Elementary School Performance Tasks and Student Thinking for Mathematics

1. Elementary SchoolPerformance Tasks andStudent Thinking for Mathematics CFN 609Professional Development | April 2, 2012 RONALD SCHWARZMath Specialist, America’s Choice,| Pearson School Achievement Services

2. Got change? Try to figure out a way to make change for a dollar that uses exactly 50 coins. Is there more than one way?

3. Performance Tasks and Student Thinking AGENDA • Reflection • Progressions • Properties • Misconceptions • Tasks • Looking at Student Work

4. Practice and Experience Have you tried out any of the strategies, tasks or ideas from our previous sessions, and if so, what were the results?

5. Reflection and Review What are one or two ways that US math instruction differs from that in higher-performing countries?

6. Reflection and Review Describe and list some of the Standards for Math Practice.

7. Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. 2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique the reasoning of others. 4 Model with mathematics. 5 Use appropriate tools strategically. 6 Attend to precision. 7 Look for and make use of structure. 8 Look for and express regularity in repeated reasoning.

8. Expertise and Character Development of expertise from novice to apprentice to expert The Content of their mathematical Character

9. Reflection and Review How are levels of cognitive demand used in looking at math tasks?

10. Levels of Cognitive Demand Lower-level • Memorization • Procedures without connections Higher-level • Procedures with connections • Doing mathematics

11. Reflection and Review What do we mean by formative assessment and what are some strategies involved in it?

12. Some Habits of Mind • Visualization, including drawing a diagram • Explanation, using their own words • Reflection and metacognition • Consideration of strategies

13. A train one mile long travels at a rate of one mile per minute through a tunnel that is one mile long. How long will it take the train to pass completely through the tunnel?

14. Some More Habits of Mind • Listening to each other • Recognizing and extending patterns • Ability to generalize • Using logic • Mental math and shortcuts

15. Getting Comfortable with Non-Routine Problems

16. Pleasure in Problem-Solving

17. Some Strategies for Approaching a Task • Make an organized list • Work backward • Look for a pattern • Make a diagram • Make a table • Use trial-and-error • Consider a related but simpler problem

18. Grid Lock

19. And Some More Strategies • Consider extreme cases • Adopt a different point of view • Estimate • Look for hidden assumptions • Carry out a simulation

21. Bananas If three bananas are worth two oranges, how many oranges are 24 bananas worth?

23. Nine properties are the most important preparation for algebra Just nine: foundation for arithmetic Exact same properties work for whole numbers, fractions, negative numbers, rational numbers, letters, expressions. Same properties in 3rd grade and in calculus Not just learning them, but learning to use them

24. Using the properties • To express yourself mathematically (formulate mathematical expressions that mean what you want them to mean) • To change the form of an expression so it is easier to make sense of it • To solve problems • To justify and prove

27. Linking multiplication and addition: the ninth property Distributive property of multiplication over addition a × (b+c) = (a×b) + (a×c) a(b+c) = ab + ac

28. What is an explanation? Why you think it’s true and why you think it makes sense. Saying “distributive property isn’t enough, you have to show how the distributive property applies to the problem.

29. Mental Math 72-29= ? In your head Composing and decomposing Partial products Place value in base 10 Factor x2 +4x + 4 in your head

30. NYSED Grade 4 Sample Tasks

31. Misconceptions How they arise and how to deal with them

32. Misconceptions about misconceptions • They weren’t listening when they were told • They have been getting these kinds of problems wrong from day one • They forgot • Their previous teachers didn’t know the math

33. More misconceptions about the cause of misconceptions • In the old days, students didn’t make these mistakes • They were taught procedures • They weren’t taught the right procedures • Not enough practice

34. Whatever the Cause • When students reach your class they are not blank slates • They are full of knowledge • Their knowledge will be flawed and faulty, half baked and immature; but to them it is knowledge • This prior knowledge is an asset and an interference to new learning

35. Place Value in Grade 2

36. Dividing Fractions 1 3 1 2 ÷

37. Dividing Fractions “Ours is not to question why, just invert and multiply.” 1 3 1 2 ÷

38. Dividing Fractions WHY? 1 3 1 2 ÷

39. Stubborn Misconceptions • Misconceptions are often prior knowledge applied where it does not work • To the student, it is not a misconception, it is a concept they learned correctly… • They don’t know why they are getting the wrong answer

40. Second grade When you add or subtract, line the numbers up on the right, like this: 23 +9 Not like this 23 +9

41. Third Grade 3.24 + 2.1 = ? If you “Line the numbers up on the right “ like you spent all last year learning, you get this: 3.2 4 + 2.1 You get the wrong answer doing what you learned last year. You don’t know why. Teach: line up decimal point. Continue developing place value concepts

42. Frequently, a ‘misconception’ is not wrong thinking but is a concept in embryo or a local generalization that the pupil has made. It may in fact be a natural stage of development. Malcolm Swan

43. Multiplying makes a number bigger.

44. Percents can’t be greater than 100%.

45. Graphs are pictures.

46. To multiply by ten, put a zero at the end of the number.

47. Numbers with more digits are bigger.

48. x means the number that you have to find.

49. Teach from misconceptions • Most common misconceptions consist of applying a correctly-learned procedure to an inappropriate situation. • Lessons are designed to surface and deal with the most common misconceptions • Create ‘cognitive conflict’ to help students revise misconceptions • Misconceptions interfere with initial teaching and that’s why repeated initial teaching doesn’t work

50. Social and meta-cognitive skills have to be taught by design • Beliefs about one’s own mathematical intelligence • “good at math” vs. learning makes me smarter • Meta-cognitive engagement modeled and prompted • Does this make sense? • What did I do wrong? • Social skills: learning how to help and be helped with math work => basic skill for algebra: do homework together, study for test together