1. Gary Greer greerg@limestone.on.ca http://napaneedss.limestone.on.ca/greer Materials Available Online http://napaneedss.limestone.on.ca/greer/teachers /cm/cmap.html

2. Concept Maps in Mathematics Education Gary Greer greerg@limestone.on.ca Cape Breton University Memorial University of Newfoundland

3. Part 1: What are Concept Maps?

4. Concept Maps?

5. 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

6. 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?

7. Concept Maps: Structure the classic Novak map is hierarchical but this may not be necessary in all situations

8. Sample Concept Map: Numbers since concept maps are idiosyncratic, each person may construct their own valid representation what does this say about assessment?

9. 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

10. 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, costInspiration/Kidspiration CMapTools - free, powerful, collaborativeCMapTools SmartIdeas - provincially licensedSmartIdeas

11. 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

12. Part 2: Analysis of Research into the Effectiveness of Concept Maps in Mathematics Education

13. 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

14. 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?

15. 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 PaperResearch Paper

16. 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 over 5 months interview & problem solving Elementary51Hasemann and Mansfield (1995)‏ Overall Assessment Type Interaction Duration Corroborating Measures Student Level nStudy

17. 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

18. 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

19. 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)‏ Reliability Estimate Validity: Criterion (C), Internal (I), External (E)‏ Learning with Concept Mapping Study

20. 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

21. Part 3: Using Concept Maps in the Mathematics Classroom

22. 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

23. Example: Solving (level 1)‏

24. Example: Solving (level 2)‏

25. Example: Solving (level 3)‏

26. Example: Solving (level 4)‏

27. 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

28. 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?

29. 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

30. Example: Factoring Polynomials

31. Example: Exponents

32. Example: Review

33. Example: Part of Large Project

34. 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

35. Activity: Area and Volume fill-in the concept map available at http://napaneedss.limestone.on.ca/greer/teachers/ cm/measurement.ipr 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

36. Measurement Concept Map when complete, compare your maps…

37. 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

38. Create a Concept Map Using… ratio parallel function tangent infinity perpendicular inverse zero equation limit absolute value similar gradient angle variable bisector

39. One Possible Representation are there misconceptions found here?

40. 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

41. Materials Available Online Thank you http://napaneedss.limestone.on.ca/greer/teachers /cm/cmap.html Gary Greer greerg@limestone.on.ca http://napaneedss.limestone.on.ca/greer