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February 2011.  Noni J Bamberger  Christine Oberdorf  Karren Schultz-Ferrell Publisher: Heinemann 2010

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Presentation on theme: "February 2011.  Noni J Bamberger  Christine Oberdorf  Karren Schultz-Ferrell Publisher: Heinemann 2010"— Presentation transcript:

1 February 2011

2  Noni J Bamberger  Christine Oberdorf  Karren Schultz-Ferrell Publisher: Heinemann 2010

3 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 1) *Number and Operations 2) *Algebra 3) *Geometry 4) *Measurement 5) Data Analysis and Probability 6) Assessing Children’s Mathematical Progress

5 * 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 * 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 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 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 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 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 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 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 Take the next 5 (five) minutes to mark the places in your materials where students work on counting.

14 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 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 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 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 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 Take the next 5-8 minutes to mark the places in your materials where students work on addition and subtraction concepts.

20 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 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 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 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 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 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 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 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 Take the next 5-8 minutes to mark the places in your materials where students work on two-digit place value concepts.

29 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 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 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 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 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 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 Take the next 5-8 minutes to mark the places in your materials where students work on multiplication and division concepts.

36 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 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 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 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 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 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 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 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 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 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 Take the next 5-8 minutes to mark the places in your materials where students work on two-digit by two-digit multiplication.

47 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 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 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 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.

51 What to Look For * Check to see if students have a procedure that will always work. * As students use their own division algorithm, make sure that they can articulate why it works and how they know their answer makes sense.

52 Questions to Ponder 1) How do you help students move from representing a division problem with manipulatives to using a paper-and-pencil algorithm? 2) How do you get a student who has an effective but inefficient algorithm to adopt one that is more efficient?

53 Take the next 5-8 minutes to mark the places in your materials where students work on division algorithms.

54 What the Research Says Sherman, Richardson, and Yard (2005) suggest several reasons for student difficulties in learning about fractional concepts and skills: * They memorize procedures and rules before they have developed a conceptual understanding of the related concepts. * Early instruction in mathematics focuses on whole numbers so children over-generalize what they know about whole-number computation and apply this knowledge to fractions.

55 What the Research Says Sherman, Richardson, and Yard (2005) suggest several reasons for student difficulties in learning about fractional concepts and skills: * Estimating rational numbers is more difficult than estimating whole numbers. * Recording fractional notation is difficult and confusing for students if they do not yet understand what the top and bottom numbers represent. Knowing which is the numerator and which is the denominator and what those numbers mean is critical.

56 What the Research Says Chapin and Johnson (2000) list four critical interpretations of fractions necessary for computing successfully: 1) Part of a whole or parts of a set. 2) Fractions as a result of dividing two numbers. 3) Fractions as the ratio of two quantities. 4) Fractions as operators.

57 What to Do * Provide students multiple opportunities to share various objects that support them in thinking more flexibly about fractions. Present nontraditional shapes (such as triangles) for students to divide. * Involve students in discussions following their work with fractions. Introduce fraction vocabulary and talk about fractional parts rather than fractional symbolism. Then expect students to explain what the symbolic representation means. * Encourage students to “fair share” a grid outline in different ways.

58 What to Do (continued) * Provide opportunities for students to solve word problems involving both area and set models of fractions. * Provide students with opportunities to develop understandings about fractional concepts in a variety of real-life connections. Reach beyond the “pizza” connection. * Present sharing problems that include the “set” model of fractions to help students establish important connections with many real-world uses of fractions.

59 What to Do (continued) * Provide opportunities for students to explore fractions such as sixths and eighths. Their understandings about “halving” will help them as they work with a variety of fractions. * Count fractional parts with students so they see how multiple parts compare to the whole.

60 What to Look For * Are students able to fairly share an area in different ways? * Are students able to divide a variety of shapes or objects accurately? * Are students able to represent fractional parts in a variety of nontraditional ways?

61 Questions to Ponder 1) What specific difficulty have your students had or what overgeneralization have they made about fractions? Discuss this misconception with members of your teaching team or with an instructional support teacher. 2) How will you plan instruction so that students can develop a better understanding of this fractional concept?

62 Take the next 5-8 minutes to mark the places in your materials where students work on fraction concepts.

63 What the Research Says * “In terms of instructional approaches, lessons are too often focused on procedures and memorizing rules rather than on developing conceptual foundations prior to skill building” (Sherman, Richardson, and Yard 2005, 139). * Many researchers have concluded that the complex topic of fractions is more challenging for elementary students than any other area of mathematics (Bezuk and Bieck 1993).

64 What the Research Says (continued) * Before students study how to add and subtract fractions, they need to understand the meaning of fractions through various models, as well as how to use the language of fractions. * Watanabe (2002, 457) delineates three models frequently used in elementary materials – the linear model, the area model, and the discrete (set) model.

65 What the Research Says (continued) Van de Walle (2007) provides some important “big ideas” that students must understand for computational understanding: * Fractional parts are equal-size portions or equal shares of a whole or unit. They don’t necessarily look alike. * The special names for the numbers that make up a fraction tell how many equal-size parts make up the whole (the denominator) and how many of the fractional parts are being considered (the numerator).

66 What the Research Says (continued) * The National Mathematics Advisory Panel (2008) notes that one key instructional strategy to link conceptual and procedural knowledge of fractions is the ability to represent fractions on a number line.

67 What to Do * Allow students to create their own materials or draw their own representations when adding and subtracting fractions. * Introduce activities in which children count by fractions. Begin by using manipulatives with which children are already familiar—pattern blocks, for example. * Have students use fraction pieces to count by halves, thirds, fourths, sixths, eighths, and even twelfths. If they get good at this, have them combine fraction strips.

68 What to Do (continued) * Introduce story problems that reinforce what it means to add and subtract fractions. Don’t have students record the equation until they have shared their strategies for getting their answers. Ask “What do you notice about how fractions are added or subtracted when the denominators are the same?” * Give students opportunities to compare fractions. This opportunity to visualize the value of a fraction will help in making sense of the computation when finding sums and differences.

