We are still early in our math journey. my girls are in elementary school, but I have become a student of math over the past couple of years. I am excited by the principles I have learned and have tried to apply them in our daily life.
I would like to share the principles I have used to design our math philosophy. They are largely shaped by Charlotte Mason, a British educator who lived in the late 1800′s and early 1900′s, and Louis Benezet, Superintendent of Schools in Manchester, NH in the 1920-30′s.
Charlotte Mason’s thoughts on math
Charlotte Mason’s principles of arithmetic can be found in The Original Homeschooling Series, Volume 1, pages 253-264 and Volume 6, pages 230-233.
“Of all his early studies, perhaps none is more important to the child as a means of education than that of arithmetic. . . The chief value of arithmetic, like that of the higher mathematics, lies in the training it affords to the reasoning powers, and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders.” (Vol 1, p 253-254)
It is not helpful to the child to know how to complete a pages of problems if he does not know what rule to apply in different situations.
Charlotte Mason suggested these principles when teaching mathematics:
- As with all lessons, keep arithmetic lessons short – 15-20 minutes for grades 1-3.
- Make sure the problems are within the child’s grasp, but are challenging enough to cause mental effort.
- Use word problems as a means of helping the child learn which rules to apply.
- Demonstrate everything demonstrable.
- Only after mastering a concept with manipulatives should the child write the problem with mathematical notation with pencil and paper.
- Allow the child to use manipulatives at each new stage or learning concept. As the child begins to understand the concept, encourage him to use imaginary manipulatives instead of concrete ones, but allow him to use manipulatives as long as necessary.
- Mathematics should not be given an undue proportion of time teaching.
She concludes the section in Volume 1 (p 264) by saying,
“I do not think that any direct preparation for mathematics is desirable. The child, who has been allowed to think and not compelled to cram, hails the new study with delight when the due time for it arrives.”
Louis Benezet’s thoughts on math
Benezet published a three part series of articles in the Journal of the National Education Association in 1935. He was responding to a question of what to cut out when the schools are constantly being asked to add more to the curriculum. Here are two paragraphs from the article:
“In the first place, it seems to me that we waste much time in the elementary schools, wrestling with stuff that ought to be omitted or postponed until the children are in need of studying it. If I had my ways I would omit arithmetic from the first six grades. I would allow the children to practice making change with imitation money, if you wished but outside of making changes where does an eleven-year-old child ever have to use arithmetic?
I feel that it is all nonsense to take eight years to get children thru the ordinary arithmetic assignment of the elementary schools. What possible needs has a ten-year-old child for a knowledge of long division? The whole subject of arithmetic could be postponed until the seventh year of school, and it could be mastered in two years’ study by any normal child.”
He decided that he should put into practice what he suggested. He removed all formal instruction in arithmetic below seventh grade in five test classrooms and “concentrated on teaching the children to read, to reason, and to recite – my new Three R’s.” The children were encouraged to do a lot of oral composition or narration. He said it was a joy to enter these classrooms. “The children were no longer under the restraint of learning multiplication tables or struggling with long division. They were thoroughly enjoying their hours in school.”
There are three main points of Benezet’s articles.
- He proposed cutting out formal instruction in mathematics in grades 1-6.
- He encouraged a focus on English expression through narration about books, visits, and experiences.
- He encouraged a focus on reasoning and estimating.
No formal instruction in grades 1-6
“In the first place, it seems to me that we waste much time in the elementary schools, wrestling with stuff that ought to be omitted or postponed until the children are in need of studying it.”
“For some years I had noted that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child’s reasoning faculties.”
Focus on English expression
“Meanwhile, I was distressed at the inability of the average child in our grades to use the English language. If the children had original ideas, they were very helpless about translating them into English which could be understood.”
He went into an eighth grade classroom and asked the children to describe two fractions. They were not able to coherently provide an answer. He said it was not the children or the teacher to blame but the curriculum. “If the course of study required that the children master long division before leaving the fourth grade and fractions before finishing the fifth, then the teacher had to spend hours and hours on this work to the neglect of giving children practice in speaking the English language.”
