Tuesday, June 30, 2015

Teaching mathematics part IV - laying the foundation

In parts II and III, I discussed why mathematics needs a good teacher, and what that good teaching entails.

Now, let's look at the how.  And let me point out that, at this point, we are mostly concerned about how to teach the subject from the beginning, in the early years.  This is when good teaching is so critical and can make or break a child's attitude toward and understanding of math.  A young math student needs to have a solid foundation upon which to build if he or she is going to be successful in the later years.

Children should not be rushed through mathematics, but should be allowed to take it slowly - slowly enough for each individual child to fully grasp the topics.
"...In arithmetic, above all other studies of the common school course, it is of the utmost importance that one step shall be thoroughly understood before the next is attempted.  The first two years' training is of more importance than all the rest the child receives." (The Eclectic Manual of Methods, p.107) - emphasis mine
Charlotte Mason also understood the importance of laying a solid foundation in the early years.
"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." (Home Education, p.257)
"If the child does not get the ground under his feet at this stage, he works arithmetic ever after by rule of thumb." (Home Education, p.259) 
So where do we start?  How do we lay this so-important solid foundation?  What steps do we take to begin our children in the study of mathematics?

It all begins with numbers - learning numbers as whole quantities and learning all the different combinations that make up that quantity.  This is how the child's mental math ability is built up.  He is taught good number sense, mentally, which at the same time teaches him to think more deeply about mathematics.  A child who does not come away in the first year (or two, if needed) with good number sense has not really been trained to think on a deeper level and understand how math is unfolding.  And that will be detrimental to the rest of his math instruction.  If he doesn't learn to think early on, he'll struggle.  Period.

So, let's go through the method of teaching the first year and we'll see how Ray's Arithmetic mirrors Charlotte Mason's writings on teaching math.

Here we go!

1.  Teach through oral instruction, focusing on the concrete.

"Do not teach the figures in the first lessons, and do not allow the children to do any written work; but teach orally, illustrating every operation, at first, by means of various objects. -- The instruction should be entirely oral, and should deal altogether at first with concrete numbers." (Eclectic Manual of Methods, p.107-108) 
CM also advocates oral instruction and, as we've already discussed, beginning with the concrete.
"Give him short sums, in words rather than in figures..." (Home Education, p.261)
If we begin with written instruction, then we're really beginning with abstract ideas.  Having a child complete a math problem like 2 + 4, on a worksheet or on the board, is an abstract idea, which, as I discussed in an earlier post, is much more difficult for a child to grasp than asking the child how many are 2 beans and 4 beans.  The latter is a concrete idea and is much easier to understand.

2.  Teach through the use of objects to manipulate.

"Begin the teaching of arithmetic, then, with objects, -- blocks, balls, marbles,...etc." (Eclectic Manual of Methods, p.108)
 CM also recommended using objects in the beginning.
"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." (Home Education, p.256)
This not only allows the child to understand the concrete ideas first, which will eventually pave the way to understanding the abstract, but will get the student involved and, in turn, get him to grasp the rationale of numbers and how they work himself, (rather than just being spoon-fed by the teacher).

3.  Teach numbering, first with objects, then without.

"The first step is to teach numbering; that is, so to train the child that he can instantly give the number of any group of objects not exceeding ten, at sight, and without counting...Do not allow a child to count by ones to find how many objects there are in a group, but teach him to recognize the group as a whole. -- Teach what three means by repeatedly combining two and one, and one and two, into groups of three apples, three blocks,...etc,...teach four in the same manner...As soon as the class is sufficiently advanced, have the children do the combining and separating of objects for themselves." (Eclectic Manual of Methods, p.108-109)
This teaching of numbering is so important - it is the first step toward training a child to think rapidly and accurately.  It is strengthening a child's mental math ability from the get-go.  The child sees all the different combinations that make up a particular number, which prepares him for addition and subtraction.  He also learns to recognize each quantity as a whole rather than teaching him to rely on counting.  A child who learns to rely on counting is not learning to think mentally.  There is also a good chance that the child will carry this habit of counting over to adding numbers - he'll rely on counting to add instead of his mental ability.

