Math for Older Students

One autumn afternoon, I (Richele) was in the kitchen when my teen breezed in and grabbed an apple. He paused on his way out the door and said, “Hey, mom, thanks for algebra today.” My voice and my heart trailed after him as I called out, “You’re welcome!” And then I sat for a moment in quiet wonder.

A few days later at the end of a math lesson, my son expressed his appreciation again, but this time I asked him for his reason. He explained that he and his buddies were discussing school when their conversation turned to math class. “One said that although he’s in an honors program and gets good grades, he doesn’t really understand what he’s doing. The other said he’d gotten left behind so long ago he has no hope of ever catching up. I thought about it and told them you never make me move on until I really understand.”

As heartwarming as it was, the credit didn’t belong to me as I’d simply been using Charlotte Mason’s approach for teaching mathematics with my teens. If you already know Charlotte Mason’s approach to teaching arithmetic, you will be relieved to find that it doesn’t really change that much as the math becomes more complex. If your older students have never experienced Charlotte Mason math, you may be pleasantly surprised at how straightforward it is to incorporate a living teaching of mathematics into their days. Like many other areas of study in a Mason education, it begins with the “three educational instruments” of principle 5: “the atmosphere of environment, the discipline of habit, and the presentation of living ideas.”^[1] Let’s explore these in reverse order.

The Presentation of Living Ideas

My son’s comment to me highlighted perhaps the most important element of math instruction: the presentation and comprehension of ideas. While my teen’s friend could do really well on his honors tests, I was aiming for something even better for my son: that he would actually understand. The fact that I never lost sight of the primary importance of ideas is, perhaps, the main reason that he thanked me those two days.

Charlotte Mason signaled the great importance of ideas in one of her earliest statements about math instruction, in 1886:

Multiplication does not produce the “right answer,” so the boy tries division; that again fails, but subtraction may get him out of the bog. There is no must be to him; he does not see that one process, and one process only, can give the required result. Now, a child who does not know what rule to apply to a simple problem, within his grasp, has been ill-taught from the first, although he may produce slatefuls of quite right sums in multiplication or long division.^[2]

Mason’s words about multiplication and division apply just as much to factoring and substitution. I didn’t want my son to produce “slatefuls of quite right” algebraic equations. I wanted him to be able to grasp the meaning and the beauty of each mathematical concept he learned.

This theme recurs throughout the writings on Charlotte Mason mathematics. Irene Stephens, for example, had children performing “sharing” or “dividing” activities well before they ever saw the symbol for division.^[3] The idea always came first. Only when children understood the concept did they begin to apply the notation to express the ideas in succinct mathematical form.

But even these ideas are not simply abstract, ethereal concepts. They are concepts filled with meaning and relevance because they accurately model and explain countless elements of our natural world and way of life. Irene Stephens recognized this and it was one of the reasons that she incorporated calculations involving money very early in mathematical instruction. She introduced money situations that the children found “interesting”^[4] — interesting because it related to relevant aspects of their everyday life.

In the same way, living ideas in upper math are made more interesting and inspiring when they are linked to reality. Observing the relationships between angles and arcs can seem like dry stuff until the student learns that through this relationship, Eratosthenes was able to calculate the circumference of the earth in the third century BC — most likely without having to set one foot outside of his home city of Alexandria.

It is important to remember that Mason was very intentional when she chose the word “idea” to describe this living and dynamic aspect of education. She did not mean simply “concept,” “symbol,” or “competency.” She meant something that is active:

An idea is more than an image or a picture; it is, so to speak, a spiritual germ endowed with vital force—with power, that is, to grow, and to produce after its kind. It is the very nature of an idea to grow: as the vegetable germ secretes that [which] it lives by, so, fairly implant an idea in the child’s mind, and it will secrete its own food, grow, and bear fruit in the form of a succession of kindred ideas.^[5]

As hard as it may be for some to believe, higher math is filled with such ideas. Ideas that do not merely direct the computer-like manipulation of symbols according to rigorously defined rules and procedures. Rather, ideas that excite the imagination, and prompt a “succession of kindred ideas.”

