Helping Students with Proving: A Tale of Two Whole Class Teaching Experiments Annie Selden New Mexico State University Department of Mathematical Sciences
[email protected] September 19, 2013 University of Oklahoma Colloquium 1
This work was done in collaboration with other team members John Selden, Kerry McKee, Milos Savic Funded in part by Educational Advancement Foundation
2
Outline of the talk • The first teaching experiment • Aspects and types of proofs • Our view of behavioral schemas • Examples of student work
3
• • • • • • •
The second teaching experiment Theoretical perspective Proving supplement description Surprising student difficulties An example of a supplement proof Effect of the supplement on students A categorization of students’ proving difficulties 4
First Teaching Experiment • A one-semester, 3-credit, special topics course for advanced undergraduate and beginning math graduate students. • Meets 2 times/week for 75 min. • Purpose is to teach proof construction • Significant modification of the R. L. Moore Method (Mahavier, 1999)
5
• Notes with statements of theorems, definitions, requests for examples, but no proofs and only minimal explanations. • No lectures. • Students work outside of class and present their proofs at the blackboard. • We read and check each proof, “thinking aloud” so students can see what we are checking. 6
• We offer, sometimes extensive, criticism and advice. • Halfway through the course, we have students validate each other’s proofs. • Course has practical value because other professors assess students’ understanding by asking them to prove theorems. • The course covers sets, functions, a little real analysis, abstract algebra, and topology.
7
• Everything is video recorded: both the classes and our planning sessions. • Field notes are taken. • This information is analyzed in planning sessions between class meetings in order to plan for the next class and make initial observations. • Later, we often reanalyze the data for research purposes.
8
Nota bene: At least some beginning mathematics graduate students do need this course. • We have had students with math bachelors from UC Berkeley, Ohio State, & Univ. of Colorado take this course and benefit from it. • We also had a student with an applied math masters from UNM take it and benefit from it. • Also students coming from computer science or engineering can definitely benefit from it. 9
Aspects of proofs We treat proof as having two parts or two aspects: • the formal-rhetorical part and • the problem-centered part.
10
The formal-rhetorical part of a proof is the part that one can write based only on logic, definitions, and sometimes theorems, without recourse to conceptual understanding, intuition, or genuine problem solving. We have also called this the proof framework. We call the remainder of the proof the problemcentered part and it does require conceptual understanding and genuine problem solving. (Selden & Selden, 2009)
11
Writing a Proof Framework Theorem. For all a ∊ ℝ and all functions f :ℝ→ℝ and g :ℝ→ℝ, if f and g are continuous at a, then f +g is continuous at a.
12
Writing a Proof Framework Theorem. For all a ∊ ℝ and all functions f :ℝ→ℝ and g :ℝ→ℝ, if f and g are continuous at a, then f +g is continuous at a. Proof. Let a ∊ ℝ and f :ℝ→ℝ and g :ℝ→ℝ be functions continuous at a.
13
Writing a Proof Framework Theorem. For all a ∊ ℝ and all functions f :ℝ→ℝ and g :ℝ→ℝ, if f and g are continuous at a, then f +g is continuous at a. Proof. Let a ∊ ℝ and f :ℝ→ℝ and g :ℝ→ℝ be functions continuous at a.
Therefore f +g is continuous at a. 14
Writing a Proof Framework Theorem. For all a ∊ ℝ and all functions f :ℝ→ℝ and g :ℝ→ℝ, if f and g are continuous at a, then f +g is continuous at a. Proof. Let a ∊ ℝ and f :ℝ→ℝ and g :ℝ→ℝ be functions continuous at a. Let ɛ ∊ ℝ, ɛ ˃ 0. Let d = [ · · ? · · ]. Note that d ∊ ℝ, d ˃ 0. Let x ∊ ℝ. Suppose |x – a| ˂ d.
Therefore f +g is continuous at a. 15
Writing a Proof Framework Theorem. For all a ∊ ℝ and all functions f :ℝ→ℝ and g :ℝ→ℝ, if f and g are continuous at a, then f +g is continuous at a. Proof. Let a ∊ ℝ and f :ℝ→ℝ and g :ℝ→ℝ be functions continuous at a. Let ɛ ∊ ℝ, ɛ ˃ 0. Let d = [ · · ? · · ]. Note that d ∊ ℝ, d ˃ 0. Let x ∊ ℝ. Suppose |x – a| ˂ d. Then |f +g (x) ̶ f +g (x)| = [ · · · · ? · · · · ] ˂ ɛ. Therefore f +g is continuous at a. 16
Writing a Proof Framework Theorem. For all a ∊ ℝ and all functions f :ℝ→ℝ and g :ℝ→ℝ, if f and g are continuous at a, then f +g is continuous at a. Proof. Let a ∊ ℝ and f :ℝ→ℝ and g :ℝ→ℝ be functions continuous at a. Let ɛ ∊ ℝ, ɛ ˃ 0. Let d = [ · · ? · · ]. Note that d ∊ ℝ, d ˃ 0. Let x ∊ ℝ. Suppose |x – a| ˂ d. Then |f +g (x) ̶ f +g (x)| = [ · · · · ? · · · · ] ˂ ɛ. Therefore f +g is continuous at a. 17
• We first concentrate on having students write the formal-rhetorical parts of proofs. • Doing this, exposes the “real problem.”
