Homeschooling math by Don Cohen
Is your child doing lots of arithmetic while doing important mathematics?
Is your child learning how to find patterns?
Is your child able to visualize- draw a picture or diagram, or graph - the problem they are working on?
Is your child making up her/his own math problems?
Is your child encouraged to ask questions about math?
Does your child enjoy doing mathematics?
Can your child solve one problem (like 64-28=?), many different ways?
Is your child learning how to learn?
Why
Don's materials
fit into any homeschooling mathematics curriculum you use, at any grade level:
1. Don's work with children from ages 3 to 73, of all abilities, over the past 48 years, gives him a unique view of what mathematics is important for children to know. And this has nothing to do with grade level or age. It only has to do with what we expect children can learn and introduce them to it in a way they can understand it. One can always go back to fill in holes or go ahead. Don has had students for 10-12 years, who score a 5 on the AP Calculus test. Not all of Don's students are going to do that or even go on to college. Don even had a student whose parents said their daughter is "dumb as a box of rocks"! But Don treats all students with the same respect, and expects them to work to their highest level- and this comes out in his materials.
2. Don has had a not insignificant number of students who hate math from 2nd grade on. Part of this is due to the school's emphasis on timed tests. Part is because the way it is presented is boring. Another part is the abundant number of examples, of the same kind, they have to do for homework. They may work slowly and have to spend hours every night on these. Don would rather a student do 2 problems, but do each one 4 different ways. Knowing how to do something different ways gives one the ability to check ones work, gives one the ability to use different concepts, and this can carry over as an adult trying to solve problems.
3. Most homeschooling parents know their child better than anyone else.
4. Don's emphasis is on the child being an active learner- making up problems, making up equations to solve, making up graphs to get an equation for it, making up functions to play "Guess My Rule", making up shapes on a geoboard to find the area within it, make up equations to graph, make up geometry problems, guessing the answer,..
5. As a teacher, Don wants to enjoy his work, and see results. He thinks parents want the same thing.
6. From a
parent: "..do
your students need to have already studied Algebra and all those other courses
that seem to come before calculus?"
There are a lot of myths in the study of mathematics; for example "you need
to division of whole numbers before you study fractions"; "you
need to know division of whole numbers before doing negative
numbers"; we could go on and on.. Most of these myths come from the schools
of education at the universities.. (why are there so few people taking calculus
in high school?). My idea is to start young people working on important
mathematics- infinite sequences, infinite series, functions, graphs, algebra.. ,
then when they study calculus they will really understand it! And in the process
they will do a lot of arithmetic! I started a 4th grade girl working on
quadratic equations last week. When dad came in to pick her up this week, he
told me he had had a teacher conference in which he was told the class was going
back to basics (heard that before?). I then showed him what his daughter was
doing in the process of solving the quadratic equations (see ch. 8 in the sample
problems)- she was multiplying whole numbers, learning how to square numbers,
subtracting, using negative numbers, substituting in a variable, using the rules
of order of operations, learning what it means to solve an equation (make it
true), looking for patterns, and doing something that looks very complicated,
but is simple. I do a lot of algebra in chapters 6, 8, 9 and 11. The infinite
series in chapter 1 involves mostly counting. As does the cookie-sharing in ch. 2. It's all in how
one approaches the math.
7. And don't worry about your child making mistakes. The following is a quote in his little book, from MATHEMATICS, THE LOSS OF CERTAINTY by Morris Kline:
"In his first paper on the Calculus (1669), Newton proudly introduced the use of infinite series to expedite the processes of the calculus...
As Newton, Leibnitz, the several Bernoullis, Euler, d'Alembert, Lagrange, and other 18th-century men struggled with the strange problem of infinite series and employed them in analysis, they perpetuated all sorts of blunders, made false proofs, and drew incorrect conclusions; they even gave arguments that now with hindsight we are obliged to call ludicrous."
If one starts at a young age, working on these ideas, allowing for all the mistakes, one will really understand things better later on. Don has seen children do just that and it has been very encouraging for him. As a parent or teacher, don't feel you need to know everything before the children see the problem. Don feels if he doesn't expect a child to solve a problem a certain way, they will pleasantly surprise him with a new way of doing it, and he will learn something new!
Hi Don,
Yes, we do have your books, and think they're
wonderful. I have two boys who love math, and we're looking forward to the
videotape. My
husband has a PhD in physics, and he thinks your books are a great way to look
at mathematical ideas -- so different from the usual textbooks.
(In fact, one of the reasons we started homeschooling is
that my younger son is gifted in math, and that didn't fit with our local
public-school mentality. When he was in first grade, the teacher wanted the kids
to write different ways to get "8", like 4+4, 5+3, 10-2, etc. When
he wrote things like 20/2.5, she marked it wrong!!!)
Thanks so much, Laura (
Please see the following pages:
Sample
problems from Don's books
The important mathematics in Don's materials
On thinking about and doing mathematics
Patterns in Mathematics (in number, in graphs, in nature,..)