Decimals: Connections to the Common Core and the IES Practice Guide Tim Boerst, Ph.D. Meghan Shaughnessy, Ph.D. University of Michigan May 21, 2015

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Analyzing student thinking about comparing decimals

Decimal subtraction Do the subtraction problem yourself. Think about the strategies that fifth graders could use to solve this problem.

Focus questions How does the student solve the subtraction problem? What does the student appear to know about decimals? What questions might you like to ask the student to learn more about his understanding?

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Focus questions How does the student solve the subtraction problem? What does the student appear to know about decimals? What questions might you like to ask the student to learn more about his understanding?

Possible observations The student does not need representations of the quantities beyond numbers to reason about them The student appears to understanding that 10/10 makes one whole It is unclear whether he understands the symbolic representation of the problem because it is read to him (i.e., Does he know that .7 is “seven-tenths”?) The student appears to prefer working with the fraction representation over the decimal representation

Using representations to support the teaching and learning of decimals (Mis)Understanding decimals Challenges using generalizations from work with whole numbers Common student misconceptions Making connections with fractions Representing decimals Choosing and using representations Comparing decimals Modeling the comparison Choosing numerical examples Representing and making sense of computation Analyzing common student errors Modeling computation with decimals

(MIS)Understanding decimals

What is mathematically challenging about decimals? Example of error Underlying misconception .34 > .5 .75 > .7500002 .20 is ten times greater than .2 Hundreds, tens, ones, oneths, tenths, hundredths... Point 21 has a value of 21 Longer decimals are greater (overgeneralizing from whole numbers) Longer decimals are lesser (overgeneralizing new insights into decimals) Adding a zero to the right makes a number ten times larger (overgeneralizing from whole numbers) Lack of understanding of the “specialness” of one in the place value system Decimal are read like whole numbers Most examples from (Irwin, 2001)

Common mis-understandings about connecting fractions and decimals How might a student produce this answer? Separating the numerator and denominator with a decimal point

Common mis-understandings about connecting fractions and decimals How might a student produce this answer? Writing the numerator as the decimal

Common mis-understandings about connecting fractions and decimals How might a students produce these answers? Dividing the denominator by the numerator (and not knowing how to express the remainder)

Common mis-understandings about connecting fractions and decimals How might a student produce these answers? Ignoring whole numbers in mixed numbers

Common mis-understandings about connecting fractions and decimals Separating the numerator and denominator with a decimal point Writing the numerator as the decimal Dividing the denominator by the numerator Ignoring whole numbers How can we help student understand decimals as numbers?

representing decimals

Commonplace/Typical decimal representations used in teaching What representations do teachers use to support students’ understanding of decimals? What contexts do teachers use to support students’ understanding of decimals? Pick a representation or context and describe why teachers use it.

How do teachers represent decimal to students? (Glasgow et. al, 2000)

Commonplace/Typical decimal representations used in teaching Use several representations to show the following numbers: 1 .6 .007 .60 2.8 What are the affordances and limitations of the representations?

The power of the number line Represents decimals as numbers Addresses these (mis)-understandings and others: Longer decimals are greater Longer decimals are lesser Adding a zero to the right makes a number ten times larger Density of the rational numbers Infinitely many names for any point on the line

Connecting decimals and fractions using the number line Write as a decimal.

Connecting decimals and fractions using the number line Write as a decimal.

The challenges of using the number line It is challenging for students to generate number line representations with parts smaller than tenths Pre-partitioned number lines can show parts smaller than tenths, but often require additional work to make the partitioning meaningful to students The amount of decimal places that you want to represent can constrain the span of numbers you are able to use

From the IES practice guide Recommendation #2: Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Things to try : Use representations to support meaningful connection of fractions and decimals Analyze the affordances of different representations / contexts in light of mathematical purposes

Comparing decimals

Ordering decimals Put the following lists of decimals in order from least to greatest however you usually do it: 2.3 0.23 0.8 0.08 .23 b) 0.4 1.4 .55 .0098 11 0.40

Ordering decimals with number lines Put the first string of decimals in order using a number line. 2.3 0.23 0.8 0.08 .23 If you finish, try the second string: 0.4 1.4 .55 .0098 11 0.40 What did you notice about putting decimal numbers in order using the number line? What do the number lines show, and how do they help with the typical difficulties we discussed?

