Problem solving opportunities in frontal classes: �Inquiry in teaching practices and learning strategies

Boris Koichu

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ITEM Symposium

March 2022

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Eman Atrash

Ofer Marmur

Facts about Israel

• About 9000000 citizens, 1500000 students
• Two official languages, Hebrew and Arabic
• The state matriculation certificate: about 50%
• Entrance exams: by two-step external examination (school achievements 50%, aptitude test (IQ) 50%)
• Most of the students begin their university study 3-4 year after graduation from the secondary school; preparatory departments at the university
• Specialization begins from the first year of study

Premises

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• Overall, learning should be active and interactive
• In-class and out-of-class learning complement each other
• The relationship between teaching and learning in a classroom is complicated (e.g., “well-intentioned pedagogy” vs. “traditional pedagogy”)
• Each teaching format has affordances for some institutionally-valued learning, otherwise it would not be there in the long run
• Even when lecturers appreciate interactive modes of teaching, they are frequently forced/choose to lecture.

Consequently,

The role of frontal teaching should be considered in the context of other teaching/learning activities;

Suggestion that frontal teaching is incompatible with active learning should not be taken for granted, but further explored.

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Questions

• What can interested and resourceful lectures do in order to create for their students opportunities for active learning during frontal large-size classes, and while teaching under strict institutional constraints?

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• What activities, besides listening* and making notes, university students are (or can be) involved in during frontal mathematics instruction in large-size classes?

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*Listen: Give one’s attention to a sound;

Make an effort to hear something;

Take notice of and act on what someone says;

(Oxford Dictionary)

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Background information

• The Technion
• Linear Algebra (3+2h), Calculus (5+2h), 14w (service courses)
• Several lecturers teach in parallel
• No. of students: 150-300; tutorials: 50-100
• The courses are demanding even for well-prepared students
• Extensive out-of-class support:

- video-library of lectures and tutorials

- rich collection of course materials

- on-line forums, with and without the presence of TAs

- students form teams to study together out of the classes

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Linear Algebra Lecturers

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• A mathematics lecturer
• Involvement in math-ed. research
• Over 20 years of teaching experience
• Excellent feedback

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Lecturer M.

Lecturer B.

• A research mathematician
• Involvement in math-ed. research
• Over 35 years of teaching experience
• Excellent feedback

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Expert Lecturers

Expert university lecturers are:

• Well prepared and organized
• Stimulate students' interest and motivation in studying the material through their enthusiasm/expressiveness
• Have positive rapport with students
• Show high expectations for the students
• Generally maintain a positive classroom environment

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(Hativa et al., 2001)

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Teaching dilemmas of B. and M.

B.: When you teach a course in parallel with other lecturers, you have to cover specific material before a midterm or an exam, and this causes a dilemma between my will to be flexible and answer the students' questions and my duty to finish the material required.

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M: I am a little stressed because I will lose two hours next week. The stress is because of the TAs, I don’t want a situation in which they don't have material to exercise because I am late [with the syllabus].

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The data �(Eman Atrash’s Ph.D. thesis)

• Lecture observations

168 h of videotaped lectures; two semesters (28w); occasional use of “clickers” during the 2^nd semester

• Interviews

Two interviews with each lecturer about their pedagogical style and preferences

• Stimulated recall conversations

Ten stimulated-recall conversations with each lecturer, about specific events that occurred during the lectures

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Focus on pedagogical practices

1. Which pedagogical practices, aimed at helping LA students to overcome their difficulties with the exposed material (PPOD), do typically appear in frontal lectures by B. and M.?�
2. Which considerations do the lecturers take into account when acting as they do during the lectures? 

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Analysis

• 50% of videotaped lessons were analyzed.
• The focus was on those episodes, lasting from several seconds to several minutes, in which the lecturers alternated the flaw “definition – theorem – proof” (cf. Davis & Hersh, 1981) by additional actions, which seemed us to be driven by their intention to highlight points of potential difficulty for the students or to increase the students’ participation.

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The analysis was stimulated by �The Knowledge Quartet

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Goulding, Rowland & Barber (2002); Rowland & Turner (2007)

Categories

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Lecturers' PPOD in “Systems of Linear Equations” lecture

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Lecturers' pedagogical practices in �“Linear Transformations” lecture

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Lecturers' PPOD �in “Vector Spaces” lecture

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Example: Making an intentional mistake

Lecturer B. writes on the board:

Given two subspaces U and W:

is a subspace, and is a subspace.

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B. writes a proof for the first claim on the board.

Then B. modifies the proof by changing some signs, claims that he “proved” the second claim, and looks at the students.

Silence for about 10 sec.

B. tells the students that the second proof is wrong and ask them to find a mistake.

Silence for 20 sec.

B. answers the question.

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Considerations underlying specific PPOD

B. about making intentional mistakes

B: "I learn from my experience. With time, I realized that the students in the lectures mainly focus on writing down the material and don’t really think about what they are writing. I want them to think more and be aware of mistakes that even the lecturer can do.“

M. about encouraging to keep working

M: “Obviously, not everyone understands everything during the lecture, and it is not harmful to review the material at home. In my opinion, I should do everything I can to make things clear, so that most of the students understand, and then at home they would try to understand deeper.”

