What is Scholarship of Teaching and Learning:

Working examples

Curtis D. Bennett

Loyola Marymount University

What is the Scholarship of Teaching and Learning?

• Four Scholarships (Boyer, 1990)• Discovery• Integration• Application• Teaching (and Learning)

• Scholarship is:• Public• Subject to Critical Review• Accessible for exchange and use.

• Thinking about teaching as a scholarly inquiry.

Situating the Scholarship of Teaching and Learning (SoTL)

Teaching TipsMath ed research

SoTL

Stages of Scholarship of Teaching and Learning

• A teaching problem.

• Frame/refine the problem into a researchable question.

• Investigate and gather evidence in a systematic way.

• Draw Conclusions.

• Go Public.

Teaching Problem – types of questions.

• What works?• Does this method work in improving student

learning.

• What is?• What are the students’ doing, thinking, etc.

when they approach the class, problem, etc.

• What could be?• A vision of the possible: often an example of

what can be achieved with one student or in one class.

A teaching problem.

Driven by a question/concern/desire• I wish my students would read their textbook.

(Laura Graff, Dustin Culhan, and Felix Marhuenda-Donate, College of the Desert)

• Only 15% of our pre-algebra students successfully complete the basic skills math sequence. (Jay Cho – Pasadena City College)

• Students have difficulty moving beyond examples to argumentation/proof? (CB)

College of the Desert

College of DesertTeaching Problem

When students enroll in liberal arts classes, they expect to attend lectures and take notes, read the textbook, and study questions the instructor provides to help them prepare for exams, but …

these same attitudes and expectations do not seem to apply to math class.

Researchable questions

• Will outlining math textbooks get students in the habit of using their textbooks, i.e., read them effectively beyond getting problems from them?

• Will being in the habit of using their textbooks make them more likely to make their own connections and become more active learners?

College of the DesertThe Classroom Change

• Students performing under 75% on first test are required to include “chapter outlines” of each section. (Variant of “think aloud” taught at Leadership Institute in Reading Apprenticeship).

• Intention: Teach students how to read book.

College of DesertThe Evidence

• Copies of the student outlines

• Student Test Scores (i.e., grades)

• Student Retention Rates

• Student Interviews

College of the DesertOutcomes (expected)

• Test scores went up for some students.

• Higher retention rates.

• Students became more adept at identifying key concepts over the term.

• Students found outlines helped them complete their homework more quickly.

College of the DesertUnexpected Outcomes

• More personal relationship with students as students apparently had more trust in instructors.

• One student said she “wanted to finish the book over the summer”

• Greater retention because students were not as afraid to learn material on their own when they missed class - Less likely to give up.

Student interviews

College of the DesertGoing Public

• KEEP Toolkit (www.cfkeep.org) - the KEEP Toolkit, developed by the Carnegie Foundation for the Advancement of Teaching provides a compact way to make results available to others.

• http://www.cfkeep.org/html/stitch.php?s=14832740290866&id=34947815104339 But easier if you go to KEEP toolkit and go to Gallery then Community Colleges then Math

Why do the Scholarship of Teaching and Learning?

• The intentional gathering of evidence makes us much more aware of what is happening in the classroom.

• Avoid pedagogical amnesia.

• Creates richer knowledge for others to build on.

• It’s fun!

My teaching problem(s)

• In a first proof course for math majors, students accept multiple examples as sufficient proof that a claim is true. How do we help them move forward on this?

• Getting students to improve their mathematical skills is hard, particularly when it comes to problem solving/explaining. (They seem to “get it” or not.)

Researchable Question

I needed to know what was going on in the brains of the students: a what is question.

• What do students at various levels consider acceptable evidence for a claim?

• What do students do as they problem solve?

Evidence (layered)

• Student Faculty Survey

• Later:– “Proof Aloud” with 12 students– Focus group with 5 of 12 students– Faculty “Proof Aloud.”

Result of Survey5 Examples Convinces Me

0%

20%

40%

60%

80%

100%

0 Sems 1-2 Sems 3-4 Sems >4 Sems Faculty

SDDNASA

Can there exist a counterexample to a proven statement

Faculty Explanation

• ‘Convinced’ does not mean ‘I am certain’…

• …whenever I am testing a conjecture, if it works for about 5 cases, then I try to prove that it’s true

• Sometimes we find holes in proofs.

We needed more and better evidence

• Proof aloud (Built off of a think aloud)– Investigate a statement (is it true or false?)– State how confident, what would increase it– Generate and write down a proof– Evaluate 4 sample proofs– Respond - will they apply the proven result?– Respond - is a counterexample possible?

Proof-aloud Task and Rubric

• Elementary number theory statement– Recio & Godino (2001): to prove– Dewar & Bennett (2004): to investigate, then prove

• Assessed with Recio & Godino’s 1 to 5 rubric– Relying on examples– Appealing to definitions and principles

• Produce a partially or substantially correct proof

• Rubric proved inadequate

Richer Representation/Rubric Needed

• Student progression toward proficiency– Using a K-12 classroom-based expertise theory

(P. Alexander across all academic domains)

• Typology of mathematical knowledge– 6 cognitive components (R. Shavelson in science)

– 2 affective components (C. Bennett & J. Dewar)

Proof generation/Problem Solving: complex tasks

• Requires several types of knowledge– Mathematical content of statement– Appropriate logical procedures– How to write/explain

• Can be approached in many ways– Involves strategic choice of method

• Requires persistence in face of uncertainty• Writing explanations/proving requires additional

knowledge and motivation to produce a polished result

Multi-faceted Student Work

• Insightful question about the statement• Poor strategic choice of (advanced) proof

method• Exhibit advanced mathematical thinking, but

had undeveloped proof writing skills• Confidence & interest influence performance• Even our expert was concerned before

knowing the task!

