To some extent I see the transition that Siegler & Opfer (2003) suggest -- a developmental progression in students’ representations of number and magnitude, moving from an earlier fuzzy representation which has been described as logarithmic – bigger numbers get more closely clumped together – to a more linear model among older students with more equal intervals regardless of magnitude. However, it’s interesting to me that while in many applications my 5th grade (accelerated to 6th grade curriculum) math class students seem to function with an equal interval representation, logarithmic thinking seems to “overstay” it’s typical developmental timeline when it comes to graphing.
In fifth/sixth grade students create a wide variety of graphs in both math and science. The focus is using graph representations to better understand change. A key concept in line graphing, of course, is that students need to represent quantities at equal intervals. Otherwise, they are not able to observe trends in change – linear vs. non-linear, increased change vs. decreased change. In the past, I have viewed this as being an issue more about clustered data…when you have more data, you need more “room” to separate and organize it. However, after reading this week’s work and reflecting on how many of our examples have clustered data of lower magnitudes, I wonder whether this “unequal interval” problem I see a lot is related possibly to a lingering logarithmic representation – the intervals get closer as the numbers get larger – moving out along the x, or more typically the y, axis. For example, if a student were graphing a series of temperature readings of ice melting (let’s say degrees Celsius) of 0, 1, 1, 3, 4, 3, 4, 8, 10….Then the student would put a lot of effort into making room to distinguish between the more highly clustered numbers (0-4 degrees) and then put 8 and 10 shortly after them. That could be considered logarithmic – in a sense – the larger numbers have smaller differences.
There are other situations in which complexities of magnitude concepts – like its application in graphing -- may re-trigger or prompt use of less accurate representations – e.g. the logarithmic model. Siegler, in another paper (a very useful paper that I will be sharing with colleagues), makes specific suggestions for mathematics education based on what was at that time current “cognitive science research” (2003) Specifically, he talks about issues of magnitude when it comes to understanding fractions. He wrote, “Much of children's difficulty in fractional arithmetic arises from their not thinking of the magnitude represented by each fraction” (Siegler, 2003, p. 222). He notes similar issues when it comes to decimal portions. He implies that students work with the numbers, perhaps effectively utilizing procedures, but often don’t recognize that they are representing magnitudes. That leads to errors. This is part of his larger argument in the paper that conceptually-oriented instruction, as opposed to heavily procedural guidance, is important for math understanding and achievement. It is also interesting to me that teachers may be working with fractions in the classroom without clarifying – making explicit – that these different-looking numbers are just different magnitudes.
For me in the classroom, these ideas about magnitude and possible “lingering, less accurate” models of magnitude will lead me to be more explicit about the concept of magnitude with students and discuss with them their individual models of it. As we prepare for state tests in a week, students are reviewing placing numbers – including fractions – on number lines. This gives me a good opportunity to talk about the extended number line – as it continues into space and numbers get larger, how do students think about those numbers? My students are at an age where they are more aware first, of the importance of making representations, and second, that the ability to represent quantities and operations in multiple ways gives them deeper understanding and more power to use and apply those ideas. What do their own models look like? I look forward to being surprised at what this conversation will teach me about their thinking.
Siegler, R. S. (2003). Implications of cognitive science research for mathematics education. In Kilpatrick, J.,Martin, W. B., & Schifter, D. E. (Eds.), A research companion to principles and standards for school
mathematics (pp. 219-233).
Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: evidence for multiple representations of numerical quantity. Psychological Science, 14, 237-243.