69 What to Do (continued) * Use the number line to represent fractions, compare the magnitude of fractions, and to add or subtract fractions with like denominators. * Reveal patterns on the multiplication chart as an example of equivalent fractions.

70 What to Look For * The denominator names the total number of pieces needed to form the whole. * The numerator indicates a specific number of pieces of the unit. * While the numerator changes when adding and subtracting fractions with like denominators, the denominator remains the same in the sum or difference.

71 Questions to Ponder 1) What big ideas about fractions must students understand before they are able to build a conceptual knowledge of equivalence? 2) What opportunities or supports can you use to empower students to manipulate values by using their own number sense rather than simply relying on procedures?

72 Take the next 5-8 minutes to mark the places in your materials where students work on adding and subtracting fractions.

73 What the Research Says * A simple yet powerful introduction to decimals is to ask students to represent two related decimal numbers using several representative models (Van de Walle and Lovin 2006). * Chapin and Johnson (2000) state, “Finding examples of decimals, explaining what the decimal numeral means in the context of its use, indicating the general value of the decimal numeral, and then stating what two whole numbers the decimal is between helps students recognize that the decimal amount is the sum of a whole number and a number less than one.”

74 What the Research Says (continued) * The Chapin and Johnson (2000) approach reaffirms the importance of avoiding “naked” mathematics and instead teaching mathematics skills and concepts with a context. * Decimal number sense should be a focus during instruction so that students recognize an unreasonable answer (Sherman, Richardson, and Yard 2005).

75 What to Do * Correctly name the decimal fraction. When a decimal fraction is read correctly, the name reinforces the place value of each digit. Prevent students from getting into the habit of saying “six point three” rather than “six and three- tenths” when reading a decimal fraction. * Use a variety of concrete models to represent decimal fractions. Students need multiple representations for decimal fractions.

76 What to Do (continued) * Provide opportunities to reinforce place value. With experience, students will recognize the relationship among adjacent values and see that moving to the left (by one digit) means ten times larger and moving to the right denotes one-tenth of the value. Additionally, students must have opportunities to recognize that a value can be named using different units.

77 What to Look For * Students verbalize the decimal fraction correctly. * Students are able to state the value of a specific digit within a decimal fraction. * Students can construct more than one visual representation for a decimal number.

78 Questions to Ponder 1) How can you reinforce the notion that the quantity represented by a digit is the product of its face value and its place value? 2) What comparisons would you expect students to identify between operations with whole numbers and operations with decimals numbers?

79 Take the next 5-8 minutes to mark the places in your materials where students work on representing decimals.

80 Working in Groups of 3, take the packet of Number and Operations Activities. Go through the Activities, talking, taking notes, planning how activities of this sort might be used in conjunction with your curriculum materials. Discuss how these activities can Undo Misconceptions

81 * Understanding Patterns * Meaning of Equals * Identifying Functional Relationships * Interpreting Variables * Algebraic Relationships

82 What the Research Says * “When students identify patterns furnished by the teacher, books, or the classroom environment or when they memorize—store various patterns and recall them—they internalize the concept of pattern and realize that it is the same irrespective of the changes in the periodic themes that create different patterns.” (Hershkowitz and Markovits 2000, 169)

83 What to Do * As the study of patterns begins, be sure to make students aware that there are patterns that repeat as well as patterns that grow. * Use cubes, links, square tiles, and other manipulatives to show challenging repeating patterns that students can identify, extend, and then create their own.

84 What to Do (continued) * Present students with materials they see every day and ask them to look for patterns within these things. * Expose students to patterns that appear in nature and within their environment. * Introduce games and activities in which students need to use patterns in order to complete a task or win a game.

85 What to Do (continued) * Examine each multiplication sequence for patterns that repeat (both in the ones place and in the tens place). For example: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, and so on. The “pattern unit” in ones place is: 3, 6, 9, 2, 5, 8, 1, 4, 7, 0. The “pattern unit” in the tens place is: 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, and so on. Students can make predictions about what will come next in the ones and tens place and then extend this to include the hundreds place.

86 What to Do (continued) * Students should use a chart to extend the idea of noting patterns that they’ve begun looking at in the early primary grades with a chart. * Introduce games and activities that require students to use patterns in order to complete the task and win the game.

87 What to Look For * Students are able to describe or name the core unit or pattern core of a repeating pattern. * Students are able to extend a repeating pattern that has a somewhat simple core pattern (AB, ABC, AAB, ABB, …). * Students are able to create a repeating pattern and extend it.

88 Questions to Ponder 1) Where in your academic curriculum do you introduce and reinforce patterns? 2) How do you reinforce repeating and growing patterns with your students?

89 Take the next 5-8 minutes to mark the places in your materials where students work on ideas about repeating and growing patterns.

90 What the Research Says * Teachers and curriculum materials view arithmetic and algebra as distinct and different. This impedes student understanding of critical ideas such as equality, and they encounter difficulties in later grades. * A nonmathematical sense of equals sign is “one of the major stumbling blocks for students when they move from arithmetic to algebra” (Falkner, Levi, and Carpenter 1999).

91 What the Research Says (continued) * We must be sure that students understand that a balance must exist on either side of the equals sign—that it represents the relationship of equality. * “A concerted effort over an extended period of time is required to establish appropriate notions of equality” (Falkner, Levi, and Carpenter 1999, 233).

92 What to Do * Provide multiple part-whole experiences to strengthen number sense. * Allow students to represent two-digit numbers in a variety of ways using connecting cubes. * Provide pairs of students with a two-pan balance and weighted teddy bear counters to explore equality. Encourage students to create multiple equations using the relationship among the bears.

93 What to Do (Continued) * Provide opportunities for students to explore with a number balance. This manipulative helps students develop an understanding of equality and inequality, number comparisons, addition, and subtraction. Symbolically representing the number balance equation connects the concrete to the more abstract.

