Materials Available Online http napaneedss limestone on cagreerteachers
Materials Available Online http: //napaneedss. limestone. on. ca/greer/teachers /cm/cmap. html Gary Greer greerg@limestone. on. ca http: //napaneedss. limestone. on. ca/greer
Concept Maps in Mathematics Education Gary Greer greerg@limestone. on. ca Cape Breton University Memorial 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 offloading • 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 CMap. Tools - free, powerful, collaborative Smart. Ideas - provincially licensed
Smart. Ideas • 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
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
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 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 or 2. 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 http: //napaneedss. limestone. on. ca/greer/teachers /cm/cmap. html Thank you Gary Greer greerg@limestone. on. ca http: //napaneedss. limestone. on. ca/greer
- Slides: 41