# Math

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What is mathematics?

As one online source puts it, mathematics is the study of representing and reasoning about abstract objects. In the early years, those "abstract objects" will consist largely of numbers, but as students progress, they will learn to apply their reasoning skills to ideas such as points, spaces, sets, and more. Some teachers and parents view mathematics as a utilitarian, life-skills topic: necessary for counting coins or cows, measuring fields, or doing one's taxes. Others go in the opposite direction: they value the abstract nature of mathematics so much that all their teaching is for those gifted in those areas. In either case, much of the potential joy and wonder of the subject can be lost.

I'm very well acquainted, too, with matters mathematical,
I understand equations, both the simple and quadratical,
About binomial theorem I'm teeming with a lot o' news,
Hmmm... lot o' news, lot o'news... Aha!
With many cheerful facts about the square of the hypotenuse.

I'm very good at integral and differential calculus;
I know the scientific names of beings animalculous:
In short, in matters vegetable, animal, and mineral,
I am the very model of a modern Major-General.
(Gilbert and Sullivan, The Pirates of Penzance)

Is mathematics an entirely objective, unbiased subject?

Thus, for instance, it often is with 'instruction' and 'education,' Cannot we 'instruct' a child, it is asked, cannot we teach it geography, or arithmetic, or grammar, quite independently of the Catechism, or even of the Scriptures? No doubt you may; but can you 'educate' without bringing moral and spiritual forces to bear upon the mind and affections of the child? (Trench, Study of Words)

We agree with Charlotte Mason and Trench that, while it is possible to offer an adequate, though somewhat lifeless, mere instruction in arithmetic (and some of us may look back at our own schooling in exactly those terms), we are, as Miss Mason said, "jealous for the children," and we believe that they deserve not simply mathematical instruction, but that it should be a vital part of their education. As Miss Mason also said, we also believe that all areas of learning are under the inspiration of the Holy Spirit, and that as those "moral and spiritual forces" are a part of all we do, they must also be reflected in our approach to teaching "matters mathematical."

The giant math tree

(A story from Anne) When I was in elementary school, my arithmetic grades were quite decent, and I even managed to survive the full course of high school mathematics. However, I viewed math mainly as something that people had to get through in school, and which seemed to consist mainly of putting numbers through painful contortions, and later doing the same to points on graphs. Why anyone would want to pursue that subject any further, or find it enjoyable, was a mystery.

Much later, when our oldest daughter was quite small but I was beginning to acquire school-related books for her, I picked up a large book on the history of mathematics (I can't remember the title now). On one of its pages was a drawing of a tree, with all its large and small branches labelled with various kinds of mathematics. Looking at this picture, I had an a-ha moment when I noticed that "Arithmetic" was just one small branch of a very large and complex tree (something like the effect of Monty Python's Galaxy song). My reaction wasn't one of intimidation ("I don't even know what most of those things are!"), but rather of reassurance. There is a great big world of mathematics out there, I thought, and how one does at childhood arithmetic tasks may, but also may not be, significant in predicting how one interacts with that larger tree, or even with other possible unknown trees, during the rest of one's life.

(As a P.S. on that, I found it useful to keep that thought in mind during the next few years, as that same oldest daughter found things like memorizing times tables to be quite difficult, but at the same time showed a strong ability to reason abstractly, and solve puzzles that didn't require remembering how to regroup in subtraction. In fact, she went on to get a degree in math and a job working with computers.)

Consider what Charlotte Mason wrote about early Bible teaching:

Let us have faith and courage to give children such a full and gradual picture of Old Testament history that they unconsciously perceive for themselves a panoramic view of the history of mankind typified by that of the Jewish nation as it is unfolded in the Bible. Are our children little sceptics [sic], as was the young Goethe, who take a laughing joy in pulling their teachers with a hundred difficulties? Like that wise old Dr. Albrecht, let us be in no haste to explain. Let us not try to put down or evade their questions, or to give them final answers, but introduce them as did he to some thoughtful commentator who weighs difficult questions with modesty and scrupulous care. If we act in this way, difficulties will assume their due measure of importance, that is to say, they will be lost sight of in the gradual unfolding of the great scheme whereby the world was educated. (Philosophy of Education, p. 162)

To misquote this, "Let us have faith and courage to give children a full and gradual picture of mathematics." The fullness of the tree.

