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Teaching Math: Subject by Subject, Part 17
(I am so pleased to welcome a guest writer who excels at presenting an insightful summary of math the Charlotte Mason way. Richele Baburina has extensively researched Charlotte Mason’s approach to math and compiled her findings in her excellent book, Mathematics: An Instrument for Living Teaching, and helpful DVD series, Charlotte Mason’s Living Math: A Guided Journey.)
Charlotte Mason valued a generous curriculum. Though emphasis on the three R’s alone makes for a meager education, these subjects did have their rightful place of study.
Charlotte valued the study of arithmetic primarily for its use in training mental and moral habits, including accuracy, attention, careful execution, neatness, and truthfulness. Though its use in daily life was important, it was the “beauty and truth” of mathematics, that awakening of a sense of awe in God’s fixed laws of the universe, that afforded its study a rightful place in Charlotte’s curriculum.
Let’s take a brief look at how mathematics are taught in a CM education—because without living teaching, that sense of wonder might not be awakened nor the desired habit training take place.
The Early Years
The study of arithmetic falls well within Charlotte’s definition that “education is an atmosphere, a discipline, and a life.” Before the age of six, a child’s education is by means of his senses, natural environment, and unstudied games. Direct preparation for mathematics in these years is considered not only undesirable but detrimental.
The formal study of arithmetic begins at about six years of age and is characterized by thorough, careful work in which the children make discoveries for themselves. Its study follows Charlotte’s basic principles of short lessons with concentrated attention.
Though the term math manipulative did not exist in Charlotte’s time, the use of concrete objects as aids in conveying ideas is significant in her method of teaching arithmetic.
Some important points to remember:
- All the manipulatives you need can be found in your own home—beads, buttons, and craft sticks to name just a few. A variety of simple objects should be used rather than a single specially-designed manipulative so the child doesn’t form a hard-and-fast connection between the math facts and the manipulative.
- Manipulatives are only a tool to the presentation or investigation of an idea. If a manipulative’s use requires too much teaching, it becomes more important than the idea it is to represent.
- Arithmetic tables should not be memorized until the child proves the facts first through the use of manipulatives.
- Allow your child enough time to work with the manipulatives but then progress to working with imaginary objects. Once the child can mentally picture the number, or has grasped the abstract, put away the manipulative until the introduction of a new concept.
“A bag of beans, counters, or buttons should be used in all the early arithmetic lessons, and the child should be able to work with these freely, and even to add, subtract, multiply, and divide mentally, without the aid of buttons or beans, before he is set to ‘do sums’ on his slate” (Vol. 1, p. 256).
Mental Arithmetic and Oral Work
In Charlotte’s methods of teaching mathematics, written work is used sparingly. Mental arithmetic and oral work help reinforce math facts and vocabulary, plus they are instrumental in the training of good habits.
“Give him short sums, in words rather than in figures, and excite in him the enthusiasm which produces concentrated attention and rapid work. Let his 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).
While children advance in their understanding, the oral questions should always remain within their ability.
“Engage the child upon little problems within his comprehension from the first, rather than upon set sums” (Vol. 1, p. 254).
“Now he is ready for more ambitious problems: thus, ‘A boy had twice ten apples; how many heaps of 4 could he make?’ ” (Vol. 1, p. 257).
Some points to consider:
- The oral questions we give our children should be engaging. For example, “How old will you be when your sister is four” will be more apt to fix your child’s attention than the same question given as, “Add 4 + 5.”
- Require your child to give fully worded answers in complete sentences for the most benefit.
- Along with oral work throughout the math lesson, consider following Charlotte’s schedule of five minutes of rapid drill at the end of the lesson or ten minutes for older children at another time in the daily schedule.
Careful Teaching vs. Careless Teaching
Charlotte felt that careless teaching—which includes offering crutches and failing to pronounce sums wrong—fosters habits of carelessness in children. In contrast, carefully graduated lessons, along with Charlotte’s methods already mentioned, foster the training of good habits.
“Arithmetic is valuable as a means of training children in habits of strict accuracy, but the ingenuity which makes this exact science tend to foster slipshod habits of mind, a disregard of truth and common honesty, is worthy of admiration! The copying, prompting, telling, helping over difficulties, working with an eye to the answer which he knows, that are allowed in the arithmetic lesson, under an inferior teacher, are enough to vitiate any child; and quite as bad as these is the habit of allowing that a sum is nearly right, two figures wrong, and so on, and letting the child work it over again. Pronounce a sum wrong, or right—it cannot be something between the two.” (Vol. 1, p. 260).
“Therefore his progress must be carefully graduated; but there is no subject in which the teacher has a more delightful consciousness of drawing out from day to day new power in the child. Do not offer him a crutch; it is in his own power he must go” (Vol. 1, p. 261).
