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Guess My Rule and Discovery of the Properties of Linear Equations

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Text

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Title

Guess My Rule and Discovery of the Properties of Linear Equations

Identifier
(type = doi)

doi:10.7282/t3-292j-7p88

Abstract

This VMCAnalytic was created as the fourth “video story” in a set of four. The four VMCAnalytics were used as the central tool of professional development in the Teachers Algebra Workshop (TAW), August, 2016. The TAW was created for a study that examined the use of video stories in the professional development of algebra teachers of minority students with low socio-economic status (Leslie, 2019).

In this study, teacher participants’ responses to this video story were audio-recorded. The teacher participants discussed the mathematical reasoning, student engagement and pedagogy that they noticed in the video story. The teacher participants were guided by questions designed to focus the discussion on these three areas. The discussions were audio-recorded and analyzed in the author’s dissertation (Leslie, 2019).

The source video for this analytic was produced by a longitudinal study of children’s mathematical thinking as they worked on mathematical tasks. The clips in this VMCAnalytic/video story show Teacher/Researcher (T/R) Robert Davis working with 6th grade students at the Harding Elementary school in Kenilworth, New Jersey. Many of the students that participated in the clips used here had been art of the longitudinal study since 1st grade. They had been working with teacher/researchers for 4-5 years prior to the work seen on these clips.

In this video story, T/R Davis works with a full classroom of students and begins by engaging them in a discussion of scientists and the secrets they discover. In doing so, T/R Davis connects mathematical problem-solving in algebra to scientific explorations on any question. In both cases the explorers, be they students in math class or scientists in the laboratory, are searching to uncover secrets.

T/R Davis begins the lesson by telling his students that they will be continuing their discussion of secrets and he explains, “because I think that’s an interesting part of what we do.” Note the collaborative use of “we”; Davis lets the students know that he is collaborating with them.

The context in this video story is a Guess My Rule lesson, where, in several examples, the students are using the points on a line to find the equation of the line. They are also looking for patterns that will reveal “secrets” about all their equations. T/R Davis’ introduction to the lesson suggests that, like scientists in the laboratory working on discovering a cure for cancer, the students will seek to uncover some secrets about their equations.

Note that T/R Davis uses Box and Triangle notation in his equations instead of the traditional “x” and “y” variables. Doing an example with T/R Davis, the students demonstrate that that they know the difference between legal and illegal equations and between true equations and false equations. A legal equation is when all the Box values in the equation are equal and all the Triangle values in the equation are equal to each other. An illegal equation is when all the Box and/or Triangle values do not have the same value. A true equation is when the left and right sides of the equation have the same value and a false equation is when they do not. They utilize truth sets [points that produce legal and true equations] in the Guess My Rule problems.

There is no evidence that the students have been provided with the vocabulary or semantics for the slope and the y-intercept in a linear equation. As they work on several Guess My Rule problems, they start to notice some patterns for these special values.

They notice that their rules all follow this pattern:

[Special Number 1] Box + [Special Number 2] = Triangle. Note how this corresponds to mx + b = y.

They start to call Special Number 2 [the y-intercept], “the plus number.” In the assigned Guess My Rule problems, the first point always has Box = 0. Note that the corresponding Triangle value is, by definition, the y-intercept. The students notice that it is always the “plus number.” They say the “plus number” comes from the “first point” in the truth set they are given for a particular equation.

They also notice that the difference between adjacent Triangle values [the y-values] is always the same and that it is Special Number 1. They have trouble articulating this and, in event #7, they physically point to it when showing T/R Davis their discovery.

This video story includes many aspects of T/R Davis’ pedagogy that may be considered to relevant to the success of the lesson in the story. Consider these questions and consult the Reference list for some of Davis’ work.

• Is Box and Triangle notation better [than x and y variables] to begin teaching students about algebraic equations?

• Why might it be helpful to introduce the concepts of legal/illegal equations and true/false equations this early in algebra learning?

