Charlotte Mason Math
My friend Lacy over at Play, Discover, Learn recently wrote this post about Charlotte Mason math. The result was a sudden increase of folks into the Facebook group and a tiny bit of controversy about Cuisenaire Rods and Charlotte Mason. Evidently, there are some in the Charlotte Mason camp that feel Cuisenaire Rods are a contrived manipulative, while beans and buttons are more natural. One of my group members asked me to address this and here we are.
The first parent training I did was titled, "Charlotte Mason and Gattegno have Tea: How the Habit of Attention Gifts Awareness." One of the things I lamented in that training was how if Gattegno and Mason would have met, Miss Mason would have been more consistent with her own philosophy of education. I think the two would have been fast friends as there is so much they agree on. Since I tend in the Charlotte Mason direction, it pleased me immensely that Miss Mason informed my understanding of Dr. Gattegno and vise versa.
What Gattegno And Charlotte Mason Agree On
First who is Gattegno? Caleb Gattegno was the most prolific math educator of the last century. He popularized Cuisenaire Rods and has influenced teachers from around the world. You can detect his influence on many of the current great math teachers that I respect. Anne Fetter, Denise Gaskins, Mike Ollerton, Simon Gregg, and all my heroes at the Bronx Charter School for Better Learning- especially Dr. Hajar, Dr. Powell, and Dr. Swartz.
I don't want to go too deeply into philosophy in this post. Maybe some other time. But the underlying philosophy that both agree on is that children are persons. They are not vessels into which adults shove information and that there are ways to educate that are humane (fit for humans) and ways that are not.
Both agree that children arrive with great intellectual powers and not only have the ability to learn, but they are learning all the time - you can't stop it. Formal education of children should be consistent with their humanity. Both focus on developing the powers children already possess: the powers of attention, observation, and reflection. We do this through certain modes: narration - orally relating back what you heard or otherwise observed and composition - relating back in writing what you have learned. It isn't from worksheets or tests that we know what a child knows, but what the child can generate on his own.
There are many finer points of agreement as well, since both understand how attention relates to learning. There is no endless questioning and talking. The less the teacher speaks the better. Interrupting a child's thought with explaining and questions is physically and mentally exhausting. Exercises should be within the child's grasp and stretch them at the same time. Students should be able to relate new information to old, and they should come to the understanding for themselves. The list could go on, but you get the point.
Gattegno And Charlotte Mason Math: 2 Points of Contention
After reading everything Charlotte Mason wrote on math, it seems there are two main sticking points. Once those two issues are addressed, the Cuisenaire Rods or some other base-ten blocks make sense. Charlotte tried to use something like the rods in her classrooms, but found them time consuming and unhelpful. So it's not like she was opposed on principle. She did try to use them.
Until Gattegno came along, there wasn't an integrated system for using the rods. Since Miss Mason wasn't a mathematician, she can be forgiven for not being able to use them in a helpful way. She was no different than many
current teachers who use the rods for place value and building numbers 1-10. Beyond that, the rods gather dust. In that situation, I can see how beans and buttons are easier and cheaper. But easier and cheaper are a different type of descriptor than contrived.
Issue One: Is Math Really Different Than Other Subjects?
Charlotte Mason saw math as something different than other subjects and highly dependent on the knowledge of the teacher. Gattegno disagrees.
The same skills used in all other subjects should be used here. The child comes with incredible powers of learning and can be taught exactly the same way she teaches other subjects with an emphasis on observation, reflection, oral narration and compositions.
Those who use Charlotte Mason will have an easier time with the idea of "Silent Teaching" than those coming from another background as Charlotte emphasizes not interrupting a child's thoughts with endless questions and talking. Questions should be few and they need to be high quality questions.
The habits of observation developed in nature and picture study will be developed even further in mathematics as children learn to observe and make sense of mathematical structures. Students and teachers learn math together and parents with little math background will find they can play math alongside their children and recover a sense of wonder and excitement over how numbers work. Math is no mystery - it is a language for describing relationships of quantity. Those relationships are rooted in what is true and that makes them discoverable by those who seek to find them.