An extremely important aspect of teaching math the Charlotte Mason way, then, is to identify the root ideas that organically prompt the rest. She referred to these as “guiding” or “captain” ideas:

Every study, every line of thought, has its ‘guiding idea’; therefore, the study of a child makes for living education in proportion as it is quickened by the guiding idea ‘which stands at the head.’^[6]

According to Mason, math instruction without these captain, guiding, living ideas falls horribly short of the wondrous adventure that mathematics should be:

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.^[7]

Why don’t teachers take the time to ensure that their students understand the captain ideas which enable their minds to build on and elaborate a “succession of kindred ideas”? One reason may be that parents and teachers often use an external standard to set the pace of learning, rather than adjusting the pace to the progress of the child. Of course, you should be aware of any district or state requirements, but your student’s progress is best measured against her own achievements rather than a rubric set by the government, the neighbors, or even a sibling. Allowing her to work at her own pace builds her confidence, and  that can help her delight in the study of math for its own sake. In fact, giving her the sense that she is somehow behind can be a violation of the very personhood we are charged to protect:

There are, however, many slow by nature to grasp mathematics or grammar—what is the teacher to do about them? Keep the work relevant, suited to the child’s power of understanding. Give him a programme easier than that of his Form; easy enough for the confidence to return which Miss Mason wanted, and give him a sense of mastery—and, no doubt, with it the assurance that this teacher can teach after all! Or let the French Dictée chosen be easy enough for some to get it all right, and none to feel defeated and silly; for that is to offend against their integrity.^[8]

Now as important as ideas are, it is essential to realize that ideas must be consumed. They cannot be pushed in. As Charlotte Mason said:

Therefore, teaching, talk and tale, however lucid or fascinating, effect nothing until self-activity be set up; that is, self-education is the only possible education; the rest is mere veneer laid on the surface of a child’s nature.^[9]

Just as with every other subject in a Charlotte Mason education, the child must work to grasp the living idea. There must be self-activity. Very few children will grasp a living idea by watching a brilliant lecture or demonstration, either live or on video. Instead, children must work to “meet” an idea. They must make their own “discovery,” differing only in degree and not in kind to the original discoveries by Euclid, Al-Khwarizmi, and Newton.

“Discovery” in Charlotte Mason math doesn’t mean the child is put in a canoe without any oars. The teacher makes sure to give her the tools and vocabulary necessary while at the same time being sure she’s the one doing the work. One great way to introduce a new concept is to use the example problems from the math text, but not letting the student see the explanation or the answer worked out. Often the concrete nature of these examples leads to the student figuring out the concept and making the discovery.

We see a similar model in a sample algebra lesson developed by a student at the House of Education. The lesson opens not with a fancy lecture or a tour de force presentation, but with the students getting to work:

Let one of the girls work a multiplication sum on the board, and then find out from them the connection between multiplication and division, and how division is the reversal or undoing of multiplication.^[10]

I love that phrase: “find out from them” — a phrase that occurs three times in that one-page lesson plan. For now, in their own small way, the children are being like Euclid and like Newton.

This is one reason why we should avoid having students memorize definitions and formulas. As with the living method of learning a foreign language, terms are grasped as they are used. If we miss this point, we run the risk of what Charlotte Mason warned against:

The child may learn the multiplication-table and do a subtraction sum without any insight into the rationale of either. He may even become a good arithmetician, applying rules aptly, without seeing the reason of them; but arithmetic becomes an elementary mathematical training only in so far as the reason why of every process is clear to the child.^[11]

Mason’s statement applies to algebra, trigonometry, and calculus just as much as to arithmetic. Just replace “multiplication-table” with “quadratic equation” and “subtraction sum” with “derivative.” My calculator can solve for the unknown and integrate a formula. But my calculator will never know why. The “why” is where the magic of math begins.

The Discipline of Habit

But this is not where the magic ends. Ideas, when fully consumed, become habits. When speaking of “The Idea which Initiates a Habit,” Mason says that “the spiritual force of the idea has its part to play” — as we’ve discussed above — and then “a habit is set up by following out an initial idea with a long sequence of corresponding acts.”^[12]

The living ideas must be worked on by the child until they “leave their mark upon the quite palpable substance of the brain.”^[13] As with elementary math in the Charlotte Mason method, this is done by following the simple formula “new, review, mental math too.” New ideas are immediately worked on via exercises and problems. And then review and mental math complete the path to fluency. By mental math we mean work that is strictly taken orally. Mental math is instrumental in training good habits, such as steadfast thinking, fixed attention, and promptness. This is as true in algebra and trigonometry as in multiplication and sums. In fact, some math textbooks explicitly incorporate this. For example, here’s an excerpt from an Oral Practice section of Paul A. Foerster’s Algebra I, chapter 9 (“Properties of Exponents”):

Simplify each expression.

