During the last decades of the 20th century, the National Council of Teachers of Mathematics (NCTM) decried the claim by many females that they didn’t have the “math gene” required to study mathematics at an advanced level. However, when Miriam Mirzekhani was awarded in 2014, a Fields Medal for her “outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces”, and became the first woman to win this prestigious award, it was clear that woman can do mathematics. However, as most teachers of mathematics will attest, not all people have an equal potential for learning mathematics.
After I finished my doctoral work in mathematics, I decide to teach high school for a year while I decided whether to take a university post or enter the business world. I was assigned to teach several classes of grade 9 students preparing for the community college stream. I enjoyed the connection with these young students who were full of energy and responded well to my stupid jokes and mathematical stories. However, we struggled together to get them to understand concepts like how to change a decimal number to a percent. By the time my lessons on a topic such as this were finished, I would administer each Friday, a test to determine their level of knowledge. After marking the test, I would delight in the fact that almost all the students performed at a mastery level.
Then, on Monday morning, I would administer the same test and discover that many of them had failed the same test they had mastered before the weekend. Deeper scrutiny revealed that they had learned a process (called instrumental learning) without understanding the underlying concepts (called relational learning.) Consequently, they were unable to retain the learning that they had demonstrated on Friday.
A few months after my work with these students, the chair of the mathematics department, who was teaching the grade 9 students in the stream destined for university, became ill and he asked me to teach some algebra to his students. I was used to introducing a topic with a motivational application, explaining the concept, showing how to perform a procedure, giving examples, and then having students explore and practice. However, these students seemed like geniuses. They picked up the concepts immediately, and were ready to move on to greater generalizations. I suddenly understood why the department head had taken the university stream classes and assigned me the others. There was a remarkable difference in the speed at which members of the two different groups could understand, assimilate and generalize mathematical concepts.
In describing the results of this research during the 1960’s and 1970’s, USSR mathematician Vladimir Krutetskii noted innate differences in the cognitive capacities of children, while carefully sidestepping the Soviet prohibition on linking talent to inheritance:
The difference between capable, average, and incapable pupils, as our research permits us to conclude, comes down to the following. In able pupils these associations can be formed “on the spot”; in this sense they are “born,” if one can so express it, already generalized, with a minimal number of exercises. In average pupils these associations are established and reinforced gradually, as a result of a whole series of exercises. They form isolated, concrete associations, related only to a given problem, “on the spot.” Through single-type exercises these associations are gradually transformed into generalized associations. In incapable pupils, even the isolated, concrete associations are formed with difficulty, their generalizations are still more difficult, and sometimes such generalizations do not occur at all.
Though many people would like to believe that we are all equal in our ability to learn mathematics and are distinguished only by the effort we expend, both research and experience teach us that this is not so. Even Bill Gates who achieved an almost perfect SAT score in mathematics, and Jeff Bezos who was identified as gifted in mathematics, acknowledged that there were others with greater gifts in this subject. Einstein also acknowledged his difficulties in mathematics and sought help from colleagues. Indeed, mathematics is difficult for all of us, but some have to struggle less than others.