How and why did Charlotte Mason teach mathematics?

You will find much ease in your labors from carefully reading and pondering her insights for teaching math in Volume 1, pages 253-264, and Volume 6, pages 110-112; 151-152; 230-231.

We encourage you to initiate discussion of these passages with the friendly and experienced members of our forum, among whom there is an astonishing wealth of wisdom and experience in teaching with CM methods.

And what do we say about it?

Is there a "correct Charlotte Mason approach" to math teaching?

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. (p. 254)

But here's something else Miss Mason said about teaching:

The objects which bore us, or the persons who bore us, appear to wear a bald place in the mind . . . (Home Education, p. 263)

First of all, and this has been widely discussed, Miss Mason differentiates between training the reasoning powers and developing them (in the Herbartian sense). Our reasoning is not a seed to be watered by educators in the hopes that it will suddenly spring from the ground; it is, perhaps, more like an already-sprouted plant that can be either trained to grow along certain lines, or left to climb wildly.

Carefully graduated teaching and daily mental effort on the child's part at this early stage may be the means of developing real mathematical power, and will certainly promote the habits of concentration and the effort of mind (Home Education, p. 257)

Second, when discussing the benefits of mathematics, we tend to focus on "accuracy" and "intellectual truthfulness," and perhaps "habits of concentration," but Miss Mason also mentions "habits of insight." What did she mean by that? In-sight sounds a lot like vision, doesn't it? So to be successful at any more than minimal mathematics skills, we need to develop insight and agility. We need to hone our ability to see what is relevant in any situation (including a math problem), and let go of the rest.

Mathematics are a necessary part of [everyone's] education; they must be taught by those who know; but they may not engross the time and attention of the scholar in such wise as to shut out any of the score of 'subjects,' a knowledge of which is his natural right. (Philosophy, p. 233)

Curriculum: Do we even need one?

In the early years in particular, it may be entirely possible to teach math without a formal curriculum, especially one which requires written work by the student. As in learning to read, students' capacity to comprehend mathematical ideas may outstrip their ability to remember in which direction the "6" goes, and the one should not place a limit on the other. The use of hundred charts, concrete objects, card games, dominoes, coins, and real-life counting and measuring opportunities may form the greatest part of some children's early (or later) math education.

But when we do want a purchased, or otherwise predetermined curriculum, what do we look for? What are some guidelines?

Look for something that explains not too little, not too much, but just enough to make students think. That provides enough practice problems, but not too many. That recapitulates and reviews material regularly, but not to the point of boredom.

And something that doesn't cause your students (or you) frequent tears or tantrums.

Simple, right?

Some of the conflict we have over how to teach math (or how much of it to teach) comes from confusion over words, especially words like "discovery learning" and "rote." The problem is that, like that swinging gate that Miss Mason once mentioned, people on either end assume that there's no middle point. Either you teach math facts, math facts, math facts, and then maybe flounder when students rebel at long division and can't quite grasp fractions; or you grab onto the latest discovery-based fads, such as forcing students to explain every thinking strategy they used in a probability question. In both cases, you just hope that the concepts will eventually click.

In terms of curriculum, for some families the "right" approach might be nothing more than buying a solid textbook or workbook series, or even a set of video-based lessons, and working through each lesson sequentially and at the student's own pace. Other families or classes might be looking at innovations such as ungraded curriculum, or math journaling or hands-on projects.

(Homeschoolers may be given a checklist of topics by an umbrella group, or have to cover certain material to pass examinations. If you are in that sort of situation, of course you will need to consider those additional requirements when planning the year's math curriculum.)

(From Anne) Years and years ago, I think before I even had children, I was shown a school math textbook where all the word problems seemed to be about the animals on Noah's ark. When I later came across Charlotte Mason's passage about not having to correlate math lessons with unit studies, I was reminded of that book. There's nothing at all wrong with trying to come up with interesting word problems in math--I don't think Miss Mason was saying that at all--but real-world connections shouldn't be artificial, and we don't need to fill math class with Bible characters to make it more "Christian." It can stand quite well on its own.

