Posted in Teaching mathematics

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
Posted in Algebra

Fun with ‘definitions’ in algebra

WARNING:  use the following definitions with great caution.

  • number phrase is a mathematical phrase which does not express a complete thought.
  • An arithmetic expression is any grammatically sensible expression made up of numbers and (possibly) arithmetic operations (like addition, division, taking the absolute value, etc). Notice that it only has to be grammatically sensible; an undefined expression like 5/0 is still an arithmetic expression, but something like ‘5)+/7?’ is just nonsense. You can always work out an arithmetic expression to a specific value, unless it’s undefined (in which case you can work that out).
  • An algebraic expression is any grammatically sensible expression made up of any or all of the following:

– specific numbers (called constants);
– letters (or other symbols) standing for numbers (called variables); and
– arithmetic operations.

  • By an algebraic expression in certain variables, we mean an expression that contains only those variables, and by a constant, we mean an algebraic expression that contains no variables at all.
  • polynomial is an algebraic sum, in which no variables appear in denominators or under radical signs, and all variables that do appear are raised only to positive-integer powers.
  • monomial is an algebraic expression made up only from any or all of these:

– Constants;
– Variables;
– Multiplication;
– Taking opposites (optional);
– Division by nonzero constants (optional);
– Raising to constant whole exponents (optional).

  • An algebraic expression is made up of the signs and symbols of algebra. These symbols include the Arabic numerals, literal numbers, the signs of operations, and so forth.
I got these definitions from where else, www. Of course we just want to simplify things for students but … . Anyway, just make sure that you don’t start your algebra lessons with definition of terms, be they legitimate or not legitimate.
Posted in Algebra, Geometry, Misconceptions, Teaching mathematics

Mistakes and Misconceptions in Mathematics

Misconceptions are very different from the mistakes students make. Mistakes are not consciously made. Misconceptions are. Mistakes are usually one-off, while misconceptions, the gods forbid, could be for keeps. Mistakes are made by a few, misconceptions are made by many and, repeatedly.

Students can figure out their mistakes by themselves because mistakes are usually due to carelessness. They cannot do the same for misconceptions. Misconceptions are committed because students think they are correct.

How can misconceptions be addressed? By undressing them, carefully exposing them until the students see it. It cannot be corrected by simply marking them x because misconceptions are usually made with full knowledge.

The following are common misconceptions in arithmetic, algebra and geometry:

1. Did we not learn that multiplication is repeated addition? So, -3 x -4 = -3 + -3 + – 3 +-3 = -12?

2. Didn’t we learn that to multiply fractions we simply multiply numerators and we do the same with the denominators? Didn’t the teacher say multiplication is simply repeated addition so {\frac{3}{5}}+{\frac{2}{3}}={\frac{5}{8}}?

3. Did not the teacher say x stands for a number? So in 3x – 5, if x is 5, the value of the expression is 35 – 5 = 30?

4. Did not the teacher/book say to always keep the numbers and decimal points aligned? So if Lucy is 0.9 meters and her friend Martha is 0.2 taller, Martha must be 0.11 meters in height?

5. Did we not learn that the more people there are to share a cake the smaller their portion? So {\frac{10}{16}}<{\frac{4}{5}}<{\frac{3}{4}}<{\frac{1}{2}}?

6. Did we not learn that by the distributive law 2(a+b) = 2a + 2b? So, (a+b)^2=a^2+b^2?

7. Did not the teacher show us that (x-3)(x+1) = 0 implies that (x-3) = 0 and (x+1) = 0 so x = 3 and x = -1? Hence in (x-3)(x+4) = 8, (x-3) = 8 and x +4 = 8 so x = 11 and x = 4?

8. Did we not learn that the greater the opening of an angle, the bigger it is? So, angle A is less than angle B in the figure below.

9. Did we not learn that you if you cut from something, you make it smaller? Hence in the diagram below, the perimeter of the polygon in Figure 2 is less than the perimeter of original polygon?

10. Isn’t it that the base is the one lying on the ground?

There’s nothing a teacher should worry about mistakes. There’s everything to worry about misconceptions. Good teaching practice exposes misconceptions, not hide them.

You might want to check out this book: