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Lessons Learned from Our Research in Ontario Classrooms

INTRO

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• The cultural activity of teaching - the ways in which the teacher and students interact about the subject - can be more powerful than the curriculum materials that teachers use. (Lewis, Perry & Hurd, 2004)

• Most reforms stop short at the classroom door; all available evidence suggests that classroom practice has changed little in the past 100 years. (Lewis, Perry & Hurd, 2004)

We noticed that…

• students had a fragile and sometimes conflicting understanding of fraction concepts

• probing student thinking uncovered misconceptions, even when their written work appeared correct

• ‘simple’ tasks required complex mathematical thinking and proving

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DIFFERENT MEANINGS OF FRACTIONS

Sorting Activity

• Match the situation to the representations and symbolic notation.

• What was your solution?

• Can you think of a representation that would be even better suited to the situation?

NUMBER LINES

Ordering Fractions on a Number Line• supports student acquisition of an

understanding of the relative quantity of fractions (beyond procedural)

• allows students to understand density of fractions

• reinforces the concept of the whole

Justification for Using the Number Line

Lewis (p.43) states that placing fractions on a number is crucial to students‘ understanding. It allows them to:

•To further develop their understanding of fraction size•To see that the interval between two fractions can be further partitioned•To see that the same point on the number line represents an infinite number of equivalent fractions

•Number Line vs. the Pie Chart and other representations

Purpose: To explore how children’s understanding of rational number develops over the elementary grades using the number line. Specifically, examining children’s knowledge of the existence of numbers between 0 and 1, their ability to partition equally, and the onset of their understanding of equivalent fractions and how to place them on the number line

What the research already says about fractions representationsUsing pre-partitioned shapes in teaching and assessing can mask an incomplete or incorrect appreciation of fractions as relational numbers. Many teachers are unaware that students are adopting only part of a regional “part of a whole” model of fractions. That is, some students focus on the “number of pieces” named by a fraction and others the “number of equal pieces” named, without addressing the relationship between the area of the parts compared to the area of the whole region.

Gould, Outhred, Mitchelmore

What’s wrong with circles?

1. They don’t fit all situations

2. They are hard to partition equally (other than halves and quarters)

3. It can be hard to compare fractional amounts.

Over-reliance on circles to compare can lead to errors and misconceptions…

No matter what the situation, children defaulted to pizzas or pies…

We had to teach another method for comparing fractions to move them forward…

Misconception:

Analysing Student Work

Fraction situationsStudents were given different situations and asked to represent them. This one in particular struck us…

Lucy walks 1 1/2 km to school. Bella walks 1 3/8 km to school. Who walks farther? What picture would help represent this fraction story?

Fractions on Stacked Number Lines

Implications for teaching

• Expose students to a range of representations

• Get students to connect representations with stories in context to make better decisions about which representation(s) to use when

Implications for teaching

• Lots of exposure/discussion/class math-talk to enhance the language of fractions

• Enable students to make precise drawings when they want/need to (e.g., provide grid paper)

• Think more about how to teach equivalent fractions

• Think more about the use of the number line

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)

Some instructional moves included:

• having students compose and decompose fractions with and without concrete materials

• revisiting the same concept in a number of different contexts

• exploring how different representations were more appropriate to some contexts

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Which representation, when?

• Insert chart here

• Selecting and using particular methods to check understanding to check student

• Ask, not feed• Go back and edit, CMT strategies etc

And what did the teachers think?• Let’s hear the debrief after the lesson.

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What are we learning?

• Over use of circle representations• Enormously difficult to teach and difficult to

learn• Under use of linear models and set

models

CONSOLIDATION

Tad Watanabe, 2002 TCM article

What we are learning:

• We know for sure that students are using circle representations even when it doesn’t make sense mathematically or contextually.

• That fractions is an enormously difficult area to teach partly because it is incredibly difficult to learn (misconceptions of students)

• Certain representations seem to be underutilized even though they appear to be helpful to students. Linear models for representing fractions have an important place in Ontario classrooms – when and how does it help kids (discrete, linear, area models)

Talk-time

Knowledge Creation

• Build knowledge at Camppp just like teachers learning together in classroom contexts and with one another