• Why might it be helpful for students to discover the slope and the y-intercept of a line in a Guess My Rule context?

• What do you notice about T/R Davis’ pedagogy and collaborative approach to working with students?

Teachers may also want to look at a published analytic on Dr. Robert Davis’ work with linear and quadratic equations in terms of box and triangle: Using Questioning to Promote Conceptual Understanding: Robert B. Davis Introduces Algebra Ideas to Sixth Graders at http://dx.doi.org/doi:10.7282/T34Q7WS9

REFERENCES

Agnew, G., Mills, C., & Maher, C. M. (2010). VMCAnalytic: Developing a collaborative video analysis tool for education faculty and practicing educators. Proceedings of the 43rd Hawaii International Conference on System Sciences, Honolulu, HI.

Boaler, J., (2016). Designing mathematics classes to promote equity and engagement. The Journal of Mathematical Behavior, 41, 172-178. https://doi.org/10.1016/j.jmathb.2015.01.002

Borko, H., Koellner, K., Jacobs, J., & Seago, N. (2011). Using video representations of teaching in practice-based professional development programs. ZDM Mathematics Education, 43, 175–187.

Davis, R. B. (1992). Understanding ‘Understanding.’ The Journal of Mathematical Behavior,11, 225- 241.

Grey, S. (2015). Using Questioning to Promote Conceptual Understanding: Robert B. Davis Introduces Algebra Ideas to Sixth Graders. Retrieved from http://dx.doi.org/doi:10.7282/T34Q7WS9

Leslie, Joyce (2019). Investigating a model using video stories for professional development for algebra teachers of low SES minority students. Dissertation, Rutgers. New Brunswick, NJ.

Maher, C. A. (1998). Constructivism and constructivist teaching, Can they co-exist? In Ole Bjorkqvist (Ed.), Mathematics teaching from a constructivist point of view (pp. 29-42). Proceedings of Topic Group 6 at the International Congress on Mathematical Education [Report on Proceedings] (8th, Seville, Spain, July 14-21, 1996). Faculty of Education Report No. 3.

Steele, C. M. (1997). A threat in the air: How stereotypes shape intellectual identity and performance. American Psychologist, 52(6), 613-629.

In this study, teacher participants’ responses to this video story were audio-recorded. The teacher participants discussed the mathematical reasoning, student engagement and pedagogy that they noticed in the video story. The teacher participants were guided by questions designed to focus the discussion on these three areas. The discussions were audio-recorded and analyzed in the author’s dissertation (Leslie, 2019).

The source video for this analytic was produced by a longitudinal study of children’s mathematical thinking as they worked on mathematical tasks. The clips in this VMCAnalytic/video story show Teacher/Researcher (T/R) Robert Davis working with 6th grade students at the Harding Elementary school in Kenilworth, New Jersey. Many of the students that participated in the clips used here had been art of the longitudinal study since 1st grade. They had been working with teacher/researchers for 4-5 years prior to the work seen on these clips.

In this video story, T/R Davis works with a full classroom of students and begins by engaging them in a discussion of scientists and the secrets they discover. In doing so, T/R Davis connects mathematical problem-solving in algebra to scientific explorations on any question. In both cases the explorers, be they students in math class or scientists in the laboratory, are searching to uncover secrets.

T/R Davis begins the lesson by telling his students that they will be continuing their discussion of secrets and he explains, “because I think that’s an interesting part of what we do.” Note the collaborative use of “we”; Davis lets the students know that he is collaborating with them.

The context in this video story is a Guess My Rule lesson, where, in several examples, the students are using the points on a line to find the equation of the line. They are also looking for patterns that will reveal “secrets” about all their equations. T/R Davis’ introduction to the lesson suggests that, like scientists in the laboratory working on discovering a cure for cancer, the students will seek to uncover some secrets about their equations.

