clothes? What is the mass of the suitcase?

67 The old (only) way: Let S = the weight of Siti’s clothes Let R = the weight of Rahim’s clothes Let X = the weight of the suitcase S = 3R S + X = 29 R + X = 11 so by substitution: 3R + X = 29 and by subtraction: 2R = 18 so R = 9 and X = 2

68 Or using a model: 11 kg Rahim Siti 29 kg

A Tale of Two Mindsets (and the alternate approaches they generate) Remember How vs. Understand Why 69

Mathematics A set of rules to be learned and memorized to find answers to exercises that have limited real world value OR A set of competencies and understanding driven by sense-making and used to get solutions to problems that have real world value

Number facts 71

Ready?? What is 8 + 9? 17 Bing Bang Done! Vs. Convince me that = 17. Hmmmm…. 72

8 + 9 = 17 – know it cold – add 1 to 9, subtract 1 from – decompose the 8 into 7 and 1 18 – 1 – add 10 and adjust – double plus 1 20 – 3 – round up and adjust Who’s right? Does it matter? 73

= How did you do it? Who did it differently? 74

Adding and Subtracting Integers 75

Remember How 5 + (-9) “To find the difference of two integers, subtract the absolute value of the two integers and then assign the sign of the integer with the greatest absolute value” 76

Understand Why 5 + (-9) -Have $5, lost $9 -Gained 5 yards, lost 9 -5 degrees above zero, gets 9 degrees colder -Decompose 5 + ( ) -Zero pairs: x x x x x O O O O O O O O O - On number line, start at 5 and move 9 to the left 77

Let’s laugh at the absurdity of “the standard algorithm” and the one right way to multiply 58 x 47 78

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How nice if we wish to continue using math to sort our students! 80

So what’s the alternative? 81

Multiplication What is 3 x 4? How do you know? What is 3 x 40? How do you know? What is 3 x 47? How do you know? What is 13 x 40? How do you know? What is 13 x 47? How do you know? What is 58 x 47? How do you know? 82

3 x 4 Convince me that 3 x 4 is Three threes are nine and three more for the fourth

3 x 40 3 x 4 x 10 (properties) with a 0 appended

3 x 47 3 (40 + 7) = 3 40s + 3 7s or

58 x x

Multiplying Decimals 87

Remember How 4.39 x 4.2  “We don’t line them up here.”  “We count decimals.”  “Remember, I told you that you’re not allowed to that that – like girls can’t go into boys bathrooms.”  “Let me say it again: The rule is count the decimal places.” 88

But why? How can this make sense? How about a context? 89

Understand Why So? What do you see? 90

Understand Why gallons Total Where are we? 91

Understand Why 4.2 gallons Total How many gallons? About how many? $ 92

Understand Why 4.2 gallons $ 4.39 Total About how much? Maximum?? Minimum?? 93

Understand Why 4.2 gallons $ 4.39 Total Context makes ridiculous obvious, and breeds sense-making. Actual cost? So how do we multiply decimals sensibly? 94

Solving Simple Linear Equations 95

3x + 7 = 22 How do we solve equations: Subtract 7 3 x + 7 = x = 15 Divide by Voila: x = 5 96

3x Tell me what you see: 3 x Suppose x = 0, 1, 2, 3….. 3.Let’s record that: x 3x How do we get 22? 97

3x + 7 = 22 Where did we start? What did we do? x 5 x 3 3x 15 ÷ x

3x + 7 = 22 X X X IIIIIII IIII IIII IIII IIII II X X X IIIII IIIII IIIII 99

Let’s look at a silly problem Sandra is interested in buying party favors for the friends she is inviting to her birthday party. 100

Let’s look at a silly problem Sandra is interested in buying party favors for the friends she is inviting to her birthday party. The price of the fancy straws she wants is 12 cents for 20 straws. 101

Let’s look at a silly problem Sandra is interested in buying party favors for the friends she is inviting to her birthday party. The price of the fancy straws she wants is 12 cents for 20 straws. The storekeeper is willing to split a bundle of straws for her. 102

Let’s look at a silly problem Sandra is interested in buying party favors for the friends she is inviting to her birthday party. The price of the fancy straws she wants is 12 cents for 20 straws. The storekeeper is willing to split a bundle of straws for her. She wants 35 straws. 103

Let’s look at a silly problem Sandra is interested in buying party favors for the friends she is inviting to her birthday party. The price of the fancy straws she wants is 12 cents for 20 straws. The storekeeper is willing to split a bundle of straws for her. She wants 35 straws. How much will they cost? 104

So? Your turn. How much? How did you get your answer? 105

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Pulling it all together Or Escaping the deadliness and uselessness of worksheets 114

115 You choose: = = 2 x 4 = etc. Vs. SALE Pencils 3¢ Pens 4¢ Limit of 2 of each!

116 OOPS – Wrong store SALE Pencils 3¢ Pens 4¢ Erasers 5¢ Limit of 3 of each! SO?

Your turn Pencils 7¢ Pens 8 ¢ Erasers 9 ¢ Limit of 10 of each. I just spent 83 ¢ (no tax) in this store. What did I purchase? 117

118 Single-digit number facts More important than ever, BUT: - facts with contexts; - facts with materials, even fingers; - facts through connections and families; - facts through strategies; and - facts in their right time.

