Developing a More Conceptual Understanding of Matrices & Systems of Linear Equations through Concept Mapping and Vee Diagrams

By Afamasaga-Fuata'I, Karoline | Focus on Learning Problems in Mathematics, Summer 2006 | Go to article overview

Developing a More Conceptual Understanding of Matrices & Systems of Linear Equations through Concept Mapping and Vee Diagrams


Afamasaga-Fuata'I, Karoline, Focus on Learning Problems in Mathematics


Abstract

The paper discusses one of the case studies of a multiple-case study teaching experiment conducted to investigate the usefulness of the metacognitive tools of concept maps and vee diagrams (maps/diagrams) in illustrating, communicating and monitoring students' developing conceptual understanding of matrices and systems of linear equations in an undergraduate mathematics course. The study also explored the tools' role in scaffolding and facilitating students' critical and conceptual analyses of problems in order to identify potential methods of solutions. Data collected included students' progressive maps/diagrams, journals of reflections and justifications of revisions, and final reports and researchers' annotated comments on students' maps/diagrams and anecdotal notes from presentations. Findings showed that students developed more enriched, integrated and connected understandings of matrices and systems of linear equations as a result of continually organizing coherent groups of concepts into meaningful networks of propositional links, critically reflecting on the results against feedbacks from critiques and negotiations for shared meanings, and crystallizing these conceptual changes and nuances where appropriate as revised or additional propositional links. Verifying and justifying solutions were greatly facilitated through the combined usage of concept maps and vee diagrams. Findings suggest that students' classroom experiences in working, thinking and communicating mathematically can be enhanced by incorporating these metacognitive tools into students' repertoire of effective learning strategies.

Introduction

Current emphases in national and state curricular frameworks urge the promotion of deep knowledge and deep conceptual understanding of students as well as enhancing students' abilities and skills in working, thinking and communicating mathematically. To achieve these content and process outcomes, mathematics teachers are encouraged to be innovative, investigative and explorative in their pedagogical approaches to designing and developing learning activities (NCTM, 2000; NSW 2002). External examination reports (MANSW, 2005) indicate that a high proportion of students have difficulties comprehending the meanings of key concepts in the context of problems, justifying solutions, and presenting coherent mathematical arguments. Furthermore, first year university students' mathematical performances (Mays, 2005) in diagnostic tests show that most have mathematical misconceptions with fractions, percentages and multi-digit subtraction. Similarly, national surveys in Samoa confirm that learning by rote-memorization is quite prevalent in most schools (DOE, 1995). Such findings resonate with recurring comments in examiners' reports concerning students' obvious inabilities to effectively apply existing knowledge to successfully answer exam questions (Afamasaga-Fuata'I, 2001, 2002a, 2002b, 2002c, 2003, 2005a, 2005b).

In foundation and undergraduate mathematics classes in Samoa, students find it difficult to explain and justify their answers mathematically in terms of the conceptual structure of relevant topics. Instead their verifications are often in terms of sequences of steps of procedures. Whilst this may work for familiar problems, this procedural view constrains them when solving qualitatively and structurally different problems (i.e., novel problems). According to Richards (1991), this manifestation is typically a communication problem resulting from students' inability to understand the meaning of a language (i.e., concepts, principles, theorems and theories) used in mathematical discussions and dialogues of more mathematically literate others. Subsequently, less mathematically literate students are unable to make sense of such conversations, offer conjectures or evaluate mathematical assumptions. When doubtful, students tend to use any procedure to get an answer without really checking whether an algorithm is suitable to the problem (Schoenfeld, 1996). …

The rest of this article is only available to active members of Questia

Sign up now for a free, 1-day trial and receive full access to:

  • Questia's entire collection
  • Automatic bibliography creation
  • More helpful research tools like notes, citations, and highlights
  • Ad-free environment

Already a member? Log in now.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this article

This article has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this article

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this article

Cited article

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited article

Developing a More Conceptual Understanding of Matrices & Systems of Linear Equations through Concept Mapping and Vee Diagrams
Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Full screen

matching results for page

Cited passage

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited passage

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

Already a member? Log in now.