8.4.10Posted by Jeanne
...and what do they spell?
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.I'm not really qualified to talk to you about very much except the genetics of blotchy and brindled mice. If you want to know about murine mottled mutants - brindled, Mo br, and blotchy, Mo blo, - and why they play a valuable role in the study of mammalian copper metabolism, then I'm your girl. Otherwise, you'll probably find a better expert than me to tell you about almost anything.
How living would Geometry become in the light of the discoveries of Euclid as he made them!
...Mathematics are a necessary part of every man'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.
Charlotte Mason A Philosophy of Education p233
I certainly can't tell you much about homeschooling. Don't look to me for advice on whether your kids would be better off using an unschooling or Montessori or Thomas Jefferson or Enki or Waldorf method in your homeschool, because frankly I don't know a thing about them. To be honest I only know their names because I plugged 'homeschooling methods' into google and up they came. I read The Well-Trained Mind: A Guide to Classical Education at Home by Jessie Wise and Susan Wise Bauer, and the amount of work required for the Classical Method of homeschooling terrified me, so don't ask me about homeschooling using the Trivium either.
It's probably providential then that the first book I read about homeschooling was Susan Schaeffer Macaulay's For the Children's Sake: Foundations of Education for Home and School. A quick google one day introduced me to Ambleside Online and my path was set. Jemimah was three years old.
It was through Carol Hepburn's posts in the Ambleside Online yahoo group that I first heard about the Mathematics Enhancement Programme, MEP developed by the CIMT - Centre for Innovation in Mathematics Teaching - at the University of Plymouth in the United Kingdom. I respected Carol's opinions, but the real reason I first looked at MEP was that it was free online.
Apart from the Math-U-See demonstration CD, this is the only homeschool curriculum I've ever seen. So don't ask me about homeschool maths programmes. I know next to nothing about Saxon, Singapore, MUS, Miquon or Horizons math. I can't compare spiral and mastery learning either, for the same reasons - I've never seen nor used a mastery system.
Can you see why I'm not qualified then, to talk to you of such things?
What I do feel qualified to talk to you about is our journey through a Charlotte Mason education using MEP maths and how they fit together. In answer to the question, "Are MEP and CM compatible?" I believe the answer is 'yes'! I'd like to tell you why.
Mastery or spiral?
Firstly let's tackle the mastery/spiral question. I'd like to start here because this is most commonly cited as the reason that MEP is not in line with Miss Mason's recommendations for a maths programme. So what is the difference?
As far as I see it, and remember that I am far from an expert, teaching to mastery is based on the idea that learning is sequential and that each new idea builds upon ideas that are already fully understood. Rather like in a staircase one stair is built upon the stair below, the child must master the subject the first time it is presented because that particular topic will not be presented again, and the next topic is dependent upon the knowledge gained before. Not only mastery of the skill but also the reasoning behind the topic are expected first time round.
With a spiral programme a subject is presented over and over again with increasing complexity each time. If a fact is not understood first time then it will be covered again in subsequent passes in a few weeks or a few weeks or even a few years.
Silvia's reviews here and here of mastery versus spiral programmes are worth a sticky if this topic interests you.
MEP is a spiral programme. Negative numbers are covered for the first time only a six weeks into year one by introducing a thermometer. Certainly my daughter looked at me as if I had horns when I told her at 5 years of age that the number after 2,1,0 was 'minus 1'. At this stage we were only working with numbers between 1 and 10, remember. A child living in Alaska where the temperature regularly goes below zero might have understood this concept at that age, but not my Australian daughter. No matter, nothing hinges on this understanding - it was merely incidental knowledge. Jemimah knew then that there were numbers below zero, nothing more.
They're covered again in year two, but it is only in year three that the children are expected to understand how negative numbers work. In year three children learn to add and subtract numbers below zero on the number line. Whole lessons are spent on explaining and practicing this difficult concept, and most children will understand. If they don't then no doubt it will be covered again in year 4.
MEP lessons move fast, and each day contains one or two new concepts and several of cumulative review. Even if the child doesn't understand something when it is first introduced, then they generally will after much review.
Okay. This is all well and good, but what does Miss Mason say? Of course she doesn't use the terms 'spiral' versus 'mastery', so what does she say even obliquely about this subject?
Engage the child upon little problems within his comprehension from the first, rather than upon set sums. Care must be taken to give the child such problems as he can work, but yet which are difficult enough to cause him some little mental effort.I need to encourage you here to read Miss Mason's own words on this subject. The quotation above is summarised from a long section of Home Education beginning on page 253.
Subtraction is worked out simultaneously with addition. As he works out each line of additions, he goes over the same ground, taking away instead of adding, until he is able to answer quite readily, 2 from 7? 2 from 5? After working out each line of addition or subtraction, he may put it on his slate with the proper signs, that is, if he have learned to make figures. It will be found that it requires a much greater mental effort on the child's part to grasp the idea of subtraction than that of addition, and the teacher must be content to go slowly until he knows what he is about.
When the child can add and subtract numbers pretty freely up to twenty, the multiplication and division tables may be worked out as far as 6x12.
