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)
SoTL
Teaching Tips
Math ed
research
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 Desert
Teaching 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 Desert
The 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 Desert
The Evidence
•
•
•
•
Copies of the student outlines
Student Test Scores (i.e., grades)
Student Retention Rates
Student Interviews
College of the Desert
Outcomes (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 Desert
Unexpected 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 Desert
Going 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=14
832740290866&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 Survey
5 Examples Convinces Me
100%
80%
SD
D
N
A
SA
60%
40%
20%
0%
0 Sems
1-2 Sems
3-4 Sems
>4 Sems
Faculty
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.
Mathematical
Knowledge Expertise Taxonomy
Affective
Acclimation
Competence
Proficiency
Students are motiv ated to learn by
Students hav e both internal and external
external (of ten grade-oriented) reasons Students are motiv ated by both internal
motiv ation. Internal motiv ation comes
that lack any direct link to the f ield of (e.g., intrigued by the problem) and
f rom an interest in the problems f rom
Interest
external reasons. Students still pref er
study in general. Students hav e greater
the f ield, not just applications. Students
concrete concepts to abstractions, ev en
interest in concrete problems and
appreciate both concrete and abstract
special cases than abstract or general if the abstraction is more usef ul.
results.
results.
Students are unlikely to spend more Students spend more time on problems.Students will spend a great deal of time
than 5 minutes on a problem if they They will of ten spend 10 minutes on a on a problem and try more than one
cannot solv e it. Students don't try a new
problem bef ore quitting and seeking
approach bef ore going to text or
Conf idence approach if f irst approach f ails. When external help. They may consider a
instructor. Students will disbeliev e
giv en a deriv ation or proof , they want second approach. They are more
answers in the back of the book if the
minor steps explained. They are rarely comf ortable accepting proof s with some
answer disagrees with something they
complete problems requir
steps "lef t to the reader" if they hav f eel they hav e done correctly . S
Cognitive
Factual
Procedural
Acclimation
Competence
Proficiency
Students hav e working knowledge of the
Students start to become aware of basic
Students hav e quick access to and
f acts of the topic, but may struggle to
f acts of the topic.
broad knowledge about the topic.
access the knowledge.
Students hav e working knowledge of the
Students can use procedures without
Students start to become aware of basic
main procedures. Can access them
ref erence to external sources or
procedures. Can begin to mimic
without ref erencing the text, but may
struggle. Students are able to f ill in
procedures f rom the text.
make errors or hav e dif f iculty with more
missing steps in procedures.
complex procedures.
Schematic
Students hav e put knowledge together
Students begin to combine f acts and Students hav e working packets of
in packets that correspond to common
procedures into packets. They use
knowledge that tie together ideas with
theme, method, or proof , together with
surf ace lev el f eatures to f orm schema.comon theme, method, and/or proof .
an understanding of the method.
Strategic
Students use surf ace lev el f eatures of
Students choose schema to apply
problems to choose between schema,
based on a f ew heuristic strategies.
or they apply the most recent method.
Epistemic
Social
Students choose schema to apply
based on many dif f erent heuristic
strategies. Students self -monitor and
abandon a nonproductiv e approach f or
an alternate.
Students begin to understand the
Students recognize that proof s don't
common notions 'ev idence' of the f ield.Students are more strongly aware that hav
a e counterexamples, are distrustf ul of
They begin to recognize that a v alid v alid proof cannot hav e
5 examples, see that general proof s
proof cannot hav e a counterexample, counterexamples. They use examples to
apply to special cases, and are more
they are likely to believ e based on 5 decide on the truth of a statement, but likely to use "hedging" words to describe
examples, howev er, they may be
require a proof f or certainty .
statements they suspect to be true but
skeptical at times
hav e not y et v erif ied.
Students will struggle to write a proof Students are likely to use an inf ormal
Students in this stage write proof s with
and include more algebra or
shorthand that can be read like
complete sentences. They use clear
computations than words. Only partial sentences f or writing a proof . They may
sentences will be written, ev en if they employ connectors, but writing lacks concise sentences and emply correct
terminology . They use v ariables
say f ull sentences. Variables will seldom
clarity of ten due to reliance on pronouns
correctly .
be def ined, and proof s lack logical
or inappropriate use or lack of
connectors.
mathematical terminology .
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