I’m in the middle of teaching my girls how to conceptualize proportions. They can solve them with no problem, but they don’t really understand what they mean…and that all lies in their (in)ability to truly understand fractions.
Typically, fractions literacy begins in grade 3 ( according to the online version of NCTM’s Principles and Standards ) and continues to develop until grade 6. However, as any middle grades ( or high school or college ) math teacher can tell you, students never fully grasp the concept of fractions as comparing a part of something to a whole ( or in the case of rates/ratios comparing a part to a part, part to a whole, etc ).
I discovered just how bad off my students were in this regard when I tried ( unsuccessfully ) to show them the visual representation of proportional relationships.
I was using an activity involving scaling down recipes to illustrate the physical aspects of proportional reasoning….and fractions. The activity required us to take 3 cups of flour (required for the original recipe) and divide them each in half, thus leaving us with 6 piles of flour, each measuring a half of a cup. The questions that followed included:
How many 1/2 cup piles have you created ? ( 6 )
How many of the 1/2 cup piles are need for the scaled down recipe? (3)
How many scaled down versions of the original recipe can you make? (2)
I thought the process would be much smoother than it actually was, but the students had trouble understanding the visual representation of each pile being divided in half. Even more difficult, was understanding that the concept of half is relative to what is considered a whole. Essentially, we couldn’t get though the discussion of dividing the piles of flour into 6 halves and that the 6 half cup piles are equivalent to the 3 whole cup piles….or that the 3 half cup piles of flour is the same as 1 and 1/2 cups of flour.
But…they could do all of the above with numbers…and I was stumped, frustrated, and slightly annoyed.
How do you teach students who have internalized a process to understand a concept rooted in physical reality ?
1 response so far ↓
Jovan, it is interesting that you used the scaling of a recipe as a context for a lesson on proportionality. Are you familiar with the controversy regarding Keith Devlin’s post on multiplication is not repeated addition. He says multiplication is scaling. Teachers pushed back hard, and Devlin wrote two more posts on the topic.
I think the main reason that difficulty with fractions and rational numbers is so pervasive is because students are not taught to think in terms of multiplicative relationships, which begins with multiplication. Students are taught to compute a product; they learn procedures to manipulate numbers. They don’t learn the concept of scaling, of working with two or more different unit sizes, of comparing the unit sizes as ratios thinking along the lines that there is x of this for every one of that, and becoming comfortable switching their point of reference regarding the unit size—i.e. what is one whole or 100%. Conceptually, they are stuck on additive relationships, which require the same unit size since you can’t directly add or subtract quantities that are based on different scales. They are only comfortable with the scale as the standard unit 1.
Alison Blank has two posts on the meaning of multiplication and concludes that it is a change of unit or a change of scale. Some of her reader comments relate multiplication to proportionality. Multiplication explicitly or implicitly always involves a ratio. In rate problems and unit conversion problems the ratio is explicit, such as miles/hour. In set problems, such as 3 six-packs of juice contains how many cans, the ratio is implicit, 6 cans/1 pack. In area problems, such as the area of a 5″ x 9″ rectangle, the ratio is implicit: 5 sq. in. /1 row, or 9 sq. in. /1 column. The structure of multiplication, the relationship among the numbers, is proportionality. In the juice example, the proportion is based on the ratio of 6 cans for every six-pack, or 6 to 1. The counts must follow that same ratio: 18 cans in 3 six-packs, or 18 to 3, which is equal to 6 to 1. Stated as a proportion, 6 cans is to 1 six-pack as 18 cans is to 3 six-packs.
Too often we ignore conceptual meaning in favor of computing an answer. Repeated addition is a computational strategy, not a meaning of multiplication. If you believe that repeated addition is a meaning of multiplication, then your logic would mean that continuous counting and counting up are conceptual meanings of addition. We need to focus on conceptual meaning and create activities that embody the concept so students develop an intuitive understanding of the concept before we formalize it. It seems to me that Mr. K is doing that with fractions on this lesson.