Describing Analysis of Change
Speed is commonly used to introduce children to the idea of rates of change. (See also Rate of Change.) It is sometimes difficult for students to understand that speed is composed of two varying quantities: time and distance. This document describes common misconceptions and ways to learn about speed as a rate of change.
|I:||How can you determine how fast a mouse is running?|
|S:||You would need to know how long the mouse is and how long the room is.+|
|I:||Suppose that turtle travels at 20 ft/sec. and rabbit travels at 30 ft./sec.|
|S:||I think that the rabbit will always be 10 feet ahead of the turtle.|
|I:||Suppose that turtle and rabbit are racing to a pole and back. Suppose turtle travels at 30 ft./sec. on the way there and 50 ft./sec. on the way back . How fast would rabbit have to travel [at constant speed] to tie with turtle.|
|S:||I think that rabbit should travel at 40 ft/sec, because it is half way in between.**|
|I:||How fast do you need to go in order to go 200 feet in 12.4 seconds?|
|S:||You are breaking the total time into 12.4 pieces, so you are breaking 200 feet into 12.4 pieces. Each piece of distance will be 200÷12.4 feet long, so you will go that many feet each second.*
In the beginning stages of understanding, many students have trouble understanding that speed is a ratio of two quantities: distance and time.
Once students understand speed as involving speed and time, many students see speed in terms of chunks of distance (or speed-lengths): a speed of 60 feet per second is understood as a chunk of 60 feet that can be traveled in 1 second. Although this is not incorrect, it only allows children to reason in one direction: given a speed and an amount of time, the student can add up “speed-lengths” to find the distance. However, in order to find the amount of time it would take to cover a given distance, the only method available for students is to break the distance up into “speed-lengths” and count the number of speed-lengths in that distance. A better understanding of speed includes the understanding that there is a direct proportional relationship between distance and speed. Partitioning the total time to travel a given distance implies a proportional partition of the distance traveled, and vice versa.
Just stating that speed is “distance divided by time” is not helpful for students, because students do not really understand how the division relates the quantities of distance and time. To teach the concept of speed, a student should form an image of two number-lines, one for distance and the other for time. The number-lines should be proportionally segmented so that a unit of time corresponds with a segment of distance. A student must connect that any change along one of the number-lines results in a proportional change in the other number-line. So if 1/7 of the total time has elapsed, that means that 1/7 of the total distance was traveled. This helps students to have a conceptual understanding of the relationship between time, distance and speed, so that when a student is given two of those quantities, they are able to visualize the third value.
Although speed is a familiar context for students, students do not automatically transfer what they know about distance and time to other types of rates. For this reason, teachers should make sure to include contexts involving distance and time and other contexts so that students can learn to transfer their thinking. (See also Understanding Rate of Change.)
+ Lobato, J., & Thanheiser, E. (1999). Rethinking slope from quantitative and phenomenological perspectives. In 21. Presented at the North American Chapter of the Psychology of Mathematics Education Conference.
Thompson, P. W., & Thompson, A. G. (1992). Images of rate. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA.
Thompson, P. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 181-234). Albany, NY: SUNY Press. **
Thompson, A. G., & Thompson, P. (1996). Talking about rates conceptually, Part II. Journal for Research in Mathematics Education, 27(1), 2-24. *
- What have you learned about your students’ thinking regarding this topic?
- What have you learned that is an effective teaching strategy to help students understand this topic?