Teaching Arithmetic Through Physical Manipulation: A Creative Approach

[anything385]

i want to teach my boy arithmetic.
my current 'best' idea, is to have a board with a few columns with nine spaces in each column. headings from right to left would be (units,tens, hundreds etc.).
nine small plastic discs showing value of +1 unit. nine discs showing value +10 units.
nine discs showing value +100 units, et al, would allow numbers to 999
a 'bank' with spare discs showing the above values, plus spare discs showing the above values but in a negative form, (to be used for subtraction by 'cancelling'.)
the idea is, that a number such as six hundred and fifty four could be physically shown by
four discs in the bottom four spaces in the units column of the board, each llabelled (+ 1 unit). five, (+ 10 unit) discs similarly placed in the tens column of the board. six (+ 100 units) in the hundreds column.
addition and subtraction could be performed adding discs from the 'bank' into the columns.
i feel that physical manipulation of numbers would lead naturally to the concept or mental manipulation.
do you think this is a good method of teaching arithmetic and can you suggest any improvements.

 

[Mentallic]

My 6 year old sister came home one day from school and showed me how she learned addition. Of course it was with her fingers for simple numbers less than 10. I went about teaching her subtraction with the similar method using her fingers as a guide.

Of course, mental manipulation of numbers is a little too advanced for their minds at this age. Without enough exposure to the physical manipulation of numbers - like you have stated - they will struggle in this field. However they learn to do arithmetic, this method will be embedded into their brains and they will strictly try to turn to using these apparatus for any arithmetic they encounter at school in the near future.

This is why I suggest starting your son off by becoming familiar with using his portable tools, his fingers. Once you feel he is ready for the 'big' numbers, move him on to your proposed teachings with the different shapes for units, tens and hundreds.

 

[physics girl phd]

This idea basically sounds like using an abacus:

http://www.presents-4-kids.co.uk/images/hibba2007/giant_abacus.jpg

You should be able to find various types (with learning guides) online.

 

[pancho_bsas]

When i was a child, I used a 30cm ruler to add and substract. Now(as an adult who study Engineering) I consider that´s a very smart way to teach addition and substraction.

 

[Moonbear]

When i was a child, I used a 30cm ruler to add and substract. Now(as an adult who study Engineering) I consider that´s a very smart way to teach addition and substraction.

We used all of the above. The number line you're describing (it wasn't technically a ruler, but a line of numbers that stuck to our desk), fingers, and objects to manipulate (including being taught the rudiments of using an abacus). I think using a variety of tools helps, because it ensures that at least one of them sticks...not every kid learns exactly the same way. The good part of including objects to manipulate (I'd recommend things like paper circles be included) is that when it is time to teach fractions, the kid is already comfortable with the tool that will represent the whole units so when you start cutting it in halves or quarters, the concept becomes easier to grasp (I don't recommend continuing to use fingers to teach fractions...otherwise they will have a harder time grasping everything else :uhh: ).

It's a good idea to think a few steps ahead of what you're currently wanting to teach. You want to start out teaching how to add and subtract single digits. But you're also eventually going to want to move on to double digits (same tools get used, just working one column at a time), fractions, multiplication, etc. If you can keep returning to the same learning tools that the child is familiar with, each new concept will more naturally build upon the previous one for them rather than seem like a foreign new idea. I'd suggest adding something like dried beans to the mix of objects used...when you get to multiplication, you'll need something in more plentiful supply.

Young children are not particularly good at generalizing ideas, so the more examples you use, the more situations they can start to apply the concepts...and somewhat generalize what they're learning.

Of course, to keep it fun for them at this age, you can use some cut-out shapes from paper and let them decorate the paper as a separate activity. You can then sneak in questions like, "How many sides does the triangle have?" "If you have two triangles, how many sides are there all together?" They can count sides, not just whole objects.

 

[Borek]

Young children are not particularly good at generalizing ideas

Moonbear touches very important thing here. Too young kids can't grasp some ideas, due to the way human brain develops.

The idea of sum may be too abstract to be taught to 4 yo. Probably when your boy will deal with a real objects - like "you have two marbles, I will give you two marbles - how many marbles you have" - he can be able to give correct answer, but arithmetic (2+2=4) may be completely alien to him.

But probably the best person to ask is Math Is Hard...