Note that T/R Davis uses Box and Triangle notation in his equations instead of the traditional “x” and “y” variables. Doing an example with T/R Davis, the students demonstrate that that they know the difference between legal and illegal equations and between true equations and false equations. A legal equation is when all the Box values in the equation are equal and all the Triangle values in the equation are equal to each other. An illegal equation is when all the Box and/or Triangle values do not have the same value. A true equation is when the left and right sides of the equation have the same value and a false equation is when they do not. They utilize truth sets [points that produce legal and true equations] in the Guess My Rule problems.

There is no evidence that the students have been provided with the vocabulary or semantics for the slope and the y-intercept in a linear equation. As they work on several Guess My Rule problems, they start to notice some patterns for these special values.

They notice that their rules all follow this pattern:

[Special Number 1] Box + [Special Number 2] = Triangle. Note how this corresponds to mx + b = y.

They start to call Special Number 2 [the y-intercept], “the plus number.” In the assigned Guess My Rule problems, the first point always has Box = 0. Note that the corresponding Triangle value is, by definition, the y-intercept. The students notice that it is always the “plus number.” They say the “plus number” comes from the “first point” in the truth set they are given for a particular equation.

They also notice that the difference between adjacent Triangle values [the y-values] is always the same and that it is Special Number 1. They have trouble articulating this and, in event #7, they physically point to it when showing T/R Davis their discovery.

This video story includes many aspects of T/R Davis’ pedagogy that may be considered to relevant to the success of the lesson in the story. Consider these questions and consult the Reference list for some of Davis’ work.

• Is Box and Triangle notation better [than x and y variables] to begin teaching students about algebraic equations?

• Why might it be helpful to introduce the concepts of legal/illegal equations and true/false equations this early in algebra learning?

• Why might it be helpful for students to discover the slope and the y-intercept of a line in a Guess My Rule context?

• What do you notice about T/R Davis’ pedagogy and collaborative approach to working with students?

Teachers may also want to look at a published analytic on Dr. Robert Davis’ work with linear and quadratic equations in terms of box and triangle: Using Questioning to Promote Conceptual Understanding: Robert B. Davis Introduces Algebra Ideas to Sixth Graders at http://dx.doi.org/doi:10.7282/T34Q7WS9

REFERENCES

Agnew, G., Mills, C., & Maher, C. M. (2010). VMCAnalytic: Developing a collaborative video analysis tool for education faculty and practicing educators. Proceedings of the 43rd Hawaii International Conference on System Sciences, Honolulu, HI.

Boaler, J., (2016). Designing mathematics classes to promote equity and engagement. The Journal of Mathematical Behavior, 41, 172-178. https://doi.org/10.1016/j.jmathb.2015.01.002

Borko, H., Koellner, K., Jacobs, J., & Seago, N. (2011). Using video representations of teaching in practice-based professional development programs. ZDM Mathematics Education, 43, 175–187.

Davis, R. B. (1992). Understanding ‘Understanding.’ The Journal of Mathematical Behavior,11, 225- 241.

Grey, S. (2015). Using Questioning to Promote Conceptual Understanding: Robert B. Davis Introduces Algebra Ideas to Sixth Graders. Retrieved from http://dx.doi.org/doi:10.7282/T34Q7WS9

Leslie, Joyce (2019). Investigating a model using video stories for professional development for algebra teachers of low SES minority students. Dissertation, Rutgers. New Brunswick, NJ.

Maher, C. A. (1998). Constructivism and constructivist teaching, Can they co-exist? In Ole Bjorkqvist (Ed.), Mathematics teaching from a constructivist point of view (pp. 29-42). Proceedings of Topic Group 6 at the International Congress on Mathematical Education [Report on Proceedings] (8th, Seville, Spain, July 14-21, 1996). Faculty of Education Report No. 3.

Steele, C. M. (1997). A threat in the air: How stereotypes shape intellectual identity and performance. American Psychologist, 52(6), 613-629.

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Joyce Leslie

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