Issue Two: Must We Start In The Concrete?
When I think about Charlotte Mason math the thing that strikes me is her insistence that we begin math with the concrete and relate numbers to what children experience in everyday life. This is probably where most of the consternation over math rods comes from. If it is true that we must start in the concrete, then it makes sense to start with familiar objects: buttons and beans or in our house - lego minifigs.
Let's assume, for now, that Charlotte was correct and we must start in the concrete. I
will give you your buttons and beans. However, at some point, you must surely come to recognize that buttons and beans are inferior math tools. We can look at Ray's arithmetic (of which, I am a huge fan) and discover Ray's uses the area model to explain math concepts.
What is the area model? Well, it is a rectangle that allows students to see what is happening with the math. It is effective for helping students understand multiplication, division, polynomials, fractions, percents, and ratios - I can teach all of those concepts in one lesson from the image below. Pick your rods, base ten blocks work the same (mostly). They are a beautiful tool for leading students to understand that all roads are leading to a singular mathematical truth. So why would we trade the only available manipulative that displays these relationships for buttons and beans?
What if Charlotte Mason was wrong? What if we don't need to start in the concrete and relate numbers to the things children already know?
What if we start with generalizations? I know. Crazy, right? What if we start with algebra? Are you scandalized yet?
Let me make the case. Can your 5-year old speak fluently? Yes, you say. Then she is using algebra daily and abstracts just fine. I've made my case. Can we dispense with the buttons already?
Ok, I won't be sarcastic. Three-year olds know how to use the word me. They know that when they speak, the word "me" applies to them. When you speak, the word "me" applies to you. That is a huge abstraction. They also know that dogs come in all kinds of sizes from mini to huge, and all kinds of colors from white to black, and smooth to fluffy and cuddly. Yet they are all still dogs.These abstractions are more difficult to comprehend than the abstractions found in elementary and some advanced mathematics. In math, there are rules that can be counted on that children discover on their own. Those rules don't change. Animals get a little tricky - tell me how you explain the difference between cats and tiny dogs to three year old? You don't know, because you didn't explain it. Yet, your three year old figured it out.
What Does It Look Like When We Start With Algebra?
There are serious limitations to keeping children working in the concrete for too long. This is a long discussion. I will refer you to Madeleine Goutard's book Mathematics and Children as this isn't the place for it. But concrete representations shape the way students think about numbers - we don't want them to come away with the idea that numbers are objects.
Given the problems with the concrete and that to Gattegno, you could not function without algebra, or as he said, "algebra is life" - we begin there.
What is algebra? According to I. N. Herstein, Topics in Algebra, "algebra is the study of mathematical symbols and the rules for manipulating these symbols. . . it is the unifying thread of all mathematics." If we take that a bit further, it is the ability to learn the rules and manipulate anything for learning. Language, math, music. It is all algebra.
At the elementary mathematics level, we are are talking about the four basic operations (addition, subtraction, multiplication, division) plus fractions. So what if we start there? What if we start with the operations themselves instead of numbers? What if we start with the unifying thread?
It is too difficult for small children. Right? That is why we don't do that. That's why we wait until students are nearly high school age to start generalizing the rules. But Gattegno says hogwash. And I agree and so do my kids. Even the one with Down Syndrome didn't get the memo that algebra is too hard for 7-year olds.
For algebra, you know, the study of how to manipulate the symbols - the unifying thread in all of mathematics, base-ten blocks are a superior manipulative in every respect. I'm not even sure what to do with beans and buttons in this context.
Addition and Difference
In the above images, we can see that only a green rod will fit in the space that is empty under the brown rod. We could replace the red with dark green and determine that only a red will fit in that space. What do we learn? We learn what difference means. We learn that we can switch out the rods, but we can't switch them out with just any rods. The rods have a fixed length so there are limitations to how you can manipulate them. We learn that addition means bringing two lengths together.
