NWSC Math Cohort Meeting

1. NWSC Math Cohort Meeting February 2011

2. Math MisconceptionsFrom Misunderstanding to Deep UnderstandingPreK-5 • Noni J Bamberger • Christine Oberdorf • Karren Schultz-Ferrell Publisher: Heinemann 2010 www.heinemann.com

3. Math MisconceptionsFrom Misunderstanding to Deep UnderstandingPreK-5 PreK and Elementary educators are by and large nurturing and supportive and have students interest in mind. We want the students to enjoy learning and end each year with all the skills and concepts they should have. But these positive characteristics sometimes lead us to unwittingly encourage some serious error patterns, misconceptions, and overgeneralizations.

4. Math MisconceptionsFrom Misunderstanding to Deep UnderstandingPreK-5 • *Number and Operations • *Algebra • *Geometry • *Measurement • Data Analysis and Probability • Assessing Children’s Mathematical Progress

5. Math MisconceptionsNumber and Operations * Counting with Number Words * Thinking Addition Means “Join Together” and Subtraction Means “Take Away” * Renaming and Regrouping When Adding and Subtracting Two-Digit Numbers * Misapplying Addition and Subtraction Strategies to Multiplication and Division

6. Math MisconceptionsNumber and Operations (continued) * Multiplying Two-Digit Factors by Two-Digit Factors * Understanding the Division Algorithm * Understanding Fractions * Adding and Subtracting Fractions * Representing, Ordering, and Adding/Subtracting Decimals

7. Math MisconceptionsCounting with Number Words What the Research Says Five principles of Counting [Gelman and Gallistel (1986)] 1) One-one principle. Each item to be counted has a “name,” and we count each item only once during the counting process 2) Stable order principle. Every time the number words are used to count a set of items, the order of the number words does not change. 3) Cardinal principle. The last number counted represents the number of items in the set of objects. 4) Abstraction principle. “Anything” can be counted and not all the “anythings” need to be of the same type. 5) Order-irrelevance principle. We can start to count with any object in a set of objects; we don’t have to count form left to right.

8. Activities to Undo Math MisconceptionsCounting Grades 3-5 What to Do * Ask students to skip-count from different numbers. * Read counting books. (In Moira’s Birthday by Robert Munsch for instance, students can count to 200 in a variety of ways since 200 kids are invited to Moira’s party.) * Support students in applying critical-thinking skills to the counting sequence by presenting number-logic riddles. When students are familiar with format of logic riddles, allow them to create riddles for classmates to solve.

9. Activities to Undo Math MisconceptionsCounting Grades 3-5 What to Do (continued) * Encourage students to create number logic riddles for three-digit through seven-digit numbers, depending on students’ grade level. * Provide small groups of students with a large box of objects to count. Challenge students to determine a way to count the objects in the box. Each group must have a different way of counting. When the task is complete, ask each group to present its counting strategy and to justify why the method is efficient.

10. Activities to Undo Math MisconceptionsCounting Grades 3-5 What to Do (continued) * Present opportunities for students to count both common and decimal fractions. * Provide experiences for students in which they place either common fractions or decimal fractions on a number line. This exercise also supports students’ understanding of relative magnitude (the size relationship one number has with another).

11. Activities to Undo Math MisconceptionsCounting Grades 3-5 What to Look For * Is the student able to count using a variety of strategies? * Does the student use logical thinking skills effectively to solve riddles about counting? * Do students use reasoning to justify why their method of counting is the most efficient for counting a large amount of objects?

12. Math MisconceptionsCounting with Number Words Questions to Ponder 1) What common path games can help students develop their understanding of the one-one counting principle? 2) What additional activities or strategies can you use to help your students become successful counters?

13. Activities to Undo Math MisconceptionsCounting Grades 3-5 Take the next 5 (five) minutes to mark the places in your materials where students work on counting.

14. Math MisconceptionsThinking Addition Means “Join Together” and Subtraction Means “Take Away” What the Research Says Problem Types – Basis of Cognitively Guided Instruction [Carpenter and Moser 1983; Carpenter, Carey, and Kouba 1990] 1) Join problems. 2) Separate problems. 3) Part-Part-Whole problems. 4) Compare problems.

