Thursday, June 20, 2013

How to Choose a Mathematics Curriculum Part 1

Charlotte Mason’s Key to Teaching Mathematics

“Of all his early studies, perhaps none is more important to the child as a means of education than that of arithmetic.”  
Charlotte Mason vol. 1 pg 253-54

Mason’s statement is bold and elevates mathematics to the top of the priority list as a means of education.  This puts choosing the curriculum for mathematics something of the utmost importance and not to be done without considerable thought and research.  Can you make a mistake and choose an ineffective curriculum or one that is very difficult to work with?  Yes, I did in my sons first year of homeschool, kindergarten.  Like most American’s, I grew up and went to college in the public school system.  Although I originally went to college to be a chemist, I eventually switched majors to graduate with a teaching degree and a science major and mathematics minor.  I had been so trained in my thinking about mathematics by this history, that I had made my choice with that mindset. Fortunately, I realized early on, my grave error.  I hope to share with you, the insights I have gained by reading what Charlotte Mason had to say about teaching arithmetic, so that you are fully equipped to make a well informed choice.

“That he should do sums is of comparatively small importance; but the use of those functions which 'summing' calls into play is a great part of education so much so, that the advocates of mathematics and of language as instruments of education have, until recently, divided the field pretty equally between them.”  
Charlotte Mason vol. 1 pg 255

The ability to make a calculation has little importance according to Mason.  In today’s high tech, fast paced culture it is even truer than in Mason day.  Everyone has access to calculators on their phones, computers, tablets, or iPods.  If you can push buttons, you can make a computation.  It requires very little thought or ability.  She goes on to explain that it is the use of these calculations that hold the value to the education of the person.  While easily measured, computational skills are not the goal of mathematics education.

What I am about to share with you I learned, not from my teacher training in college, but instead from reading chapter XV- Arithmetic, in volume 1 of Charlotte Mason’s Homeschooling Series, which in turn caused a monumental paradigm shift in how I understood the teaching of mathematics.

Think about how you were taught to add and then how your child is taught the same concept in a traditional mathematics programs today.  Not much has changed.  If you open a math workbook today, you probably see problems like 2+5=___.  To a young child, addition is a difficult, abstract concept to understand.  He must first understand what the numbers, addition sign and equal sign mean.   In the workbook there may be two objects printed under the 2 and five objects printed under the 5 which the child then counts and finds the sum.  With the new hands on approach to learning, the student is given counters to represent each of the numbers and then he counts them all together.  Either way he is expected to find the sum of seven.  The pictures or counters are the means to make this abstract number sentence more concrete and to help the child understand what the number sentence means.  On an average math page you may see up to 20-30 computation problems.  The child labors and struggles to understand the idea of addition some with more difficulty than others.  This is the traditional approach to teaching abstract math concepts and how, you may agree, mathematics is taught.

Through decades of observing children, Charlotte Mason discovered that this is not really how children learn math naturally. She discovered the missing key to teaching mathematics.

Imagine trying to teach a person to read music.  You show her the notation and tell her the names of each of the notes and symbols, but you never give her an instrument to actually play the notes.  You may be able to teach her to read the notes on the page with much struggle along the way, but does she really know music?  In the same way you may be able teach him to do math calculations with much struggle along the way, but does he really know math?  In the case of the music, the playing of the instrument is the proper context for the understanding of the abstract symbols that represent the music.  Presenting abstract ideas and concepts within their proper context is the key to understanding knowing the ideas or concepts.  Properly framing the abstract ideas or concepts in the real world setting is what makes learning something abstract so natural.  What then is the proper context for knowing mathematics?  Charlotte Mason went directly to real life application.

“Engage the child upon little problems within his comprehension from the first, rather than upon set sums.” 
Charlotte Mason vol. 1, p. 254

All mathematics starts with a real life problem that requires the use of numbers in order to solve it. You cannot go through a day without using math in one form or another.  We use it all of the time. This idea of starting with a real problem is a crucial foundational concept to the teaching of mathematics.  It is also the exact opposite of the traditional understanding of mathematics.    In traditional teaching of mathematics, the symbolic is the beginning and then manipulatives are used to make the symbols concrete.  Remember those long pages of ciphers with two word problems at the end?  This is not how we do math in the real world and this is not how children learn math concepts naturally.  First, start with the problem and offer the use of manipulatives to help solve the problem and make the real world math problem concrete.
For example:

You have two gerbils.  The mama gerbil gives birth to five baby gerbils.  Now how many gerbils do you have?

Next, give him counters of some sort in order to solve the problem.  It is after he has solved the problem and discovered there are seven gerbils that the symbolic representation of the problem he just solved is presented.


It is very easy at this point for the child to understand the meaning of the numbers and symbols.  Do you see the difference?  The symbolic representation now has meaning because it is representing something the child already knows.  The knowing (solving a real world math problem) must precede the representation (the corresponding number sentence).  Mason goes on to explain the need to demonstrate what needs to be demonstrated and here would require the skill of the math teacher to be able to present a concept in different ways if the child does not immediately grasp it in the way you initially presented it.

“The practical value of arithmetic to persons in every class of life goes without remark. But the use of the study in practical life is the least of its uses. The chief value of arithmetic, like that of the higher mathematics, lies in the training it affords the reasoning powers, and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders.” 
Charlotte Mason vol. 1 pg 255

Mason eloquently lays out the chief end of mathematics, the true value of it for the child.  The result of teaching children in the way they learn naturally as she has described is the training in reasoning.   The habits of understanding, willingness to work, accuracy, and being intellectually truthful are also developed.

When children are taught in a way that is compatible to the nature of learning, it is like paddling a canoe down a river along with the current.  The way is gentle and joyful.  There may be challenging problems to solve, but the struggle is with many rewards.  On the other hand, teaching in a way contrary to how learning is done naturally is like paddling that canoe against a turbulent current.  The journey is filled with frustration and confusion.  Some of you may have experienced this yourselves with the way you were taught mathematics.  The result of teaching contrary to nature is that you and many other people relate feeling that they are not good at math, but what is not good was the way in which you were taught.

Now that you understand the way in which mathematics should be taught and the goal of that education, when you analyze the curriculum choices you will be able to filter the choices by looking for a curriculum that matches this natural way of teaching children.

Look for part 2, my next article on how to analyze the curricular.

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