Now he is ready for more ambitious problems: thus, 'A boy had twice ten apples; how many heaps of 4 could he make?' He will be able to work with promiscuous numbers, as 7+5-3. If he must use counters to get his answer, let him; but encourage him to work with imaginary counters, as a step towards working with abstract numbers. 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 effort of mind.
When the child is able to work pretty freely with small numbers, a serious difficulty must be faced, he must be made to understand our system of notation. Introduce him to the notion of tens and units, being content to work very gradually.
Let the child work with tens and units only until he has mastered the idea of the tenfold value of the second figure to the left, and would laugh at the folly of writing 7 in the second column of figures, knowing that thereby it becomes seventy. Then he is ready for the same sort of drill in hundreds, and picks up the new idea readily if the principle have been made clear to him, that each remove to the left means a tenfold increase in the value of a number. Meantime, 'set' him no sums. Let him never work with figures the notation of which is beyond him, and when he comes to 'carry' in an addition or multiplication sum, let him not say he carries 'two,' or 'three,' but 'two tens,' or 'three hundreds,' as the case may be.
On the same principle, let him learn 'weights and measures' by measuring and weighing; let him work with foot-rule and yard measure, and draw up his tables for himself. Let him use his judgment on questions of measure and weight. The sort of readiness to be gained thus is valuable in the affairs of life, and, if only for that reason, should be cultivated in the child. While engaged in measuring and weighing concrete quantities, the scholar is prepared to take in his first idea of a 'fraction,' half a pound, a quarter of a yard, etc.
If the child do not get the ground under his feet at this stage, he works arithmetic ever after by rule of thumb.
The fundamental truths of the science of number all rest on the evidence of sense but, having used eyes and fingers, the child has formed the association of a given number with objects, and is able to conceive of the association of various other numbers with objects. In fact, he begins to think in numbers and not in objects, that is, he begins mathematics. Therefore I incline to think that an elaborate system of staves, cubes, etc., instead of tens, hundreds, thousands, errs by embarrassing the child's mind with too much teaching, and by making the illustration occupy a more prominent place than the thing illustrated.
Dominoes, beans, graphic figures drawn on the blackboard, and the like, are, on the other hand, aids to the child when it is necessary of him to conceive of a great number with the material of a small one; but to see a symbol of the great numbers and to work with such a symbol are quite different matters.
So what does she say? This is how I read it:
- Single digit addition and subtraction
- Multiplication and division up to 6x12
- Introduce place value.
- Work all four operations as each new place is introduced.
- Practice place value using money, weights and measures.
- Introduce fractions.
- Begin to phase out concrete manipulatives by encouraging virtual manipulatives instead as a preparation for real 'mathematics'.
- Beware the trap of over-using manipulatives and making them more important than the mathematical concept being illustrated. (As an aside, Miquon and Math-U-See may be guilty of this perhaps?)
Pretty well, it seems to me. MEP covers addition and subtraction of numbers from 1-20 in year one. In year two the number line is extended to 100 with money as an example. Addition and subtraction followed by multiplication and later division are covered in year two. Weights and measures are introduced. Gradually over year three the number line is extended to 1000, with a brief foray into larger numbers near the end of this year. The four operations are revisited using these larger numbers, and length, capacity, mass and time are extended as new mathematical concepts are reached.
Despite being a spiral programme , the basics that Miss Mason discusses are covered to mastery at each stage before a new place value is introduced and the four operations are covered again. I think this is exactly what Miss Mason recommends. You can read my post on why I like MEP's spiral approach here.
Demonstrate everything demonstrable
The next point is to demonstrate everything demonstrable. 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. 2+2=4, is a self-evident fact, admitting of little demonstration; but 4x7=28 may be proved.This is MEP's position as well. From the Notes on the Lesson Plans:
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.
Visualising mentally and through the use of models and manipulatives, and relating concepts to real life situations where relevant, are important aspects. Children should be encouraged to build up their own set of manipulatives eg shells, pebbles, beads, buttons, lolly sticks, marbles, used match sticks etc in a container so that their maths becomes more meaningful to them. In each lesson, children should work with manual aids, number cards, counters, sticks, rods, dominoes, items from their collection etc wherever possible.Problems
Now he is ready for more ambitious problems: thus, 'A boy had twice ten apples; how many heaps of 4 could he make?' He will be able to work with promiscuous numbers, as 7+5-3. If he must use beans to get his answer, let him; but encourage him to work with imaginary beans, as a step towards working with abstract numbers. 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 effort of mind.MEP begins with mental maths in year one where they say:
Children should count and carry out operations mentally, sometimes as a class, sometimes individually...Also, we have planned some thought-provoking logic problems to form a basis for the development of problem solving skills and creativity.Notation
Let the child work with tens and units only until he has mastered the idea of the tenfold value of the second figure to the left, and would laugh at the folly of writing 7 in the second column of figures, knowing that thereby it becomes seventy. Then he is ready for the same sort of drill in hundreds, and picks up the new idea readily if the principle have been made clear to him, that each remove to the left means a tenfold increase in the value of a number. Meantime, 'set' him no sums. Let him never work with figures the notation of which is beyond him, and when he comes to 'carry' in an addition or multiplication sum, let him not say he carries 'two,' or 'three,' but 'two tens,' or 'three hundreds,' as the case may be.I think we've dealt with this above. MEP spends some significant time ensuring that a child understands a place value before moving onto operations. Once the child has been introduced to 'hundreds' for example, the four operations as well as related examples of measurement, time and the like, are reviewed using the newly extended number line.