94 What to Do (Continued) * Provide student experiences in which they create equivalent representations using Cuisenaire Rods. For example, assign a value of 10 to the orange rod. Then ask students to find different ways to represent that value with different rods, such as 10 white rods are as long as one orange rod.

95 What to Do (Continued) * Let students explore unknowns in equations by placing number squares to make an equation true. * Ask “Is this true?” regularly and present equations that are recorded in nontraditional ways (7=2+5 or 11+3=20-6). Expect students to support their answer with an explanation.

96 What to Look For * Are students able to represent numbers in different ways? * Are students able to demonstrate a variety of representations of different numbers? * Are students able to represent equations in a variety of nontraditional ways?

97 Questions to Ponder 1) “Is it true?” is one strategy you can use to help your students understand the meaning of the equals sign. Create several examples of number sentences your students could discuss. How well can they discuss and demonstrate true or untrue? 2) What other instructional experiences can you introduce that will help your students better understand the equals sign represents the relationship of equality?

98 Take the next 5-8 minutes to mark the places in your materials where students work on ideas about the equals sign and equality. Also use this time to share what you remember about the September 2010 Cohort Meeting and the presentation on Equality.

99 What the Research Says * NCTM (2000) defines two specific expectations for grades Grades 3-5 students with regard to understanding patterns, relations, and functions:  Describe, extend, and make generalizations about geometric and numeric patterns.  Represent and analyze patterns and functions using words, tables, and graphs.

100 What the Research Says (continued) * The inclusion of “words, tables, and graphs” emphasizes the notion that students need experiences with multiple representations of functional patterns. * “It is important to see that each representation is a way of looking at the function, yet each provides a different way of looking at or thinking about the function” (Van de Walle 2007).

101 What the Research Says (continued) * Teachers must expose students to a variety of methods of communicating functional relationships, including physical models, pictorial models, symbolic models, and verbal models. * Providing varied representations gives students a comprehensive look at this component of algebraic thinking.

102 What to Do * Provide opportunities for students to explore growing patterns. Growing patterns are precursors to functional relationships (in a functional relationship, any step can be determined by a step number, without calculating all the steps in between). Students observe the step-by-step progression of a recursive pattern and continue the sequence.

103 What to Do (continued) * Choose a meaningful context for functional relationships. - Money spent on candy -Ingredients needed for a recipe -Time required to finish a race -Fuel needed for a vacation

104 What to Do (continued) * Allow students to construct physical models of functional relationships using tiles, toothpicks, connecting cubes, or other hands-on materials. The act of placing toothpicks in a specified pattern or connecting cubes in a sequence can provide insight into the relationship between the two variables.

105 What to Do (continued) * Compare physical models with pictorial or symbolic representations. * Model the language of the dependent relationship and encourage students to describe the relationship. -The amount of money I spend depends on how much candy I buy. -The amount of fuel we use is directly related to the miles traveled.

106 What to Do (continued) * Reinforce number sense through estimation. When students are able to articulate the intuitive understanding of the relationship, they may estimate and solve the function simultaneously. Estimation may also help a student more readily recognize an error. * Have students graph the relationships revealed in a function as a visual picture while learning about rates of change.

107 What to Look For * Students test their rule for the pattern among many terms to confirm their rule is correct. * Students are able to describe a rule verbally as well as pictorially. * Students can extend increasing and decreasing patterns.

108 Questions to Ponder 1) What varied representations can be used to illustrate a functional relationship? 2) Using the sequence of triangles or other combinations of geometric figures, what questions can you pose to students to assess their current level of understanding?

109 Take the next 5 minutes to mark the places in your materials where students work on patterns and functional relationships.

110 What the Research Says * “Students may have difficulty if they view algebra as generalized arithmetic. Arithmetic and algebra use the same symbols and signs but interpret them differently” (Billstein, Libeskind, and Lott 2007, 40). * This can be very confusing to students, particularly if their arithmetic concepts and skills are weak.

111 What the Research Says (continued) * “Many students think that all variables are letters that stand for numbers. Yet the values a variable takes are not always numbers, even in high school mathematics” (Usiskin 1988, 10). * In middle school a variable can be used to represent identifying points on polygons.

112 What the Research Says (continued) * In high school logic, p and q are used to stand for propositions. * The idea that a letter can replace a number only is a misconception many students have—one that is supported by educators who view problems like 5+x=12 as algebra, but 5+ =12 as arithmetic.

113 What the Research Says (continued) * In elementary grades letters appear as abbreviations. The letter m is used to represent the word meter. * And even when students realize that a letter is being used to replace a numerical value, many still assume this is a unique value rather than a general number (Kuchemann 1981).

114 What to Do * Have students identify the elements in a repeating pattern with letters of the alphabet as well as other descriptors. Students will probably already know how to do this, but reinforce that a circle, square, triangle core unit may also be called an ABC pattern.

115 What to Do (continued) * Label polygons with letters identifying each vertex of the shape. Middle school students aren’t confused when they see this and neither should third- through fifth-grade students. When learning about angles in geometry class it seems perfectly sensible to have fifth graders identify these angles with letters that correspond.

116 What to Do (continued) * Have students look for places where a letter is used to represent some word. “Mathematical Equations” can be created in which students replace the letter with a word that makes the equation true. * Begin replacing the “box” in an arithmetic equation with a letter.

117 What to Do (continued) * Point out how letters are used in formulas that are being learned. Before students learn formulas to determine the area and perimeter of polygons or the volume of solid figures, they should know the words that the letters represent.

118 What to Look For * Students aren’t just looking at the digits and the sign and then following the sign. Be sure they are trying to make sense of the open expression. * Students are using a letter to label things that might have been assigned a numeral (length of a rectangle). * Students can use a number balance to determine the missing addend or sum.

119 Questions to Ponder 1) What manipulatives could you use to help young students understand how to solve for a missing number? 2) Once formulas have been introduced into the curriculum, how can you help students see the different uses for variables?

