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Teaching without understanding

We all remember from our school days some instances when we were taught without understanding. We recall the act of having learned, but not what was taught. Whatever is learned in this way will be forgotten easily and will not contribute to the understanding of our world, either natural or cultural.

Every day, research on the teaching for understanding increases1. When learning is relevant, we can apply our knowledge to new situations and make connections among diverse fields of knowledge; we strengthen our competence to use our knowledge2 and we are able to connect our knowledge to our day-to-day experiences.

Children show interest in mathematical ideas form an early age3. Through their day-to-day experience, children develop informal ideas about numbers, quantities, patterns, shapes, and size, among other concepts. The learning of mathematical ideas begins much earlier than children’s formal school experience4. Even after entering school, students of all ages develop mathematical ideas on a daily basis5.

A problem with contemporary teaching is that it does not integrate these experiences into the formal classroom learning environment. When one of the researchers was working on verbal problems with fifth graders (10–11 years old), she had the following exchange with them after posing this problem: A man bought 20 oranges. If the oranges are 5 for a dollar, how much did he spend?

Student: I don’t know how to do the problem. I know the answer is $4.00, but I don’t know how to do it.

Researcher: But, how do you know that the answer is $4.00?

Student: Well, if they give 5 oranges for a dollar, for $2.00 they’ll give 10, so that 20 oranges will be $4.00.

We see that students separate two types of knowledge they possess to solve the problem. On the one hand, they have the informal learning of day-to-day life in and out of school. On the other, they carry out the arithmetic operations, which they do not relate to their informal knowledge. They think that solving a problem equals translating it into a single arithmetic operation (addition, subtraction, multiplication, or division), as is demanded from them at school. Being unable to do this translation, they feel that they cannot solve the problem, though they have done so with their informal knowledge.

It is imperative to bridge the informal math that students do daily with the formal math taught in schools. In fact, the math that students learn through their daily interactions is learned with meaning and can be used in diverse contexts, as shown in the example above. The math they learn in school is for the most part a set of meaningless rules that they barely use outside of school. An area of research that college professors and school teachers must share is that of identifying activities that, springing from the students’ experiences, support the construction of mathematical concepts.

Mathematics is one of the most misunderstood disciplines. The understanding among a great portion of the population is that mathematics is a series of rules used for numerical calculations. Responding to this interpretation, mathematics is taught as a set of formulas to do various calculations. Mathematics, however, as most fields of knowledge, developed as a result of the human need to understand and interpret the world6

The teaching of mathematics must therefore spring from contexts that are meaningful for the students and build on their informal knowledge. For example, if we want to introduce the concept of measurement in second grade (ages seven to eight years), we can start with the question, “How much do we grow in a year?” We start by discussing with students their notions of measurement, and from these build up to the more sophisticated notions that mathematics has developed.


1. Bruner, 1990; Cohen et al., 1993; Quintero et al., 2006; Wiske, 1998

2. Brans ford et al., 2000

3. McCrink and Wynn, 2004; Whalen et al., 1999

4. Gelman and Gallistel, 1978; Resnick, 1987

5. Bransford et al., 2000

6. Davis and Hersh, 1972; Kline, 1973

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