This is very similar to CM's method.
"He may arrange an addition table with his beans, thus --
OO   O          = 3 beans
OO   OO       = 4 "
OO   OOO    = 5 "
and be exercised upon it until he can tell, first without counting, and then without looking at the beans, that 2+7=9, etc." (Home Education, p.256) - emphasis mine
The place where the two methods differ is when to have the students learn to recognize the combinations of imaginary objects and eventually abstract numbers.
"From combining and separating objects they can see, lead them to combine and separate groups of objects that they cannot see, but can readily imagine, such as animals, houses, trees, tools, toys, or any objects with which they are familiar." (Eclectic Manual of Methods, p.110-111)
So the Manual of Methods recommends introducing imaginary objects after mastering the combinations of objects before their eyes; then abstract numbers after mastering imaginary objects.
"When you are satisfied with the results of the work thus far, take the next step by simply dropping the names of the objects, and teach the abstract digital numbers orally in the same order that you taught the concrete." (Eclectic Manual of Methods, p.111)
It seems that CM, however, recommends teaching with imaginary and abstract along with using objects that the children can see.  Continuing with the above CM quote from p.256 in Home Education:
"Thus with 3, 4, 5, - each of the digits:  as he learns each line of his addition table he is exercised upon imaginary objects, '4 apples and 9 apples,' '4 nuts and 6 nuts' etc.; and lastly, with abstract numbers - 6+5, 6+8." 

4.  Teach addition and subtraction together.

"The one is the reverse of the other, and when taught together they help the child to understand each process more readily than if they were taught separately." (Eclectic Manual of Methods, p.110)
"A subtraction table is worked out simultaneously with the addition table.  As he works out each line of additions, he goes over the same ground, only taking away one bean, or two beans, 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 has learned to make figures." (Charlotte Mason, Home Education, p.256-257) 

5.  Teach the written characters last.

"...when the children are thoroughly proficient in the preceding steps, teach them the written characters that stand for the numbers which they have learned to use orally." (Eclectic Manual of Methods, p.111)
It seems that CM would agree with this as well - mastering the material mentally through oral teaching before any written work.
 "...the child should be able to work with these freely {objects}, 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." (Home Education, p.256)
Note, however, that this does not mean to teach the children to write the numbers themselves.  It's just teaching them to recognize the written numbers.
"It is granted that much greater apparent advance can be made at this time, and that children can be taught the names of numbers as high as a hundred or more, and to write the figures representing them; but the learning of names and the making of figures do not of themselves imply the gaining or developing of ideas, and classes forced too rapidly over the preliminary ground without thoroughly understanding each step as they advance, will sooner or later show the bad effect of this method of teaching." (Eclectic Manual of Methods, p.113-114)
All that is required in the first year of mathematics is mastering groups of numbers, and addition and subtraction of numbers, through 10.

What this does is present the concrete ideas of mathematics which the child can understand, as opposed to presenting mathematics abstractly, which he cannot.  Instead of beginning with the abstract idea of 2 + 1 = 3, for example, we begin with the child combining and separating 2 beans and 1 bean.  And in allowing the child time to manipulate and work freely with these concrete ideas, he develops an understanding of how mathematics works - its logical nature and the truth and beauty and exactness of it.  Using this method to teach subsequent topics - addition and subtraction with larger numbers, place value, multiplication, division, fractions, weights and measures, etc. - will hopefully develop in the child an appreciation and respect for the subject as an immutable law of the universe, worthy to be studied deeper.


So, we can see how Ray's Arithmetic so far seems to be very conducive to the way CM taught mathematics.  I think that as long as these general steps are taken early on, allowing the child enough time to fully understand each step, we'll see our children begin to flourish in the subject.

One last question!  How will we know that our children are progressing the way they should be?
"Accuracy and rapidity are the important aims, and the children should be drilled until they can give the answers to all the possible combinations and separations instantly, and apparently without stopping to think." (Eclectic Manual of Methods, p.111)
 "...excite him in 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." (Charlotte Mason, Home Education, p.261)
Accuracy and clear, rapid thinking from the child is how we'll know that they're on the right track, and that they're building that solid foundation so important to future success in mathematics.

Other posts in this series:
Part I - why study math?
Part II - good teaching
Part III - good teaching continued
Part IV - laying the foundation <---you are here
Materials, manipulatives, and activities


  1. This series has been such a help to me, Angela. I have shared with you about my childrens' attitudes towards Math (they don't hate it, but they say "it's boring", and sometimes have a bad attitude while doing their Math lessons), and I guess I am to blame for that. Next year, things will be different around here. I am turning over a Math leaf, so to speak, with a new appreciation for this subject. So thank you for this eye-opening series.
    I do have a question for you...my 16-yr-old completed Saxon Calculus last year, and will taking a Consumer Math course this year. He loves the practicial stuuf and can't wait to dive in! :)
    But my 3 middle children (ages 14 - in Saxon Algerbra 1, age 13 - in Saxon Algebra 1/2, and age 11 in Saxon 7/6)...they are already in those higher Maths and how would I go about introducing concrete before abstract? Isn't it too late for that? Any ideas about how to make up for lost time with them?
    My youngest is 8 yrs. old, and ready to begin Saxon yr 3 (IF I don't switch to a different Math curriculum before the new year begins..still pondering that), so I feel as if she will get the most benefit from my new Math-it-tude. ;)

    1. No, it's not too late. Usually when you get to the higher-level math, teaching concretely before abstract means to teach a concept using numbers and objects - usually imaginary objects - first, before using variables.

