|Intelligence and Learning|
If learning to read and write is difficult, then learning to become proficient in mathematics is much more difficult or impossible for some children. One problem is that the term, mathematics, refers to a wide range of intellectual skills that are more divergent than coherent. In the neurological literature, there are reports of brain damaged patients who cannot count or calculate, but the more abstract reasoning that advanced mathematicians celebrate remains somewhat of a mystery.
In schools, arithmetic is taught to everyone but mathematics is hard to learn and attracts fewer and fewer students as the grades advance. The innate basis of arithmetic lies in the spontaneous tendency to measure and to recognize clusters of objects. For example, Barth et al showed that that preschool children can compare and add sets of elements without counting and without prior instruction. They could integrate quantity information presented as sound or visual dot arrays. They concluded that abstract knowledge of number and addition precedes and guides language-based instruction in mathematics.
Children should learn arithmetic and become familiar with methods of measurement and calculation. In the best case, children should become familiar with numbers and even fascinated with some of intriguing features of numbers. When I was a student I took many math courses for many years and developed a romantic version of number theory that involved a tense interaction with odd versus even numbers. I was 7, a potential hero, always thwarted by 8 and his even-number allies. The feminine principal was 6, a real beauty, that I hoped to marry. My competitor was 8. Interesting relationships occurred with 2 ( a tricky even number), for example, multiplying 3 (my friend ) becoming six (my true love). Numbers such as 327 were full of hope that 6 and 7 would be intimately related. Other numbers such as 246 were very threatening since 2X4 would bring 8 adjacent to 6. The worst case numbers had 6 surrounded by 8. This meta-world of numerical meaning added drama to the thousands of calculations I performed over many years.
I have no doubt that arithmetic must be learned by rote and practiced every day until proficiency is achieved. Arithmetic should be taught as a set of basic life skills including money management, budgeting, investment planning, and the evaluation of numerical data. Early education should equip children with the ability to do simple arithmetic calculations and then move to developing proficiency in the use of calculators and computers. All the calculations I have made in the past 20 years were done by a calculator or on computer. Several programs such as spreadsheets and accounting software make collecting data and doing multiple calculations easy. My mental math now is usually limited to factoring larger numbers to approximate the expected result of a calculation.
Arithmetic learning is similar to language learning that requires memorizing facts, rules and practicing calculation over many years to acquire reliable skills. In the best case, children should become familiar with numbers and even fascinated with some of intriguing features of numbers. When I was a student I took many math courses for many years and developed a romantic version of number theory that involved a tense interaction with odd versus even numbers. I was 7, a potential hero, always thwarted by 8 and his even-number allies. The feminine principal was 6, a real beauty that I hoped to marry. My adversary was 8. Interesting relationships occurred with 2, for example, multiplying 3 (my ally) and becoming six. Numbers such as 327 were full of hope that 6 and 7 would be intimately related. Other numbers such as 246 were very threatening since 2X4 would bring 8 adjacent to 6. The worst case numbers had 6 surrounded by 8. This meta-world of numerical meaning added drama to the thousands of calculations I performed over many years.
When children go to school, they are confronted by a teacher and a curriculum that will confuse and frustrate them. Family members seldom have math skills and cannot help their child. Tutors come in different shapes and sizes and many of them have no insight into what the child needs to learn and how they should learn it. In the US, only 39% of fourth graders and 34% of eighth graders scored at or above the proficient level on a nationwide math test in 2009. Bisk, a math professor suggested: "To give students a firm foundation in math, we must start in the elementary grades by providing three things: a substantial improvement in elementary teachers’ knowledge of mathematics; a more focused curriculum that emphasize core concepts and skills; and more challenging textbooks that teach for mastery and not just exposure. Many elementary teachers have strong backgrounds in reading and writing, but will readily admit their discomfort with math. Typically they have taken little mathematics in their teacher training programs. Too often, the math they studied has little connection to the math they teach in the classroom."
I have described gender differences in a previous chapter. Geary’s conclusion is that males outperform females in solving mathematical word problems and in geometry. He explains that male and female specializations early in hominid evolution favored males with superior navigation skills, one possible source of the male advantage in geometry. The gender differences are appreciated in population studies but are not reliable when assessing individual abilities. You cannot argue that girls cannot learn math. Some girls excel in math and become math-using scientists or engineers in adult life. The male advantage is selective; about 60% of males have math disabilities. It is to argue that males and females often do best with different teaching methods and with modifications to curricula. Overall, girls have a intellectual advantage in language and social skills that persist through high school.