MODULE
Patterns and Functions: How can we connect informal thinking and observations to algebraic notation?
BACKGROUND
Students need support with seeing how visual representations are constructed if they are to make generalizations about the representation. If they are given opportunities to construct or draw the pictorial growth pattern they are better able to “see” the important relationships (i.e., the number of chips from one figure to the next, the figure number and the number of chips).
1) SET: Engage with a problem or problems that help teachers/candidates consider students' algebraic thinking (teachers’ prior knowledge)
Construct the figures shown above. Investigate and report all you can about this growing pattern.
Construct the figures shown above. Investigate and report all you can about this growing pattern.
[1]
Depending on your groups’ experiences, you may want to pose the growing pattern that is a linear function rather than a quadratic function.
2) STUDENTS: Watch video clips of students describing their thinking as they engage with problems
[The “Christmas Tree” task above was recommended as an item to use for interviews this spring, do we have video clips or transcripts from interviews that we can access?]
Tree size / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10Number of lights / 9 / 13 / 17 / 21 / 25 / 29 / 33 / 37 / 41
How many lights would you need for a Size 16 tree? (16 • 4) + 1 = 65
How many lights would you need for a Size 100 tree? (100 • 4) + 1 = 401
Explain in words how you can work out the number of lights when you know the tree size. There are rows of four lights repeated the number of times as the size you are on plus one on the top. Students may also describe the total number of lights as a 4-by-n array or 4 groups of n for each column of lights. Students may also find the number of times they are adding 4 beyond a certain size (i.e., size 2 is 14 away from size 16 so add 14 fours to the 9 lights on size 2.
Students may not see the relationship between the size number and the total number of lights. They may only see that if they continue adding fours they can find the number of lights. This is an opportunity to discuss both recursive (nnow + 4 = nnext, start = 5) and closed form equations (see below).
Use algebra to write an equation that relates the tree size and the number of lights. [Use s for the size and n for the number of lights.] (n • 4) + 1 = 4n + 1 = s... students may not see that the first tree pictured is size “2” rather than 1, so their formulas may be shifted 1 to be [(n + 1) • 4] + 1 = 4n + 5 + 1 = 4n + 6 = s. [This error would result in f(16) = 69 and f(100) = 405.
3) RESEARCH: Examine/discuss research (encyclopedia entries)
4) ASSESSMENT: Consider assessments (formative assessment database)
The “Christmas Tree” (Figure 1) task was posed to 7th through 10th graders that were prompted to provide arithmetic, verbal, and symbolic representations. The findings (MacGregor & Stacey, 1993) indicate that students who could provide a symbolic representation were more likely to also provide a correct verbal description. They add that their “findings suggest that the verbal description is an important and perhaps necessary part of the process of recognising a function and expressing it algebraically” (p. 1-187).
Table 1 presents the correct responses given arithmetic, verbal, symbolic, and both verbal and symbolic represents. MacGregor and Stacey (1993) found that if students were able to provide a correct verbal description, they were more likely to also generate an algebraic representation. They argue that using language to link the two variables is an important step in the learning process.
A similarly structured problem is shown in Figure 2 below along with student performance results in Table 3.
5)SUGGESTIONS FOR TEACHING: Consider strategies based on research (including apps)
●Knowing that students have difficulty expressing generalizations symbolically, what are implications for instructional practice?
●What scaffolds may be necessary in supporting students with connecting their informal thinking and observations to algebraic notation?
Possible responses include: recording arithmetic expressions/equations prior to algebraic; describing in words how one could find the total [xxx] (tiles, cubes, chips, lights) for any figure in the pattern then linking the verbal to the algebraic by replacing phrases such as “two times the figure number” with “2n” when n is the figure number; asking what stays the same (constant)? what changes (variable)? how does it change from one figure to the next? how does [xxx] change as [xxx] changes?
6) Did the preservice teachers understand? How do you know? Evidence
REFERENCES:
Ferrini-Mundy, J., Lappan, G., & Phillips, E. (1997). Experiences with patterning. Teaching Children Mathematics, 3(6), 282-289.
MacGregor, M. & Stacey, K. (1993). Seeing a pattern and writing a rule. In I. Hirabayashi, N. Nohda, K. Shigematsu, & F. Lin (Eds.), Proceedings of 17th Conference of the International Group for the Study of the Psychology of Mathematics Education,1, 181-188. Tsukuba, Japan.
Quinlan, C. (2001). From geometric patterns to symbolic algebra is too hard for many. Paper presented at the 24th Annual Mathematics Education Research Group of Australasia Conference, Sydney, Australia.
[1]See complete version of the task in part 4 below, Figure 1, Test Item A.