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 6
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 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? 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…
Fraction situations Students 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 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 MisconceptionsTranslates 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 doubling1/2 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 1/10 th because 2 fives are 10 and 1 ten is 10 Note - 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 22
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)