Possible observations The magnitudes of the decimals are visible The equivalence of multiple decimal representations is visible (it is the same point even when the partitioning looks different) Decimals need to be selected strategically A number line does not produce answers– students need to learn about its features and properties and develop ways of using them to do mathematical work

Developing tasks that illuminate big ideas of decimals Develop a set of five numbers that would require that students grapple with decimal challenges and misconceptions through the use of number lines. Longer decimals are greater (or longer decimals are lesser) Adding a zero to the right makes a number ten times larger You can ignore all zeros to the right of the decimal point 1.20 1.02 1.2 1.020 10.2 2.1 0.4 2.10 0.40 0.04 0.123 0.4 .35 .456 0.5

From the IES practice guide Recommendation #2: Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Things to try : Generate numerical examples to support “productive struggle” with key decimal ideas and likely misconceptions Analyze the affordances of different representations / contexts for particular numerical examples

representing and making sense of computation

IES practice guide recommendation #3 – Help students understand why procedures for computations with fractions make sense. Use area models, number lines, and other visual representations to improve students’ understanding of formal computation procedures Provide opportunities for students to use estimation to predict or judge the reasonableness of answers to problems involving computation with fractions. Address common misconceptions regarding computational procedures with fractions. Present real-work contexts with plausible numbers for problems that involve computations with fractions.

Operations with decimals: Common errors For each piece of student work: Solve the problem yourself Describe the steps the student likely took to reach his/her answer Indicate why students might make this error.

Operations with decimals: Common errors vs. 5.63 vs. 1.03 vs. 40.29 vs. .08 Students make these errors because they are: not making sense of the numbers involved in the processes overgeneralizing from previous experiences relying on rules

Multiplication as repeated addition a x b is the result of adding b repeatedly, where a is the number of b’s that are added 2 x 4 = 4 + 4 Counting model Often used to motivate multiplication -- a “short cut” for adding many of the same number a x b can be interpreted where a represents “number of groups” and b represents “size of group”

Modeling multiplication as repeated addition

7 x 1 (repeated addition) 7 x 1 how many size of groups one group

7 x .1 (repeated addition) or 7 x .1 how many size of groups one group

.7 x .1 (repeated addition) .7 x .1 how many size of groups one group

.7 x .5 (repeated addition) .7 x .5 how many size of groups one group

Comparing representations of decimal multiplication What similarities and difference do you notice? How, if at all, are the differences important?

Representing multiplication of decimals Calculate. Model each problem using the unit square, matching the factors and the product to the model. If you have time, switch the position of the factors and notice how it changes the unit square representation

7 x 1 (repeated addition) 7 x 1 how many size of parts one part

7 x .1 (repeated addition) 7 x .1 how many size of parts one part

.7 x .1 (repeated addition) .7 x .1 how much size of of a part one part

Revisiting the IES practice guide Recommendation #3: Help students understand why procedures for computations with fractions make sense. Things to try : Use the representations to support sense making of computational procedures and solutions Don’t race to commutativity, while answers will end up the same, the representations of the process will look different Develop/select problems that expose and work through common misconceptions

Improving instruction to support understanding of decimals To help students recognize that fractions (and decimals) are numbers and that they expand the number system beyond whole numbers, try to: Use representations to support meaningful connection of fractions and decimals Analyze the affordances of different representations / contexts in light of mathematical purposes Generate numerical examples to support “productive struggle” with key decimal ideas and likely misconceptions Analyze the affordances of different representations / contexts for particular numerical examples

Improving instruction to support understanding of decimals To help students understand why procedures for computations with fractions (and decimals) make sense. Use the representations to support sense making of computational procedures and solutions Don’t race to commutativity, while answers will end up the same, the representations of the process will look different Develop/select problems that expose and work through common misconceptions