Comparing example generation with classification in the learning of new mathematics concepts

Figures & data

Table 1. Summary of the three example-generation strategies identified by Antonini (2006).

Strategy	Description
Trial and error	The example is sought among some recalled objects.
Transformation	An object that satisfies part of the requested properties is modified through one or more successive transformations until it is turned into a new object with all the requested characteristics.
Analysis	Assuming the constructed object, and possibly assuming that it satisfies other properties added in order to simplify or restrict the search ground, further properties are deduced up to consequences that may evoke either a known object or a procedure to construct the requested one.

Figure 1. Overview of the course content (left) with the experimental tasks and their ordering for the four experimental groups (right). Task names include G where the task involves generation, and C where it involves classifying.

Table 2. Content of the initial learning tasks in Study 1.

Task	Contents
C1-10	For each of the following sequences, decide if they are increasing, decreasing, both or neither.
	[10 sequences, shown as a list of the first 8 terms – see Figure 2 for details]
C11-13	What are the properties of each of the following sequences?
	[3 sequences, shown as a formula for the nth term – see Figure 2 for details]
C14-15	For the sequences shown in the graphs below, decide if they are increasing, decreasing, neither or both.
	C14: Plot of first 8 terms of $u_n=3+(-1{)}^n$
	C15: Plot of first 8 terms of $u_n=5-\frac{3}{n}$
G1	Give the formula for the nth term of a sequence $u_1,u_2,u_3,\dots$ which is:
	(a) increasing.
	(b) increasing and bounded above.
	(c) increasing, bounded above, and has $u_4=16$.
G2	Give the formula for the nth term of a sequence $u_1,u_2,u_3,\dots$ which is:
	(a) decreasing.
	(b) decreasing and bounded below.
	(c) decreasing, bounded below, and has $u_4=1$.
G3	For each of the following, find an example of a sequence with the given property. Enter the first 8 terms of your sequence as a list, e.g. [a,b,c,d,e,f,g,h].
	(a) increasing, and different from the examples above.
	(b) not increasing.
	(c) not increasing, and not decreasing.
G4	Drag the points $u_1,\dots ,u_8$ so that they show the first 8 terms of
	(a) an increasing sequence.
	(b) a decreasing sequence.
	(c) a sequence which is neither increasing nor decreasing.

Figure 2. Distribution of first responses to the classification task. Raw numbers are available in Table A1 in the appendix.

Table 3. Content of the subsequent tasks in Study 1, and the points awarded to each (based on one point for each distinct response).

Task	Contents	Points available
NearC	The sequence defined by $u_n=\frac{1}{3n+3}$ is: ° Increasing ° Decreasing ° Neither increasing nor decreasing and is: ° Bounded ° Unbounded	2
NearG	Give an example of a sequence which is increasing and bounded above.	1
	(Give your answer as a formula for the nth term.)
MidC1	The arithmetic sequence defined by $u_n=2n+2$ is: ° Increasing ° Decreasing ° Neither increasing nor decreasing	2
	and is: ° Bounded above ° Bounded below ° Bounded above and below° Unbounded
MidC2	The geometric sequence defined by $u_n=-2(-3{)}^{n-1}$ is: [same options as MidC1]	2
MidG	In each case below, give an example of an arithmetic sequence with the stated property, by entering an expression for the general term. If it is not possible, enter none.	4
	(a) Increasing
	(b) Decreasing
	(c) Bounded above
	(d) Decreasing and bounded below
FarC1	Copy of MidC1	2
FarC2	Copy of NearC	2

Notes: Note that the numerical values appearing in the C questions were randomised.

Figure 3. Distribution of correct and incorrect first responses to the generation task.

Figure 4. Distribution of correct and incorrect first responses to the subsequent tasks.

Figure 5. Marks attained in the subsequent tasks by each student in Study 1 (pale dots) together with the estimated mean for each group (solid dots with error bars), represented by the median and 95% HDI of the posterior distribution.

Figure 6. Estimates for the contrasts between different groups, as both raw scores and effect sizes (Cohen's d). The horizontal lines show the median and 95% HDI for the estimates, with the posterior distribution illustrated by 100 grey dots. The highlighted region around 0 is the region of practical equivalence, corresponding to a standardised effect size smaller than 0.1.

Figure 7. Marks attained in the subsequent tasks by each student in Study 2 (pale dots) together with the estimated mean for each group (solid dots with error bars), represented by the median and 95% HDI of the posterior distribution.

Figure 8. Scores on the Near and Mid classification and generation tasks across both studies, with students grouped according to the order the tasks were presented in, together with the estimated mean for each group (solid dots with error bars), represented by the median and 95% HDI of the posterior distribution.

Table 4. Types of examples given by students in answer to NearG, broken down by group.

		Example type
Group	N	$1/2^n$	1/n	other
Study 1 G-	48	17 (35%)	13 (27%)	18 (38%)
Classify first	61	2 (3%)	24 (39%)	35 (57%)
Generate first	73	6 (8%)	23 (32%)	44 (60%)

Note that the classify/generate first groups include students from Study 2 as well as students from the C- groups in Study 1.

Table A1. Distribution of first responses to the classification task in Study 1.

Item	Both	Increasing	Neither	Decreasing
1	18	28
2			46
3		1		45
4	28	17		1
5		46
6	21	11	14
7	1	19	26
8	21	19	6
9			41	5
10			1	45
11			42	2
12		42	2
13		1	28	15
14	1	44
15		1	44

Antonini, S. (2006). Graduate students' processes in generating examples of mathematical objects. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th conference of the international group for the psychology of mathematics education (pp. 57–64). Charles University.

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Data availability statement

All data and analysis code is available at https://doi.org/10.17605/osf.io/gry6v.