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Part 1: What are Concept Maps?
Citation preview
Gary [email protected]://napaneedss.limestone.on.ca/greer
Materials Available Onlinehttp://napaneedss.limestone.on.ca/greer/teachers/cm/cmap.html
Concept Maps in Mathematics Education
Gary [email protected]
Cape Breton UniversityMemorial University of Newfoundland
Part 1: What are Concept Maps?
Concept Maps?
What are Propositions?• concept maps constructed of propositions
(two concepts connected by a short linking phrase) which are the unit of meaning (perhaps analogous to sentence)
• the brief linking phrase describes the relationship between the two concepts
• anyone can draw lines between words, but connecting these with a phrase creates the window into understanding
A Mathematical Example
• here are two common ideas from secondary school math
• what linking phrase should we use?• “is a”? “represents”? • should the arrow go the other way?• Is there some common root that would better
define the relationship?
Concept Maps: Structure• the classic Novak map is hierarchical but
this may not be necessary in all situations
Sample Concept Map: Numbers
• since concept maps are idiosyncratic, each person may construct their own valid representation
• what does this say about assessment?
Theoretical Basis
• concept mapping is grounded in Ausubel’s assimilation theory– new knowledge is assimilated into previously held
knowledge structures• use is supported by the idea of cognitive off-
loading• also supported by constructivist approaches to
learning and by the idea of a zone of proximal development
Software• abundance of applications• creative, colorful, links to resources• students may be more willing to improve it• easy to edit• can create web pages• support collaborative learning• Inspiration/Kidspiration - widely used, cost• CMapTools - free, powerful, collaborative• SmartIdeas - provincially licensed
SmartIdeas• experiment…typing anywhere will create a
concept• clicking once on a concept enables the blue
linking arrow– click, hold and drag to link
• clicking twice should enable link editing• to change the appearance, click once on the
concept to select it and click on the desired design at the left
• right clicking presents you with menus• try creating a concept map explaining adding and
subtracting integers
Part 2: Analysis of Research into the Effectiveness of Concept Maps in Mathematics Education
My Study
• my final project of M.Ed. (I.T.) investigated whether maps are effective in mathematics
education as suggested in literature• concept maps developed by Novak in the 1980’s
and widely used in Science education• motivation:
– increasing concerns over achievement in Math– shift from procedural to conceptual learning with
increased emphasis on relationships, connectedness – availability of powerful concept mapping software– personal interest in using technology
Research Questions• Overall
– Will a review of literature provide supporting evidence that concept maps are effective tools in facilitating learning and assessing student knowledge in mathematics education?
• Specific Questions– Are concept maps valid and reliable
instruments for assessing mathematical knowledge?
– Can improvements in student learning be attributed to the use of concept maps?
The Study• an analysis of 11 available peer-reviewed
research papers was undertaken• these were analyzed with respect to validity and
reliability as well as links between the use of concept maps and learning
• 5 to 108 subjects, • 1 elementary, 3 secondary and 7 post-secondary• assessment varied between qualitative,
quantitative, holistic, and combinations of these• see Research Paper
Summary of Studies
qualitative3 maps over a semester
traditional exam & interview
Post-Secondary26McGowen and Tall (1999)
quantitativeone timeinterviewsPost-Secondary17Mwakapenda (2005)
holistictwicePost-Secondary19Roberts (1999)
holisticone timePost-Secondary28Williams (1998)
qualitativeone timeunsupervised examPost-Secondary5Zwaneveld (2000)
holisticone timeessay & traditional exam
Post-Secondary108Bolte (1999)
quantitativesemestervee diagram & collaboration
Post-Secondary16Afamasaga-Fuata’i (2004)
holisticone timehypermedia, interview & exam
Secondary20Flanagan (2002)
quantitativeone timetraditional examSecondary48Barolos (2002)
quantitativepre & post instruction
traditional examSecondary23
qualitativetwice over5 months
interview & problem solving
Elementary51Hasemann and Mansfield (1995)
Overall AssessmentType
InteractionDuration
Corroborating Measures
StudentLevel
nStudy
Findings: • generally correlate with other instruments• great variability, but there is some support for
assessment validity• great variability, but there is some support for
concept maps as reliable instruments• while learning was observed to have occurred,
there was insufficient effort to limit the effect of external factors – no control groups
• no clear empirical link was established between the use of concept maps and learning
• indirect evidence was supportive however
Findings: Participant Perspective
