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How can we measure teaching and learning in mathematics?

Did you know you can measure whether students have fragmented or cohesive conceptions of mathematics, or whether they learn at a surface- or deep-level? Did you know there are concept inventories available for research purposes at the level of prealgebra

Maria Andersen

on 18 August 2013

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Transcript of How can we measure teaching and learning in mathematics?

How can we measure
teaching and learning
in mathematics?

Student Conceptions of Mathematics
Maria H. Andersen, Ph.D.

How do students approach their learning?
How do instructors
approach teaching?
Entwistle & Ramsden, 1983; Biggs, 1987; Ramsden 1991, 1992; Marton et al, 1997
More likely to be associated
with higher quality learning
How do students perceive
their learning environment?
Students perceive a heavy workload and less freedom in learning.
Students perceive that there is a choice in what is learned and a clear awareness of goals and standards.
Trigwell & Prosser, 1991
Student Conceptions
of Learning
Quantitative increase in knowledge
Abstraction of meaning
Understanding of reality
Marton and Saljo, 1997
Crawford et al 1994, 1998
How do you build a good inventory?
2. Look for patterns in the statements
3. Develop a pool of questions to represent categories
4. Multiple researchers classify items and compare results. Revise as needed.
5. Pilot the inventory. Check for internal consistency reliability and perform factor analysis.
6. Revise. Pilot again (twice with same group) for reliability.
1. Understand the Problem
Ask population to respond to open-ended questions
7. Final tweaking.
tasks are "imposed" on them
study without purpose or strategy
can see relevance of learning new things
seek to develop new understanding
Five conceptions
of Mathematics
1. Math is numbers, rules, and formulas
2. Math is numbers, rules, and formulas which can be applied to solve problems
3. Math is a complex logical system: a way of thinking
4. Math is a complex logical system which can be used to solve complex problems
5. Math is a complex logical system which can be used to solve complex problems and provides new insights used for understanding the world
Example: Think about the math you've done so far. What do you think mathematics is?
Conceptions of
Crawford et al., 1998
19 items
Likert scale
Fragmented and Cohesive scales
"Mathematics is about calculations"
"Mathematics is a logical system which helps explain the things around us"
"Mathematics is like a universal language which allows people to communicate and understand the world"
Approaches to Teaching Inventory
Prosser and Trigwell, 1999
16 items
Two scales
Conceptual change / student-focused
Information transmission / teacher-focused
"I feel it is important to present a lot of facts in classes so that students know what they have to learn for this subject."
"In lectures for this subject, I use difficult or undefined examples to provoke debate."
"I feel a lot of teaching time in this subject should be used to question students' ideas."
Mathematics Concepts Test
Jerome Epstein ,
NY Polytechnic University
Credits for Photos (all licensed under Creative Commons):
Ruler: http://www.flickr.com/photos/917press/456443078
EEG: http://www.flickr.com/photos/squashpicker/1480124942/
Puzzle pieces: http://www.flickr.com/photos/myklroventine/3261364899/
Puzzle on face: http://www.flickr.com/photos/eli-santana/2933926582/
Wikipedia Concept Map: http://www.flickr.com/photos/juhansonin/407874864/
Water Surface “Overflate”: http://www.flickr.com/photos/randihausken/1877810147/
Scuba diver: http://www.flickr.com/photos/30691679@N07/2891679952/
Thinker: http://www.flickr.com/photos/tmartin/32010732/
Construction signs: http://www.flickr.com/photos/15708236@N07/2754478731/
Computer classroom: http://www.flickr.com/photos/phoenixdailyphoto/1782001450/
Classroom with desks: http://www.flickr.com/photos/25312309@N05/2829580870/
Lecture hall: http://www.flickr.com/photos/kitsu/404092967/
Guy with barcode: http://www.flickr.com/photos/jaumedurgell/740880616/
B&W Stressed woman: http://www.flickr.com/photos/librarianbyday/3181780269/
Head pillars: http://www.flickr.com/photos/jannem/376980800/
Guy studying: http://www.flickr.com/photos/tripu/3441921187/
Approaches to
Entwistle & Ramsden, 1983
Teacher-focused strategy with the intention of transmitting information to students
Teacher-focused strategy with the intention that students acquire the concepts of the discipline
A teacher/student interaction strategy with the intention that students acquire the concepts of the discipline
A student-focused strategy aimed at students developing their conceptions
A student-focused strategy aimed at students changing their conceptions
Math Instructional Practices
Emphasis on
Collaborative Lecture
Cooperative Learning
Inquiry-Based Learning (IBL)
Application Problems
Mastery Learning
Formative Assessment
Communication Skills
Multiple Representations
Project Based Learning
Designing and assigning project work that requires students to solve a non-standard problem that requires a longer period of time than problems that would typically be assigned for homework or in class. There is often a research component where students must actively seek data, background knowledge, or formulas. Often the students work on projects in pairs or small groups. The final result of a project might include a written paper or a presentation on the findings.
Including class time for students to solve problems based on data from real-world situations (present or past) problems that come from the partner disciplines of mathematics (e.g. Engineering, Chemistry, Biology, Physics, Economics, Business).
Designing and using activities where students learn new concepts by actively doing and reflecting on what they have done. The guiding principle is that instructors try not to talk in depth about a concept until students have had an opportunity to think about it first (Hastings, 2006).
Including class time for learning that engages students in working and learning together in small groups, typically with two to five members. Cooperative learning strategies are designed to engage students actively in the learning process through inquiry and discussions with their classmates (Rogers et al., 2001).
Teaching by giving a series of short, focused lessons intermixed with student-centered activities that solidify the concepts of the lessons or serve to introduce the next short lesson (DeLong and Winter, 2002). The interaction during the activities is primarily between students.
Teaching by giving a presentation on some subject for a time period longer than 20 minutes. This instructional method includes the exchange of questions and answers between the instructor and students. The key characteristic is that the students rarely interact with each other during this learning process.
Teaching by including multiple ways (e.g. graphs, diagrams, algebra, words, data, manipulatives) to represent mathematical ideas whenever possible. The rule-of-four (representing a function visually, algebraically, numerically, or with words) is an example of multiple representations.
Providing opportunities for students to practice their ability to communicate mathematical and quantitative ideas using both written and oral communications.
Designing summative assessment check-points into the instructional program where the student is tested on their mastery of a single topic (or subtopic). The instructor may coach students during class time or outside of class to help students who struggle with understanding the concepts while they are intensely focused on learning. Note that the students do not receive partial credit for partially correct responses on mastery-based assessments.
Making use of instructional strategies in the learning environment that assess where students are having problems so that students can learn more and learn better. (Gold, 1999)
Andersen, 2009
Physics Education Research
FCI: Force Concept Inventory
Interactive-Engagement (IE) sections had higher normalized gain on FCI than Traditional Lecture (TL)
Hake et al, 1998
"The FCI provides a reproducible and objective measure of how a cours improves comprehension of principles, not merely how bright or prepared the studetns are, nor what they have memorized."
- Jerome Epstein
If you know that a function f(x) is positive everywhere, what can you conclude from that about the derivative, f '(x)?