Miss Mason made it clear that she did not approve of certain mathematical gizmos because they seemed to do students' thinking for them. Whenever possible, in any subject, she preferred improvised, noncommercial models. Drawing a quick and simple map in the dirt was considered better than showing children something elaborate and pre-made. How might this relate to curriculum, to commercial math manipulatives, and particularly to number tools such as Cuisenaire rods?

(From Anne) As one who used Cuisenaire rods from the earliest days of teaching my own children, I was of course a bit disgruntled when I came across that passage. However, I finally settled in my own mind that, depending on how these are used, sets of colored rods can encourage mathematical thinking, and particularly the building of number relationships, in a way that is quite within C.M. parameters, and as productively as, say, piles of beans or other individual counters. The key for us was in not marking the rods with numerals, but in allowing them to represent any numbers that were needed for the lesson. (For example, the longest, 10 cm rod might represent 10, but it might also be 100, 1/2, or really any number at all; then the shorter rods would be named in relation to that one. These rods are necessary to use Miquon Math, which listed under Curriculum). Another point would be (as with any manipulative) not to overuse or become completely dependent on them; variety is important.

As in learning a musical instrument, a certain amount of this will usually be necessary in arithmetic, and a short (but important) time in lessons should be given to oral or written practice in things such as skip counting and times tables.

What part should we allow electronic devices to play in our everyday lives, and, more specifically, what place they will take in our approach to math teaching? Should we allow the use of calculators at some point to relieve the drudgery of long division and square roots? What are the pros and cons of a completely digital math curriculum? How about the occasional or regular use of online games to practice math facts, or more sophisticated software to model higher math concepts? Space here prevents much discussion of this, but again we must point back to Charlotte Mason's core values: nurturing wonder and curiosity, and using tools well and wisely, but also sparingly, in the hope that we may preserve, as far as possible, our ability to think.

Do We Have to Keep Math Lessons to 15 or 20 Minutes?

For young children, shorter is better, especially when you're homeschooling. Even if you are following a K-3 curriculum that suggests longer lessons, they can often be broken up, or busywork can be skipped. It's also possible, though not always practical, to have separate math-related timeslots, such as a "drills" time or a "math reading" time. And of course lots of math learning can happen in other subjects and in daily activities.

However, back in our early days of C.M. homeschooling, some of us struggled with the misperception that all math teaching should be limited to such short periods, including lessons for older students. Even in Miss Mason's day, that was not the case: the time allowed for math lessons increased as the students grew, and in the upper years a geometry or algebra lesson would probably take three quarters of an hour.

The important point seems to be to keep things in proportion. In cultures with a heavy emphasis on utilitarian education, math and science courses may be given a disproportionate amount of time (plus homework). Miss Mason herself was a bit of a rebel for her insistence that there was more to a quality education than Latin, Greek, and mathematics. Similarly, some homeschoolers claim to focus on the "3 R's," spending a great deal of time on grammar and arithmetic, and less on not only the "fun" fine arts subjects, but on the vital ideas taught in the "social economy" subjects (but that's another curriculum page). No matter what the age of our students, most of them (with, to paraphrase Miss Mason, the exception of a few born math geeks) need a balanced educational diet, and it's up to us as parents/teachers to make sure that happens.

An unpleasant elephant in the room

As arithmetic in particular is a "disciplinary subject," it can become a testing ground, or, more unhappily, a battle ground between teachers/parents and students. Conversely, math class can be a beautiful opportunity for many vital habits, such as perseverance to take root; and for teachers to demonstrate the proper use of authority (see, for example, "Docility and Authority in the Home and the School," in School Education). Although mathematics may not end up being everyone's favourite subject, it should never be allowed to become the most dreaded part of the day. While it cannot and should not be presented as nothing but play, while sloppy work or dawdling should not be tolerated; and while wrong answers cannot be passed off as correct, math teaching also cannot be maintained for long as an uphill climb to a place with no oxygen. It may be time for a change of scenery (or presentation); for examination of the amount of time being given (too long) or the learning environment (too many distractions); or even, at certain times, for a break from formal math teaching. Something that seems impossible today may be learned quickly six months from now.

