Mathematics for Teaching Teaching mathematics What is proportional reasoning? Does cross multiplication help learn it?

What is proportional reasoning? Does cross multiplication help learn it?


proportionProportional reasoning is a capstone of children’s elementary school arithmetic and a cornerstone of all that is to follow (Lesh & Post, 1988). But for some reason, the teaching of elementary school mathematics topics either become an end in itself or has become more a preparation for learning algebra. Proportional reasoning is not being given its due attention. Solving proportion problems has become an exercise of applying routine procedure than an opportunity to engage students in proportional reasoning.

What is proportional reasoning? Why is it important?

Proportional reasoning is a benchmark in students’ mathematical development (De Bock, Van Dooren, Janssens, & Verschaffel, 2002). It is considered a milestone in students’ cognitive development. It involves:

  1. reasoning about the holistic relationship between two rational expressions such as rates, ratios, quotients, and fractions;
  2. synthesis of the various complements of these expressions;
  3. an ability to infer the equality or inequality of pairs or series of such expressions;
  4. the ability to generate successfully missing components regardless of the numerical aspects of the problem situation; and
  5. involves both qualitative and quantitative methods of thought and is very much concerned with prediction and inference.

Proportional reasoning involves a sense of co-variation and of multiple comparisons. In this sense it is a ‘subset’ of algebraic thinking which also give emphasis on structure and thinking in terms of relationship.

What is cross multiplication? Does it promote proportional reasoning?

Cross multiplication is a procedure for solving proportion of the type A/B = x/D. It solves this equation by this process: A*D = x*B. This algorithm is not intuitive. It is not something that one will ‘naturally generate”. Studies have consistently shown that only very few students understand it although many can carry out the procedure. I know many teachers simply tell the students how to do cross multiplication and use specific values to verify that it works without explaining why the algorithm is such.

Many mathematics textbooks and lessons are organized in such a way that students are taught to do cross multiplication before asking them to do problems involving proportion. This practice deprives the students from understanding the idea of proportion and developing their proportional thinking skills. Research studies recommend to defer the introduction of cross multiplication until students have fully understood proportion and have had experiences in solving proportion problems using their knowledge of operation and their understanding of fraction, ratios, and proportion.

References and further readings:

  1. Number Concepts and Operations in the Middle Grades
  2. Proportional reasoning tasks and difficulties
  3. Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics: Grades 6-8

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