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Lessons Learned from Our Research in Ontario Classrooms. 1. Intro. 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).

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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
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
sorting activity
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?
ordering fractions on a number line
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 representations
What the research already says about fractions representations

Using 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
What’s wrong with circles?
  • They don’t fit all situations
  • They are hard to partition equally (other than halves and quarters)
  • It can be hard to compare fractional amounts.
over reliance on circles to compare can lead to errors and misconceptions
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…

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?
implications for teaching
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 teaching1
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
some instructional moves included
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

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

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)

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