The learning and teaching of school algebra.

Serious attempts are being made to improve the students' preparation for algebra. However, without a clear-cut demarcation between arithmetic and algebra, most of these undertakings merely provide either an earlier introduction of the topic or simply spread it out over a longer period of instruction. The present study investigates the upper limits of the students' informal processes in the solution of first degree equations in one unknown prior to any instruction. The results indicate the existence of acognitive gap between arithmetic and algebra, a cognitive gap that can be characterized asthe students' inability to operate spontaneously with or on the unknown. Furthermore, the study reveals other difficulties of a pre-algebraic nature such as a tendency to detach a numeral from the preceding minus sign in the grouping of numerical terms and problems in the acceptance of the equal symbol to denote a decomposition into a difference as in 23=37−n which leads some students to read such equations from right to left.

A cognitive gap between arithmetic and algebra

This article describes test data showing that a large proportion of science-oriented college students were unable to solve a very simple kind of algebra word problem. Several thinking-aloud protocols are presented in order to (a) formulate hypotheses for why the errors occurred, (b) examine the phenomenon of shifts between inconsistent approaches within a single solution, and (c) illustrate a method of diagramming thought processes. The data indicate that relatively advanced students can experience serious difficulties in symbolizing certain meaningful relationships with algebraic equations. Paige and Simon (1966) have shown that syntactic methods-that is, methods not dependent on comprehending the meaning of the described problem situation-are adequate for solving some algebra word problems in one variable. They point out, however, that these methods can produce incorrect or meaningless results in other problems. They also present evidence for the idea that some students use an internal "physical representation" that encodes qualitative features of a situation in solving word problems. A number of misconceptions concerning the meaning of algebraic equations have also been discussed by Wagner (Note 1), Collis (1978), Herscovics and Kieran (1980), Galvin and Bell (Note 2), Kiichemann (1978), and Matz (Note 3). Several of these researchers have noted a tendency on the part of some students to treat numerical variables as if they stood for objects rather than numbers. This study concentrates on one particular type of misconception in order to obtain a deeper understanding of its cognitive sources. Research attempting to describe thought processes in this area must be considered exploratory, since the processes involved are rather complex.

Algebra Word Problem Solutions: Thought Processes Underlying a Common Misconception

https://onlinelibrary.wiley.com/doi/pdf/10.1080/03640210701863933

Trade-Offs Between Grounded and Abstract Representations: Evidence From Algebra Problem Solving

https://files.eric.ed.gov/fulltext/ED543090.pdf

Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples

Children's Understanding of Numerical Variables.

A common approach to generalization activities in school algebra is to set them in the context of figurative patterns (e.g., a row of touching squares formed of matchsticks, with a relationship between the numbers of squares and matchsticks; or a row of touching tables surrounded by chairs, with a relationship between the numbers of tables and chairs). This approach allows students to see and explain numerical relationships in terms of the figurative pattern's structure. However, patterns are all too often presented, in textbooks and by teachers and researchers, in the form of a systematic sequence of elements, with the result that students generate a systematic set of ordered pairs for which they try to induce an empirical relationship, divorced from the structure of the pattern that produced them. In this paper, I argue that one reason that students appear to have difficulty in seeing structure is that they are not sufficiently initiated into this way of thinking. Further, I argue that the search for st...

Using patterns generically to see structure

Over the past 30 years, there has been a great deal of work directed at, firstly, understanding children's difficulties in mathematics and, secondly, examining ways of tackling these difficulties. ...

Children's understandings of algebra 30 years on

Over the past 30 years, there has been a great deal of work directed at, first, understanding students' difficulties in algebra and, second, examining ways of tackling these difficulties. There is a vast research literature detailing successful ways of teaching algebra in experimental and laboratory settings. Yet, there is no clear evidence that this work has had a significant effect in terms of improving either attainment or engagement in algebra in ordinary, or non-experimental, classroom settings. Indeed, in England, current attainment in algebra appears to be no better than that of 30 years ago. In this paper, we analyse the algebra topics from two textbooks currently in widespread use in England, focusing on Grade 7 (ages 12–13). We examine the extent to which these textbooks draw on the research literature to support the teaching of algebra. Finally, we discuss the implications of this study.

Textbooks for the teaching of algebra in lower secondary school: are they informed by research?

Increasing Student Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS) is a four-year research project funded as part of an initiative aimed at identifying ways to increase ...

Dietmar Kuchemann

Papers

Children's Understanding of Numerical Variables.

Using patterns generically to see structure

Children's understandings of algebra 30 years on

Textbooks for the teaching of algebra in lower secondary school: are they informed by research?

Lower secondary school students' attitudes to mathematics: evidence from a large-scale survey in England