Charlotte Mason Math: Reason and Proof

The two ideas that Mason considered important in math — rightness and reason — are connected. It is our reasoning that convinces us an answer is right or wrong. How do we know we got a sum correct? We can take the numbers apart and add them another way, to see if we get the same answer. Or we can subtract one of the numbers from the sum and see if we get the other number. Or … well, how would you prove it?

More than anything else, Mason wanted her students to discover in math a sense of immutable truth, a truth that stands on its own, apart from anything we say or do, a truth we can explore and reason about but can never change.

This sense of rightness, of solid, unalterable truth, inspires a feeling of wonder and awe — she calls it “Sursum corda,” a call to worship — that delights our minds. It’s that “Aha!” feeling we get when something we’ve been struggling with suddenly fits together and makes sense.

From the very beginning, children should be doing this sort of informal proof, explaining how they figured things out. Don’t wait until high school geometry to let your children wrestle with ideas.

“The behaviour of figures and lines is like the fall of an apple, fixed by immutable laws, and it is a great thing to begin to see these laws even in their lowliest application. The child whose approaches to Arithmetic are so many discoveries of the laws which regulate number will not divide fifteen pence among five people and give them each sixpence or ninepence; ‘which is absurd’ will convict him, and in time he will perceive that ‘answers’ are not purely arbitrary but are to be come at by a little boy’s reason.”

 — Charlotte Mason, Towards a Philosophy of Education

This is why stories and manipulatives are so important when working with elementary children. Do not rush to abstract math notation, because children cannot reason with it. Very young children need the physical presence of objects to count, while older ones may rely on the mental images of a story — whatever it takes to let them recognize and avoid absurdity.

Proof at a Child’s Level

“But 4×7=28 may be proved. He has a bag of beans; places four rows with seven beans in a row; adds the rows thus: 7 and 7 are 14, and 7 are 21, and 7 are 28; how many sevens in 28? Therefore it is right to say 4×7=28.”

 — Charlotte Mason, Towards a Philosophy of Education

Students may use manipulatives (blocks or beans) if they like, or draw pictures, or make tally marks, or just explain their reasoning orally.

Perhaps the child has played around with doubling numbers and feels confident working with mental math: “2 sevens is 14, and then 2 fourteens is 28.”

Or the child may break the sevens apart into small numbers that feel more friendly, easier to work with, like chunks of 5+2: “4 fives is twenty and 4 twos make eight more”

The point is not to demonstrate the problem in a certain way, but to show a logical justification.

The Power of Imagination in Math

Even teenagers and adults can struggle to reason using only abstractions. As mathematician and teacher W. W. Sawyer wrote:

“Earlier we considered the argument, ‘Twice two must be four, because we cannot imagine it otherwise.’ This argument brings out clearly the connexion between reason and imagination: reason is in fact neither more nor less than an experiment carried out in the imagination.

“People often make mistakes when they reason about things they have never seen. Imagination does not always give us the correct answer. We can only argue correctly about things of which we have experience or which are reasonably like the things we know well. If our reasoning leads us to an untrue conclusion, we must revise the picture in our minds, and learn to imagine things as they are.

“When we find ourselves unable to reason (as one often does when presented with, say, a problem in algebra) it is because our imagination is not touched. One can begin to reason only when a clear picture has been formed in the imagination. Bad teaching is teaching which presents an endless procession of meaningless signs, words and rules, and fails to arouse the imagination.”

 — W. W. Sawyer, Mathematician’s Delight

Our school experience leads most people to think of mathematics as a list of facts and procedures, tools we can use to solve problems. But in reality, math is a whole world of ideas and concepts, a diverse ecosystem of living knowledge just waiting to be discovered.

We can explore this world of mathematics with our children, arousing their imagination through the playful exploration of numbers, shapes, and patterns, noticing connections, wondering about relationships, finding delight in new understanding.

Thus we build a solid foundation that will be able to support our children’s reasoning, even into higher mathematics.

Read the Whole Series

• Introduction to Charlotte Mason Math
• Reason and Proof

To Be Continued: Next time, putting Mason’s principles into practice…