A. (x³)(x²)
B. (x³)²
C. (x⁴)(x⁵)
D. (x⁴)^{5 [14]}

How many problems a student needs to “internalize” (or “habitualize”) a concept can vary with the individual. The wise teacher will assign enough work to form a habit but not so much work that the student becomes bored. And then just as any new habit must be reinforced over time, sufficient periodic review must continue to ensure the concept is solidified in the child’s mind — and brain.

However, just as with elementary math, upper level math involves several other habits. Having students put written work in a gridded notebook helps them to establish habits of neatness and organization. Every new notation introduces new ways to arrange and record equations and operations, and it is important to ensure that neatness and order are consistently applied each step of the way. It may or may not be necessary for the student to write out every step in the solution to a problem, but the written work should be expansive enough that the student can use it to explain his line of thinking to obtain the answer.

Sadly, it is usually the parent who undermines habit training by allowing sloppy work, giving excessive explanations, teaching to the test, or immediately helping a student over a difficulty rather than allowing him time to investigate, think, and do the work himself. If your teen gets an answer wrong due to carelessness, then a do-over only reinforces the behavior, but if it’s because he doesn’t understand the concept, then you need to secure his understanding before giving a new problem. Either way, it is the teacher who has something to rectify.

One way to maintain habits of order, neatness, and attention is to continue to embrace short lessons. Our teens can accomplish more in a short and focused math lesson than they can in a long and dawdling one. The PNEU time-tables for even the oldest teens never allowed more than 30 minutes for a math lesson, and we are wise to follow that precedent. A lot can be accomplished in a half-hour lesson, especially with the use of oral work and mental math. Also, be sure to schedule the math lesson when you know your particular student is at his freshest. Mental fatigue can also be avoided by alternating it with something that uses a different part of the body or brain, such as literature, a foreign language, or even some time outside.

The Atmosphere of Environment

Charlotte Mason is clear that atmosphere includes not just the physical environment but also the attitudes of the parents and others who surround the child. Often these attitudes are not explicitly stated, but are still conveyed in a variety of ways, consciously or not. One important attitude to address is the common assumption that the enjoyment and study of math is only for a small number of gifted people, the “mathy-kids” or those intended for a STEM career. But Charlotte Mason was very clear that this is not the proper basis for math in the curriculum:

We take strong ground when we appeal to the beauty and truth of Mathematics; that, as Ruskin points out, two and two make four and cannot conceivably make five, is an inevitable law. It is a great thing to be brought into the presence of a law, of a whole system of laws, that exist without our concurrence,—that two straight lines cannot enclose a space is a fact which we can perceive, state, and act upon but cannot in any wise alter, should give to children the sense of limitation which is wholesome for all of us, and inspire that sursum corda which we should hear in all natural law.^[15]

All teens should be regularly brought into the presence of that law — not only the ones who want to go on to a career in science or engineering. In the same way, few children are going to grow up to become accomplished artists. But every child’s artistic sense can be cultivated through picture study and brush drawing lessons. Similarly, every child can be taught to think mathematically to some extent when the subject is approached in a living way. Irene Stephens made the same point with regard to music appreciation:

It may be argued that this attitude towards mathematics, or the appreciation of some of its fascination and beauty, is a gift only given to the very few: but we venture to think that such an argument is more a habit of thought than a proven truth. In days not long past we were accustomed to think that music was a heaven-sent gift to a chosen few and a hard exercise in discipline and the sterner virtues to those not of this blessed minority. In this present-day we are told that every child can be a musician of some degree; that, given the right beginning, every child can be taught to think musically, to improvise for himself, and to appreciate intelligently the music of others. Though the world may never produce another Beethoven, the teachers of music are learning how to put their pupils into the right relationship with this subject, so that it is a gift to every one of them of opportunity for happy and joyful experiences; not, as it has been for the majorities of the past, a grinding between the millstones of drudgery and despair.^[16]

If we clear the atmosphere of conflict and anxiety, we have an amazing opportunity to draw power out of our older children, excite their enthusiasm, and allow them the time they need to wonder and grapple with math. A healthy atmosphere can result not only in a happy relationship with beginning numbers but also with more complex mathematics.

Atmosphere also includes the physical environment. Charlotte Mason urged teachers to use real-world objects in the place of contrived manipulatives. While manipulatives for algebra and beyond are rare, they are still to be eschewed on the same principle. There are endless applications of algebra and trigonometry to the real world; there is no need to invent an artificial setting for these intensely relevant concepts.