MEP expects correct mathematical vocabulary and terminology. I do not recall whether they refer to carrying 'two tens' or not, but a parent can use this terminology with any programme. (I have written a reminder to myself to ensure I am doing that with Jemimah.)
Weighing and measuring
Let him learn 'weights and measures' by measuring and weighing; let him have scales and weights, sand or rice, paper and twine, and weigh, and do up, in perfectly made parcels, ounces, pounds, etc. The parcels, though they are not arithmetic, are educative, and afford considerable exercise of judgment as well as of neatness, deftness, and quickness. In like manner, let him work with foot-rule and yard measure, and draw up his tables for himself. Let him not only measure and weigh everything about him that admits of such treatment, but let him use his judgment on questions of measure and weight. The sort of readiness to be gained thus is valuable in the affairs of life, and, if only for that reason, should be cultivated in the child. While engaged in measuring and weighing concrete quantities, the scholar is prepared to take in his first idea of a 'fraction,' half a pound, a quarter of a yard, etc.From MEP:
It is very important to use pupils' knowledge, customs and information from their home and 'out of school' lives and to build bridges between that life and what they have learned in school. Use the playground, park, woods, market, shops, railway station, airport, countryside etc as contexts whenever possible.The MEP lesson plans detail the way measuring of capacity, length and mass should be demonstrated. I'm afraid they do not explain the perfectly made parcels - you will need to do that yourself!!
You should demonstrate the discussion with real-life models...
Putting Mathematics in its place.
Mathematics are a necessary part of every man'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.Miss Mason was frustrated by the - to her - excessive amount of time that mathematics consumed in a child's education.
In a word our point is that Mathematics are to be studied for their own sake and not as they make for general intelligence and grasp of mind. But then how profoundly worthy are these subjects of study for their own sake, to say nothing of other great branches of knowledge to which they are ancillary! Lack of proportion should be our bête noire in drawing up a curriculum, remembering that the mathematician who knows little of the history of his own country or that of any other, is sparsely educated at the best.If were are going to spend time on mathematics when we could be studying more edifying subjects then at very least we need to make it come alive. Much like we have living books we need living maths. I'm not convinced that this means the unorthodox approach to mathematics seen in such sites as Living Math!, however, although maths literature and history certainly bring out the excitement of great minds like Euclid's in a way that a study of Pure Mathematics never will. We have used many of the books included in Living Math's excellent book list as part of our maths programme over the years, and I recommend a judicious use of a few of these books regardless of which curriculum you eventually settle on.
But... why should a boy's success in life depend upon drudgery in Mathematics? That is the tendency at the present moment to close the Universities and consequently the Professions to boys and girls who, because they have little natural aptitude for mathematics, must acquire a mechanical knowledge by such heavy all-engrossing labour as must needs shut out such knowledge of the 'humanities' say, as is implied in the phrase 'a liberal education.'
The claims of the London Matriculation examination, for example, are acknowledged by many teachers to be incompatible with the wide knowledge proper to an educated person.
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.
How living would Geometry become in the light of the discoveries of Euclid as he made them!
And so, in conclusion and from the website, what of MEP? Let's finish with their words shall we?
- MEP aims to make all pupils mathematical thinkers and to make mathematics lessons challenging and fun for both teachers and pupils.
- It has very high expectations of both teachers and pupils.
- MEP continually revises facts and concepts.
- Lessons anre highly interactive and have many activities.
- It stresses the logical foundation of mathematics and correct consise mathematical notation and language are used at all times.
- Visiualising mentally and through the use of models and manipulatives, and relating concepts to real life situations where relevant are important aspects.
- Above all, MEP is fun.
I think Charlotte Mason would have approved of MEP, I really do. But then again, I don't really know. As I said at the beginning, I'm not an expert. I do know that in our homeschool, MEP dovetails neatly with the philosophies and aims of our Charlotte Mason education. More importantly, as my 8yo daughter reaches the end of MEP year three in the next few weeks, she continues to thrive. Her approach to mathematics is positive and she continues to enjoy what for many children is a much disliked subject. I am so glad that I found this programme, and I encourage you to give it some consideration.
This post is written for Kathy and Phyllis. I do hope it is what you were after, ladies. If not, do yell. I'd be glad to answer your questions...for what my opinion's worth!! Heather has also tackled the topic of MEP and CM here, for more on this topic. You can read my article MEP 101 here if you are interested in MEP and would like to give it a burl.
Now anybody want to know about blotchy and brindled mice? Anyone? Coz I'm your man.