18
Proofs requiring a previous result We have begun distinguishing 3 types of proofs, beyond those following immediately from definitions: 1. Those requiring a result in the notes. 2. Those requiring a result not in the notes, but easily articulated and proved. 3. Those requiring a result not in the notes that is not easily articulated and proved.
19
We try to provide experiences with all three of these, as we want “looking back” to become a habit.
This example of a Type 1 proof occurred with Theorem 24 of the notes that states polynomials are continuous. Theorems 19-23 stated that sums and products of continuous functions are continuous and that the identity and constant functions are continuous. This is enough to prove Theorem 24 by induction, but there were students who did not notice this and could not prove it.
20
For an example of a Type 3 proof, we turn to the semigroups portion of our notes. We define a semigroup to be a nonempty set S together with an associative binary operation, and we define an ideal I of S to be a nonempty set so that IS U SI is contained in I. Theorem 46 states that if S is a commutative semigroup with no proper ideals then S is a group.
21
The first result needed is that if a e S, then aS is an ideal (and hence aS = S). The second result needed is that if aS = S, then the equation ax = b can be solved for x. After that students still need to “explore” to see what one can “get out of” such equations in order to produce an identity and inverses. 22
These results (lemmas) were not in the notes in order that students could experience a Type 3 proof.
23
Actions in the Proving Process We see actions in the proving process as responses to (inner) situations. After similar situations occur in several proof constructions with the same resulting action, a mental link is built between the situation and the action. 24
Example of an action In a situation calling for C to be proved from A or B, one constructs 2 independent subproofs arriving at C, one supposing A, the other supposing B. • If one has had repeated experience with such proofs, one does not have to think about doing or justifying this action, one just does it. • The action in this case is setting up the proof this way. 25
We call such persistent (often small grain-size) linked situation-action pairs, behavioral schemas. We see behavioral schemas as a form of (often tacit) procedural knowledge that yields immediate (mental or physical) actions. They call for knowing how to act. They are similar to what Mason & Spence (1999) have called “knowing to act in the moment.” 26
Six-point theoretical perspective of the genesis and enactment of behavioral schemas
1. Within a broad context, behavioral schemas are always available – they do not have to be searched for or recalled. 2. Behavioral schemas operate outside of consciousness. One is not aware of doing anything immediately prior to the resulting action. 27
3. One becomes aware of the resulting action of a behavioral schema as it occurs or immediately afterwards. 4. Behavioral schemas cannot be “chained together” outside of consciousness so that one only becomes aware of the final action. E.g. If the solution to a linear equation would take several steps, one cannot give the answer without being conscious of some of the intermediate steps. 28
5. An action due to a behavioral schema depends in large part on conscious input. 6. Behavioral schemas are learned through practice. To acquire a schema, a person should carry out the appropriate action (correctly) a number of times. Changing a detrimental schema requires similar, perhaps longer, practice. (Selden & Selden, 2008; Selden, McKee, & Selden, 2010)
29
• Enacting behavioral schemas does not require conscious processing and reduces the burden on working memory. This allows working memory to be better applied to the rest of proof construction. Working memory is definitely a limited resource. (Selden & Selden, 2009) 30
Taking a more external, or third person, view and perhaps a larger grain-size, behavioral schemas may also be seen as habits of mind. (Margolis, 1993) We want to encourage good habits of mind for proving and discourage detrimental ones.
31
Examples of how we try to encourage helpful behavioral schemas and discourage unhelpful ones
32
• Focusing too soon on the hypotheses Moore (1994) described undergrad transitionto-proof students who could not prove on the final exam: If f and g are functions from A to A and f o g is 1-1, then g is 1-1. He said that students started in the wrong place, with the hypothesis, instead of supposing g(x) = g(y).
33
Like (Bob) Moore, we have found that a number of our students habitually focus on the hypothesis immediately, instead of unpacking the conclusion and trying to prove that.
