1 Lessons Learned from Our Research in Ontario Classrooms

SHELLEY YEARLY CATHY BRUCE 3 out of 2 people have trouble with fractions…. 2

Why Fractions? Students have intuitive and early understandings of ½ (Gould, 2006), 100%, 50% (Moss & Case, 1999) Teachers and researchers have typically described fractions learning as a challenging area of the mathematics curriculum (e.g., Gould, Outhred, & Mitchelmore, 2006; Hiebert 1988; NAEP, 2005). The understanding of part/whole relationships, procedural complexity, and challenging notation, have all been connected to why fractions are considered an area of such difficulty. (Bruce & Ross, 2009) 3

Why Fractions? Students also seem to have difficulty retaining fractions concepts (Groff, 1996). Adults continue to struggle with fractions concepts (Lipkus, Samsa, & Rimer, 2001; Reyna & Brainerd, 2007) even when fractions are important to daily work related tasks. “Pediatricians, nurses, and pharmacists…were tested for errors resulting from the calculation of drug doses for neonatal intensive care infants… Of the calculation errors identified, 38.5% of pediatricians' errors, 56% of nurses' errors, and 1% of pharmacists' errors would have resulted in administration of 10 times the prescribed dose." (Grillo, Latif, & Stolte, 2001, p.168). 4

We grew interested in… What types of representations of fractions are students relying on? And which representations are most effective in which contexts? We used Collaborative Action Research to learn more

Data Collection and Analysis AS 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

Your Turn: “Matching Game” TWO FIFTHS OF THE PETALS ARE WHITE

Match again: “Triplets” Part-whole relationship Part-part relationship Linear relationship Quotient relationship Operator relationship Do your third match, and discuss

In our study… We focused particularly on these three Tad Watanabe, 2002

Early 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 13

Represent 2/5 or 4/10

Early ratio thinking?

Multiple Meanings Simultaneously

Fraction Situations 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?

Circles are just easier

But it simply isn’t true… 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.

Typical Partitioning Challenges

Show video of kid trying to partition a circle to show 2/5ths >

Over-reliance on circles to compare fractions can lead to errors and misconceptions… No matter what the situation, students defaulted to pizzas or pies… We had to teach another method for comparing fractions to move them forward…

Beyond Counting Using pre-partitioned shapes in teaching and assessing can mask an incomplete or incorrect appreciation of fractions as relational numbers. 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

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

Fractions on Stacked Number Lines

Exploratory Lesson: 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

Your Turn: “Number line 0-4” ½ 100% 7/18 2 5/ Sticky arrows with fractions, percents and decimals

Exploratory Lesson Video Shelley look for number line video

Other Challenges & Misconceptions Encountered 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 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 1/10 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 31

Implications for Teaching Expose students to precise representations in precise contexts Get students to connect representations with stories in context to make better decisions about which representation(s) to use when Lots of exposure to representations other than part-whole relationships (discreet relationship models are important as well as continuous relationship models, as early as possible)

Implications for Teaching Discussion/class math-talk to enhance the language of fractions Use visual representations as the site for the problem solving (increased flexibility) Think more about how to teach equivalent fractions Think more about the use of the number line

Which Representation When? Number Lines Fraction as number Density of fractions Comparing Ordering Operations (+, -, x, /) Circle Area Models with circle friendly fractions, such as ½, ¼, 1/8, 1/3 Comparing Ordering Operations (+, -, x, /) Rectangle Area Models Using factors to generate equivalent fractions Comparing Ordering Operations (+, -, x, /)

Teacher Learning Let’s hear the debrief after the lesson. 35

Knowledge Creation: CAR Build knowledge at CAMPPP just like teachers learning together in classroom contexts and with one another