120 Take the next 5-8 minutes to mark the places in your materials where students work on variables.

121 What the Research Says * NCTM (2000) says “instructional programs from prekindergarten through grade 12 should enable all students to:  create and use representations to organize, record, and communicate mathematical ideas  select, apply, and translate [from] among mathematical representations to solve problems  use representations to model and interpret physical, social, and mathematical phenomena”.

122 What the Research Says (continued) * It is critical that students have numerous opportunities to represent problem solving using concrete and pictorial representations before using abstract representations. * Ennis and Witeck (2007) propose that using abstract representations such as numbers and equations requires a deep understanding of a topic. *.

123 What the Research Says (continued) * Moving students too quickly to abstract representations encourages them to perform certain procedures by rote without understanding why these procedures work or what they mean. * “However, we would be negligent as well if we did not help students make the connection between ideas and equations and see how equations can help us solve problems and visualize ideas” (Ennis and Witeck 2007). *.

124 What the Research Says (continued) * Choosing from a variety of representations (concrete, pictorial, equations) helps students understand that some representations are more useful than others when solving a particular problem. * “Little understanding is being developed when a representation is used in a procedural way (Van de Walle and Lovin 2006). *.

125 What the Research Says (continued) * By encouraging students to use a representation in a way that makes sense to them, we allow them to think and reflect about the mathematical idea involved in solving the problem. *.

126 What to Do * Provide opportunities for students to explore, and then talk about, a variety of manipulatives. Model the correct vocabulary that is specific to each manipulative. Plan lessons in which students use these manipulatives. * Encourage students’ use of multiple representations. Create and model an environment in which all explanations and representations are honored and respected.

127 What to Do (continued) * Allow students to freely select from different representations to use in solving any problem. Initially, model conventional ways of representing mathematical situations, but eventually allow students opportunities to choose representations that they are comfortable using. Knowing which type of representation is useful in which situation is an important milestone in mathematical understanding and reasoning for students.

128 What to Do (continued) * Ask students to explain and show how they are thinking about a problem during and following a problem-solving task. When students hear how others use representations to show how they are thinking about a mathematical idea, it helps them to consider other perspectives. Communicating their thinking requires students to reflect on their problem solving and reasoning; listening to students’ explanations enables teachers to determine what students know and can do at any point in time.

129 What to Do (continued) * Provide opportunities for students to solve many open-ended tasks with a representation they have chosen. Follow these problem-solving tasks with classroom discussions. * Model for students how to record their way of solving a problem using numbers. Be sure to record the students’ methods both horizontally and vertically.

130 What to Do (continued) * Observe when students use a representation to determine if they understand representation and how to use it effectively to solve the problem. * Use literature to engage students in problem solving and ask them to represent their solutions in a representation that makes sense to them.

131 What to Look For * Students’ flexibility in their use of representations to show their thinking. * Student use of appropriate vocabulary in describing a strategy or representation.

132 Questions to Ponder 1) Why is it important to model many ways to use representations? Why is it important for students to explain and show how they used a representation to communicate their thinking about how to solve a problem? 2) Think about the types of representations your students are now using. Do they use the same representation for every problem? Do they choose one representation over another one because it more efficient in helping them think about the task?

133 Take the next 5-8 minutes to mark the places in your materials where students work on algebraic representations for their problem solving.

134 Working in Groups of 3, take the packet of Algebra, Patterns, and Functions Activities. Go through the Activities, talking, taking notes, planning how activities of this sort might be used in conjunction with your curriculum materials. Discuss how these activities can Undo Misconceptions

135 * Categorizing Two-Dimensional Shapes * Naming Three-Dimensional Figures * Navigating Coordinate Geometry * Applying Reflection * Solving Spatial Problems

136 What the Research Says * Although NCTM recommends that elementary curricula ask students to use” concrete models, drawings, and dynamic geometry software so that they can engage with geometric ideas” (NCTM 2000, 41), “research continues to indicate that, regrettably, little geometry is taught in the elementary grades, and that what is taught is often feeble in content and quality” (Fuys and Liebov 1993).

137 What the Research Says (continued) * The works of Pierre and Dina van Hiele (Van de Walle 2007, ) has led the way for other mathematics researchers to better understand the different levels of geometric understanding and the variety of experiences that students need in order to move comfortably to the next level. We know that levels have nothing to do with the grade students are in or with their age, but rather relate to experiences to which students are exposed and in which they participate.

138 What to Do * Go on a shape hunt and have students identify shapes in their classroom, school, and home environment. * Combine geometry with number concepts by having students find different shapes on activity pages. * Select math-related literature that shows children accurate plane figures. * Develop “concept cards” of examples and nonexamples.

139 What to Do (continued) * Develop some “best examples” (“clear cases demonstrating the variation of the concept’s attributes” [Tennyson, Youngers, and Suebsonthi 1983, 282]) for each of the two- and three- dimensional shapes included in your curriculum. * Ask students questions about these examples to determine whether they recognize the important properties of each.

140 What to Do (continued) * Encourage students to describe, draw, model, identify, and classify shapes as well as predict what the results would be for combining and decomposing these. * Take care in selecting posters, math-related literature, and other commercial displays. Often these items include inaccurate examples of shapes (show rectangles with only two long and two short sides) and incorrect shape names (ellipses are labeled “ovals” and rhombuses are labeled “diamonds”).

141 What to Do (continued) * Allow students to create shapes from a variety of materials so they see regular as well as irregular shapes. *Have students use Venn diagrams to list common attributes and to classify figures. * Play games like “Guess My Shape,” where clues are given, students draw a shape after each clue, and then determine the shape being described after all the clues have been read.

142 What to Do (continued) * Incorporate other areas of geometry into activities with shapes (for example, creating tessellations and transforming shapes through rotations, translations, and reflections, as well as combining shapes) to give students opportunities to spend more time manipulating and exploring with plane figures.