How are fractions and difference related? You probably don't know do you? Well, I'm not going to tell you here. If you have a second grader who can't tell you how fractions and difference are related, you need to reconsider how you are teaching them.
Let's talk about the equal sign. So do you know why the equal sign looks like it does - two line segments that are the same length? In the arithmetical theorems in Euclid's Elements VII-IX, numbers were represented by line segments to which letters had been attached. Rods are not contrived at all. They follow in the line of traditional mathematics. What do you see on the left? Two lines that, though the look different are, in fact, the same length. And they look like an equal sign. The rods represent lengths, just as numbers in traditional math represented lengths.
Two weeks ago, I taught 2 classes of 22 homeschoolers. I asked a simple question, "What does the equal sign mean?" Every one of them gave me a variation of this, "It means you do something like add or subtract and then after the equal sign you write the answer." So I asked, "So are you saying that the equal sign means, 'Write the answer here?' " And they thought for a bit and then EVERY. SINGLE. ONE. said, "Yes."
The equal sign means we have a quantity on one side of the equal sign and another quantity on the other side and those quantities, though they don't look the same, ARE the same. They don't become the same. They are just another way of saying the same thing. 2 +1 doesn't become 3, it doesn't make 3, it IS three. It is the same thing as three. It's just a different way of saying it. Maybe they should know that before they start using the symbol. And it's not such an easy thing to grasp. They need time to think about it before they start developing ideas like the equal sign means "turns into" or "write the answer here" or "makes".
Relationships of Multiplication
Any child can make a train of a single color that equals another rod. This helps students SEE fractions as a multiplicative relationship. We see that 4 reds are in brown and red is 1 of the 4 reds in brown. A red is 1/4 of brown. Brown is 4 times as long as red. That isn't hard. Students can and do understand that.
Are Base Ten Blocks A Contrivance?
My patience for this is question is limited. If you go to GeoGebra - you will find the area model used for modeling all kinds of math. When we set up long division, we use a symbol that is shorthand for the rectangle. The only manipulative that can model that exact symbol are base ten blocks. Same with the square. Ever wonder why we call square numbers squares? Because they form a shape that makes, of all things, a square.
If base-ten blocks are a contrivance, what do you do with all the mathematical models used by mathematicians?
I read once that all manipulatives lie. Some lie worse than others. What this means is that some are more accurate at modeling math than others - nothing is perfect because math is an abstraction. While beans and
counters will show you numbers, you will be hard pressed to get them to reveal relationships, which is what math is about.
Base-ten blocks are the most accurate model available and they last from preschool to some pretty advanced algebra. If you want to use beans and buttons, knock yourself out. I don't want to be snotty, but the contrivance thing shows a bit of ignorance for what math is.
So What's The Bottom Line?
Even if you don't choose to use Gattegno curriculum, it would do you good to learn how to apply Charlotte Mason's principles of education to mathematics and ignore a bit of what she actually says about teaching mathematics. Particularly the bits about not all children being gifted - that is like saying children aren't gifted at picture study or nature study. Also ignore the bits about speed and accuracy.
Of course, we want the student's to be accurate with calculations. But that is not the main goal of mathematics. Math is a weird and wonderful place full of all kinds of crazy things to wonder and explore and if your kids don't get that, something is wrong with what you are doing.
Even if you don't want to do algebra first, there are plenty of ways to remain consistent with Charlotte Mason's whole philosophy without getting stuck in what she says about math. We have to take this part with a bit of skepticism. She wasn't a mathematician and she considered taking math out of the curriculum. This is not to fault her. There were plenty of people who, at the time, didn't think students should waste their time with math as it was primarily for the merchant class.
Why should we study math? I believe we should study math for the same reasons we study other liberal arts. We want to love, and teach our children to love, what is lovely. That it may have other uses in no way takes away from the beauty inherent in the subject.
Mathematicians don't get into math because it makes them speedy and efficient. They get into math because they are caught up in the crazy, weird, and wonderful world of quantity that is waiting for you and your kids to discover.