Algebra seems to be a subject that some find trivial but most find it peculiarly difficult. My aim is to develop theory to explain why this is so, and then to consider how to help individuals to make sense of algebraic thinking.
It is possible to learn procedures to pass exams, but this may give only short-term success and be fraught with difficulty. Working as a schoolteacher in 1982, I introduced my class of eleven-year-olds to programming in BASIC and found that after experiencing that when one types A=3 and then PRINT A+2 causes the computer to print 5, all the students were able to predict what would happen in other cases, such as B=2 followed by PRINT B-1 or PRINT 2*B. My collaborative research in algebra began with Michael Thomas using the language BASIC to program expressions such as 3*A+6 and 3*(A+2), to provide the experience that the two expressions always give the same answer. We had success in teaching algebra by this method, not only in being able to carry out procedures, but in giving meaning to the notation.
Analysis of whether the students are carrying out a mental procedure, or manipulating concepts helps to clarify why some algebra problems cause more difficulty than others. The equation 3x+1=7, for example, can be seen as a procedure: start with a number, multiply it by 3 and add one to the result to get 7. Undoing by taking off one and dividing by 3 gives the input number as x=2. However, the equation 3x+1=4x-1 cannot be undone in the same way. It requires the two expressions 3x+1 and 4x-1 to be thought of as having the same result for some value or values of x. This kind of problem requires manipulation of expressions as mental concepts, rather than mental procedures. The theory of procepts (symbol as process and/or concept) proves helpful to analyse mental activities not only in algebra, but in arithmetic (whole number, negatives, fractions, decimals), algebra, vectors, calculus (Tall et al, 2001).
Subsequent theoretical developments include the distinction between evaluation algebra where algebraic expressions are evaluated numerically, as in programming in BASIC or using spreadsheets, manipulation algebra (traditional school algebra) and axiomatic algebra (groups, rings, fields, vector spaces etc) (Thomas & Tall, 2001) and the theory of met-befores, in which experiences that were met before act as underpinning for interpretation of new ideas sometimes in a positive helpful sense, but also in ways that act as obstacles to learning.
1983b Introducing algebra on the computer: today and tomorrow, Mathematics in School, 12, 4 37-40.
1986d (with Michael Thomas) Playing algebra with the computer, Exploring Mathematics with Microcomputers (ed. Nigel Bufton) C.E.T., 59-74.
1986i (with Michael Thomas) The value of the computer in learning algebra concepts, Proceedings of P.M.E. 10, London 313-318.
1987d Algebra in a Computer Environment, Proceedings of the Eleventh International Conference of P.M.E., Montreal, I, 262-271.
1988c (with Michael Thomas) Longer Term Effects of the Use of the Computer in the Teaching of Algebra, Proceedings of P.M.E.12. Hungary, 601-608.
1989b New Cognitive Obstacles in a Technological Paradigm, Research Issues in the Learning and Teaching of Algebra, N.C.T.M., 87-92.
1989f (with Michael Thomas) Dynamic algebra [computer program and lesson plans], in Secondary Mathematics with Micros - A Resource Pack, Mathematical Association.
1989g (with Frank Knowles) Using the Algebraic Calculator in the Sixth Form [computer program and text] in Secondary Mathematics with Micros - A Resource Pack, Mathematical Association.
1991c (with Michael Thomas) Encouraging Versatile Thinking in Algebra using the Computer, Educational Studies in Mathematics, 22 2, 125-147.
1992d Success and failure in arithmetic and algebra, Mathematics Teaching 1991, Edinburgh University, September 1991, 2-7.
1993c School Algebra and the Computer, Micromath, 9 1, 38-41.
1993f The Transition from Arithmetic to Algebra: Number Patterns or Proceptual Programming?, New Directions in Algebra Education, Queensland University of Technology, Brisbane, 213-231.
1993g Success & Failure in Arithmetic and Algebra, New Directions in Algebra Education, Queensland University of Technology , Brisbane, 232-245.
1994d (with Lillie Crowley & Michael Thomas), Algebra, Symbols, and Translation of Meaning, Proceedings of PME18, Lisbon, II, 240-247.
1999d (with Lillie Crowley), The Roles of Cognitive Units, Connections and Procedures in achieving Goals in College Algebra. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of PME, Haifa, Israel, 2, 225-232.
2001b (with Eddie Gray, Maselan Bin Ali, Lillie Crowley, Phil DeMarois, Mercedes McGowen, Demetra Pitta, Marcia Pinto, Michael Thomas, and Yudariah Yusof). Symbols and the Bifurcation between Procedural and Conceptual Thinking, Canadian Journal of Science, Mathematics and Technology Education 1, 80-104.
2001e Reflections on Early Algebra, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, Utrecht, The Netherlands.
2001i (with Eddie Gray) Relationships between embodied objects and symbolic procepts: an explanatory theory of success and failure in mathematics, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education 3, 65-72. Utrecht, The Netherlands.
2001n Thomas, M. O. J. & Tall, D. O. The long-term cognitive development of symbolic algebra, International Congress of Mathematical Instruction (ICMI) Working Group Proceedings - The Future of the Teaching and Learning of Algebra, Melbourne, 2, 590-597.
2004c David Tall. Reflections on research and teaching of equations and inequalities. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway.