“Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If they aren’t beautiful, nothing is”

Paul Erdos

It seems that feelings towards mathematics goes in one of two directions: you either love it, or hate it. How can one person develop an appreciation for math, and another despise the same thing? And why should every person love, or at least appreciate, mathematics? As Galileo said “Mathematics is the language with which God has written the universe.”

Math is everywhere. It’s in everything. I didn’t comprehend this until I began learning math with my kids – reading mathematician biographies and seeing the “living ideas” behind the mathematical principles. Math can also be fun. My boys love logic puzzles and playing with math in other ways. The living ideas are rich with truth that is satisfying to the soul. One day my 9-year old son came to me “Did you know that all prime numbers are odd, but not all odd numbers are prime?” My 7-year old will regularly comes to me and tell me some simple mathematical truth he discovered, and showed visible satisfaction when taught division. It is truth, and it feeds his soul. 

But math is more than just interesting ideas and puzzles to solve. “The Principality of Mathematics is a mountainous land… People who seek their work or play in this principality find themselves braced with effort and satisfied with truth.”
Ourselves, Book 1, pg. 11

“Braced with effort…”

Many of you are nodding your heads at this phrase. Mastering math does require effort, and the mental blood, sweat, and tears may be all you remember from your past math lessons. 

Charlotte recommended that the “… arithmetic lesson be to the child a daily exercise in clear thinking and rapid, careful execution, and his mental growth will be as obvious as the sprouting of seedlings in the spring.”
Vol. 1, p. 261

Math is meant to provide exercise for the mind. It strengthens reasoning, logic, and clear thinking. Its benefits extend to more than just knowing how to double  a recipe and balance checkbooks. 

“…the use of the study in practical life is the least of its uses. The chief value of arithmetic, like that of higher mathematics, lies in the training it affords to the reasoning powers, and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders.”
Vol. 1, p. 254

So, how does one nurture a love and appreciation of math? How do we help our children see the living ideas, but not neglect math as a discipline that requires mental exercise? In other words, how should we teach math according to Charlotte Mason’s principles?

First, math requires living (or “Captain”) ideas

“Mathematics depend upon the teacher rather than upon the text-book and few subjects are worse taught; chiefly because teachers have seldom time to give the inspiring ideas, what Coleridge calls, the ‘Captain’ ideas, which should quicken imagination.”
Charlotte Mason, Vol. 6, p. 233

My favorite way to introduce these captain ideas is through living books, specifically biographies about mathematicians. These books show the questions or problems the person had, and how they discovered solutions. Stories of people who love math have nurtured a love of math in my own kids (and myself!). Other ways of introducing captain ideas is giving your children a problem –real life or story problem–and giving them a chance to solve it before you introduce the method of solving it. 

Teaching children math is so much more than simply following a curriculum. It means you must understand the principles and know how to teach it, not just follow a script. My two favorite resources for this is Let’s Play Math! by Denise Gaskins and Arithmetic for Parents by Ron Ahroni. 

Second, start with the concrete and move to the abstract.

“One must mentally pass through all the stages from concrete to the abstract, The teacher’s role in this process is to guide the students so that he experiments with the principles in the correct order.” Ron Aharoni p. 9

Math is a skill-based subject. This means that the principles and skills build on one another. It also means that the principles must be learned concretely, then mentally (or imaginary), and finally abstractly. 

This would look like a child counting, adding, and subtracting objects. After they are able to count and have a solid idea of what each number means, then they can be asked mental math problems, like “what is 5 apples plus 3 apples?” The child isn’t using objects to solve math, but is imagining the apples in their mind. Once they can imagine the objects being adding together they are ready for abstract math, which is the numbers represented symbolically on paper:  5 + 3 = 8. 

I don’t begin written math problems until my children are late into first grade, sometimes even second grade. All their math is done with objects or mentally. And this is how all math concepts should be learned! Even geometry and algebra. 

Third, utilize the tools of learning – narration, retrieval, interleaving, and spacing.

Being forever fascinated by how people learn, I’ve read quite a few books on the science of learning. And I’m always delighted to see the near identical learning methods described in Charlotte Mason’s books and current research. Here is a quick rundown of the most important ones and how to use them in math lessons:

• Retrieval (aka narration). Ask your child to tell back in their own words what they learned in their math lesson. They can draw it out, explain it, or create their own story problems (and answers!) to show they understand the principle. 
• Spacing. This is related to retrieval in that you are asking your child to retrieve knowledge, but instead of doing it everyday, allow some time to pass (space the lessons) so that the child’s mind has to work a little bit harder to retrieve the knowledge. I reserve 5 minutes of each math lesson for reviewing past math concepts. It could be a week, it could be a month! The point is to set some time aside to solve problems that you learned to keep those neural connections strong and make it easy to retrieve it when needed. 
• Interleaving. Science on learning shows that we learn better when we do a mixture of problems (interleaving) rather than practicing  the same problem or skill  over and over again (blocked). When I give my son problems for review, I give him a variety of problems. For example, my 6th grade son’s review might look like this: 3-digit multiplication problem, long division, adding within 100,000, 2-digit multiplication, adding fractions, substracting within 10,000, etc. 
• Generation. I mentioned earlier that I like to give my kids a new problem without telling them how to solve it. I allow them time to experiment, ask questions, and grow a healthy sense of curiosity. This is called “generation” because children generate their own solutions and questions. 

Fifth, teach different mathematics in stages. 

“…knowledge is established in layers, each relying on the preceding one. The secret to proper teaching lies in recognizing these layers and establishing them systematically.”
Ron Aharoni, Arithmetic for Parents, p. 18

Charlotte Mason began with arithmetic for the first years. Beginning in 5th grade practical geometry (the concrete version of geometry) was added. Then algebra and geometry are added in middle and high school.  Beauty and Truth Math has a great visual of this method, along with a lot of different options. 

Finally, remember that math is  learned line upon line, precept upon precept.

Forget about “grade level”  and what “should” be learned in a prescribed grade year. Focus on mastering each concept before moving on to the next once. Your child may learn quickly and master a whole school year of material in a month, or they may need a whole school year to master only a couple concepts. The important thing 1) consistent, daily lessons, and 2) mastering concepts before moving on.