(From Anne: This is as true for older students as for younger ones. My oldest daughter, the eventual math major, did very little math during what should have been her ninth-grade year. She was attending public high school part time and had a lot of homework, so the math I had planned to do with her kind of fell through the cracks. The next year, however, she wanted to take math at school, and the counselor placed her in a grade ten class even without her taking the grade nine credit. The rest, I guess, is history.)

To Sum Up

No matter what approach is chosen, there are a few things to keep in mind.

We avoid a purely utilitarian approach, but take our cue from Charlotte Mason's educational principles such as "Children are born persons" and "Education is the science of relations."

Consecutive learning mattered to Miss Mason. Not wasting time, in any sense, was important. (Baking cookies is often touted as a great "natural" way to teach math, but there are other less convoluted ways to present those concepts. Miss Mason would probably advise us to keep the math out of the baking, except as it comes up incidentally.)

Accuracy was important, but so was "awe." Presenting things in context, building connections, seeing relationships, were vitally important. That might mean exploring the history of how people understand numbers, or how they have tried to deal with them on clay tablets and blackboards and paper over the years. It might mean providing books like Kenn Amdahl's Algebra Unplugged, a simple overview that might have changed my own outlook on high school math. It might mean the creative use of word problems, and teaching good strategies for solving them, including the estimation of reasonable answers. It might mean doing a mixture of multiplication, division, and fraction questions to help students see that those concepts are all interchangeable. (Once you understand that, arithmetic gets so much easier.)

With those principles supporting us, we may confidently invite our students to join this vital part of the Great Conversation.

## Suggested Resources

There are many good math programs available to homeschoolers. When selecting the best fit for your family, consider your own math proficiency, and how much time you can realistically devote to teaching math every day. We list here (in alphabetical order) the resources recommended by AO users (and which have often been reviewed and discussed at length in the Forum). Please read reviews, obtain samples, and otherwise check to see that any resource is appropriate for your needs.

We would also suggest reading the "Important Threads" posts in the AO Math subforum.

Note: Math programs generally follow either a mastery formula or spiral formula. In spiral programs, skills are cycled through lessons for review over multiple years, which is fine as long as the skills are truly mastered when they cycle through the lessons. When selecting a math curriculum, and when planning how you will assess skill proficiency with that curriculum, it is helpful to know from the start whether it is a mastery program or a spiral program.

Books and Websites for Teachers:

Mathematics: An Instrument for Living Teaching, by Richele Baburina, available at Simply Charlotte Mason

An Easy Start in Arithmetic, by Ruth Beechick -- This pamphlet is packed with teaching wisdom for grades 1-3. It's part of the Three R's packet. (\$amzn) Highly recommended.

Living Math by Julie Brennan-- math resources with a CM focus.

Let's Play Math: How Families Can Learn Math Together--and Enjoy It, by Denise Gaskins contains many, many resources, including book suggestions. Sold at Tabletop Academy or Amazon. (\$amzn) (K) Denise Gaskins' website is a fabulous resource for all kinds of math activities.

Seashell Math by Lynn Hocraffer -- A helpful model for teaching multiplication using CM methods, by the leader of the original cmason yahoo group.

Games for Math, by Peggy Kaye (\$amzn) A book of fun and effective math games using household objects; for grades K-4, approximately.

Knowing and Teaching Elementary Mathematics, by Liping Ma (\$amzn) (K)

The Myth of Ability: Nurturing Mathematical Talent in Every Child, by John Mighton (\$amzn) (K)

Read Any Good Math Lately?: Children's Books for Mathematical Learning, K-6, by David J. Whitin and Sandra Wilde (\$amzn)

## Parents' Review Articles About Math

Children's Arithmetic by the Rev. R. H. Quick
Home Arithmetic by Mary Everest Boole
The First Stage in Arithmetic pt 1 by Rev Quick (part 2 is here)
Mental Arithmetic by Amy Pridham
Teaching Arithmetic pt 1 by C. H. Wilkinson (part 2 is here)
Notes of Lessons has a sample math lesson.
Nursery Examples of Fractions discusses math for very young children.
See Index of Articles for more math articles.