      For example, when teaching to combine like terms such as 4a + 6a + 3a, we could begin by posing the following problem:
      John has three baskets. One basket contains 4 apples, another 6 apples, and the last 3 apples. To find out how many apples John has, we simply add 4 apples + 6 apples + 3 apples which equals 13 apples.
      That's teaching concretely.
      Then with that understood, we move to abstract: Instead of using "apples," we could use the letter (or variable) "a". So, 4a + 6a + 3a= 13a.

      That's the basic method - beginning with numbers (since the child *should* have pretty good numbers sense) to illustrate a concept (and possibly imaginary objects, if needed), then generalizing the concept (usually using variables in Algebra). I've heard good things about Saxon - I'm not familiar with it - so I bet it teaches this way.

      I hope that makes sense!

      Also, in my part III post, I responded to Melissa's comment - you may want to read that if you haven't already. She seems to *kind of* be in a similar situation. And when I said to her that what we want is for our kids to be successful and confident, success to me means that our children have learned to become thoughtful individuals, able to ask questions and seek that which is good and beautiful in the world, whether they like math or not (or any other subject, for that matter)!!

    2. Thanks again for the tips, Angela. We do Math weekly through the summer to keep it fresh in their minds, and I am hearing the "when will we ever use this??" question often from my 13 and 14 yr-old boys. I am trying to get them to understand that we are doing not because they might use it later (although that would be a plus!), but for the order and beauty of it, as you mentioned in an earlier post. I mentioned to them that God is a God of order; and that they can see order everywhere in the universe, and Math is no exception.
      I can see that it's going to take alot of work (and prayer!) on my part to help change this mindset in our home. But I think it can be done. :)

    3. Hi Angela,
      Another Math ?...my 11 yr-old daughter is nearing the end of Saxon 7/6. She has dislikes Math, but would do well if she would just apply herself. When I check her lesson, she might miss 7 or 8 problems. So I make her correct each one. By the end of the corrections, she is almost in tears because she is so frustrated, and it has taken so long to get through the process. Also, if any of my children miss too many problems, I make them do the entire lesson over again the next day, which, I admit, causes more frustration and anxiety.
      So my ?........is asking them to correct the missed problems too much? How else will they find their mistakes and remember to not make them again?
      I admit that for our family, Math makes homeschooling very hard and sometimes I feel like sending them to public school just so I don't have to deal with it. And, of course, therein lies the problem - me. But it's hard to have patience with my daughter in this area when it has become a daily struggle! I have tried implementing what you have posted, but it's not helping. I must be missing - or forgetting - something!
      Any tips (or refreshing of my memory from your posts on Math) would be appreciated!!
      I would leave you my email address for easier correspondence, but I don't want to publish it on here.

    4. Lisa, sorry it has taken me so long to respond. Our computer crashed!

      The first thing that comes to mind is the question: why is your daughter missing so many problems to begin with? A couple of missed problems here and there would probably mean simple mistakes are being made, but 7 or 8 would tell me that she didn't understand the topic to begin with. OR that she's rushing to get through the lesson. If - she's not understanding, I wouldn't have her correct every problem - I would start over and reteach the lesson. It's about the child understanding, not making it through a certain number of lessons by the end of the year. (But, like I mentioned in a previous post, the student should be efficient in their lessons, which goes back to habit training.) If she's rushing through just to get done, then I guess that's a habit training issue :)

      Also, how many problems are being assigned for each lesson? It doesn't take 20 or even 10 problems of the same type for a student to show that he or she understands the concept. A handful - maybe 4 or 5 - of problems done efficiently and correctly should be sufficient (and again, I'm referring to 4 or 5 problems of the same type - drill problems are one type, word problems another, covering the same concept).

      And again, keeping lessons short (according to age) is important. If they need to finish the next day, so be it.

      My email is anwilhite at gmail dot com. Feel free to email me with any other questions, or if something doesn't make sense!!

  2. Thank you for this series! My daughter has decided she hates math, doesn't understand it, and can't wrap her head around it. She actually does prety well just following the rules but the lack of understanding frustrates her and makes math time dreadful. I felt much the same way in school. My grades told my teachers I understood, when in reality, I could just follow rules and get the right answer without knowing why. So, after much consideration, I've decided to move to Ray's and kinda start over. Your series has been very helpful to me to understand what I am doing. Thanks!

    1. I'm so glad it has been helpful for you, Kimberly. That is a big problem nowadays - just learning to follow the rules, which is not beneficial at all. It really goes against all that Charlotte Mason advocates - allowing the child to do the mind work. I plan to write a little more about that in the future.

      Thank you for your comment!