By patiently guiding students to first write the formal-rhetorical parts of proofs, this detrimental schema, or bad habit, can be overcome.
34
Willy On the 26th day of the course, Willy was asked to prove:
Theorem 29: Let X and Y be topological spaces and f: X Y be a homeomorphism of X onto Y. If X is a Hausdorff space, then so is Y. 35
On the left side of the board, Willy wrote:
Proof. Let X and Y be topological spaces. Let f be a homeomorphism of X onto Y. Suppose X is a Hausdorff space. ... Then Y is a Hausdorff space.
36
Then, on the right side of the board, he listed: homeomorphism one-to-one onto continuous (is open mapping) Then he looked perplexedly back at the left side of the board. 37
Even after two hints to look at the final line of his proof, Willy said, “And, I was just trying to think, homeomorphism means one-to-one, onto …” After some discussion of homeomorphism, we said, “There is no harm in analyzing what stuff you might want to use, but there is more to do before you can use any of that stuff,” meaning that the conclusion should be unpacked and examined first. 38
We inferred that Willy was enacting a behavioral schema in which the situation was having written little more than the hypotheses, and the action was focusing on the meaning and potential uses of those hypotheses before examining the conclusion. 39
We conjectured that, had Willy not been distracted by focusing on the meaning of homeomorphism, he might have written more of the formal-rhetorical part of the proof, e.g., inserting into the middle: Let y1 and y2 be two elements of Y. ... Thus there are disjoint open sets U and V so that y1 e U and y2 e V.
40
This is what we had expected: Proof. Let X and Y be topological spaces. Let f be a homeomorphism of X onto Y. Suppose X is a Hausdorff space. Let y1 and y2 be two elements of Y. ...
Thus there are disjoint open sets U and V so that y1 e U and y2 e V. Then Y is a Hausdorff space. 41
Willy did not make further progress that day. As there was little time left, we asked Willy to do the proof next time, and he constructed a proof the way we had expected. Other students also showed a reluctance to examine the conclusion, preferring instead to “plunge ahead” by examining the hypotheses immediately. This led us to infer they had a feeling of discomfort or inappropriateness regarding this action. 42
•Proving universally quantified statements One often starts the proof of a statement “For all (numbers) x P(x)” by writing “Let x be a number,” meaning x is “fixed, but arbitrary.” Some students are reluctant to write this in their arguments. Students eventually come to do this as if they were enacting a behavioral schema.
43
Mary Mary was a returning grad student taking beginning real analysis with Dr. K, who assigned 3 or 4 weekly proofs, graded them very thoroughly, and allowed them to be resubmitted. He emphasized things like writing “let e be a number > 0” into proofs. Mary recalls feeling this requirement was not particularly important or appropriate. She did so to get full credit. 44
Near the middle of the course, Mary came to feel that writing things like “let e be a number > 0” into proofs “made sense and it was the way to do it.” She reported to us, two years later, that she cannot think of any other way to write (this aspect of) proofs. For Mary, this positive behavioral schema took long to develop, but has now become a welldeveloped habit of mind, with an associated feeling of appropriateness. 45
An “unreflective guess” schema Sofia was a diligent student; however, as the course progressed, an unfortunate pattern in her proving attempts emerged. When she did not know how to proceed she often produced what one might call an “unreflective guess” or more colloquially, she was “grasping at straws.” 46
We inferred that Sofia was enacting a behavioral schema that depended on a feeling of not knowing what to do next. This situation was linked in an automated way to the action of just guessing any approach that we could not see as helpful in making a proof, nor did she seem to reflect on, or evaluate her guesses. 47
• Showing an object is in a set Here is an example of a tutor leading Sofia towards constructing a beneficial behavioral schema to replace her detrimental one.
48
Our intervention consisted of trying, during tutoring sessions, to prevent Sofia from enacting the “unreflective guess” schema by suggesting substitute actions, such as draw a figure, look for inferences from the hypotheses, reflect on everything done so far, do something else for a while.
49
The following tutoring session occurred in the middle of the course, and was devoted to helping Sofia prove: Theorem 20: Let (X,𝒰 ) be a topological space and Y a subset of X. Then (Y, {U I Y | U e 𝒰 } ) is a topological space (called the relative topology on Y).
50
Sofia said she didn’t know how to prove the theorem. At the tutor’s suggestions she wrote the first and last lines (the formal-rhetorical part) and drew a sketch. With guidance, she unpacked what was to be proved into 4 parts -- the defining properties of a topology.