143 What to Look For * Listen to see if students are able to classify shapes in a variety of ways. * Look to see if students can name shapes regardless of their position. * Watch to see if students can create both regular and irregular polygons.

144 Questions to Ponder 1) How can you use geometry activities in other content areas to help students see the purpose of understanding shapes and concepts surrounding shapes? 2) How does your knowledge about shapes impact your comfort level in teaching geometric ideas?

145 Take the next 5-8 minutes to mark the places in your materials where students work on two-dimensional figures.

146 What the Research Says * Fuys and Liebov (1993) identify various misconceptions children have about geometric shapes. When they undergeneralize, students include irrelevant characteristics. When they overgeneralize, they omit key properties. * Language-related misconceptions occur when they create their own inaccurate definitions (for example, diagonal means slanty).

147 What the Research Says * Clements and Battista (1992) found that children often form a geometric concept by noticing characteristics and developing an “average representation” for any new example. * Research on visual discrimination by Hoffer (1977) indicates that young children sometimes lack the ability to distinguish similarities and differences between objects.

148 What to Do * Look at various objects in the classroom and name these figures correctly. The geometry words should be placed on a mathematics word wall, along with various pictures or other real objects and wooden or foam objects that are made from these solid figures.

149 What to Do (continued) * Offer students a range of activities in which they find, color, name, and discuss the solid figures they need to learn. This enables them to generalize the characteristics of a solid figure so that size, color, and other unimportant attributes aren’t confusing. * Ask questions such as, “Is it OK for this to be a cone even though it’s smaller than this cone?” “Can this still be a cylinder even though it’s not red?” “Would this still be a called a sphere even it was made out of plastic?”

150 What to Do (continued) * Ask students to determine the number of vertices, faces, and edges each figure has, and be sure to have them name the plane figures making up these faces/surfaces. Unfolding and then refolding “nets,” made of cardstock, allows students to decompose these figures to better see what each is made out of. * Play “Guess the Shape”, which provides students with a logical thinking activity while it reinforces the name of the solid figure and the plane figures as well as other characteristics.

151 What to Do (continued) * Provide students with geometric analogies so that they begin to look at the attributes of specific solid figures and see the difference between these and plan figures.. * Teach students how to draw various three- dimensional figures. Be sure to discuss the attributes of these figures, naming the surfaces as plane figures.

152 What to Do (continued) * Have students listen to The Important Book, by Margaret Wise Brown. Give every four students a different labeled picture of a solid figure. Use a sphere, cube, cone, cylinder, pyramid, and rectangular prism. Have students brainstorm things that the shape reminds them of and attributes of the shape. Reread The Important Book and ask students what style the author is using on each page. Have them then work together to create their own page in a book called The Important Thing About Solid Figures.

153 What to Look For * Are students able to describe and classify geometric solids in a variety of ways? * Can students name geometric solids regardless of their position?

154 Questions to Ponder 1) How might you reinforce the names and characteristics of solid figures while still introducing the names of plane figures? 2) How can you use what you know about teaching letter and word recognition to support teaching shape recognition?

155 Take the next 5-8 minutes to mark the places in your materials where students work on three-dimensional figures.

156 What the Research Says * NCTM (2000) expects students to be able to specify location and describe spatial relationships using coordinate geometry. Students in grades Grades 3-5 should be able to:  Describe location and movement using common language and geometric vocabulary.  Make and use coordinate systems to specify locations and describe paths.  Find the distance between points along horizontal and vertical lines of a coordinate system..

157 What the Research Says * “Coordinate systems are an extremely important form of representation” (Van de Walle 2007, 437). This important idea translates into students’ ability to analyze other geometric ideas such as transformations. Later, coordinate geometry plays a crucial role in the representation of algebraic equations.

158 What to Do * Encourage students to articulate descriptions of location, direction, distance related to current of future positions. * Generate a list of vocabulary words for location, direction and distance.  Location: over, under, behind, between, above, below  Direction: left, right, up, down, north, south, east, west, diagonal, clockwise  Distance: near, far, long, short, inches, miles

159 What to Do (continued) * Create a series of steps using the location, direction, and distance words, and allow students to act them out in order to reach a specific destination. When students generate directions for others to follow, or attempt to follow the directions of their peers, they become aware of the importance of direction and distance when seeking location.

160 What to Do (continued) * Have students work in small groups with a large piece of graph paper. Let each student select a game piece and place it on the grid. Students should name the location and then respond to questions such as:  How would you describe your location compared to mine?  How far are you from the origin?  Which is the shortest path for you to reach another student?  Do you share a coordinate with anyone else?

161 What to Do (continued) * Provide activities for students to plot points and additional activities that require students to name spaces. * Make connections to real-world applications using stories and maps. For example: Plan field trips, plan scavenger hunts, use city street maps to explore multiple ways to get to the same location. * Expand the coordinate grid to include negative values when students are ready.

162 What to Look For * Are students able to correctly identify the coordinates for a specific space/point?. * Can students navigate the coordinate grid by describing location, distance, and direction?

163 Questions to Ponder 1) How can you provide meaningful opportunities for students to navigate a coordinate grid and describe location, direction, and distance? How can they be involved in the construction of such systems? 2) Think about the errors your students make when they are naming or plotting points? What questions can you ask to diagnose the misconception?

164 Take the next 5 minutes to mark the places in your materials where students work on foundational ideas about coordinate geometry.

165 What the Research Says * NCTM (2000) says that in prekindergarten through grade 2 all students should be able to:  Recognize and apply slides, flips, and turns.  Recognize and create shapes that have symmetry.

166 What the Research Says (continued) * NCTM (2000) says that all students in grades Grades 3-5 should be able to:  Predict and describe the results of sliding, flipping, and turning two-dimensional shapes.  Describe a motion or a series of motions that will show that two shapes are congruent.  Identify and describe line and rotational symmetry in two- and three-dimensional shapes and designs.

