1. 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

2. This information is being provided as part of a Research to Practice Bridge Workshop administered by the Regional Educational Laboratory Southeast. Information and materials mentioned or shown during this presentation are provided as resources and examples for the viewer's convenience. Their inclusion is not intended as an endorsement by the Regional Educational Laboratory Southeast or its funding source, the Institute of Education Sciences (Contract ED-IES-12-C-0011). In addition, the instructional practices and assessments discussed or shown in these presentations are not intended to mandate, direct, or control a State’s, local educational agency’s, or school’s specific instructional content, academic achievement system and assessments, curriculum, or program of instruction. State and local programs are free to use any instructional content, achievement system and assessments, curriculum, or program instruction they wish, insofar as they support the goals and objectives of their state and local education agencies.

3. Analyzing student thinking about comparing decimals

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

5. 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?

6. Video

7. 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?

8. 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

9. 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

10. (MIS)Understanding decimals

11. What is mathematically challenging about decimals?
Example of error
Underlying misconception
.34 > .5
.75 >
.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)

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

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

14. 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)

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

16. 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?

17. representing decimals

18. 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.

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

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

21. 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

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

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

24. 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

25. 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

26. Comparing decimals

27. Ordering decimals
Put the following lists of decimals in order from least to greatest however you usually do it: b)

28. Ordering decimals with number lines
Put the first string of decimals in order using a number line.
If you finish, try the second string:
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?

29. 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

30. 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

31. 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

32. representing and making sense of computation

33. 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.

34. 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.

35. Operations with decimals: Common errorsvs vs vs
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

36. 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”

37. Modeling multiplication as repeated addition

38. 7 x 1 (repeated addition) x
how many size of
groups one group

39. 7 x .1 (repeated addition) or x
how many size of
groups one group

40. .7 x .1 (repeated addition) x
how many size of
groups one group

41. .7 x .5 (repeated addition) x
how many size of
groups one group

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

43. 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

44. 7 x 1 (repeated addition) x
how many size of
parts one part

45. 7 x .1 (repeated addition) x
how many size of
parts one part

46. .7 x .1 (repeated addition) .7 x .1 how much size of
of a part one part

47. 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

48. 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

49. 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