1. Margaret M. Flores, Ph.D., BCBA-D
Increasing Understanding and Fluency in Multiplication and Division with CRA
Jessica H. Milton, M.Ed.
Margaret M. Flores, Ph.D., BCBA-D
Auburn University
KUCRL 2018
Multiplication with Regrouping

2. Session Objectives
Describe how multiplication and division instruction can be structured to address conceptual understanding of operations.
Describe how you can readily use available materials for teaching multiplication and division at the concrete or representational levels of CRA
Multiplication with Regrouping

3. Background Foundations of mathematical thinking
Additive reasoning is understanding that quantities are joined
Multiplicative reasoning is more complex; numbers act upon each other
Example 3x2: for each of the units in three, there will be a set of two
Multiplicative reasoning assists students in understanding division
To establish conceptual understanding of operations, instruction should include hands-on methods
Concrete-representational-abstract (CRA) sequence is an example
Multiplication with Regrouping

4. Purpose of Intervention
Students in study (grades 4-6) had mastered most multiplication facts. However, they struggled with division.
Although students had memorized multiplication facts, they may not have developed conceptual understanding of the operation or multiplicative reasoning
Purpose: investigate the effects of an alternating sequence of CRA instruction in both multiplication and division.
Research Questions
What are the effects of alternating CRA multiplication and division instruction on students’ accuracy and fluency in unknown facts?
What are the effects of alternating CRA multiplication and division instruction on students’ understanding of the division operation and its inverse relation to multiplication?
Multiplication with Regrouping

5. Cognitive Ability a (IQ)
Participants
Student
Age
Grade
Cultural
Back-ground
Disability
Cognitive Ability a (IQ)
Mathematics
Calculation b
Math Facts
Fluency c
Tyler 9 4
White
OHI d 84
SS=94
SS=89
Julia 10 5
SLD e
100
SS=97
Wyatt 13 6
SLD 96
SS=81 NA
Sam
Latino 91
SS=65
Antoni 73
SS=63
SS=56
a. standard score Wechsler Intelligence Scale for Children V
b. standard score Calculations subtest of Woodcock Johnson Tests of Achievement IV
c. standard score Math Facts Fluency subtest of Woodcock Johnson Tests of Achievement IV
d. OHI = Other Health Impairment
e. SLD = Specific Learning Disability
Multiplication with Regrouping

6. Instructional Materials
Miller and Mercer’s (2010) Strategic Math Series
Multiplication Facts (0-81)
Division Facts (0-81)
Changed learning sheets
Same format, except problems included unknown facts
2x9, 8x3, 6x3, 9x3, 7x3, 4x6, 4x8, 7x4, 9x4, 6x6
Teacher used manual and its suggested scripts, but equations and problems included unknown facts
Multiplication with Regrouping

7. Instructional Methods
Explicit Instruction
Concrete-representational-abstract (CRA)
Concrete
Computation and identification of numbers using objects
Representational
Computation and identification of numbers using pictures and drawings
Abstract
Computation and identification of numbers without visual aids (just numbers)
Strategy Use
DRAW
Discover the sign; Read the problem; Answer, or draw
and check; Write the answer
Multiplication with Regrouping

8. Multiplication and Division
Teach multiplication with regrouping using base-ten blocks
Demonstrate understanding of relations between operations
Implement multiplication and division together, alternating lessons
Lesson 2: concrete division
18 ÷ 6
Given 18, how many groups of six can be made?
Set out 18 objects
Make groups of six and place each on plate
Count the number of groups (plates).
Lesson 1: concrete multiplication
6 x 3
Six groups of three
Set out plates to represent groups
Place three objects on each plate
Multiplication with Regrouping

9. Assessment Materials 3 Types of Quantiative Assessment Probes
One-minute multiplication probes (30 facts, random order)
Untimed division probes
One-minute division probes (30 facts, random order with varied divisors)
Qualitative Data
Teacher interviewed each student before and after the intervention
Teacher presented a sheet with 3 division facts. Used script that included:
What is eighteen divided by nine?
Explain your answer. How did you find the answer?
If no student response, the teacher asked, Can you show me in the space below?
If the student’s response did not include reference to multiplication, the teacher asked, How can you find the answer using multiplication?
Multiplication with Regrouping

10. Research Design
Quantitative: Multiple probe across students for each skill (mult fluency, div accuracy, div fluency)
First student began intervention after stable baseline (five data points with no more than 10% variability across the last three)
Other students remained in baseline until the first student demonstrated progress (60% of facts division facts correct). If the second student’s data were stable, she began the intervention. Repeated procedures for third group.
Mastery defined as writing at least 30 correct digits across three consecutive timed probes and 100% accuracy for three consecutive untimed probes.
Maintenance data collected every two weeks after student met mastery criteria
Qualitative: Deductive thematic approach to assess the extent to which students’ explanation of division was similar to the definition of division and its relation to multiplication (Braun & Clarke, 2006).
Researchers analyzed data collected from videos of student interviews in which students described how they solved division equations.
Multiplication with Regrouping

11. Quantitative Results
Multiplication Fluency

12. Quantitative Results
Division Fluency

13. Quantitative Results
Division Accuracy

14. Qualitative Results
Prior to intervention, none of the students provided a verbal explanation of the division operation, its relation to multiplication, or constructed drawings.
Example responses: I don’t know, I divided eighteen divided by nine, Eighteen divided by two is three, I divided.
After the intervention, all students provided a verbal explanation and drawing
When presented with 24÷8, Tyler said, there are 24 in all, and proceeded to draw tallies, stopping the drawing process after eight and starting the next group further away from the last group of eight. Tyler continued until he drew 24 tallies and said, There are three groups of eight.
When presented with 28÷7, Julia, Sam, Wyatt, and Antoni drew tally marks and circled same-sized groups. The following is an explanation that each student used, but with different problems. You could make 28 tally marks and have four in each group. You will have four tally marks in each one and then you count how many groups there are, 1, 2, 3, 4, 5, 6, 7.
When given the problem, 24÷3, Wyatt said he would keep subtracting three until there were zero.

15. Qualitative Results continued
After the intervention, four students spontaneously indicated knowledge of an inverse relation between division and multiplication.
Example responses:
I would say nine times what is 36.
You would do nine times blank equals 18.
I know that 18 divided by three is six because three times six is 18.
Antoni required a specific prompt from the teacher asking how to solve the problem with multiplication.
After the teacher asked, Antoni said, Three times what is 18; three times six is 18.

16. Social Validity
Students reported that lessons were helpful, skills increased and they enjoyed instruction
Students’ comments about division and their increased participation general education mathematics classes may have indicated that their confidence increased
Teacher reported that lessons were easy to implement and that students’ skills improved
General education teachers commented on students’ increase in multiplication and division fact fluency

17. Implications
Implementation occurred during short sessions which allowed for instruction in other skills
Students’ conceptual understanding as well as fluency improved
Better prepared to more advanced mathematics concepts and skills
Fluency intervention can address conceptual understanding
Implementation could occur in most settings
Materials accessible
Short amount of time

18. Contact Margaret M. Flores, Ph.D., BCBA-D MMF0010@auburn.edu
Jessica H. Milton, M.Ed.

19. or scan the evaluation QR code (above or on your conference program).
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