Different Representations in Instruction of Vertical Angles

By Tsamir, Pessia | Focus on Learning Problems in Mathematics, Winter 2003 | Go to article overview

Different Representations in Instruction of Vertical Angles


Tsamir, Pessia, Focus on Learning Problems in Mathematics


It is widely accepted that teachers should be aware of and familiar with students' mathematical ideas and conceptions, and it is believed that such awareness and knowledge should play a significant role in planning and carrying out instruction (e.g., Kilpatrick, Swafford, & Findell, 2001; NCTM, 1989; 1991; 2000). One source of information regarding students' ways of thinking and their common errors in various mathematical topics is the vast amount of research findings reported in the literature. However, the shift from awareness of and familiarity with students' ways of thinking to designing research-based instruction, i.e., instruction that takes into consideration the findings reported in the literature, is not a trivial one. An important step in this direction might be the identification of general factors that play a role in students' mathematical reasoning. One such factor is the specific way in which the mathematical concepts included in the problems are represented. It has been reported that students tend to provide different and even conflicting solutions to different representations of the same mathematical problem (e.g., Janvier, Girardon, & Morand, 1993; Tirosh & Tsamir, 1996). Clearly, incompatible solutions to the same mathematical problem are not acceptable.

This article demonstrates a way of using knowledge about students' incompatible reactions to different representations of angles to raise students' own awareness of their intuitive ways of thinking, and to guide them towards proving the equality of vertical angles. More specifically, two main questions are addressed.

(1) How can research-based instruction promote students' awareness of their own intuitive thinking? and (2) How can research-based instruction promote students' appreciation of formal proof?

First, students' reactions to different representations of vertical angles are presented by means of a brief description of two related studies. Then, the Vertical Angles Conflict Activity and students' reactions to it are described. Finally, some concluding comments are made.

Students 'Reactions to Different Representations of Vertical Angles

Tsamir (1995) examined the responses of 204 students (Grades 4, 6 and 9) to tasks on comparing vertical angles. The vertical angles were presented in different representations, including two types of "equal arms" representations and one type of "different arms" representation (see Figure 1). In the "equal arms" representations the lengths of the arms of one vertical angle were equal to the lengths of the arms of the other. In the "different arms" representation the arms of one vertical angle were longer than the arms of the other.

It was found that both types of "equal arms" representations of vertical angles triggered "equal" responses (95% and 90% on average, to the "four equal arms" and the "two pairs of equal arms" representations, respectively). The "different arms" representation, however, triggered higher percentages of "unequal" responses (about 40%, 35% and 25% of the 4, 6, and 9th graders respectively).

On average, 45% of the participants correctly justified their correct responses to the two "equal anus" representations. Since only 9th-grade participants had studied the theorem regarding vertical angles, it was expected that only they would use the theorem in their justifications. The younger students' correct justifications were based on measurements and on the claim that the two angles had "the same opening" between their arms. Incorrect justifications were mostly based either on length or on area considerations. When relating to the 'equal sides' representations, typical claims were "the angles are equal because their arms are equal," or "the angles have an equal area enclosed by the angle's arms and the segment connecting the endpoints." When relating to the 'different sides' representation those students explained that "the angle with the longer arms is larger", "the angle with the longer segment connecting the end points of the drawn anus is larger", or "the angle with the larger area enclosed by the angle's arms and the segment connecting the end points is larger". …

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

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.
Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

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

Different Representations in Instruction of Vertical Angles
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?

Help
Full screen

matching results for page

    Questia reader help

    How to highlight and cite specific passages

    1. Click or tap the first word you want to select.
    2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
    3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

    OK, got it!

    Cited passage

    Style
    Citations are available only to our active members.
    Buy instant access 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.

    Buy instant access to save your work.

    Already a member? Log in now.

    Oops!

    An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.