167 What the Research Says (continued) * “Younger students generally ‘prove’ (convince themselves) that two shapes are congruent by physically fitting one on top of the other. But students in grades Grades 3-5 can develop greater precision as they describe the motions needed to show congruence” (NCTM 2000, 167).

168 What the Research Says (continued) * Young children also create pictures with rotational symmetry using, for example, pattern blocks. But they will have difficulty explaining what they did and recognizing that the figure shows rotational symmetry. These informal explorations are important because they prepare students to be able to understand and describe rotational symmetry in later grades.

169 What the Research Says (continued) * Providing concrete materials and introducing paper- folding activities are important; however, technological experiences enhance students’ understanding of transformations, symmetry, and congruence.

170 What to Do * Allow students to role-play flips (reflections), slides(translations), and turns(rotations) with their bodies. * Provide students with pattern blocks, attribute blocks, or tangram puzzles. They will naturally use transformations to create designs. Ask them to explain how a design was made to reinforce the vocabulary of flip, slide, and turn. * Find all possible different arrangements for five connected squares.

171 What to Do (continued) * Let students use materials to model vertical, horizontal, and diagonal reflections across a line of reflection. * Make rotational tools. Students rotate a figure around a point to view how its position looks at different points (quarter- or half-turns).

172 What to Do (continued) * Allow students to investigate transformations on the computer. * Show examples and nonexamples of symmetrical designs and pictures. * Model how to place a mirror and/or a GeoReflector perpendicular to a design or picture to show a symmetrical reflection. Let students explore symmetry with these tools.

173 What to Do (continued) * Provide paper-folding experiences. * Provide geoboards for students to create symmetrical designs or pictures. Geoboards allow concrete experience with rotational symmetry. Students make a design on the geoboard and predict how it will look when turned or rotated. They record predictions on dot geoboard paper and check their predictions by actually turning the geoboard.

174 What to Look For * Are students able to describe and model transformations accurately? * Are students able to recognize, model, and describe symmetry and congruence?

175 Questions to Ponder 1) How will you change the way you are currently teaching transformations? If you feel a change is not necessary, explain what you are currently doing to help students understand this concept. 2) What is one task that has helped your students understand the idea of symmetry? Is there a way you will change the task to provide students with a richer understanding of symmetry? Describe your enhancement.

176 Take the next 5 minutes to mark the places in your materials where students work on transformations, symmetry, and congruence.

177 What the Research Says * The van Hieles propose instruction rather than maturation as the most significant factor contributing to the development of geometric thought (Burger and Shaughnessy 1986). * Geometric thinking can be enhance through meaningful experiences.

178 What the Research Says (continued) * “Any activity that requires students to think about a shape mentally, to manipulate or transform a shape mentally, or to represent a shape as it is seen visually will contribute to the development of students’ visualization skills” (Van de Walle 2007, 443).

179 What to Do * Provide visualization opportunities for students to develop their “mind’s eye.” * Allow students to manipulate and build using a variety of materials, such as multilink cubes, wooden blocks, and connecting cubes. These tactile experiences provide opportunities to view different transformations of figures.

180 What to Do (continued) * Present small groups of students with several nets and geometric solids. Challenge the students to match each net to its corresponding solid. Allow students to confirm their predictions by cutting and folding the nets to form the solids. Additionally, instruct students to trace each face of a solid to form their own nets. They may then cut out their nets and fold to form solids.

181 What to Do (continued) * Let students make two-dimensional representations of three-dimensional figures using Cartesian graph paper or isometric graph paper. * Ask students to draw various polygons using a ruler. Students may then cut out the polygons and draw a line segment connecting any two points on the polygon. Encourage students to name the original polygon and the two new polygons created with the line (slice).

182 What to Do (continued) * Challenge students to: - Slice a triangle to make a trapezoid and a triangle. - Slice a pentagon to make two quadrilaterals. - Slice a hexagon to make two pentagons. * Have students fold a piece of paper and make a single cut through both layers. Have them predict what will result when they unfold what they have. Then have them discuss what happened and how it compared to their prediction.

183 What to Do (continued) * Provide puzzles for students to complete using tangrams, pattern blocks, pentominoes, and GeoReflectors. (For example: make specific shapes out of tangrams and pentominoes; use GeoReflectors for symmetry and reflections.) *Expose students to real-world applications of two- dimensional drawings such as blueprints, house plans, or aerial-view photographs.

184 What to Do (continued) * Present students with sets of cards showing the top, front, and side view of a figure made with connecting cubes. Students then construct the figures using the visual clues. * Look for geometric shapes and figures in works of art.

185 What to Look For * Can students describe shapes and figures and relate them to real-world objects. * Are students able to match two-dimensional representations with the corresponding three- dimensional objects? * Are students able to describe mental images of objects?

186 Questions to Ponder 1) What tools and materials are available at your school to enhance your instruction of geometry concepts? What may be needed to enhance your collection? 2) It is important for students to develop a strong vocabulary when studying geometry. How can you effectively incorporate vocabulary development into your instruction of spatial problem solving?

187 Take the next 5 minutes to mark the places in your materials where students work on spatial problem solving.

188 Working in Groups of 3, take the packet of Geometry Activities. Go through the Activities, talking, taking notes, planning how activities of this sort might be used in conjunction with your curriculum materials. Discuss how these activities can Undo Misconceptions.

189 * Reading an Analog Clock * Determining the Value of Coins * Units Versus Numbers * Distinguishing Between Area and Perimeter * Overgeneralizing Base-Ten Renaming

190 What the Research Says * Fully understanding how to read the hour and minute hands of an analog clock demands a “conscious switching for quarter and half turns in relation to either the hour past of the hour approaching” (Ryan and Williams 2007, 99). * Later, these clockwise fractions of a turn need to be converted into fifteen-, thirty-, and forty-five minute intervals.

191 What the Research Says (continued) * To complicate matters, students need to learn to read an analog clock both clockwise and counterclockwise. * These skills, are hard to master, especially given the non-decimal nature of time. * “Measuring time causes problems for children right through the primary school” (Doig et al. 2006).

