Meta-Representation in an Algebra I Classroom

Abstract

We describe how 1 Algebra I teacher and her 8th-grade students used meta-representational knowledge when generating and evaluating equations to solve word problems. Analyzing data from a sequence of 4 lessons, we found that the teacher and her students used criteria for evaluating equations, in addition to other types of knowledge (e.g., different interpretations of the equal sign) previously reported in the literature. Moreover, the teacher and her students had trouble understanding one another's proposed algebraic models of problem situations due to differences in the criteria that each applied, and this impeded learning. These findings (a) extend an accumulating body of evidence for the role meta-representation plays in mathematics and science learning and (b) add a new dimension to researchers' growing understanding of what teachers must know in order to teach algebra and other complex mathematics and science topics effectively.

Notes

¹ Expressions are algebraic forms that do not contain an equal sign.

²Izsák (2003, 2004) used a similar frame to describe knowledge that students used when constructing algebraic expressions and equations to solve problems about a physical device called a winch.

³Names of the school and all persons are pseudonyms.

⁴Throughout our analysis, we use the term variable because that term was used by Ms. Jennings and her students during the observed lessons. How students interpreted letters was not always clear. Sometimes they explicitly interpreted letters as specific unknown numbers.

⁵Ms. Jennings had skipped over one other coins problem earlier in the unit that described a similar situation and that included a guess-and-check table with column headings for the numbers of nickels, dimes, and quarters and for the values of all the nickels, all the dimes, and all the quarters.

⁶Maria's assignment for x may have been influenced by her experience working on the age problem during the second lesson and her first interview.

⁷Ben was absent from his second interview, and the discussion of the book club lesson touched only briefly on Greg's understanding of his Equation 14.