assessing elementary students’ functional thinking skills: the case of function tables

Katherine L. McEldoon & Bethany Rittle-Johnson

Project GoalsDevelop an assessment of elementary

students’ functional thinking abilities, an early algebra math skill

Develop a model of knowledge progression

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Functional Thinking

• A type of mathematical thinking which focuses on the relationship between two (or more) varying quantities, specifically the kinds of thinking that lead from specific relationships to generalizations of that relationship across instances. (Smith, 2008)

• Encapsulates important core components of early algebraic reasoning, such as generalization and covariation. (Carraher, Martinez, & Schliemann, 2008)

77

Out = (In x 2) + 1Y = 2X + 1

The table shows how the “In” numbers are related to the “Out” numbers. When a 38 goes in, what number comes out?

A.41 B.51 C. 54 D. 77

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Functional Thinking Performance – Grade 4

The table shows how the “In” numbers are related to the “Out” numbers. When a 38 goes in, what number comes out?

A.41B.51C.54D.77

National Assessment of Educational Progress (NAEP), National Performance results in Mathematics at Grade 4; 2007

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Function TablesFocus: Functional Tables

Determining Values and Rules

Typical Tasks (Carraher & Earnest, 2001; Schliemann & Carraher, 2000)

Fill in the missing values in this table

What is the rule for this table?Asked to select a rule from several

choicesAsked to write the rule verbally or

symbolically

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Column A

Column B

2 6

3 7

4 8

5 9

6 10

14 18

21 25

41 45

Column A

Column B

2 6

3 7

4 8

5 9

6

14

21

41

Function Table Competencies

Within function table problems, we isolated required competencies and used this as a basis for our assessment

Loosely hypothesized order of difficulty:

1. Apply a Given Rule (prerequisite) 2. Determine Next Y value in Sequence3. Determine Near Y value in Sequence4. Determine Far Y value in Sequence5. Recognize a Rule (verbal/symbolic)6. Generate Rule Verbally7. Generate Rule Symbolically

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Column B = Column A + 4

B = A + 4

Wilson’s Construct Modeling Approach

Wilson’s Four Building Blocks1) Construct Map2) Item Design 3) Item Score4) Measurement Model

Assess the student performance data to evaluate your construct map and items

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Item Desig

n

Item Score

Measurement Model

Construct Map

Item Design: Assessment We designed items that tapped each of these

competenciesModified from e.g. Blanton; Schliemann; Warren, Cooper &

Lamb

Items varied in operation used in underlying function

33 responses to 11 items16 of which had an additive underlying function

Y = X + 2

10 had a combination underlying function Y = 2X + 2

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Item Design: Assessment We developed items that

tapped each of these competencies1. Apply a Given Rule

2. Determine Next Y value in Sequence

3. Determine Near Y value in Sequence

4. Determine Far Y value in Sequence

5. Recognize a Rule

6. Generate Rule Verbally

7. Generate Rule Symbolically

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5. Recognize a Rule



10

Column A

Column B

2 6

3 7

4 8

5 9

6

14

21

41






5. Recognize a Rule



11

Column A

Column B

2 6

3 7

4 8

5 9

6

14

21

41






5. Recognize a Rule



12

Column A

Column B

2 6

3 7

4 8

5 9

6

14

21

41






5. Recognize a Rule – Verbal & Symbolic



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What is a rule used in the table above to get the numbers in column B from the numbers in column A? A) Multiply the number incolumn A by 2.B) Divide the number in column A by 2.C) Subtract 2 from the number in column A.D) Add 2 to the number in column A.






5. Recognize a Rule



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Column A

Column B

2 6

3 7

4 8

5 9

6

14

21

41

“The rule is that you add 4 to the A number to get the B number”

What is a rule for figuring out what number belongs in column B?






5. Recognize a Rule



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Column A

Column B

2 6

3 7

4 8

5 9

6

14

21

41

“B = A + 4”

Write this rule as a number sentence, using “A” to stand for any number in column A and “B” to stand for any number in column B.

Item Scores: CodingCoding

Each response only tapped one competency

Each was coded as correct or incorrect

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Item Desig

n

Item Score

Measurement Model

Construct Map

Data Collection: Procedure231 second through sixth grade students

Middle class suburban community

Predominantly Caucasian population

During one 40 minute class period

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Measurement ModelBased on Item Response Theory

Item Response Theory encompasses a set of ways to mathematically model how both Student Ability Estimate and Item Difficulty are related to a student’s Item Responses

It is a useful methodology to use when evaluating an assessment instrument both in terms of its ability to accurately

estimate student ability but it also give metrics of the quality of each

item on the instrument.

