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Breakfast or dessert?

Breakfast or dessert?

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Breakfast or dessert?

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  1. Breakfast or dessert?

  2. NCTM Standards

  3. All the standards rolled up into one: • Problem Solving: What is this? What’s that white thing? • Communication: Tell the person sitting next to you. • Reasoning: How do you know? • Connections: A real rip-off ad. • Representations: A picture

  4. Compare that with…..

  5. Simplify: 45 √2 + √7

  6. So Why Bother? Look around. Our critics are not all wrong. • Mountains of math anxiety • Tons of mathematical illiteracy • Mediocre test scores • HS programs that barely work for half the kids • Gobs of remediation • A slew of criticism Not a pretty picture and hard to dismiss

  7. So….. It’s Instruction, silly

  8. Join me in Teachers’ Room Chat • 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 THEYTHEY BLAME BLAMEBLAME An achievement gap or an INSTRUCTION gap?

  9. 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.

  10. So it’s instruction, silly! Research, classroom observations and common sense provide a great deal of guidance about instructional practices that make significant differences in student achievement. These practices can be found in high-performing classrooms and schools at all levels and all across the country. Effective teachers make the question “Why?” a classroom mantra to support a culture of reasoning and justification. Teachers incorporate daily, cumulative review of skills and concepts into instruction. Lessons are deliberately planned and skillfully employ alternative approaches and multiple representations—including pictures and concrete materials—as part of explanations and answers. Teachers rely on relevant contexts to engage their students’ interest and use questions to stimulate thinking and to create language-rich mathematics classrooms.

  11. Accordingly: Some Practical, Research-Affirmed StrategiesforRaising Student Achievement Through Better Instruction

  12. K-1 Reading Gifted Active classes Questioning classes Thinking classes My message today is simple: We know what works!

  13. Our job is to extract from these places and experiences specific strategies that can be employed broadly and regularly.

  14. But look at what else this example shows us: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?

  15. Now consider mathematics:TAKE OUT YOUR HOMEWORK. - #1 19 - #2 37.5 - #3 185 (No why? No how do you know? No who has a different answer?)

  16. Strategy #1 Adapt from what we know about reading (incorporate literal, inferential, and evaluative comprehension to develop stronger neural connections)

  17. 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?

  18. Number from 1 to 6 • 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?

  19. Good morning Boys and GirlsNumber from 1 to 5 1. What is the value of tan (π/4)? 2. Sketch the graph of (x-3)2 + (y+2)2 = 16 3. What are the equations of the asymptotes of f(x) = (x-3)/(x-2)? 4. If log2x = -4, what is the value of x? 5. About how much do I weight in kg?

  20. Strategy #2 Incorporate on-going cumulative review into instruction every day.

  21. Implementing Strategy #2 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.

  22. On the way to school: • A term of the day • A picture of the day • An estimate of the day • A skill of the day • A graph of the day • A word problem of the day

  23. Ready, set, picture….. “three quarters”

  24. Why does this make a difference?Consider the different ways of thinking about the same mathematics: • 2 ½ + 1 ¾ • $2.50 + $1.75 • 2 ½” + 1 ¾”

  25. Ready, set, picture….. 20 centimeters

  26. Ready, set, picture…..y = sin xy = 2 sin xy = sin (2x)

  27. Ready, set, picture…..The tangent to the circlex2 + y2 = 25 at (-4, -3) .

  28. Strategy #3 Draw pictures/ Create mental images/ Foster visualization

  29. 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?

  30. 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

  31. Or using a model:

  32. So let’s look more deeply at alternative approaches and multiple representations

  33. Ready, set, 8 + 9 = 17 – know it cold 10 + 7 – decompose the 9 to get to 10 18 – 1 – add 10 and adjust 16 + 1 – double plus 1 20 – 3 – round up and adjust Who’s right? Does it matter?

  34. Multiplying Whole Numbers

  35. Remember How 213 X 4

  36. Understand Why 213 x 4 213 + 213 + 213 + 213 = 852 200 10 3 4 800 40 12 4 ( 200 + 10 + 3) = 852

  37. Which leads to: 4 threes 4 tens 4 two hundreds 213 X 4 12 40 800 852

  38. Multiplying Decimals

  39. 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.”

  40. Understand Why gallons $ 4.39 Total How many gallons? About how many? Max/min cost?

  41. Understand Why gallons $ 4.39 183.38 Total Context makes ridiculous obvious, and breeds sense-making

  42. Solving Simple Linear Equations 3x + 7 = 22

  43. 3x + 7 = 22 How do we solve equations: Subtract 7 3 x + 7 = 22 - 7 - 7 3 x = 15 Divide by 3 3 3 Voila: x = 5

  44. 3x + 7 = 22 • Tell me what you see: 3 x + 7 • Suppose x = 0, 1, 2, 3….. • Let’s record that: x 3x + 7 0 7 1 10 2 13 4. How do we get 22?

  45. 3x + 7 = 22 Where did we start? What did we do? x 5 x 3 3x 15 ÷ 3 + 7 3x + 7 22 - 7

  46. 3x + 7 = 22 X X X IIIIIII IIII IIII IIII IIII II X X X IIIII IIIII IIIII

  47. 73 63 Tell me what you see.

  48. Tell me what you see. 2 1/4

  49. Tell the person sitting next to you five things you see.