119 Deep dark secrets 7 x 8, x 6, since 5+4= …… 18 – 1 no, ÷ 7 = 7 x ___ = 63

120 You choose: vs. I’ve got $85. You’ve got $47. SO?

121 You choose: 1.59 ) 10 vs. You have $10. Big Macs cost $1.59 SO?

122 You choose…. The one right way to get the one right answer that no one cares about and isn’t even asked on the state tests vs. Where am I? (the McDonalds context) Ten? Convince me. About how many? How do you know? Exactly how many? How do you know? Oops – On sale for $1.29 and I have $20.

123 You choose: Given: F = 4 (S – 65) + 10 Find F when S = 81. Vs. The speeding fine in Vermont is $4 for every mile per hour over the 65 mph limit plus $10 handling fee.

124 Which class do YOU want to be in?

125 Strategy #5 Embed the mathematics in contexts; Present the mathematics as problem situations.

126 Implementing Strategy #5 Here’s the math I need to teach. When and where do normal human beings encounter this math?

127 Last and most powerfully: Make “why?” “how do you know?” “convince me” “explain that please” your classroom mantras

128 To recapitulate: 1.Incorporate on-going cumulative review 2.Parallel literal to inferential to evaluative comprehension used in reading 3. Create a language-rich classroom 4.Draw pictures/create mental images 5.Embed the math in contexts/problems And always ask them “why?” For copies: See also: “Accessible Math” by Heinemann

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130 Processing Questions What are the two most significant things you’ve heard in this presentation? What is the one most troubling or confusing thing you’ve heard in this presentation? What are the two next steps you would support and work on to make necessary changes?

People won’t do what they can’t envision, People can’t do what they don’t understand, People can’t do well what isn’t practiced, But practice without feedback results in little change, and Work without collaboration is not sustaining. Ergo: Our job, as professionals, at its core, is to help people envision, understand, practice, receive feedback and collaborate. Next Steps 131

To collaborate, we need time and structures Structured and focused department meetings Before school breakfast sessions Common planning time – by grade and by department Pizza and beer/wine after school sessions Released time 1 p.m. to 4 p.m. sessions Hiring substitutes to release teachers for classroom visits Coach or principal teaching one or more classes to free up teacher to visit colleagues After school sessions with teacher who visited, teacher who was visited and the principal and/or coach to debrief Summer workshops Department seminars 132

To collaborate, we need strategies 1 Potential Strategies for developing professional learning communities: Classroom visits – one teacher visits a colleague and the they debrief Demonstration classes by teachers or coaches with follow-up debriefing Co-teaching opportunities with one class or by joining two classes for a period Common readings assigned, with a discussion focus on: –To what degree are we already addressing the issue or issues raised in this article? –In what ways are we not addressing all or part of this issue? –What are the reasons that we are not addressing this issue? –What steps can we take to make improvements and narrow the gap between what we are currently doing and what we should be doing? Technology demonstrations (graphing calculators, SMART boards, document readers, etc.) Collaborative lesson development 133

To collaborate, we need strategies 2 Potential Strategies for developing professional learning communities: Video analysis of lessons Analysis of student work Development and review of common finals and unit assessments What’s the data tell us sessions based on state and local assessments “What’s not working” sessions Principal expectations for collaboration are clear and tangibly supported Policy analysis discussions, e.g. grading, placement, requirements, promotion, grouping practices, course options, etc. 134

135 The obstacles to change Fear of change Unwillingness to change Fear of failure Lack of confidence Insufficient time Lack of leadership Lack of support Yeah, but…. (no money, too hard, won’t work, already tried it, kids don’t care, they won’t let us)

Finally – let’s be honest: Sadly, there is no evidence that a session like today makes one iota of difference. You came, you sat, you were “taught”. I entertained, I informed, I stimulated. But: It is most likely that your knowledge base has not grown, you won’t change practice in any tangible way, and your students won’t learn any more math. And this is what we call PD. 136

Prove me wrong by Sharing Supporting Taking Risks 137

138 Next steps: Taking Risks It all comes down to taking risks While “nothing ventured, nothing gained” is an apt aphorism for so much of life, “nothing risked, nothing failed” is a much more apt descriptor of what we do in school. Follow in the footsteps of the heroes about whom we so proudly teach, and TAKE SOME RISKS

Thank you! 139

Appendix Slides 140

141 The Basics – an incomplete list Knowing and Using: +, -, x, ÷ facts x/ ÷ by 10, 100, , 100, 1000,….,.1,.01…more/less ordering numbers estimating sums, differences, products, quotients, percents, answers, solutions operations: when and why to +, -, x, ÷ appropriate measure, approximate measurement, everyday conversions fraction/decimal equivalents, pictures, relative size

142 The Basics (continued) percents – estimates, relative size 2- and 3-dimensional shapes – attributes, transformations read, construct, draw conclusions from tables and graphs the number line and coordinate plane evaluating formulas So that people can: Solve everyday problems Communicate their understanding Represent and use mathematical entities

143 Some Big Ideas Number uses and representations Equivalent representations Operation meanings and interrelationships Estimation and reasonableness Proportionality Sample Likelihood Recursion and iteration Pattern Variable Function Change as a rate Shape Transformation The coordinate plane Measure – attribute, unit, dimension Scale Central tendency

144 Questions that “big ideas” answer: How much? How many? What size? What shape? How much more or less? How has it changed? Is it close? Is it reasonable? What’s the pattern? What can I predict? How likely? How reliable? What’s the relationship? How do you know? Why is that?