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Selected results

The most frequent PPOD (aggregative):

- Giving Numerical Examples (97 occurrences in 84 hours);

• Highlighting by Asking Questions (79);
• Advising to Do or Not to Do Something (47);
• Repeating Twice (39).

Pedagogical considerations:

B: to enable students to ask questions

M: to provide students with comprehensive and relevant explanations

For each lecturer:

• the numbers of PPOD across different topics were about the same during the 1^st semester, and varied during the 2^nd one
• the numbers of PPOD identified in the second semester were considerably larger than in the first semester
• In the lectures with “clickers” the time of dealing with the students' anticipated mistakes was longer than in the lectures on the same topics, but without “clickers”.

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A story on making intentional mistakes: Design study with one problem

First year calculus tutorials; five semesters

60-80 students in a classroom

Frontal teaching: TA (Ofer) explains and demonstrates problem-solving methods

As a rule, no time is explicitly allocated for independent problem solving

The goal: to create wow-experiences for the students

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(Marmur & Koichu, 2016)

The problem

Show that the sequence below converges and calculate its limit

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Standard approach: �- prove, by induction, that the sequence is monotonous

- Show, by induction, that

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The problem

Show that the sequence below converges and calculate its limit

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Standard approach: �- prove, by induction, that the sequence is monotonous

- Show, by induction, that

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DOES NOT WORK

What does work?

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Initial teaching approach:

TA: “This is an example of a situation when induction is not helpful. If we’d had more time… But you can try it at home.” Then TA shows the solution by AM-GM inequality

As a result, no emotional response from the students

“Advising what to do or not,” “Encouraging to work”

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Further attempts (during two semesters):

TA makes a failed attempt to prove by induction the monotonicity of the sequence, and then shows the successful solution.

As a result, 2-3 students show signs of being excited (“amazing!”, “wow!”, “impressive!” body-language).

“Making an intentional mistake”

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A “wow-evoking” teaching sequence

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Emotional response

About the third of the students (20-25) showed “wow-signs”

At the end: “beautiful!” “amazing!”

At the beginning:

„It was horrible!,” „I felt frustrated,”

and even „It was infuriating”.

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Contrastive Valence Theory� (Huron, 2006)

Stimulated recall conversations

• Nine students who had expressed strong emotions at the lesson and agreed to talk about their experience
• 15-min video of the episode
• Instructions: �- Stop the video whenever you have a particular recollection of what you thought or felt at that moment. Share.

- Explain why you thought/felt this way at that moment.

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From the interviews

“I thought you'd simply make some kind of change and we would move on. That there will be some small nuance you will use and we'll be able to continue. But here it was extremely dramatic. You just erased it all!”

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“I felt a rise in self-confidence that this can happen, maybe it can also happen to you [the TA], or any student [...] who I think is smart. It can happen to anyone. . . that we try something and it doesn't work.”

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A more detailed reflection

On the video, TA: We have done several standard exercises, and now we are going to consider a non-standard exercise. BTW, it has been given as a HW some time ago. [TA writes the problem on the board.]

R (stops the video): Here I recalled that very-very similar questions have indeed appeared in HW, with these inductions, and that I’ve used induction before.

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A more detailed reflection

On the video, TA crosses-out the second failed attempt.

R (stops the video): I was shocked, no doubt. I was sure at that moment that we are about to finish [the solution], because in the past I had really solved such problems by induction. Hmm… personally, I began looking for the ways of how to continue from here. It took me about three minutes to cut off [my attempts] and come back to you…

TA stops the video (+3 min.), asks: At this moment, have you been with me?

R: Yes, yes, yes, yes! I think that because him [points to a student on the video] talked, I had a momentum, extra time [to think autonomously]. I was not listening to what he was saying. Then you said [to the student on the video] “OK, show me after the lesson”, and I came back.

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Conjecture

While participating in an emotionally-loaded episode, the students seemed to do much more than “just listening”. �They seemed to:

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• notice unclear places to come back to later
• formulate questions
• test alternative solution pathways
• construct anticipations (cf. Simon & Tzur, 2004).

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How can we name what R. did?

Self-regulated listening?

Inquiry-driven listening?

Exploratory listening?

Participatory listening?

Relevance-looking listening?

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Mathematically active listening?

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Concluding remarks: Zoom out

How can we help less skillful students than R. to become active listeners?

Heuristic-didactic discourse: lecturer’s discourse that presents heuristics monitored from an expert’s perspective, yet derived from a student’s perspective.

Key memorable events: events that are “perceived by many students in class as memorable and meaningful in support of their learning, and are typically accompanied by strong emotions, either positive or negative”

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(Marmur, 2019;Marmur & Koichu, 2021)

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Thanks for (active?) listening!���boris.koichu@weizmann.ac.il

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