## Curriculum for Elementary through Middle School

Beauty and Truth Math which uses Strayer-Upton Arithmetic books. These books can be accessed online or purchased at Yesterday's Classics.

Frank Hall Arithmetic - Free on Archive.org/Google Books for elementary/middle.

Gattegno Mathematics Textbook I, and others in the series, by Caleb Gattegno. (Uses Cuisenaire Rods) The first book can be accessed at issuu.

Khan Academy -- Free video lessons in math and science.

Life of Fred Elementary Series, Complete 10-Book Set (\$amzn) Books also available at Rainbow Resources

Math on the Level - a complete homeschool math curriculum

Math with Confidence by Kate Snow - Comment from an international user: "I appreciate the suggestions throughout on adapting content if you're not in the US, rather than having US money and measurements 'baked in'. I think it is a great combination of budget friendly, open and go, and emphasising number sense with concrete objects in the early years." You can find the various texts and workbooks by searching Amazon.com. (\$amzn)

Math U See -- Teaching videos, manipulatives and consumable texts.

Mathematical Reasoning Series, available at The Critical Thinking Co. Elementary level. Comment from a user: "I'm not sure why Mathematical Reasoning doesn't get talked about more . . . It's straightforward, open-and-go, budget-friendly, does NOT require a student book and teacher manual (an answer key is at the back of each book), can be used with things around the house (does not require special manipulatives), is visually appealing, has short lessons, and develops logic."

MEP (Mathematics Enhancement Programme) -- A free math curriculum from the University of Plymouth in Great Britain. Download the practice books, lesson plans, and answer keys (for upper years) through the University website. Password required for use of some portions of the site, but homeschoolers in the US have had no problem obtaining one when they ask. MEP has a Facebook support group.

Minimalist Math Curriculum on ResearchParent.com - recommended by Denise Gaskins of Let's Play Math

Miquon Math -- Based on Cuisenaire rods; very kinesthetic. Teacher-intensive; teacher materials can be confusing. For primary grades. Read an extensive review on Anne White's blog.

Ray's Arithmetic -- Historical, no-frills set of books for grades 1-8; recommended by Ruth Beechick. These are the texts used in US schools in the 19th and early 20th centuries, and the computational skills are advanced compared to modern math texts. Lots of word problems; develops strong mental calculation skills. Affordable; very teacher-intensive.

Right Start Mathematics -- Based on the abacus; teacher-intensive. Favored by many CM teachers. For grades 1-8.

Singapore Math -- Uses methods that have helped Singapore obtain high international math scores. May need supplementing with another math program.

Teaching Textbooks -- An interactive program using CD-roms and an optional text; virtually self-teaching. Also a favorite among CM teachers. Elementary levels through Pre-Calculus.

## Books for Math Students: Elementary through Middle School

The Number Devil: A Mathematical Adventure, by Hans Magnus Enzensberger. Review from Anne: "A math-hating boy named Robert has a series of dreams, each featuring a little red guy with horns who does amazing things with numbers. The dreams have different topics, but they're definitely sequential . . .and they come back to a starting place, usually â€˜one.' (Who knew that 'one' could be so interesting?)" (\$amzn) (K)

Alvin's Secret Code, by Clifford B. Hicks (One of Anne's childhood favourites) (\$amzn)

The Phantom Tollbooth, by Norton Juster (\$amzn) (K)

