Mathematics and your gifted child – part 1
Why Learn Mathematics
I often hear about how most mathematics we learn in school is useless. How many times have you seen a social media post mentioning how often they have used algebra since they left school, or who would ever buy 10 watermelons etc.?
As a mathematics specialist I have often thought about that. I must give the disclaimer; I love numbers and enjoy puzzles. I especially love algebra. However, the comments and perception of people who do not love mathematics has lead me on a journey to look at why we make numeracy and mathematics a compulsory subject at school.
My thinking is why target mathematics? What about humanities? Why do we have to analyse other people’s writings? Obviously, there is more to the learning that the subject matter. Why do our educational experts choose what they do for our children to learn at school? Why not make it a free choice to learn whatever they are interested in? Why make them learn to read, write and calculate? That is not the purpose I am writing for. It has interesting ideas and I do think it may be worth exploring that more.
I decided to look at what learning is. Part of learning is about developing pathways in your brain to help you to make good decisions, inform your thinking and look beyond your own experience. The more you push yourself to do things that are difficult, the more your brain grows physically, and the more neural pathways you produce. One thing I discovered while looking more at mathematics is that your brain uses a different area to process information to do with mathematics than that in learning language and subjects we would perhaps connect to literacy.
I was surprised to see such a simple answer to the question about why we should learn mathematics. Just like when you are exercising, you don’t concentrate just on your legs and ignore your stomach muscles (well you try not to). In forming your mind, you want to develop many aspects of your intellect.
I often talk to my mathematics students and tell them that mathematics is not just about the topics but is much more about a way of thinking and problem solving. It draws on logic and creative strategies to figure out a method to solve it.
I think the main problem with mathematics in schools is that often teachers (particularly those in the primary school) are not all in love with the subject and so project their own difficulties into the teaching of the subject (even unconsciously). Also, mathematics is taught in a linear and somewhat boring way in school. Most of the primary curriculum is the ABCs of mathematics, without the creativity and discovery. The more interesting mathematics is only presented in later high school after a lot of basic skills have been drilled and algorithms learned to support the more difficult mathematics. Alas, this is often to the subject’s detriment.
In my exploring this subject I also came across an interesting paper which looked at an idea of what if students had no “formal” mathematics during primary school? What if they only got a sense of how to count, how to read time and measure, and basic money understanding? In the time usually reserved for mathematics, what if they instead spent time in improving their writing and reasoning? To the writer’s great joy, he discovered that such an experiment had actually been done, in the 1930’s. I think I will let the paper speak for itself now.
This quote is taken from a paper by Brian Butterworth entitled “Mathematics and the Brain” which was the Opening Address to The Mathematical Association, Reading: April 3rd 2002.
The experiment took place in Manchester, New Hampshire, in the 1930’s. The superintendent of schools Louis P. Benezet reasoned as follows *:
In the first place, it seems to me that we waste much time in the elementary schools, wrestling with stuff that ought to be omitted or postponed until the children are in need of studying it. If I had my way, I would omit arithmetic from the first six grades. I would allow the children to practice making change with imitation money, if you wish, but outside of making change, where does an eleven-year-old child ever have to use arithmetic?
I feel that it is all nonsense to take eight years to get children through the ordinary arithmetic assignment of the elementary schools. What possible needs has a ten-year old child for a knowledge of long division? The whole subject of arithmetic could be postponed until the seventh year of school, and it could be mastered in two years’ study by any normal child.
For some years, I had noted that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child’s reasoning faculties. There was a certain problem which I tried out, not once but a hundred times, in grades six, seven, and eight. Here is the problem: “If I can walk a hundred yards in a minute [and I can], how many miles can I walk in an hour, keeping up the same rate of speed?”
In nineteen cases out of twenty the answer given me would be six thousand, and, if I beamed approval and smiled, the class settled back, well satisfied. But if I should happen to say, “I see. That means that I could walk from here to San Francisco and back in an hour” there would invariably be a laugh and the children would look foolish.
Benezet had been concerned not only that the standard of maths had been very disappointing in his school district, but the children’s ability to express themselves in speech and writing was depressingly poor. With less time wasted on long division, more time could be spent reading, writing, and thinking. He wrote,
In the fall of 1929, I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite – my new Three R’s. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language. I picked out five rooms – three third grades, one combining the third and fourth grades, and one fifth grade.
Teachers in these experimental rooms were not to teach arithmetic, but should give the children practice in estimating heights, lengths, areas, distances, and the like. Starting in 1932, children in the experimental rooms received no arithmetic teaching until 6th grade.