15. Activities to Undo Math MisconceptionsAddition and Subtraction Concepts Grades 3-5 What to Do * Before using symbols, provide students with various materials that can be used to create part-whole representations of numbers (bi-colored counters, connecting cubes, teddy bear counters, Cuisenaire Rods). * Use correct terminology for the addition and subtraction signs (+ plus) and (– minus). * Provide students with opportunities to solve story problems that include all four problem types: join, separate, part-part-total.

16. Activities to Undo Math MisconceptionsAddition and Subtraction Concepts Grades 3-5 What to Do (continued) * Provide manipulatives to model story problems, along with symbolically recording what they do. * Have students share their strategies they used to get their answers, reinforcing correct terminology. * Use dominoes to have students model part-whole addition number sentences. * Use classroom routines to generate meaningful comparative subtraction problems. * Have students generate their own story problems.

17. Activities to Undo Math MisconceptionsAddition and Subtraction Concepts Grades 3-5 What to Look For * When students share their equations, listen for the correct use of “minus” and “plus.” * When students solve comparison subtraction story problems, look to see if students create two sets. Then look to see if they use an appropriate strategy to determine how many more or fewer one set is compared to the other. * When students generate their own stories, look to see if they are developing a variety of problems based on the types and structures taught.

18. Math MisconceptionsThinking Addition Means “Join Together” and Subtraction Means “Take Away” Questions to Ponder 1) What manipulatives do you currently have to reinforce the idea of part-whole for addition and subtraction? 2) How might you communicate to families the way you’ll be teaching the concepts so that misconceptions and overgeneralizations about addition and subtraction do not occur?

19. Activities to Undo Math MisconceptionsAddition and Subtraction Concepts Grades 3-5 Take the next 5-8 minutes to mark the places in your materials where students work on addition and subtraction concepts.

20. Math MisconceptionsRenaming and Regrouping When Adding and Subtracting Two-Digit Numbers What the Research Says * “When children focus on following the steps taught traditionally, they usually pay no attention to the quantities and don’t even consider whether or not their answers make sense.” [Richardson 1999, 100] * Teaching so that students can understand the traditional algorithm seems to be a real challenge.

21. Math MisconceptionsRenaming and Regrouping When Adding and Subtracting Two-Digit Numbers What the Research Says (continued) * For students to understand place value, they need to connect the concept of grouping by tens with the procedure of how to record numerals based on this system of counting. * Counting is fundamental to constructing an understanding of base-ten concepts and procedures. * Models that are both proportional and groupable should be used before models that are proportional but not groupable.

22. Math MisconceptionsRenaming and Regrouping When Adding and Subtracting Two-Digit Numbers What the Research Says (continued) * “Given the opportunity, children can and do invent increasingly efficient mental-arithmetic procedures when they see a connection between their existing, count-by-tens knowledge and addition by ten.” (Baroody and Standifer 1993, 92) * Many mathematics educators recommend spending a good deal of time with manipulative models while simultaneously practicing mental computation before putting pencil to paper to solve expressions (Baroody and Standifer 1993, 92).

23. Activities to Undo Math MisconceptionsPlace Value: Addition and Subtraction of Two-Digit Numerals Grades 3-5 What to Do * Have students use their understanding of counting by ones to group larger quantities, making it easier to count up or back to determine a sum or a difference. * Have students use a five-hundreds chart to look for patterns, and determine simple sums and differences by moving around the chart. * Use estimation activities so students get regular practice estimating quantities and then determining the actual amount by grouping by hundreds and tens to see how many.

24. Activities to Undo Math MisconceptionsPlace Value: Addition and Subtraction of Two-Digit Numerals Grades 3-5 What to Do (continued) * Have students share their solutions and the strategies that they used to get their answers. * Play games that require students to bundle, connect, or place objects together when there are ten of the object. Have students record the quantity of hundreds, tens and ones and the number that this represents.

25. Activities to Undo Math MisconceptionsPlace Value: Addition and Subtraction of Two-Digit Numerals Grades 3-5 What to Do (continued) * Give students a three-digit number and have them represent, either through modeling, pictures, or symbols, all of the ways to show this number using hundreds, tens and ones only. * Have students use whatever strategy is efficient and effective in getting a sum or difference as long as it makes sense to them

26. Activities to Undo Math MisconceptionsPlace Value: Addition and Subtraction of Two-Digit Numerals Grades 3-5 What to Look For * Do students understand the value of each digit rather than looking at the digit in isolation? * Are students able to compose and decompose numbers? * Do students see addition and subtraction as inverse operations?

