Both the Math Literacy course and the Applications for Living course deal with two common models — linear and exponential. I’m finding it interesting to watch how different and similar the experience is.
For both students, they have not seen exponential models in their college (developmental) courses; none of the current Applications for Living students had the Math Lit course previously. (That will change as some Math Lit students take Applications for Living.) In both cases, we explore models from numeric and symbolic forms; the Applications for Living course includes more variety, and also requires active graphing of exponential models.
In both courses, students have a difficult time leaving the linear world of adding and subtracting. There is confusion about the role of slope in an adding world; during the exploring process, we take the time to show repeated adding as a multiplying, and identify the number as the slope. When we work in exponential situations, the linear view seems to dominate. During the exploring process, we show repeated multiplying as an exponent and learn about the role of the multiplier. The performance learning outcomes are not what we would want; there are some differences between numeric and symbolic problems.
For example, the final exam in the Math Lit course had a doubling problem for which students needed to write the model. Something like:
At the start, 25 people knew about the latest i-product; this number is going to double every day. Write the exponential model for N (the number who know) based on t (days since the start).
Another problem for the Math Lit final was a growth pattern from a numeric standpoint:
The cost of a machine is $400, and this is expected to grow by 10% per year. Complete the following table of values. [The table shows years 1 to 5, where the value for each year needs to be completed.]
In Applications for Living, the corresponding problems were this symbolic one:
The value of an investment is expected to grow by 6% per year. Write the exponential model for the value in terms of the number of years.
And, this numeric one:
At 3pm, 20 mg of a drug were in the body. At 4pm, 15 mg were in the body. Complete the following table of values. [The table shows hours 1 to 5, where the amount of drug needs to be completed.]
Almost half of the Applications for Living students treated the last problem as a linear one: They showed values of 10, 5, 0 and 0 (sometimes with a puzzled comment about having zero as the amount). In class, we had done drugs in both half-life and percent decrease models; we had calculated the multiplier as well. They did a little better on the symbolic form; part of this is the fact that this course also does work with finance formula, and one of those formulae is basically the answer for this problem.
The Math Lit students did well on the numeric problem; part of that success was the remediation we did earlier when most students had difficulty on all things exponential. Few of the Math Lit students wrote a correct exponential model, which is noteworthy since the problem is a slight variation of a situation we used to introduce exponential models. Most of the incorrect answers were variations on y =mx + b.
Clearly, this assessment feedback is indicating a need for an adjustment to the instructional cycles.
However, I also think that the results reflect a math curriculum that tends to treat topics in isolation. How often do students need to deal with both linear and exponential models in one assessment? Also, do we use the word “always” with students? As in: “Compare the y-values; the difference always tells you what the slope is.” Or, “If you can see how to get the next value in a table, you can always use this to complete a table.” Or, “In a function, you can always get the next function value by adding or subtracting.”
During the instructional cycles in both courses, I can see the resistance to leaving the linear model. It’s a bit like distributing, where students become fixated on one process. I want students to see the power of understanding exponential models; students want the comfort of one model for all situations.
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