Choosing and Using Textbooks

So if we understand the atmosphere, discipline, and life of Charlotte Mason math, how do we go about actually implementing it with our older students? Of course we must have a textbook. The math texts Charlotte Mason chose were those that could easily be adapted to the principles and practices of the living teaching of mathematics, rather than those that would undermine her philosophy of education. We can follow Mason’s example and choose similar textbooks today. Here are some guidelines:

• The book should be free from “bells and whistles.” The living ideas are what should attract the student’s attention, not fancy implements and media.
• It should not be founded on a different system or theory of education. The underlying principles make a difference, and you don’t want to be in a constant tug-of-war with your book.
• The problems included in the text should be concrete and practical. They should be as interesting to an older student as Irene Stephens’s money problems were for young students. Engaging questions of an interesting or practical nature spark the imagination and aid in logical thinking.
• The text should support the “new, review, mental math too” formula, ideally including review and mental math problems in the sequence of the text itself.

No matter how compatible the textbook is with Charlotte Mason’s ideas, you will still need to adapt and customize it for your lessons. If a lesson contains four sets of problems this doesn’t necessarily mean your student must work all the problems in each of the sets before moving on. For example, the first set might be comprised of review work and your student may only need to work a few from there to continue solidifying concepts already mastered, or perhaps, the fourth set is for advanced placement or honors work. Be sure to read the front matter or the teacher’s handbook to better understand your textbook and how to get the most out of it.

The biggest departure you will make from your textbook is how the living ideas are introduced. Even the best textbooks often fall back to detailed and lengthy explanations of procedures and algorithms which are best discovered and internalized by the child on her own. You will almost never want to have your child read the text that begins each new section. (Or worse, read it aloud to your teen, or have her watch a video.) Instead, you should make sure you understand the key ideas and principles, and then have your student grapple with the example problems and discover the concepts herself.

Sadly, there is apparently no upper math textbook that supplies an explicit roadmap that says: “Here are the captain ideas! And here are the ideas that flow from them!” It will require some work on your part to understand the flow of the text and identify those most important ideas. Some texts make it easier than others to find these. An example that I (Art) love is Paul A. Foerster’s introduction of trigonometry via the unit circle in his Algebra and Trigonometry (in the Prentice Hall Classics series). In sections 13-1 to 13-3, Foerster lays out a captain idea that—when fully grasped—makes trigonometry a rich and rewarding experience. (In fact, this was the turning point for my daughter, and the first time she actually said she enjoyed math.) On p. 718, Foerster even explains the origin of the word “sine,” a detail I love to include in my trigonometry immersion lesson which many parent-teachers have enjoyed.

Other excellent attributes of Foerster’s Algebra I and Algebra and Trigonometry texts are that they include explicit sets of questions for oral work (mental math) as well as review problems interwoven throughout the new learning. Also they have an overflowing abundance of interesting and relevant real-world problems which show just how complementary the concepts of algebra and trigonometry are to the processes of the natural and social worlds. Finally, Algebra and Trigonometry emphasizes functions, a captain idea in itself that lays an excellent foundation for the study of calculus.

One question that is sometimes asked is whether “living books” should be used to teach math. Charlotte Mason clearly believed that math was one subject that was not best taught via literary works:

… when the mind becomes conversant with knowledge of a given type, it unconsciously translates the driest formulæ into living speech; perhaps it is for some such reason that mathematics seem to fall outside this rule of literary presentation; mathematics, like music, is a speech in itself, a speech irrefragibly logical, of exquisite clarity, meeting the requirements of mind.^[17]

Interestingly, the first major book of algebra, written in the 9^th century by Al-Khwarizmi, was written entirely in a literary style: “Even the numbers were written out in words rather than symbols!”^[18] Fortunately, over time mathematicians developed a language “of exquisite clarity” which greatly facilitates the expression and understanding of mathematical concepts. This special language has enabled what is arguably the most famous formula of all time, Einstein’s mass-energy equivalence, to be written with the elegant simplicity of E=mc².

Although storybooks aren’t used to teach concepts, they may be consulted when introducing a new branch of mathematics or a theorem. A few sentences of the history of discovery or captain thinkers can ignite the imagination and excite interest in the subject. Outside of the math lessons, biographies of famous mathematicians or a history of math should adhere to the same standards of other living books by drawing the reader in and putting her in touch with vital ideas.