27. Math MisconceptionsRenaming and Regrouping When Adding and Subtracting Two-Digit Numbers Questions to Ponder 1) What hundreds-chart activities help students better understand two-digit numbers? 2) What manipulative materials help students add and subtract? How can you use them? 3) How can you use estimation activities to reinforce ideas of tens and ones? 4) What research supports your instruction of place value, addition, and subtraction?

28. Activities to Undo Math MisconceptionsPlace Value: Addition and Subtraction of Two-Digit Numerals Grades 3-5 Take the next 5-8 minutes to mark the places in your materials where students work on two-digit place value concepts.

29. Math MisconceptionsMisapplying Addition and Subtraction Strategies to Multiplication and Division What the Research Says * In grades 3-5, multiplicative reasoning emerges and should be discussed and developed through the study of many different mathematical topics. Students’ understanding of the base-ten number system is deepened as they come to understand its multiplicative structure. That is, 484 is 4 x 100 plus 8 x 10 plus 4 x 1 as well as a collection of 484 individual objects (NCTM PSSM 2000, [144]).

30. Activities to Undo Math MisconceptionsMultiplication and Division Concepts Grades 3-5 What to Do * Number Lines - Use number lines (rulers, yardsticks, and meter sticks) to model multiplication and division situations. * Equal groupings – Making equal groups to model multiplication allows students to create equal sets and reinforces the notion that all sets are the same size.

31. Activities to Undo Math MisconceptionsMultiplication and Division Concepts Grades 3-5 What to Do (continued) * Partial Products and Partial Quotients – The partial products strategy emphasizes the importance of place value when multiplying whole numbers and provides an alternative algorithm that emphasizes the whole number rather than isolated digits within a number. * The same can be done by pulling out equal groups for division and then finding the total quotient by adding all of the partials.

32. Activities to Undo Math MisconceptionsMultiplication and Division Concepts Grades 3-5 What to Do (continued) * Area Model of Multiplication – By using an area model in conjunction with a rounding strategy, students see the value of the rounded product and how it compares to the actual product.

33. Activities to Undo Math MisconceptionsMultiplication and Division Concepts Grades 3-5 What to Look For * All groups must be the same size when solving multiplication and division problems. * Adjustments made to any one group (for ease of computation) must also be applied to all groups when solving multiplication and division problems.

34. Math MisconceptionsMisapplying Addition and Subtraction Strategies to Multiplication and Division Questions to Ponder 1) Which multiplication and division strategies maintain the emphasis of place value? Which do not? 2) When teaching multiplication and division, how do you decide which models and representations to use so that students understand the multiplicative concept?

35. Activities to Undo Math MisconceptionsMultiplication and Division Concepts Grades 3-5 Take the next 5-8 minutes to mark the places in your materials where students work on multiplication and division concepts.

36. Math MisconceptionsMultiplying Two-Digit Factors by Two-Digit Factors What the Research Says * Children spend a good deal of time learning and then practicing multi-digit addition. Consequently, it’s not uncommon that they combine algorithms when they do not have a complete understanding of place value (decomposing numbers) as well as what it means to multiply.

37. Math MisconceptionsMultiplying Two-Digit Factors by Two-Digit Factors What the Research Says (continued) * Ruth Stavy and Dina Tirosh, in How Students (Mis-)Understand Science and Mathematics (2000), attribute some errors as based on “intuitive rules.” Schemas about concepts and procedures are formed by students. Without a firm understanding of new content, students return to “relevant intuitive rules” that they have come to rely on. They may not make sense in the specific situation that they are now in, but unless new knowledge makes sense these rules persist.

38. Math MisconceptionsMultiplying Two-Digit Factors by Two-Digit Factors What the Research Says (continued) * Jae-Meen Baek spent time with students in six classrooms in grades Grades 3-5 to observe the different algorithms that were invented, as well as to see whether students were utilizing the traditional algorithm. Since none of the teachers taught rules or formal algorithms to students, many developed procedures that made sense to them.