How did these experimental children fare under the new curriculum?
Let me give an example of Benezet’s method.
Benezet Regular Group
“Here is a wooden pole that is stuck in the mud at the bottom of a pond. There is some water above the mud and part of the pole sticks up into the air. One-half of the pole is in the mud; 2/3 of the rest is in the water; and one foot is sticking out into the air. Now, how long is the pole?”
First child: “You multiply 1/2 by 2/3 and then you add one foot to that.”
Second child: “Add one foot and 2/3 and 1/2.”
Third child: “Add the 2/3 and 1/2 first and then add the one foot.”
Fourth: “Add all of them and see how long the pole is.”
Next child: “One foot equals 1/3. Two thirds divided into 6 equals 3 times 2 equals 6. Six and 4 equals 10. Ten and 3 equals 13 feet.”
You will note that not one child saw the essential point, that half the pole was buried in the mud and the other half of it was above the mud and that one-third of this half equalled one foot. Their only thought was to manipulate the numbers, hoping that somehow they would get the right answer. I next asked, “Is there anybody who knows some way to get the length?”
Next child: “One foot equals 3/3. Two-thirds and 1/2 multiplied by 6.”
My next question was, “Why do you multiply by 6?”
The child, making a stab in the dark, said, “Divide.”
Then Benezet took the problem to a room that used the new curriculum. Here is what happened:
Benezet Experimental Group
First child: “You would have to find out how many feet there are in the mud.”
“And what else?” said I.
Another child: “How many feet in the water and add them together.”
“How would you go to work and get that?” said I to another child.
“There are 3 feet in a yard. One yard is in the mud. One yard equals 36 inches. If two-thirds of the rest is in the water and one foot in the air [one foot equals twelve inches] the part in the water is twice the part in the air so that it must be 2 feet or 24 inches. If there are 3 feet above the mud and 3 feet in the mud it means that the pole is 6 feet or 72 inches long. Seventy-two inches equals two yards.”
It amazed me to see how this child translated all the measurements into inches. As a matter of fact, to her, the problem was so simple and was solved so easily, that she could not believe that she was doing all that was necessary in telling me that the pole was 6 feet long. She had to get it into 72 inches and two yards to make it hard enough to justify my asking such a problem.
The next child went on to say, “One-half of the pole is in the mud and half must be above the mud. If two thirds is in the water, then two-thirds and one foot equals 3 feet, plus the 3 feet in the mud equals 6 feet.”
The problem seemed very simple to these children who had been taught to use their heads instead of their pencils.
So what had the experimental children been doing while the regular children were being drilled in arithmetic? Benezet’s three Rs – reading, reciting and reasoning. There was quite a lot of number work, though no formal instruction, and no specific period in which arithmetic was turned. Rather the teacher responded to the number topics pupils themselves brought up. The new curriculum included teaching the children to recognise and read the numbers up to 100, dates and times, terms like “larger”, “smaller”, “twice”, and money terms (Grades 1 and 2), measurement (Grades 3 and 4), and more on estimating (a constant theme) in grade 5. The result of focusing on the three Rs was that their spoken and written works was far more interesting and imaginative. They used a wider range of words, and their spelling was better! And, of course, they reasoned about arithmetical problems in a sensible way!
Some food for thought.
I believe that mathematics should be part of every other thing you learn as you grow, part of ordinary life. If you become passionate about number, then explore deeper, find out more, look for patterns. Mathematics is predominantly a search for pattern, a way to look at information and through observation you can then predict future outcomes. Mathematics has a great beauty in the way numbers work together and make sense of the world.
Just because I love numbers, does not mean that everyone loves numbers. In the same way some people love history or geology or music or even particular genres of books, we all have our interests and passions and that is a good thing. What a boring world it would be if everyone was the same.
If your child expresses an interest in mathematics, encourage it. If your child expresses a dislike of mathematics, first look a little to find out why. Is there something else going on? Is it a learning problem, a personality issue, or just a simple lack of desire for that topic? I do not think there are maths people and there are non-maths people. I really do believe that there is value in learning about number, just as there is value in learning to read and write.
In my next article I will try to give you a few suggestions about resources to help your more able and less able children find joy in mathematics.
Bonato, Anthony, April 2016. This is your brain on mathematics
Butterworth, Brian, April 2002. Mathematics and the Brain Opening Address to The Mathematical Association, Reading.
Durant, Mylene, May 2015. Where does the brain do math?
Hrala, Josh, May 2016. Math Has its Own Brain Region
Leado, Byjus, October 2015. The Myth named Math Brain Unfolded