• researchers generally supportive of concept maps as effective assessment instruments
• specifically useful in identifying student misconceptions
• researchers indicated that maps were useful in facilitating student learning
• students indicated that concept maps were useful in promoting understanding
Summary of Analysis
moderategood (C), moderate (I),moderate (E)
McGowen and Tall (1999)
modestmoderate (C), low (I), modest (E)
Mwakapenda (2005)
modestmoderate (C), moderate (I), low (E)
questionableRoberts (1999)
modestmoderate (C), good (I), moderate (E)
Williams (1998)
lowmodest (C), moderate (I),low (E)
no clear linkZwaneveld (2000)
moderatemodest (C), good (I),good (E)
Bolte (1999)
lowmoderate (C), modest (I), modest (E)
no clear linkAfamasaga-Fuata’i (2004)
no clear linkFlanagan (2002)
moderatemoderate (C), low (I), modest (E)
Barolos (2002)
moderatemoderate (C), moderate (I),moderate (E)
modestmoderate (C), low (I), good (E)
questionableHasemann and Mansfield (1995)
ReliabilityEstimate
Validity: Criterion (C), Internal (I), External (E)
Learning withConcept Mapping
Study
Inconsistencies, Gaps & Limitations
• student mapping ability is not accounted for• there is an unsupported assumption that a
concept map represents student knowledge• validity and reliability not firmly established• not clear how they correlate with other
assessment instruments• limited studies, particularly at the secondary
and elementary levels
Part 3: Using Concept Maps in the Mathematics Classroom
Sample of Student Work
• grade 9 academic students asked to create a concept map in pairs to demonstrate how to solve a variety of single variable equations (examples of types were provided)
• topic had been discussed in class and this exercise was used as a test review
• teacher candidate (had only used concept maps for 3 days) gave a 5-10 minute demo of building a concept map explaining adding fractions
• students had not used the tool before
Example: Solving (level 1)
Example: Solving (level 2)
Example: Solving (level 3)
Example: Solving (level 4)
Approaches
• while the basic idea for constructing knowledge is to engage in student creation of maps, there are some various approaches
– map as navigator– map as a guide or as expert knowledge– partially complete map with a list of terms– partially completed map to fill in– list of words to construct a map from– list of seed terms to use or draw from– a question to resolve
Types of Maps• processes
– how do you add fractions?• organization
– what topics have we studied in this unit?– what resources do we have?
• conceptual– how do these topics relate to each other?
• problem solving– what knowledge do we have, what steps do we take?
• brainstorming– how do we approach this task?
Suggestions
• start with more structured activities (simple organizational maps may be valuable to students)
• use mapping software• let the students be creative• decide what will work in your classroom• decide what student needs you are addressing• experiment, and try the assignments on
your own first
Example: Factoring Polynomials
Example: Exponents
Example: Review
Example: Part of Large Project
Assessing Concept Maps
• assessment may be quantitative, qualitative, holistic or combinations of these, sometimes compared to an expert map
• Novak scoring system– 1 point for valid propositions and examples– 5 points for valid hierarchies– 10 points for valid crosslinks showing synthesis and 2
for less significant ones• other criteria may include directionality, accuracy,
centrality, number of links, misconceptions, structure and improvement
Activity: Area and Volume
• fill-in the concept map available at http://napaneedss.limestone.on.ca/greer/teachers/cm/measurement.ipr
• fill-in maps provide a way to add structure, focus the activity and scaffold understanding
• there are a number of possible phrases for some of the links, these may be quite indicative of understanding
• students who are comfortable can add new geometric figures
Measurement Concept Map• when complete, compare your maps…
Collaborative Exercise
1. create a map explaining how to add fractions or2. the following page contains 16 terms from a study of
first year university students• collaboratively create a concept map using these
terms, adding terms as needed• focus on linking words and phrases• making links confronts you with understanding
the relationships• discussion is encouraged in creating a map that
makes sense to all collaborators
Create a Concept Map Using…
• ratio• parallel• function• tangent• infinity• perpendicular• inverse• zero
• equation• limit• absolute value• similar• gradient• angle• variable• bisector
One Possible Representation• are there misconceptions found here?
Why Use Concept Maps?
• Will they revolutionize learning? No.• They can help
– organize– recognize misconception– represent knowledge– build knowledge
• They provide another tool for learning• They can be a change for students• They can provide an alternative
Materials Available Online
Thank you
http://napaneedss.limestone.on.ca/greer/teachers/cm/cmap.html
Gary [email protected]://napaneedss.limestone.on.ca/greer