51
She proved Y is in the relative topology, but could not prove the empty set was in the relative topology. It became clear she did not know how to show an object is in a set, when the defining variable in the set is compound (for example, U I Y).
52
The tutor forgot about the theorem for a moment and asked Sofia whether 6 is an element of {2n | n e N} and why. She said yes, because 6 = 2 x 3 and 3 e N.
Using this as a model, Sofia was able to show the empty set was an element of {U I Y| U e 𝒰 } and do the third and fourth parts of the proof.
53
• The tutor’s guidance facilitated Sofia’s construction of a behavioral schema (habit of mind) in which the situation is needing to show an object is in a set (where the defining variable is compound), and the action is showing the defining property is satisfied.
54
By the end of the course, our intervention of having Sofia do something else, whether it be draw a diagram or review her notes, was showing promise.
For example on the in-class final exam, Sofia proved that: If f, g, and h, are functions from a set A to itself, f is one-to-one, and f o g = f o h, then g = h.
55
Also on the take-home final, except for a small omission, Sofia proved the set of points on which two continuous functions between Hausdorff spaces agree is closed. This shows Sofia was able to complete the problem-centered parts of at least a few proofs by the end of the course, and suggests her “unreflective guess” behavioral schema was weakened.
56
The Second Teaching Experiment At the invitation of a real analysis teacher, we have been offering a one-hour per week supplement devoted to improving the students’ proving skills.
57
• The real analysis course is a 3-credit junior level class which serves three populations – Math majors. – Pre-service secondary math teachers. – Graduate students needing remediation. • In an interview, the real analysis teacher said that the course tries to be all things to all students, which is virtually impossible in three hours a week. 58
Theoretical Perspective • Some actions in the proving process, such as looking up a definition, drawing a sketch, or focusing on a particular part of the proof, are not easily noticed or visible in the final written proof.
59
• Making such actions, and the reasons for them, explicit and visible facilitates reflection and the autonomous enactment of future similar actions. (McKee, Savic, Selden, & Selden, 2010)
60
• This perspective is consistent with that of social psychologists like Bargh (1997) who discuss the automated nature of everyday life. • However, to our knowledge, they do not employ a theoretical framework such as we have described.
61
Supplement Description • We are describing mainly the second iteration of the supplement. • The supplement was scheduled at a time when almost every student could attend, and those who did, did so on a voluntary basis. Of the 18 students in the course, 6-10 regularly came to the supplement. • The supplement met once a week for 75 minutes, a total of one-third of the class time for those students who chose to attend. 62
• Each week, the real analysis teacher would choose a homework problem to be graded very carefully. • We worked the problem, noting the actions. Then we selected or wrote a theorem that used many of the same actions but that was not a template problem. • Students who attended the supplement co-constructed its proof with guidance.
63
• One of the supplement teachers wrote the theorem on the board. Then the students, or teacher if need be, offered suggestions about which actions to do next. • For each suggested action, such as writing down a definition or drawing a sketch, a student was asked to carry out the action on the board. • The goal was to have students reflect on what occurred and later to perform these actions, or similar actions, autonomously. 64
• Every student was encouraged to participate in co-constructing the proof although not every student could carry out every action. • Class discussion and questions were actively encouraged. • At the end of each supplemental class, students were given a handout that went through the proof and described the actions – a hypothetical proof co-construction trajectory. (Simon, 1995) 65
• The supplement was videotaped and field notes were taken. • We and the real analysis teacher met following each supplemental class to review what happened and plan for the next supplemental class. • The real analysis teacher used misconceptions or difficulties that occurred during the supplement to inform her instruction. Further, she pointed out some of the actions in her lectures to reinforce the supplemental instruction. 66
Surprising Student Difficulties • Difficulty articulating some of the words and symbols in a proof. • Not turning the pages of their books or notes to find the appropriate definitions, theorems, etc. • Unable to copy a definition accurately. • Difficulty altering the notation in a definition or theorem to match the current proof. N.B. It sounds like these are very weak students, but they are not. That’s why the difficulties are so surprising. 67
A Supplement Proof and a Paired Homework Proof Problem • Theorem from Supplement: Let {an} and {bn} be sequences, both converging to P. If {cn} is the sequence given by cn=an when n is even and cn=bn when n is odd, then {cn} converges to P.
• Theorem from Class: Show that {an} converges to A if and only if {an- A} converges to 0.
68
• Definition of Convergence - A sequence {an} converges to a real number A iff for each e>0 there is a positive integer N such that for all n>N we have |an-A|0. As {an} converges there exists an Na such that for all i> Na,|ai-P| Nb,|bj-P|N. Case 1: Suppose n is even. Then |cn-P|=|an-P|