A final consideration is the fact that the PNEU taught algebra and geometry in parallel rather than in sequence. There is a lot to be said in favor of this approach. Taking a break from algebra for a year to learn geometry can result in the student forgetting a lot of algebra. Furthermore, the switch back from geometry can imply to the student that geometry is “done,” when in reality it remains an essential element of future mathematics. Indeed, in calculus, algebra and geometry are so intertwined that they cannot be separated. I (Art) prefer to prepare my students for this reality by teaching Algebra I and geometry simultaneously, and then teaching Algebra II and trigonometry together. In the case of Foerster’s second book, that meant my daughter and I kept two bookmarks (!), as we alternated between algebra and trigonometry in successive lessons.

Sharing the Effort to Know

A parent may see all the benefits of this living approach to teaching math and yet may feel woefully underqualified to deliver it. The main problem is perhaps that many of us did not receive upper math instruction in this manner. Thus we don’t know the living ideas or the “why” behind the formulas. Furthermore, many parents have either forgotten whatever they did learn, or were not exposed to advanced math at all. This is not a new challenge in our day, however. In 1923, a homeschool mother wrote in The Parents’ Review:

Arithmetic was certainly the hardest subject when one had forgotten all but the four rules; however, by means of keeping just a little ahead, teaching myself by means of the examples, even this difficulty was negotiated.^[19]

The mother’s name was E.M. Capron, and she identified a major principle in the Charlotte Mason philosophy. Essex Cholmondeley described it as follows:

Teaching is not a technique exercised by the skilled of behalf of the unskilled. It is a sharing of the effort to know, using all that is best in the world of books, of music, of pictures, all that can be observed and cherished out of doors, all that hand and eye can make; all that religion, history, art, mathematics and science can reveal to the active mind.^[20]

Sometimes the “sharing of the effort to know” means keeping just one step ahead of our children. More recently, David Chandler of Math Without Borders wrote:

The optimal situation… is if a parent commits to learn a new subject along with the student. Your own struggles to master something new, contributing your adult perspective while engaging with your child as a peer, and the opportunity for your child to actually teach you something you may be having difficulty with, are all components of the teaching/learning experience you have access to in the home but which rarely happen in the classroom.^[21]

A graphic and inspiring illustration of this principle is given by Mrs. W.J. Brown in her “A P.U.S. Schoolroom, By the Mother Who Runs It,” published in the 1930 Parents’ Review:

Let us take arithmetic as an example of a subject with which we were likely to have some difficulty. At school I was decidedly poor at arithmetic. I’m not brilliant now, but I have spent many interesting hours working at problems with my children—problems which baffled us, sometimes for days, but we were never beaten, we always solved them in the end—occasionally, it must be admitted, with some assistance from relatives and friends. It may be an unconventional way of doing arithmetic, but a child who has taken the trouble to walk half a mile with a problem, or written a letter to an aunt in the North of England to ask her to explain some knotty point, is not likely to forget the working out that she carefully follows on receiving the answer to her S.O.S.

Our method, if unconventional, acted well. It kept us keen and interested in our work—and is the average child in the average school really interested in arithmetic, apart  from the mark-gaining point of view? I don’t think so. At school I hated mathematics, although there were marks to be gained. None of my children dislike mathematics, though they have no marks to gain. They are none of them mathematically minded, yet they do enjoy doing their work. The school standard for arithmetic in these days is high, but in spite of our unconventional methods we managed to keep up to it.^[22]

Brown’s example shows so clearly that learning with our children offers a set of unique rewards. And in our day, help is so much easier to find. The parent or teen no longer needs to write a letter to ask for help; assistance can be as close as a Zoom call or an instant message. But either way, when effort is made to solve a problem, the solution is not easily forgotten.

But to be honest, we don’t actually have to reach all the way back to E.M. Capron and W.J. Brown to find examples of parents learning math with their children. In fact, the same story has played out in the life of both authors of this article. Richele has taught her teens levels of math beyond what she herself received in her own formal schooling. And some of my (Art’s) most rewarding experiences as a parent have been learning new levels of difficult math so that I could teach my firstborn (a STEM major). Indeed I credit the Holy Spirit with whispering the knowledge in my ears that I simply could not grasp on my own.