After eight months he returned to all fourth grade classrooms in the city. “The contrast was remarkable. In the traditional fourth grades when I asked children to tell me what they had been reading, they were hesitant, embarrassed, and diffident. In one fourth grade I could not find a single child who would admit that he had committed the sin of reading. I did not have a single volunteer, and when I tried to draft them, the children stood up, shook their heads, and sat down again. In the four experimental fourth grades the children fairly fought for a chance to tell me what they had been reading. The hour closed, in each case, with a dozen hands waving in the air and little faces crestfallen, because we had not gotten around to hear what they had to tell.”
Benezet visited seventh grade classrooms and hung a picture for them to observe. They were asked to write whatever they felt inspired to put down. The traditional classrooms, whose children came from English speaking parents, had used 40 adjectives (nice, pretty, blue, green, cold, etc.). The experimental classroom, whose children came from non-English speaking homes except three children, had used 128 adjectives (magnificent, awe-inspiring, unique, majestic, etc.).
Focus on reasoning and estimating
After a year of practice in reasoning and estimating, Benezet visited the experimental rooms and asked questions about Niagra Falls. Not only were the children able to tell him about the picture of the falls that he drew, they were able to estimate how long it would be before the falls retreated to Buffalo and drained the river based on the knowledge of what has already happened. When Benezet attempted the same line of questioning in a traditional classroom, the children were not able to recognize the drawing, estimate the height of the classroom or compare that to the height of the falls, correctly subtract two dates, or calculate how many feet per year the falls were retreating.
By 1933 he was ready to implement this new approach in all classrooms. He recognized the resistance he would have so a committee of principals agreed on the following compromise. The number represents the grade, followed by what the students would study. While this doesn’t look like they studied much math, instead “studying” math the teacher found ways to talk about mathematical concepts throughout the day.
1 Numbers < 100; comparison: more, less, higher, lower; keep a calendar
2 Continue comparing; telling time (hours and half-hours); page numbers; using an index; half, double, twice, three times
3 Bigger numbers: license plates, house numbers; money value; telling time to the minute; 60 minutes equals 1 hour; 7 days in a week; 24 hours in a day; 30 days in a month; 12 months in a year
4 Inch, foot, yard through rulers and yard sticks; estimating lengths; square inch, square foot; how to read a thermometer and significance of 32 degrees, 98.6 degrees, and 212 degrees; practice making change in denominations of 5’s only; estimating distances including mile, half mile, and quarter mile and show them the actual distance; table of time and pounds and ounces
5B Counting by 5’s, 10’s, 2’s, 4’s, and 3’s (mentally), leading to those multiplication tables; practice making change in amounts up to a dollar including using pennies; estimation games followed by checking; fractions by pictures and discover that ⅓ is smaller than ½ (the larger the denominator the smaller the fraction); mental math
5A Counting by 6‘s, 7‘s, 8‘s, and 9‘s (mentally), leading to those multiplication tables; commutative property (3×2=2×3); relative value of fractions ½, ¼, ⅕, 1/10; mental math
6 Formal arithmetic, but estimate first then check; understand the reason for the processes they use; still more mental than paper; fractions and mixed numbers; review multiplication tables
7 Lots of mental arithmetic without reference to paper or blackboard
8 More mental arithmetic; reasons for processes; explaining how to attack problems
Time and again Benezet visited classrooms taught with the traditional method and classrooms taught with the new method and had consistent results. The students who had formal arithmetic instruction delayed were better able to reason and express their thoughts articulately.
To sum it up
I’m still pondering what this means in practicality and how to implement these philosophies in our homeschool. What strikes me the most is that math needs to be a part of our daily lives. We count everything, cut sandwiches into fractional parts, use mathematical terms in our conversations, and ask arithmetic questions (such as I have four Lincoln Logs but need 9 for this building. How many more do I need? . . . If I want to put 1 2/3 cups of beans I just cooked into a bag and I’m using a 1/3 cup scoop, how many scoops should I put in each bag? . . . If I have 6 bags of cooked beans, how many cups of beans do I have?). You don’t have to use a curriculum to teach math in the elementary grades. And if you do choose to use a curriculum, you can find ways to incorporate some of these principles.
Here are resources that we have found helpful in our math journey.