Living Math Books
Charlotte believed mathematics fell outside her rule of literary presentations. She stated:
“…mathematics, like music, is a speech in itself, a speech irrefragibly logical, of exquisite clarity, meeting the requirements of mind” (Vol. 6, pp. 333, 334).
Charlotte did not employ the modern notion of “living math books” to teach mathematical concepts. She advocated acquainting the children with the “captain” ideas of math by introducing the different branches or their great thinkers through an interesting or exciting history.
The methods we’ve discussed today are not just for the teaching of elementary arithmetic; they also apply to more advanced arithmetic: geometry, algebra, and beyond. Whether you are comfortable teaching the higher levels of mathematics or rely more heavily on textbooks, a curriculum, or a tutor, be sure to ensure a living treatment of math for your older child as well.
- Guide your older child in discovery, allowing her to think for herself. Be patient and advance slowly. Allowing your older child to wonder, discover and permit ideas to germinate.
- Practical exercises should continue along deductive exercises in geometry, and the practical side of algebra should be introduced as early as possible.
- Provide a slow, steady approach with lots of practice.
- Exclude long or tedious examples for calculation.
The Mathematics: An Instrument for Living Teaching book includes all the details and practical step-by-step instructions that describe how math was taught through each advancing level of Charlotte’s classrooms—from learning numbers through algebra and geometry.
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I appreciate this helpful and thorough insight on teaching Mathematics! Many great points and reminders!
Is there a typo in the section, Careful vs Careless Teaching?
“…failing to pronounce sums wrong…”
Should that be,
“…failing to pronounce sums correctly…”
If not please explain.
Great resources. (as always!)
Good question. Charlotte was cautioning against some teachers’ tendency to give a math problem to a child, and if the child gets the sum close, the teacher pronounces it “nearly right.” Charlotte said, “Pronounce a sum wrong, or right—it cannot be between the two.”
Now that makes sense.
Sounds like a great book.
I have been considering ordering Right Start math, but have been doing some research first, including reading this post. This comment caught my attention: “All the manipulatives you need can be found in your own home—beads, buttons, and craft sticks to name just a few. A variety of simple objects should be used rather than a single specially-designed manipulative so the child doesn’t form a hard-and-fast connection between the math facts and the manipulative.” Hmmm…doesn’t this seem almost the opposite of Right Start, especially in it’s use of the abacus and teaching kids to visualize it? Can anyone who has used it comment on this?
RightStart uses a variety of manipulatives, not just the abacus. You would probably be able to substitute items in your home, but it’s handy to have the whole pack of craft sticks and colored tiles and cards and such already collected and in one place.
Any ideas on late middle school/high school math?
Richele’s tips at the end of the post are a great place to start.
In addition, you might take a look at this forum discussion where Richele talked a bit more about math in those upper grades.
Any tips on where to start with an 8 year old? He still struggles with simple addition facts like, ‘4 and six is or 4+6=’. He can only add or subtract mentally up or from five. 5+5 is even a struggle. He was already introduced to symbols at his previous school and has ‘+/plus and =\equals down’.
I pulled him and I am now homeschooling him. Should I start all over with just objects and maybe have him follow rays arithmetic from the beginning?
Reading fluency has been our main goal lately with little questions here and there that are math related but I worry he’s not getting enough at his age and I don’t want to confuse him or delay his progress anymore than it already has. On the upside, I have 18 months by law before he’s required to do any state testing and maybe longer due to his learning disability the state put in place. :/
That sounds like a good plan, Tricia. Go back and secure the ground under his feet. Since the majority of the work will be oral, his reading fluency shouldn’t be an issue with this method. The DVD set, Charlotte Mason’s Living Math: A Guided Journey, will give you a clear picture of how to proceed. It really helped me!
What about Math U See, doesn’t that use a set of manipulatives that does not change?
Yes, it does, and that’s one of the points in which MUS is not a true CM-style math curriculum. I wish there was a pure CM math program that we could recommend. Until that time comes, we try to recommend math programs that have some of the CM methods and principles relating to math, and provide tools (like the mathematics book and DVD set) to help you tweak them more toward CM methods.
Which program is closer to CM style math, RightStart Mathematics or Math-U-See? Which of the two would you recommend?
Hi, Susan. Neither is a truly CM approach to math; we don’t know of any truly CM math curriculum currently available. However, RightStart and Math-U-See both contain some of the components of a CM approach. They both use manipulatives as an integral part of the lessons, rather than just an occasional add-on. They both emphasize the student’s having an understanding of mathematical thinking, the Why behind the How. They both allow you to go at the student’s pace. We recommend both for those reasons.
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