Misconceptions and Intuitions C Bruce Fractions Think Tank October 3, 2012

Misconceptions Three examples Treating a fraction as two numbers (relationship?) The symbolic notation is the driver (who’s misconception is this?) Circles are just easier WHERE DO THESE COME FROM?

Represent 2/5 or 4/10 www.tmerc.ca/video-studies/ CATHY: INTRODUCE: GRADE 4 FEMALE STUDENTS Early understanding of what a fraction is – or LACK thereof. What DO these students seem to understand? What do these students NOT seem to understand? VIDEO GUIDE www.tmerc.ca/video-studies/

Video Studies

Ratio thinking?

Circles are just easier SHELLEY: Circles are just easier

Intuitions Two examples Using spatial reasoning to compose and decompose whole areas: ‘a half combined with another half’ Magic snakes (snake one, cut in half to make two snakes, which is cut in half to make…) How do we build on children’s intuitions? See also Watanabe (1996). Ben's understanding of one-half, TCM See also Bruce & Flynn (2010) Which is greater: One half or two fourths?, CJSSMTE

Simon’s Thinking

“Magic Snakes” Children are asked to observe and explain the relationship between the lengths of Shorty and Longy They are provided with materials to demonstrate the 1:2 relationship Questioning focus: on ‘halving’ and ‘doubling’

Magical Feats and Pellets The context provided is that the snakes can perform magical feats when fed just the right amount of pellets Children reason about how the 1:2 relationship between Shorty and Longy affects the number of pellets they would need to eat to perform different magical feats Then students think about making Mr. Really Really Short (by cutting the strip of paper that is Longy in half and half again, etc)

Halving and Density

Things I am trying to remember… 1. The importance of using models to make sense of the situation 2. The importance of establishing equivalence (and equal partitions) 3. The importance of estimation (through mathematical reasoning) The sum of 1/12 and 7/8 is closest to: a. 20 b. 8 c.. ½ d. 1 (why might a student select a, b, c, or d) – reasoning used? 4. The importance of developing a quantitative notion of fractions (relative quantity or bigness) is important (Bezuk & Bieck, 1993) Petit, Laird & Marsden, 2010

More things to remember… Procedural fluency and conceptual understanding work together to deepen student understanding of fraction addition and subtraction “Premature experiences with formal procedures (algorithms) may lead to symbolic knowledge that is not based on understanding, or connected to the real world.” (Huinker, 2002) Petit, Laird & Marsden, 2010

Interesting Contexts 1 Fraction Flags (Tom Kieren, Brent Davis, Ralph Mason) Sections of the flag – what is my equation to get to a whole flag? Composing and decomposing figures / composing and decomposing fractions)

Interesting Contexts Liquid Quantity Height 2  Liquid Quantity Height – Adding two liquid amounts Thinking about PERCENT as a starting point (Text example; Joan Moss & Robbie Case, 2003)

Interesting Contexts Ribbon Fractions (ribbon strips) 3 Partitioning the ribbon into segments – adding and subtracting ribbon segments Measure and distance from zero in segments, one third segment and another third segment, and another third segment  

Things I keep thinking about Adding bowls of candy 50% green in this bowl, 20% green in this other bowl (If we put them together, are 70% green?) OR 8 dots (4 are red) 8 dots (2 are red) 4/8 + 2/8 = 6/16 VS 6/8 of the dots are red Is this adding ratios? What is this? Let go of part-part / part-whole construct for operations? Is it relevant? How?

Does this help? Using a linear model (Petit, p. 156) 2/5 of the candies are chocolate 3/10 of the candies are mint Are there more mint or chocolate candies in the dish?

Mixed Fractions as a Starting Point? Students should be presented with a mix of addition and subtraction problems involving mixed numbers as well as only involving proper fractions as it aids in developing the students’ ability to look at the entire problem and identify the relationships that will be useful. Empson & Levi, 2011 IS… One and three fifths already addition? Can we say this as “One and three one-fifths”?