192 What the Research Says (continued) * Confusion over the hour and minute hands, the language associated with time (quarter past, quarter of, half past), and the fractions associated with periods of time create all sorts of problems. * By the intermediate grades elapsed time can be more than a challenge for many students.

193 What the Research Says (continued) * In Connect to Standards 2000: Making the Standards Work at Grade 2 (Fennell et al. 2000), the authors suggest that when first introducing how to tell time on an analog clock, teachers use only the hour hand, so that students learn its relative position as time passes. They also suggest placing the numbers on the clock on a number line to demonstrate the divisions of a clock.

194 What the Research Says (continued) * In Understanding Mathematics in the Lower Primary Years (1997), Haycock and Cockburn indicate that part of the problem students have learning to tell time and grasping the passage of time is “the multitude of words relating to time” : “how long, second, minute, hour, day, week, fortnight, month, quarter, year, leap year, decade, century, season, spring, summer, autumn, winter, weekend, term, lifetime, sunrise, sunset, past, present, future, evening, midnight, noon, earlier, prior, following, never, always, once, eventually, instantly, in a jiffy, meanwhile, sometime, sooner, during” (1997, ).

195 What to Do * Using an hour-hand-only clock, position the hand directly on a numeral and have students say the ‘o’clock time. Then position the halfway between two numerals and have students say the half- hour time. Do the same for quarter-past, quarter of, and three-quarters past. * Have students match clock faces with phrase cards to connect the time vocabulary with the face on an analog clock.

196 What to Do (continued) * Play “How Many Minutes After?” which has students making a connection between the numeral that the minute hand is pointing to and the number of minutes after the hour this represents. * Provide students with story problems that give them practice drawing the hands on an analog clock face, writing the digital time, and working on elapsed time problems where they write to explain how they got their answer.

197 What to Look For * Can students correctly match the phrase, digital time, and analog clock face? * Do students have a strategy for determining the number of minutes after the hour each of the numerals on a clock face represents? * Are students able to articulate different ways to say the time at quarter-hour intervals? * Can students solve elapsed-time problems and write to explain how their answer was obtained?

198 Questions to Ponder 1) What are some other activities that you’ve done in your classroom that help students learn how to tell time (on the hour, half-hour, quarter-hour, five-minute intervals, or to the minute? 2) How can you use the technology that is already in your classroom to help students learn to tell time?

199 Take the next 5-8 minutes to mark the places in your materials where students work on telling time on an analog clock.

200 What the Research Says * Randall Drum and Wesley Petty (1999) find that although coins are a concrete model, because their value is nonproportional to their size, they become abstract when those values are taught. * In their research, Douglas Clements and Julie Sarama (2004) found that it takes a long time to master money skills because children have to both count on and skip count in different increments. In traditional instruction, young students are often expected to use mental computation before they completely understand the concept of addition.

201 What the Research Says * Van de Walle and Lovin’s research (2006) reveals that in order for coin values—5, 10, 25—to make sense, students need to understand what they mean. Children look at a dime without thinking about the countable pennies it represents.

202 What to Do *Use children’s literature to pose problem-solving questions that support students’ understanding of counting larger amounts. * Skip-count by 5s, 10s, 25s, and 50s. Use coins to help facilitate the skip-counting. * Play “earn $1.00” with pairs of students throwing two dice and gradually earning their dollar as they make exchanges from pennies to nickels to quarters to fifty cent pieces and/or to dollars.

203 What to Do (continued) * Play “shift count” having students count by dimes, holding up your hand to shift to nickels then quarter then back to dimes, etc.. * Provide opportunities for students to place coin amounts in order according to values. * Provide sets of coins for students to count that are only a nickel and some pennies or a dime and some pennies. Later, include a quarter and some pennies, etc.

204 What to Do (continued) * Provide hundreds charts for students to visually and concretely see the amount of pennies in nickels, dimes, and quarters (Drum and Petty 1999, ). * Encourage students to mentally add numbers that represent the values of different coins. * Target an amount and challenge students to find all the different ways to make that amount using specific coins. * Provide opportunities for students to solve logic riddles about coins. Expect students to justify their answers.

205 What to Look For * Are students able to organize coin sets before determining the value? * Can students find multiple ways to make a given amount?

206 Questions to Ponder 1) What difficulties do your students encounter when they are counting coins? 2) What new strategy can you implement to help students count coins or make change?

207 Take the next 5 minutes to mark the places in your materials where students work on the value of coins.

208 What the Research Says * Constance Kamii (2006) says that teachers almost always ask students to produce a number about a single object rather than asking them to compare two or more objects (156). * The purpose for measurement is not immediately obvious when children measure numerous isolated pictures of objects. It’s not surprising they view these as procedural tasks only.

209 What the Research Says (continued) * In Engaging Young Children in Mathematics (2004), Clements and Sarama describe foundational concepts related to length measurement ( ).  Partitioning. Dividing an object into same-size units.  Unit iteration. Iterating a unit repeatedly along the length of an object.  Transitivity. The understanding that if the length of one object is equal to the length of a second object, which is equal to the length of a third object that cannot be directly compared to the first object, the first and third are also the same length.

210 What the Research Says (continued)  Conservation. The understanding that as an object moves, its length does not change.  Accumulation of distance. When you iterate a unit along an object’s length and count the iterations, the number works convey the space covered by all units counted up to that point.  Relation between number and measurement. Many children fall back on their earlier counting experiences to interpret measuring tasks. Students who simply read a ruler procedurally have not related the meaning of the number to its measurement.

211 What the Research Says (continued) * “Although researchers debate the order of the development of these concepts and the ages at which they are developed, they agree that these ideas form the foundation for measurement and should be considered during any measurement instruction” (Clements and Sarama 2004, 304).

212 What to Do * Ask students to estimate the size of an object first before measuring it. Expect students to explain why they think their estimate is reasonable. * Allow students to measure real-world objects, rather than only pictures on paper. Environmental objects force children to approximate measure, a more realistic application of measurement.