a) the derivative is positive everywhere
b) the derivative is increasing everywhere
c) the derivative is concave upward
d) you can't conclude anything about the
Example similar to the CCI
Example from http://www.flaguide.org/tools/diagnostic/calculus_concept_inventory.php
Approaches &
Study Skills
Inventory for
Marilyn Carlson,
Arizona State University
Jerome Epstein ,
NY Polytechnic University
Credits for Illustrations and Cartoons
Mat Moore, Muskegon Michigan
Freelance Illustrator (garlicandcoffee@gmail.com)
Measure of learning approach giving three orientations: surface, deep, and achieving
Biggs, 1987
Includes Surface, Deep, and Achieving subscales
Designed for Higher Ed
NOTE: For a K-12 version, look at the Learning Process Questionnaire (LPQ)
Ramsden & Entwistle, 1981
Entwistle & Ramsden, 1983
Entwhistle et al., 2000
Deep approach
Strategic approach
Surface Apathetic approach
Preferences for different types of course and teaching
Related subscales
Fear of failure
Lack of purpose
Unrelated memorizing
"Mathematics faculty will use a variety of teaching strategies that reflect the results of research to enhance student learning."
The Problem
"There are a limited number of studies that document the impact of these efforts on student learning."
There is no sufficient evidence to support all-inclusive policy recommendations of any of the math instructional practices that were studied .
- National Mathematics Advisory Panel, 2008
- Susan Ganter, 1997
(after reviewing the research from ten years of Calculus Reform)
- Beyond Crossroads, 2006
What we are told:
What we are NOT told:
52 questions
dimensions are deep, strategic, and surface learning
"Individual projects look at specific pieces of the picture, but the pieces do not make a coherent whole and, in fact, often seem unrelated."
- Burrill et al., 2002
(meta-analysis of 100s of studies on the effectiveness of teaching math with graphing calculators)
Why do math instructors continue to choose the lecture method as a major component of their courses?
What we DON'T know
What knowledge of alternate instructional practices do instructors have?
How do instructors ultimately choose instructional practices in math?
Which math instructional practices are actually effective?
... and with what types of students are these practices effective?
... and for which levels of math courses?
number of CC math students in 1 year
Roughly 1900-2000 of them are members of AMATYC
67% of these instructors work part-time
78% of the PT and 98% of the FT have graduate degrees
47-50% of these instructors are women
Data from AMATYC and 2005 CBMS Statistical Report
VASS: Views about Science Survey
Characterize student views about knowing and learning science
Purpose: Assess the relation of student views to achievement in science courses
CLASS: Colorado Learning Attitudes
about Science Survey
MPEX: Maryland Physics Expectation Survey
Links students' perception of their learning environment and their quality of learning
What we NEED
A common language and common measures of teaching and learning.
number of community college math instructors
The Instructors
The Students
The Environment
The Subject Area

How does the environment effect what instructors do?
"To improve the quality of learning, it is more important to encourage deeper approaches to study
through the creation of a context involving good teaching, clear goals, and some independence in learning
than the discouragement of surface approaches to learning."
- Prosser & Trigwell, 1997
68% of two-year college faculty reported at least some stress from teaching underprepared students
(Lindholm et al., 2005)
Control of Teaching
Appropriate Class Size
Enabling Student Characteristics
Departmental Support for Teaching
Appropriate Academic Workload
Appropriate Learning Space
Perceptions of
Prosser & Trigwell, 1997
1. We begin using common Language for
Math Instructional Practices (MIPS) in research.
3. The research is promoted and "crowdsourced" to any instructor at any level of math who would like to particpate.
4. We mine the data to identify situations or instructors who are seeing results.
2. We design a research study to measure instructors, students, the environment, and outcomes so that it is relatively painless to participate.
5. We interview those instructors and look for commonalities in environments that produce change.
6. THEN we target research at these strategies and environmental changes to see if interventions can produce results.
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