39. Math MisconceptionsMultiplying Two-Digit Factors by Two-Digit Factors What the Research Says (continued) * “Many children in the study developed their invented algorithms for multi-digit multiplication problems in a sequence from direct modeling to complete number to partitioning numbers into non-decade numbers to partitioning numbers into decade numbers” (Baek 1998, 160).

40. Math MisconceptionsMultiplying Two-Digit Factors by Two-Digit Factors What the Research Says (continued) * Not only does this observation lead one to believe that multi-digit multiplication algorithms can be invented by students, but it also leads one to believe that when this is done, students have a clearer understanding of how to multiply.

41. Activities to Undo Math MisconceptionsMultiplication: Two-Digit by Two-Digit Grades 3-5 What to Do * Before doing any computation have students estimate the product based on the numbers in the expression. Any strategy (front-end, rounding, compatible numbers) can be used to determine this estimate. * Have students “expand” the factors of the multiplication expression. 34 x 27 = (30 + 4) x (20 + 7) or = (34 x 20) + (34 x 7) or = (30 x 27) + (4 x 27)

42. Activities to Undo Math MisconceptionsMultiplication: Two-Digit by Two-Digit Grades 3-5 What to Do (continued) * Try three ways to see if different results will be achieved. 1) Build an array that matches the expression using either base-ten blocks or centimeter grid paper. 2) Dissect the array by comparing it with the expanded form of the expression so students can see all of the different factors and partial products that will form from a two-digit by two-digit expression.

43. Activities to Undo Math MisconceptionsMultiplication: Two-Digit by Two-Digit Grades 3-5 What to Do (continued) * Try three ways to see if different results will be achieved. 3) If it’s the first expanded form that’s used, have students explain where the 30x20 part of the array can be found and label this as 30x20=600. then have the students explain where the 4x2=80 part of the array can be found and label this. Do this for the 30x7 part and the 4x7 part. * Try another expression and work on it as a whole group before having students do this either independently or with a partner.

44. Activities to Undo Math MisconceptionsMultiplication: Two-Digit by Two-Digit Grades 3-5 What to Look For * As students work through your chosen activities, look to see if they are using a strategy that is both efficient and effective. Also, be sure to ask students to explain how they know that they have used all of the digits in each of the factors as they’ve multiplied.

45. Math MisconceptionsMultiplying Two-Digit Factors by Two-Digit Factors Questions to Ponder 1) What are some other common errors students make when multiplying multi-digit numbers? 2) What are some effective strategies that you’ve used to help students understand why their procedures aren’t yielding the correct answers?

46. Activities to Undo Math MisconceptionsMultiplication: Two-Digit by Two-Digit Grades 3-5 Take the next 5-8 minutes to mark the places in your materials where students work on two-digit by two-digit multiplication.

47. Math MisconceptionsUnderstanding the Division Algorithm What the Research Says * “The traditional long-division algorithm is difficult for many students. Many never master it in elementary school and fewer develop meaning for the procedure or the answer” (Silver, Shapiro, and Deutsch 1993). * First reason – the procedure contains so many steps, and for each step students need to get an exact answer in the quotient.

48. Math MisconceptionsUnderstanding the Division Algorithm What the Research Says (continued) * Second reason – the algorithm treats the dividend as a set of digits rather than an entire numeral. Students are taught to ignore place value as they routinely work through a procedure they don’t necessarily understand.

49. Activities to Undo Math MisconceptionsDivision Algorithm Grades 3-5 What to Do * Encourage students to estimate the quotient using their mental multiplication skills so they have a sense of whether their answer is reasonable. * Before using symbols, provide students with a story problem that is meaningful to them. * Provide students with manipulatives to model the story that is given, but also have them symbolically record what they’ve done. * Ask students to share the strategies they used to get their answers and discuss whether their answers make sense.

50. Activities to Undo Math Misconceptions Division Algorithm Grades 3-5 What to Do (continued) * “Try out” someone’s procedure that is both efficient and effective to see if students are able to use this same strategy. * Use number-sense activities to foster mental computation and an understanding of how to use multiples of ten to arrive at answers. * Look at ways to adjust numbers to make them easier to use for computing.