**How can we measure**

teaching and learning

in mathematics?

teaching and learning

in mathematics?

**Student Conceptions of Mathematics**

Fragmented

Cohesive

**Maria H. Andersen, Ph.D.**

busynessgirl.com

@busynessgirl

busynessgirl.com

@busynessgirl

Surface

approach

Deep

approach

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

outcome.

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

Memorization

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

Mathematics

Questionairre

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"

Example

Items

Approaches to Teaching Inventory

Prosser and Trigwell, 1999

16 items

Two scales

CCSF

ITTF

Conceptual change / student-focused

Information transmission / teacher-focused

Example

Items

"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

Study

Inventory

Course

Perceptions

Questionaaire

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

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

derivative

Example similar to the CCI

Example from http://www.flaguide.org/tools/diagnostic/calculus_concept_inventory.php

Study

Process

Questionnaire

Approaches &

Study Skills

Inventory for

Students

Precalculus

Concept

Assessment

Calculus

Concepts

Inventory

**CCI**

**PCA**

Marilyn Carlson,

Arizona State University

Elementary

Algebra

Concept

Inventory

Jerome Epstein ,

NY Polytechnic University

**EACI**

Credits for Illustrations and Cartoons

Mat Moore, Muskegon Michigan

Freelance Illustrator (garlicandcoffee@gmail.com)

SPQ

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)

ASSIST

**ASI**

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

Subscales

Related subscales

Fear of failure

Lack of purpose

Syllabus-boundness

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:**

ATI

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?

**28,000**

**1,700,000**

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

**CMQ**

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

CPQ

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**

**The**

Solution

Solution

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

**PTEI**

Perceptions of

Teaching

Environment

Inventory

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.