How *do* you teach a kid the underlying concepts of math instead of forcing rote memorization? Here’s what has been successful for me so far (note that this is aimed at parents working with their own kids. Folks working in education or group settings need a different approach. I’ll blog that later)::

It is best if you can catch them young. Really young. Like around two or three. At that age kids want to learn as much as they can. Start with counting – not just rote “1,2,3,4…” but also counting of physical objects. Once they can count physical objects fairly well, have them start to make the connection between the number of physical objects and their fingers (how many fingers is that?) Integrate the learning into their daily life in a way that they can relate to (how many peas are left? How many blocks can you stack, etc). this will accomplish a couple of things:

- It will get the kid used to using and learning about math in their day to day life. They probably won’t realize that they are learning math until they’re in the 3
^{rd}or 4^{th}grade, but at some point it will click. This is a long term investment that will help them get through the crappy math that is taught in the schools. - It shows them that math is relevant to their life. How many of you sat in your school math classes saying “but what use is this in my life”?
- It gives the kids a practical use for their math skills. This is pretty easy early on, but later as they get into geometry, trig, algebra and calculus, it can be a bit more challenging (but not impossible)

You also have to make sure you keep the kid challenged. One of my biggest complaints about the public schools is that they spend way too long on one thing without introducing something new. Once your child can count to 10, start having them add single digits (with the sum always coming out <10). At the same time, keep working on counting from 10 to 100. Addition is simple. Start out by having them count out 2 piles of objects, then put them together and have them count them all together. Once they are comfortable with that, have them count the two piles, then show them how to add them on their fingers Example:

Pile 1 has 4 items, pile 2 has 2 items. How many fingers is pile 1? (kid counts 4 fingers). OK, now on the other hand, how many fingers is the other pile? (kid counts 2 fingers). So how many all together? The first couple of times, you’ll have to help your kid count across both hands to get 6.

By using a method like this, you’re not only teaching your kid how to add, they are intuitively building an understanding of combining groups. The same approach works for subtraction (4-2: put up 4 fingers, now put 2 away (down). How many are left?). For subtraction, using “take away” instead of minus seems to help them get the concept – use language the kids know: they’ll get the jargon later as you slowly slip it in.

As your kid starts to get the concepts, you can easily expand it to include multiplication – make 3 stacks of 5 blocks. How many is that? Now make 4 stacks –how many is that. Have the kid start counting by 5s (or 2s. for starting out, these are the easiest intervals to use). Once they’ve reached the point where they can accurately count by 5 or 2, you can start introducing the concept of multiplication. What you have done is taught your child that multiplication is really just adding in groups. They’ve learned the *concept* of multiplication without ever having studied what multiplication is. At this point, your child may not be able to recite “math facts” by rote, but they will be able to solve addition and subtraction problems (I know I skipped over multi-digit and carrying problems, but the concept is the same). As you transition your child to solving problems on paper, they will easily move from the physical (object based) to the abstract problem solving skills. Let the child drive – they are curious and want to know how things work and what things do. Keep your eyes open for (or create) opportunities for teaching. Making a batch of cookies is a great way to introduce fractions. Blocks can teach shapes and basic geometry (Please hand me the octagonal column). I use my woodshop to teach geometry (hypotenuse of a triangle, angles for cuts, etc). I use Lego robotics to teach linear step thinking (My daughter is finishing up second grade and can do basic Lego programming). A lot of what is taught as “math” is really simple life skills. Get yourself an analog clock (if you don’t already have one). Spend a little bit of time teaching your child to read a clock, then make sure you have them tell you what time it is. It shouldn’t take more than a couple of weeks before a 1^{st} or second grader can read a clock accurately (within 5 minutes) – it does help if your clock chimes on the ¼ hours – the auditory cue helps a lot, and the kid will actually get up and go look at the clock to check…

I guess the bottom line is that you have to spend a lot of time with your kid. Accept that fact that making cookies (or bread, or whatever) is going to take a lot longer when you have them decide what measuring device to use, figure out how many scoops etc. Yep, they will decide to use a teaspoon to measure out 3 cups of flour (be ready with the conversion). Let them spend the time with the teaspoon, then show them that they could have used a cup a lot quicker. Accept the fact that (for now) it’s more fun to use the teaspoon…..

As for rote learning, I really don’t think its needed. The idea of memorizing the multiplication tables may make sense at first (it takes *so* long to figure a problem out), but as your child finds uses for multiplication, they’ll learn the ones they need simply by using them a lot. Same goes for learning the symbols on the periodic table, structures of sugars, foreign language vocabulary, and almost everything else that the schools teach by rote. Instead of rote learning, try application learning – here’s a chart/list/source for the facts. Now here is a bunch of stuff to do that needs those facts. The facts that are commonly used will get learned, and the ones that aren’t can always be looked up or solved. Just like in real life. Oh yeah, and by using this approach, you’re teaching your kid to use the resources around them to find the information they don’t have. Tool use is good…

Will these methods work for all kids? I suspect not. The message here is the philosophy, not the details. What worked to teach my kids algebra may not work for yours. The approach should work for anyone. You need to know your kid, use what interests them, and *always* make sure that they have new things to investigate, figure out, and discover. Forget all the crap that the teachers, shrinks, and schools tell you about your child “not being developmentally ready” for those concepts. All that really means is that your child is ahead of (or behind) the one-size-fits-all track that they have created. I know second graders that can easily handle long division – something that isn’t in their curriculum until the 4^{th} grade. You know your kid. Its your job to make sure that they keep learning.

mrschili, on June 3, 2007 at 11:46 am said:This all makes perfect sense, of course, and I have to say that Mr. Chili and I have DONE this – and continue to do things like this as often as possible; while cooking, as you suggest, or in the car (if we can go forty miles an hour, how long will it take us to go sixty miles at that speed?) and in everyday sorts of situations (if a movie ticket costs so much money, how much do we need to get the whole family into the theatre?). I’m certain that my chidren understand the CONCEPTS – the problem is that they don’t know the things that the school is expecting them to know; they don’t do well on timed tests and Punkin’ is being pressured to have a store of math facts that she can just call up at will. We’re trying to figure out how to make the success she has in understanding the underlying ideas (what we think is most important) match up with success in school. I’m not sure we can make that happen until the schools are able to adjust their priorities…

sphyrnatude, on June 3, 2007 at 5:09 pm said:mrschili: unfortunately, you’ve hit the nail right on the head. The schools don’t care if the kids understand the concept: they want the kid to be able to regurgitate the fact quickly and acurately. Remember: when you’re taking a standardized test, it doesn’t matter *how* you get the answer, as long as you get it fast.

As long as our schools are evaluated based on how well the kids do on standardized tests, there is no reason for the school to teach anything except how to score well on the test: hence fact regurgitation.

You’ve just presented one of the best arguments I know for homeschooling, school vouchers, and pretty much anything thatill protect our kids from publib education….