How Far is Far Enough

The PNEU scope and sequence ended with algebra and geometry in Form VI (roughly equivalent to our 12^th grade).^[23] A review of the texts and examinations used by the PNEU shows that while this went beyond what we would today call Algebra I, and included some elements of trigonometry, it certainly did not include pre-calculus or calculus. Does that mean that as Charlotte Mason educators today, we also should stop short of pre-calculus, except perhaps for STEM-oriented students?

This question has a pragmatic side and an idealistic side. From a pragmatic point of view, college admissions are often influenced by math scores on standardized tests such as the SAT and the ACT. Indeed, Mason herself was aware of the irresistible temptation to measure scholastic aptitude on the basis of math:

Arithmetic, Mathematics, are exceedingly easy to examine upon and so long as education is regulated by examinations so long shall we have teaching, directed not to awaken a sense of awe in contemplating a self-existing science, but rather to secure exactness and ingenuity in the treatment of problems.^[24]

Ironically, the very immutable laws which make math so delightful to the mind also make math delightful to the tester. Each problem has only one right answer.

It is important to note two things about the SAT and the ACT. First, both only require a relatively small amount of math knowledge beyond Algebra I and geometry. Second, they are not tests of mastery. They are tests designed to generate a bell curve. The result is not a test that identifies the students who have a thorough understanding of the fundamentals of algebra. Rather, it is a test that identifies students who are clever at problem-solving and have mastered certain test-taking tricks. Therefore parents need not worry that missing calculus or precalculus will affect SAT or ACT scores. That being said, a short prep course specific to these tests can have a dramatic impact on scores.

But besides the practical side there is also the ideal side. Why do I (Art) teach music appreciation and picture study? My children will never be tested on these. The knowledge will not help them in their future careers. I taught them these subjects to enrich their lives. They are persons who are sustained by a wide and varied banquet of ideas. Many of these ideas have no practical function in their day-to-day lives — unless general joy and fulfillment are considered practical benefits.

My daughter had no interest in pursuing a career in STEM. She was done with her ACT and was accepted at the college of her choice. By all pragmatic measures, we could be “done” with math. But I knew of a mountainous land, a land of beauty and wonder which “pays the climber.”^[25] Before my daughter left for college, I wanted to take her there. She had climbed many peaks in that mountainous region, but there was one she hadn’t seen yet. So during her last year at home, we packed our bags and assembled our gear. Together we ascended a snowy peak. The peak is called calculus. There was no practical purpose for these lessons. I just wanted my daughter to see the beauty. I taught her for love.

Your journey may have a different sequence and a different end. That’s fine. But remember that God designed a world that operates in harmony with mathematics. He invented it all. Every day we study math, we take one more step into the vast reaches of the mind of God. And every such step is its own reward.

Endnotes

^[1] Mason, C. (1925). Towards a Philosophy Education, p. xxix.
^[2] Mason, C. (1886). Home Education, p. 145.
^[3] Baburina, R. (2012). Mathematics: An Instrument for Living Teaching, pp. 36–37.
^[4] Ibid., pp. 31–32.
^[5] Mason, C. (1886). Home Education, p. 128.
^[6] Mason, C. (1904). Parents and Children, p. 39.
^[7] Mason, C. (1925). Towards a Philosophy Education, p. 233.
^[8] The Parents’ Review, volume 66, p. 128.
^[9] Mason, C. (1925). Towards a Philosophy Education, p. 240.
^[10] The Parents’ Review, volume 15, p. 794.
^[11] Mason, C. (1886). Home Education, p. 146.
^[12] Mason, C. (1904). Parents and Children, pp. 124–125.
^[13] Ibid., p. 123.
^[14] Foerster, P. (2006). Algebra I, pp. 377–378.
^[15] Mason, C. (1925). Towards a Philosophy Education, pp. 230–231.
^[16] The Parents’ Review, volume 40, pp. 39–40.
^[17] Mason, C. (1925). Towards a Philosophy Education, p. 334.
^[18] Boyer, C. (1991). A History of Mathematics, p. 228.
^[19] The Parents’ Review, volume 34, p. 170.
^[20] Cholmondeley, E. (1960). The Story of Charlotte Mason, p. 157.
^[21] Chandler, D. “Assessment (Testing and Grading) for High School Homeschoolers.” Retrieved 9/6/2020.
^[22] The Parents’ Review, volume 41, p. 285.
^[23] Baburina, R. (2012). Mathematics: An Instrument for Living Teaching, p. 19.
^[24] Mason, C. (1925). Towards a Philosophy Education, p. 231.
^[25] Ibid., p. 51.

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