Does generating examples aid proof production?

Abstract

Many mathematics education researchers have suggested that asking learners to generate examples of mathematical concepts is an effective way of learning about novel concepts. To date, however, this suggestion has limited empirical support. We asked undergraduate students to study a novel concept by either tackling example generation tasks or reading worked solutions to these tasks. Contrary to suggestions in the literature, we found no advantage for the example generation group on subsequent proof production tasks. From a second study, we found that undergraduate students overwhelmingly adopt a trial and error approach to example generation and suggest that different example generation strategies may result in different learning gains. We conclude by arguing that the teaching strategy of example generation is not yet understood well enough to be a viable pedagogical recommendation.

Appendix

1.1 Instrument used in the proof production study

1.1.1 Example generation tasks

The following definition is about a mathematical concept you have probably not seen before. Please spend a few minutes reading it carefully:

Let f : ℝ → ℝ be a real-valued function. Then f is called fine if it has a root (zero) at each integer. In other words, f is fine if n ∈ ℤ ⇒ f(n) = 0.

You will have 20 min to answer a series of questions about this concept. In each question, you will be asked to provide examples relevant to the concept of fine function. Each question appears on a new page. Write your answers in the space below each question. If you get completely stuck, move on to the next question. Try to answer as many questions as you can in the 20 minutes available.

• 1. A periodic function is one that repeats its values after a certain period. That is to say that f: ℝ → ℝ is periodic if there exists a P ≠ 0 such that f(x + P) = f(x) for every x ∈ ℝ. P is called the function’s period.

  • (a) Find an f which is fine and periodic with period $\frac{1}{2}$.

  • (b) Find another such f.

• 2. Let f(x) = sin(kx), where k ∈ ℝ. Give two examples of values of k for which f is fine.

• 3. Let f(x) = cos(ax + b). Give an example of a pair of values a,b for which f is fine.

• 4.

  • (a) Draw a function which is fine but not periodic.

  • (b) Draw another such function.

• 5.

  • (a) Find f: ℝ → ℝ such that f is fine and periodic with period 1.

  • (b) Find another such f.

• 6. Let f(x) = k, where k ∈ ℝ. Give an example of a value of k for which f is fine.

• 7. Give an example of a function f:ℝ → ℝ that is fine but not continuous.

• 8. Draw a fine function that takes no negative values.

• 9. Draw a fine function that always takes values between $-\frac{3}{2}$ and 1.

1.1.2 Proof production tasks

In this section, there are four questions, each on a new page. Each question involves proving a statement about fine functions.

• 1. Let f(x) = ax ² + bx + c, where a,b,c ∈ ℝ and a ≠ 0. Prove that f is not fine.

• 2. Let f: ℝ → ℝ and g : ℝ → ℝ be fine functions. Prove that g ∘ f is fine.

  [Here (g ∘ f)(x) = g(f(x))].

• 3. Let f,g : ℝ → ℝ be fine functions. Prove that f + g is a fine function [Here (f + g)(x) = f(x) + g(x)].

• 4. Let f: ℝ → ℝ be a fine function. Let g: ℝ → ℝ be a function defined by g(x) = f(x − k) for some k ∈ ℤ. Prove that g is a fine function.

1.1.3 Proof marking scheme

Question 1

• 1. f(x) = ax² + bx + c is a quadratic equation, which has at most two real roots (by the Fundamental Theorem of Algebra). [2]

• 2. Consequently, there must exist integers n such that f(n) ≠ 0 [2]

• 3. So f is not fine. [1]

Question 2

• 1. Let k ∈ ℤ. Then f(k) = 0, as f is fine. [2]

• 2. So g(f(k)) = g(0) = 0 as g is fine. [2]

• 3. So, g ∘ f is fine. [1]

Question 3

• 1. Let n ∈ ℤ. Then f(n) = g(n) = 0 as f and g are fine. [2]

• 2. So f(x) + g(x) = 0 + 0 = 0. [2]

• 3. So f + g is fine. [1]

Question 4

• 1. Let n ∈ ℤ. Then n − k ∈ ℤ as k ∈ ℤ. [1]

• 2. So f(n − k) = 0 as f is fine. [2]

• 3. So g(n) = f(n − k) = 0. [1]

• 4. In other words, g(x) = 0 whenever x ∈ ℤ. [1]

1.2 Instrument used in the example generation study

The following definition is about a mathematical concept you have probably not seen before. Please spend a few minutes reading it carefully:

Let f : ℝ → ℝ be a real-valued function. Let $A\subseteq\mathbb{R}$. Then f is preserved on A if and only if $f(A)\subseteq A$. In other words f is preserved on A if and only if $a\in A\Rightarrow f(a)\in A$.

You will have 20 minutes to answer a series of questions about this concept. In each question, you will be asked to provide examples relevant to the concept of function preservation. Each question appears on a new page. Write your answers in the space below each question. If you get completely stuck, move on to the next question. Try to answer as many questions as you can in the 20 minutes available.

• 1. Let A be the open interval (1,2).

  • (a) Find f: ℝ → ℝ such that f is preserved on A.

  • (b) Find another such f.

  • (c) Find an f which is strictly decreasing and preserved on A.

  • (d) Find an f which is strictly increasing and preserved on A.

  • (e) Find an f which is preserved on A but not continuous on A.

• 2. Let A be the open interval (1, 2) and B be the open interval (2,3). Find an f : ℝ → ℝ such that f is preserved on A but not on B.

• 3. Let A be the open interval (1, 2), B be the open interval (2,3) and C be the open interval (3,4). Find an f : ℝ → ℝ such that f is preserved on A and C, but not on B.

• 4. Let A be the closed interval [ − 1,0]. Find an f : ℝ → ℝ such that f has a local minimum in A and is preserved on A.

• 5. Let B be the open interval (1, 2). Find f : ℝ → ℝ such that f has a local minimum in B and is preserved on B.

• 6. Let A be the open interval (1, 2). Find an f : ℝ → ℝ which is not preserved on A.

• 7. Find f : ℝ → ℝ such that f is preserved on ℕ but not on the set of negative numbers (i.e. not on the set {x ∈ ℝ | x < 0}).

• 8. Find f : ℝ → ℝ such that f is preserved on ℝ ∖ {0} but not preserved on {0}.

• 9. Let f be a step function defined by f(x) =  max {n ∈ ℤ | n ≤ x}.

  • (a) Find a set A such that A has five members and f is preserved on A.

  • (b) Find a,b ∈ ℝ such that a ≠ b and f is preserved on [a, b].

• 10. Let f(x) = sinx.

  • (a) Find a ∈ ℝ such that f is preserved on {a}.

  • (b) Find a, b ∈ ℝ such that a ≠ b and f is preserved on [a,b].

• 11. Let f(x) = x ¹³.

  • (a) Find a set A such that A has two members and f is preserved on A.

  • (b) Find the largest a ∈ ℝ such that f is preserved on $[-\frac{1}{2},a]$.

• 12. Let $f(x)=\left\{ \begin{array}{rl} 0 & \text{if }x = 0 \\ \frac{1}{x} & \text{if }x \neq 0 \end{array} \right.$.

  • (a) Find a set A such that A has five members and f is preserved on A.

  • (b) Find m ∈ ℝ such that f is preserved on $\left[\frac{1}{2},m\right]$.