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Wright MapAn Wright map was generated by a Rasch model (a

type of item response model) and was used in this evaluation

Item difficulties based on the Wright maps were used in the development of our Construct map

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XXXXXXXX| XXXXXXXXXXX|Item E 3 XXXXX-3-Item C Item D XXXXXXXXXX| XXXXXXXXXXXX| 2 XXXXXXXXX-2- XXXXXXXXXXXXXX| XXXXXXXXXXXXXXXXXXXXX| 1 XXXXXXXXXXXXXXXXXXXX| XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX|Item B XXXXXXXXXXXXXXXXXXXXXXXXXX| 0 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX|Item A

Student Ability ScoresItem Difficulty Scores

This item has a difficulty level of .98, meaning that the average student has a ~0.47 probability of getting it correct

This item difficulty is 3.1logits, or the average student has a ~0.28 probability of getting it correct

Measurement ModelWright Map

Wright Map Combination Functions

---------------------------------------------------------------------------------------| 4 XXXXXXXXX| | XXXXXXXX| | XXXXXXXXXXX|4-8 | 3 XXXXX|3-6 4-7 6-9 7-10 | XXXXXXXXXX| | XXXXXXXXXXXX| | 2 XXXXXXXXX| | XXXXXXXXXXXXXX| | XXXXXXXXXXXXXXXXXXXXX| | 1 XXXXXXXXXXXXXXXXXXXX| | XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX|2-5 | XXXXXXXXXXXXXXXXXXXXXXXXXX| | 0 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX|5b-4 | XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX|5a-3 | XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX| | -1 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX| | XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX| | -2 XXXXXXXXXXXXXXXXXXXXXXXXXXX| | XXXXXXXXXXXXXXXXXXXXXXXXXXX|1-2 | XXXXXXXXXXXXXXX| | -3 XXXXXXXXX| | XXXXXXXXXXX|1-1 | XXXXXXX| | -4 XXXXXXXXX| |=======================================================================================Each 'X' represents 0.4 cases

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7) Generate Rule Symbolically

6) Generate Rule Verbally5) Recognize a Rule4) Determine Far Y value in

Sequence3) Determine Near Y value

in Sequence2) Determine Next Y value

in Sequence1) Apply a Given Rule

Construct MapConstruct Map

A representation of the continuum of knowledge that people are thought to progress through for the target construct (Wilson, 2005)

Placed competencies into a hierarchy based on We used item difficulty scores from IRT measures Their clumping on the Wright maps From theory

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Mapping of Competencies into Construct Map Levels

Level Description

Competencies

Level 4: Generate Symbolic Rule

- Generate an explicit symbolic rule

Level 3: Generate & Use Verbal Rule

- Generate an explicit verbal rule- Complete a function table with missing values

Level 2: Recognize Rule & Determine Next

- Select a correct rule out of several choices- Determine the next Y value in a function sequence

Level 1: Apply Rule

- Use a given rule to determine new Y values

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5. Recognize a Rule (verbal/symbolic)




1. Apply a Given Rule

Benefits of a Construct Modeling Approach

First, it elucidated the relative difficulty of functional thinking abilities, and at times this was not in line with our predictions.

Second, the resulting assessment is a criterion referenced measure which is particularly appropriate for assessing

Students’ ability estimate levels

Learning gains from an intervention

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Summary Identified key competencies that are important for

elementary-level functional thinking, with a focus on function table problems

These competencies were then incorporated into an assessment

Student performance data was used to develop a construct map, or proposed knowledge progression, of elementary-level functional thinking abilities

The resulting construct map provided insight into the acquisition of functional thinking knowledge in elementary-school students

This can be used as a research tool, and to guide instructional sequences for students

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Thank you

For more information:

http://peabody.vanderbilt.edu/earlyalgebra.xml

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The first author is supported by a predoctoral training grant provided by the Institute of Education Sciences, U.S. Department

of Education, through Grant R305B040110 to Vanderbilt University. The opinions expressed are those of the authors and

do not represent views of the U.S. Department of Education.

assessing elementary students’ functional thinking skills: the case of function tables

Documents

y value

given rule prerequisite

additive underlying

given ruledetermine

schliemann carraher

national performance

function tablesfocus

student performance