The Sir Cumference series of picture books, by Cindy Neuschwander (also at Rainbow Resources):
Sir Cumference and the First Round Table: A Math Adventure (\$amzn) (K)
Sir Cumference and the Dragon of Pi (\$amzn) (K)
Sir Cumference and the Great Knight of Angleland (\$amzn) (K)
Sir Cumference and the Sword in the Cone (\$amzn) (K)
Sir Cumference and the Isle of Immeter (\$amzn) (K)
Sir Cumference and All the King's Tens (\$amzn) (K)
Sir Cumference and the Viking's Map (\$amzn) (K)
Sir Cumference and the Off-the-Charts Dessert (\$amzn) (K)
Sir Cumference and the Roundabout Battle (\$amzn) (K)
Sir Cumference Gets Decima's Point (\$amzn) (K)
Sir Cumference and the Fraction Faire (\$amzn) (K)

The Adventures of Penrose the Mathematical Cat, by Theoni Pappas (\$amzn) (K)

Murderous Maths series, by Kjartan Poskitt, 10-book set (\$amzn)

G is for Googol: A Math Alphabet Book, by David M. Schwartz (and others by the same author) (\$amzn)

Number Stories of Long Ago, by David Eugene Smith (\$amzn) View for free at Archive.org

Roman Numerals I to MM, by Arthur Geisert (\$amzn)

## Curriculum for Middle through High School and Beyond

Art of Problem Solving (for middle/high school)

Algebra 1, from BJU Press

Harold Jacob's Elementary Algebra, Basic Geometry and Mathematics: A Human Endeavor -- Classic texts that are highly rated for being engaging, thorough and yet accessible. Available from various sellers online, including MasterBooks.

Key to . . . Math series of workbooks (Key to Percents (\$amzn), Key to Decimals (\$amzn), Key to Fractions (\$amzn), Key to Measurement (\$amzn), Key to Metric Measurement (\$amzn), Key to Algebra (\$amzn), Key to Geometry (\$amzn)). Designed to be used through upper elementary and middle school, and may be useful for some high school students. Recommended also for those who need extra help in target areas such as decimals. (\$earch) Rainbow Resources

Life of Fred, High School Set 1 (Beginning and Advanced Algebra) (\$amzn) High School Set 2 (Geometry and Trigonometry) (\$amzn) Books also available at Rainbow Resources

Khan Academy -- Free video lessons in math and science.

The Math Page -- Free online courses in arithmetic, Euclid-based plane geometry, and algebra, as well as topics in trigonometry and calculus.

Math U See - Algebra 1, Geometry, Algebra 2

Purple Math -- Free online algebra.

Ray's Arithmetic -- An economical upper-level math course on CD-rom. Check for availability of answer keys. (Ray's Algebra page images online) (\$amzn)

The Great Courses/The Teaching Company courses -- They offer a wealth of courses that make math come alive.

Teaching Textbooks -- CD-rom courses for Algebra, Geometry and Pre-Calculus.

VideoText -- DVD program for Algebra and Geometry

## Books for Math Students: Middle through High School and Beyond

Elements, by Euclid -- The historic, standard text of classical geometry.

String, Straightedge, and Shadow, by Julia Diggins-- A living book supplement which geometry students enjoy. (\$amzn)

Math with Bad Drawings: Illuminating the Ideas that Shape Our Reality, by Ben Orlin. (Previewing recommended.) (\$amzn) (K)

Exploring the World of Mathematics: From Ancient Record Keeping to the Latest Advances in Computers, by John Hudson Tiner (\$amzn)

How to Use an AL Abacus (instead of manipulatives) with Any Math Curriculum

Exploding Dots - Comment from a user: "Great for place value (among other things)."

[Math] Facts that Stick, by Kate Snow. Books for Addition (\$amzn), Subtraction (\$amzn), Multiplication (\$amzn), Division (\$amzn)

Balance Benders, by Robert Femiano: Level 1 (\$amzn) Level 2 (\$amzn) Level 3 (\$amzn)

Natural Math (Click Books and Goods) - not really a curriculum, but the books are a playful beginning and/or supplement

Monopoly money - Prints ten bills per sheet, which can be used to practice counting money.

From an AO parent: "There is also a wonderful facebook group called Charlotte Mason Math Together which is about using CM methods applied to math."

More detailed comparisons of math curricula:
Cathy Duffy Math Reviews (many reviews online; more in Duffy's curriculum guide books)

By Anne White, July 2022

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