 

[mathwonk]

lots of children are smart enough to learn almost anything. that does not mean it is a good idea to impose any regimen on them. i was a parent who wanted to teach my kids the things i knew they could learn, but found at last that it was a bad idea to teach them what i wanted them to learn. (my child eventually made me the main character of a tell all junior high writing project called "the math teacher".)

kids deserve and flourish best when allowed to learn what they love, not what we love. all of us here who love math are able to recall arithmetic learning events of our childhood, but i conjecture we were in the minority who enjoyed that activity and sought it out ourselves.

i suggest respectfully, give a 6 year old a break, find out what he/she wants to do and help with that. you'll be glad in the long run.

this thread reminds me of a charles dickens character who worried little children with arithmetic problems every time he met them, he came across as a sort of villain to children, a busybody no one wanted to be around.

may i suggest as parents one try to be an enabler, rather than a professor.

 

[Borek]

lots of children are smart enough to learn almost anything.

There are always exclusions, but they don't change the way most of the population develops. Have you heard about conservation experiments? Take a preschool kid, show it two identical balls of clay, ask if the are equal in size, then squash one and ask the same question. Most kids will say that now one is larger. Some ideas and some ways of thinking are just beyond their comprehension at this stage.

Piaget's stages of cognitive development are key words that you may look for if you need more details, you better don't ask me as I know very little and what I know I know in Polish

that does not mean it is a good idea to impose any regimen on them (...)
may i suggest as parents one try to be an enabler, rather than a professor.

And here we agree

 

[Moonbear]

i suggest respectfully, give a 6 year old a break, find out what he/she wants to do and help with that. you'll be glad in the long run.

Six years old isn't too young to learn arithmetic. That would be first grade, and is when we started learning arithmetic when I was a child (counting was learned in kindergarten at 5 years old).

I agree not to make it too formal in the home setting, because they will get the formal lessons in school. But, there's no reason not to make play sessions educational...hence my suggestion to use the lesson as a time for an arts and crafts activity of cutting out and coloring in shapes while talking about things like how many sides the shapes have.

Of course, every parent has to assess their own child's readiness for and interest in learning. If they start to get frustrated, back off. If they seem to learn rapidly and want to soak up knowledge like a little sponge, offer it.

And, in case it was misunderstood, I'm not suggesting to start teaching fractions and multiplication at this early age. I'm just suggesting to think about the years ahead and introduce learning tools that can be used for those future lessons as well, so that there is a familiar element present when new concepts are being taught.

For example, when it was time to learn negative numbers, we got our number lines back...except this time they added a section to them that included negative numbers. We were already familiar with how to use number lines and they weren't intimidating new tools, just something simple had been added to them.

 

[mgb_phys]

Have you heard about conservation experiments? Take a preschool kid, show it two identical balls of clay, ask if the are equal in size, then squash one and ask the same question. Most kids will say that now one is larger.

Theres an anecdote about a famous psychologist visiting as colleague who has small children.
He decides to try this on one of the kids and asks which piece is larger.
The younger one replies - I don't know, you should ask my brother ,he's six so he has conservation!

 

[mathwonk]

I also read piaget as a student and recall his claims that a child of 6 or so thinks a glass of milk holds a different amount of milk from the amount yielded by pouring out the milk into a different glass. But I respectively point out that these august opinions of his did not gibe with what I actually observed in the children near me.

The observations I am giving here are those of an ex parent, not a professor or psychologist, or reader of psychology books. Of course they are also as fallible as those may be, and may apply only to one set of children.

I am just saying that in my experience it is easy to be too eager. My own kids could read and partially understand junior high and high school algebra and geometry books starting at around 6 or 8, and thoroughly master them at 12, but they wanted to be like the other kids and not have extra work to do. Teaching them is relatively easy, knowing when not to is harder.

the key giveaway in this thread is the first sentence, which does not read "my boy wants to learn arithmetic". To overcome that fact is your first task.

The abacus illustrated above seems a good idea, since it has a manual and play feel to it.

 

[flatmaster]

I like the abbicus idea, but you could extend it to the concept of the places of powers of tens. What if the higher the power of ten, somehow the beads are harder to move either through mass or friction with the rod. Would a kid get this?

 

[mathwonk]

here is something i did successfully with 7-8 year olds, and it worked with brilliant and even retarded ones. i made cardboard or construction paper models of the regular solids, and asked them to count the faces and edges and corners. then we gradually built up to discovering euler's formula V-E+F = 2. the solids were colored and they enjoyed them.