The Nature of our Collaborative Research Project Funded by KNAER

Principal Investigator Dr. Cathy Bruce (Trent University)

In partnership with the Ministry of Education – Lead: Shelley Yearley

KPRDSB, SMCDSB, OCDSB

Data CollectionAS A STARTING POINT Literature review Diagnostic student assessment (pre) Preliminary exploratory lessons (with

video for further analysis)

Data Collection and Analysis

THROUGHOUT THE PROCESS Gathered and analysed student work samples Documented all team meetings with field notes and

video (transcripts and analysis of video excerpts) Co-planned and co-taught exploratory lessons (with

video for further analysis after debriefs) Cross-group sharing of artifacts

Data Collection and Analysis

TOWARD THE END OF THE PROCESS Gathered and analysed student work

samples Focus group interviews with team

members 30 extended task-based student interviews Post assessments

Findings

Students had a fragile and sometimes conflicting understanding of fraction concepts when we let them talk and explore without immediate correction

Probing student thinking uncovered some misconceptions, even when their written work appeared correct

‘Simple’ tasks required complex mathematical thinking and proving

FRACTIONS Digital Paper

We explored linear, area and set models…

We asked: Which representation is helpful in which situations?

Tad Watanabe, 2002 TCM article

Number Lines

So we looked closely at linear models…

How do students: -think about numbers between 0 and 1-partition using the number line-understand equivalent fractions and how to place them on the number line

Why Number Lines?Lewis (p.43) states that placing fractions on a number line is crucial to student understanding. It allows them to:

• PROPORTIONAL REASONING: Further develop their understanding of fraction size

• DENSITY: See that the interval between two fractions can be further partitioned

• EQUIVALENCY: See that the same point on the number line represents an infinite number of equivalent fractions

What We Found Out

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Growth in Achievement

Pre Percentqge

post Percentage

The project was a benefit to students

Other challenges & misconceptions we encountered with Grade 4-7 Students

Some Common Misconceptions Translates to…

Area Models: Size of the partitioned areas doesn’t matter when using an area model, just the number of pieces

Approximations in drawings (is it good enough? Understanding the need for equal parts, but approximating equal parts in area model drawings)

Set Models: Fractions cannot represent ‘parts of a set’. All representations of fractions must show the ‘parts’ as attached or touching, and all parts must be exactly the same no matter how the set might be named

Inability to see the following as a fraction relationship:

Equivalent Fractions: Always involve doubling

1/2 then 2/4 then 4/8 then 8/16(5/10ths would not be considered in this scenario)

The numerator and denominator in a fraction are not deeply related: (that the fraction has two numbers that represent a value because the numbers have a relationship)

2/5ths is equal to 1/10th because 2 fives are 10 and 1 ten is 10Note - Use of fraction language by educators may contribute to this problem: When we show a fraction, but don’t say it, the students seem to have multiple ways of naming it themselves, some of which are confusing (e.g., one over ten)

Implications for teachingExpose students to a careful selection of

representations that further student understanding.

Consider how representations can act as the site of problem solving/reasoning.

Think more about how to teach equivalent fractions.

Think more about the use of the number line.

2012-2013

5 teams (5 teachers – 5 release days)

3 teams engaged in Collaborative Action Research on addition and subtraction of fractions KPRDSB, SCDSB, TLDSB

2 teams field testing lessons from last year on compare, order, represent YCDSB, SMCDSB