Proof-aloud results

• Compelling illustrations– Types of knowledge

– Strategic processing

– Influence of motivation

• Greater knowledge = poorer performance• Both expert & novice behavior on same task

How do we describe all of this?

• Expertise Theory (P. Alexander, 2003)

• Typology of Scientific Knowledge (R. Shavelson, 2003)

School-based Expertise Theory: Journey from Novice to Expert

3 Stages of expertise development• Acclimation or Orienting stage

• Competence

• Proficiency/Expertise

Typology: Mathematical Knowledge

• Two Affective Dimensions (Alexander, Bennett and Dewar):– Interest: What motivates learning.– Confidence: Dealing with not knowing.

• Six Cognitive Dimensions (Shavelson, Bennett and Dewar):– Factual: Basic facts– Procedural: Methods – Schematic: Connecting facts, procedures, methods, reasons.– Strategic: Heuristics used to make choices.– Epistemic: How truth is known.– Social: How truth/knowledge is communicated.

MathematicalKnowledge Expertise Taxonomy

Affective Acclimation Competence Proficiency

Interest

Students are motivated to learn by external (often grade-oriented) reasons that lack any direct link to the field of study in general. Students have greater interest in concrete problems and special cases than abstract or general results.

Students are motivated by both internal (e.g., intrigued by the problem) and external reasons. Students still prefer concrete concepts to abstractions, even if the abstraction is more useful.

Students have both internal and external motivation. Internal motivation comes from an interest in the problems from the field, not just applications. Students appreciate both concrete and abstract results.

Confidence

Students are unlikely to spend more than 5 minutes on a problem if they cannot solve it. Students don't try a new approach if first approach fails. When given a derivation or proof, they want minor steps explained. They are rarely complete problems requir

Students spend more time on problems. They will often spend 10 minutes on a problem before quitting and seeking external help. They may consider a second approach. They are more comfortable accepting proofs with some steps "left to the reader" if they hav

Students will spend a great deal of time on a problem and try more than one approach before going to text or instructor. Students will disbelieve answers in the back of the book if the answer disagrees with something they feel they have done correctly. S

Cognitive Acclimation Competence Proficiency

FactualStudents start to become aware of basic facts of the topic.

Students have working knowledge of the facts of the topic, but may struggle to access the knowledge.

Students have quick access to and broad knowledge about the topic.

ProceduralStudents start to become aware of basic procedures. Can begin to mimic procedures from the text.

Students have working knowledge of the main procedures. Can access them without referencing the text, but may make errors or have difficulty with more complex procedures.

Students can use procedures without reference to external sources or struggle. Students are able to fill in missing steps in procedures.

SchematicStudents begin to combine facts and procedures into packets. They use surface level features to form schema.

Students have working packets of knowledge that tie together ideas with comon theme, method, and/or proof.

Students have put knowledge together in packets that correspond to common theme, method, or proof, together with an understanding of the method.

StrategicStudents use surface level features of problems to choose between schema, or they apply the most recent method.

Students choose schema to apply based on a few heuristic strategies.

Students choose schema to apply based on many different heuristic strategies. Students self-monitor and abandon a nonproductive approach for an alternate.

Epistemic

Students begin to understand the common notions 'evidence' of the field. They begin to recognize that a valid proof cannot have a counterexample, they are likely to believe based on 5 examples, however, they may be skeptical at times

Students are more strongly aware that a valid proof cannot have counterexamples. They use examples to decide on the truth of a statement, but require a proof for certainty.

Students recognize that proofs don't have counterexamples, are distrustful of 5 examples, see that general proofs apply to special cases, and are more likely to use "hedging" words to describe statements they suspect to be true but have not yet verified.

Social

Students will struggle to write a proof and include more algebra or computations than words. Only partial sentences will be written, even if they say full sentences. Variables will seldom be defined, and proofs lack logical connectors.

Students are likely to use an informal shorthand that can be read like sentences for writing a proof. They may employ connectors, but writing lacks clarity often due to reliance on pronouns or inappropriate use or lack of mathematical terminology.

Students in this stage write proofs with complete sentences. They use clear concise sentences and emply correct terminology. They use variables correctly.

Confidence:Proficient/Expert Stage

Students spend a great deal of time on a problem and try more than one approach before going to text or instructor. Students are accustomed to filling in the details of a problem. Can solve multi-step problems. Students will disbelieve answers in the back of the book if the answer disagrees with something they feel they have done correctly.

Using the Taxonomy

• Analyzing work of students from other classes and institutions.

• Helps guide my teaching of problem solving in all my classes.

Solving quadratics• In solving problems on quadratic equations, what types of

knowledge are necessary. Can you categorize.

– Factual: Quadratic formula, what is a quadratic, what does a missing term mean, factoring.

– Algorithmic: Completing the square, factoring methods, applying QF.

– Schematic: How do all of these fit together.

– Strategic: Which method should you choose when?

– Epistemic: How do you know check to see that you have the right answer?

Scholarship of Teaching and Learning

• Idea is to investigate teaching issues in rich systematic ways to help with understanding.

• Teachers investigating their practice with the notion of communicating to other teachers.

• It is important that it happens at all levels with all students.

How do you get started

• Start with a teaching project or problem• Frame a question that you would like answered.• Think about what evidence would help you

answer the question.• Gather the evidence (if you’re like most of us, you

will gather too much – that’s okay)• Analyze• Present