213 What to Do (continued) * Encourage students to measure the same object with a variety of nonstandard units and standard units. This reinforces the importance of a unit’s size, and that some units are more efficient for measuring an object. * Provide students who are having difficulty, with rulers that have fewer markings.

214 What to Do (continued) * Bridge nonstandard units to standard units by providing manipulatives that are “standard” size. * Allow students to make their own rulers to help them understand that it’s not just a tool used to complete a procedural task. Engage students in comparing student-made rulers to standard rulers.

215 What to Do (continued) * Use rulers with the “0” mark a short distance from the edge. Students will be engaged in thinking about both endpoints when measuring with this type of ruler. They will also be focusing on units (and not markings) when they measure. * Ask students to develop and explain strategies for measuring curved and crooked lines or other hard-to-measure objects.

216 What to Look For * Do students understand that when units are small, more are needed to measure, and when units are larger, fewer units are needed. * Can students explain why they think their estimates are reasonable? * Can students identify the starting point on each ruler?

217 Questions to Ponder 1) What difficulties have your students had in measuring length? How will you change your instruction to avoid future misconceptions? 2) What activities or strategies can you use to help your students develop the foundational concepts described in this section?

218 Take the next 5-8 minutes to mark the places in your materials where students work on understanding measurement units versus number.

219 What the Research Says * Big ideas are that area is a measure of covering and that perimeter (length around) is a measure of distance. * Area and perimeter are often taught together or in immediate succession. This combination or sequence may cause confusion, because both area and perimeter require students to consider the boundaries of shape.

220 What the Research Says (continued) * “It is common fallacy to suppose that the area of a region is related to its perimeter” (Leibeck 1984). * Haycock (2001) notes that this relationship proves a contradiction to ideas of conservation. He stresses that rearranging a constant perimeter to form a new shape conserves the perimeter, but not the area. It is therefore important to provide meaningful problems that allow students to experience this phenomenon.

221 What the Research Says (continued) * “Through problem-solving tasks, students develop an understanding of math content and ultimately use that content understanding to find solutions to problems. Problem solving is both the process by which students explore mathematics and the goal of learning mathematics” (O’Connell 2007).

222 What the Research Says (continued) * It is important for students to find their own strategies and algorithms to measure specific attributes rather than simply plug numbers into formulas presented to them without context or meaning.

223 What to Do * Allow students to explore area by covering the surface of a variety of objects with nonstandard units. Then move on to covering surfaces using congruent units such as index cards or multilink cubes. Using a consistent unit allows students to compare the areas of different shapes. * Connect the concepts of area and perimeter to meaningful scenarios like those in children’s books.

224 What to Do (continued) * Give students a set amount of squares and triangles (pattern blocks work well) to use to make a design. Compare the varying designs created. Compare the area of each. Discuss the notion of conservation of area. Get students to recognize that although the designs vary in appearance, the area is consistent for all.

225 What to Do (continued) * Have students use pattern blocks or square tiles to create a variety of designs with a constant perimeter. Instruct students to compare the areas of the shapes made with like materials. Discuss what trends they notice about the area of shapes that all have the same perimeter. * Use a multiplication chart to illustrate the connection between the area of a rectangle and multiplicative arrays. Such connections help students construct their own formulas based on their conceptual understanding rather than mimicking a formula without context

226 What to Do (continued) * Provide cutouts of rectangles, triangles, trapezoids, and circles and have students develop strategies to compare the area of each. *Have students shade, fold, or even cut square-grid paper to explore the relationship between area and the dimensions of the length and width:  What happens to the area of a rectangle if the length is doubled?  How is the area affected if both the length and width are doubled? Why?  If the length and width were cut in half, what would happen to the area?

227 What to Look For * Do students understand that the distance around the perimeter of a figure is different from the amount of space covered by a figure? * Can students cover a figure with units and count the number of units used? * Are students able to see that the size of the units affects the number of units needed to measure the area or perimeter of a figure?

228 Questions to Ponder 1) How can you help students distinguish between area and perimeter in a meaningful context? 2) What materials are available in your school to make the concepts of measurement a hands-on experience?

229 Take the next 5-8 minutes to mark the places in your materials where students work on area and perimeter.

230 What the Research Says * In a 1984 Arithmetic Teacher article, Jim Hiebert writes, “Effective instruction should take advantage of what children already know or are able to learn and then relate this knowledge to new concepts that might be more difficult to learn” (22-23). * The best way to help students see the relationship between units of measure is by having them use measuring devices during mathematics class.

231 What to Do * Give students time to explore with whatever units of measure are being used, prior to giving students problems to solve (whether they be story problems or numerical expressions). * Have students record different ways to represent the same unit of measure.

232 What to Do (continued) * Have students discuss what they might have to do if they want to know what the difference would be between two units of measure when subtracting involves renaming. Ask students to share their ideas for solving a story problem and then discuss whether the answers make sense. Try them out with units of linear measure, liquid measure, and time.

233 What to Look For * Look to see that students represent addition and subtraction of time, length, or fractions with the appropriate conversions. * Have students use illustrations, whenever possible, to represent different ways to name the same thing. * Provide students with manipulative materials to represent fractions in multiple ways.

234 Questions to Ponder 1) What other instructional experiences will help students better understand the base-ten relationship and when (and when not) to use it for renaming? 2) What other measurement tools or materials ought to be introduced to students in third through fifth grades so they can use them to solve problems?

235 Take the next 5 minutes to mark the places in your materials where students work on conversions and renaming.

236 Working in Groups of 3, take the packet of Measurement Activities. Go through the Activities, talking, taking notes, planning how activities of this sort might be used in conjunction with your curriculum materials. Discuss how these activities can Undo Misconceptions.

237 * Sorting and Classifying * Choosing an Appropriate Display * Understanding Terms for Measures of Central Tendency * Analyzing Data * Probability

238 Final Thoughts

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