This is one of a series of webpages to help students understand math, and to help parents teach their children math -- especially to help children have a good foundation. These are not pages to teach the mere recipes or algorithms for solving problems. My philosophy is that if you understand math as you go along, you will be able to do your homework well yourself and you should be able to do well on exams. And although understanding takes some work, it is usually actually easier than trying to memorize rules and recall them later, especially in situations that are slightly new and different, and especially under pressure of tests.
There ARE some things that one needs to know almost automatically and that need to be a part of your memory, but those things can be learned with practice that shouldn't be too tedious. Practice is important to help you do math more automatically and therefore to recognize possible solutions more quickly, but you should still understand the things you memorize or know automatically. Memorization and rote learning, by themselves, in math just won't help much by the time you get to algebra; you have to understand numbers and relationships between numbers -- the logic of numbers and of math. It is not all that difficult if you develop a good foundation, which requires both understanding and sufficient practice to keep your understanding sharp. This essay presents ways for parents or teachers to help children develop such a foundation.
The following are some other essays in this series.
"The Concept and Teaching of Place-Value" -- a theoretical explanation about (the problem of) teaching Place-Value to children, with a practical method given based on that explanation
"The Socratic Method: Teaching by Asking Instead of by Telling" -- an example and explanation of how to use students' (including young children's) inherent reasoning abiltiy to guide them gently and easily into an understanding of difficult ideas
"A Supplemental Introduction to Algebra" and a separate webpage "The Way Algebra Works" -- both explain some things algebra books don't tend to tell students, but which I think are important for students to know so that they can understand what algebra "is about" and see, in a sense, how it works
"Understanding 'Rate' Word Problems" -- a conceptual explanation of doing various kinds of math "rate" problems (e.g., "distance/speed/time" problems, problems involving combining work done by different agents working at different rates of speed, quantity/proportion problems, etc.).
My training is in philosophy and I use a conceptual and analytic approach to teach math. I believe it is lack of conceptual understanding that causes students to have the most difficulty, along with lack of practice manipulating quantities and numbers in certain ways to the point of being comfortable and familiar with them.
For example with regard to practice, adults often don't realize how
difficult it is for young children to associate number NAMES with the QUANTITIES
OF THINGS those names represent; adults have associated quantities with
numbers for so long, it seems second nature to them; but to children, it
is more like the following would be to an adult, and it is just as difficult:
|Imagine if I said from now on we will change all the numeral names
(that is 0-9) to the ten letters of the alphabet below, which you already
know in order, so this should make the tasks following them even easier:
k, l, m, n, o, p, q, r, s, t.
How long do you think it would take you
[m, o, q, s, lk, lm, lo, lq, ls, mk, mm, ... , tq, ts, lkk], or
The same is true for little children with regard to learning numbers because number names don't mean anything more to children about quantities than the above letter names for quantities do to you.
I think it is important for children and their teachers to accept and believe in the following three teaching/learning principles:
1) When the student tells the parent or teacher what s/he doesn't understand or cannot do, s/he should ALSO TELL what s/he tried to do and why. S/he should tell what s/he understands about the problem and how s/he tried to go about solving it, and why s/he tried that. S/he should tell as specifically as s/he can where s/he thinks s/he is getting stuck. Obviously this will be more difficult for young children to verbalize, but they should have the opportunity and the coaxing to try to communicate in some way how they are thinking and seeing a concept or idea. It is important for parents and teachers to try to help children explain their own ideas and reasoning as much as possible.
2) It is important for the parent or teacher to (try to) understand where the child is going wrong and how the child got there, so that the teacher can correct misconceptions and so that he or she can know what sort of answer might serve the student best.
3) When a parent or teacher believes a child has understood an idea or concept, the parent or teacher should present the child with a problem that requires a slight modification of it, in order to see whether the child recognizes something new needs to be done and can see how to take it into account. If a child can do this, that presents better evidence the child does understand what s/he been doing and has not just hit on a pattern that works for particular situations or has not just learned by rote how to do what you were working with them on previously.
While teachers should be looking for any indication a student does not understand something, it is also important that the student should say if s/he does not understand a parent or teacher's explanation or answer or some part of it. There may be more the parent/teacher needs to say, or there may be some other way they need to say it so that the student can see it. "Seeing" someone else's explanation about something, particularly in math, is not always easy; and it does not mean a student cannot learn or come to understand a principle or relationship, or that a student is not smart, just because s/he doesn't see it the same way the teacher sees it, or the way the teacher says it the first time they give what they THINK is an adequate explanation. The teacher will just have to try a different approach to help, or will just have to explain more about his/her initial approach. But the student needs to let the teacher know, because otherwise the teacher may not realize s/he doesn't understand, and may keep on going in a way that gets the student really lost.
The following are some of the aspects of arithmetic where I believe a good foundation is particularly important. If you are the parents of young children, you can practice some of these things with your children while you are in the car with them; I found that to be a good time to work with them:
1) Naming whole numbers in order (sometimes called counting, even though one may not actually be counting anything). As young children get older, they can add more numbers to the list. Notice, naming numbers in order is different from counting, though counting sometimes involves naming numbers in order. But you can name numbers in order without counting and you can count without naming numbers in order. (You, as an adult, can count without naming numbers because you can sometimes see five objects immediately as five, without having to "count out" each one of them -- as when playing dominoes; or you might multiply to get a total, or you might count by two's or by 10's, etc.) But before kids can count objects by naming the numbers one at a time in order as they point to objects, they need to learn the number names in order. So you can start with "one, two" and then add numbers as kids are able to absorb them. You can name numbers while pointing to fingers, or just reciting the numbers, or by using nursery rhymes such as "One, two, buckle my shoe...." Give little kids plenty of practice naming numbers as high as they can go, helping them and making it fun for them, and applauding them when they learn them. (See #5 and #6 below for typical particular problems about learning number names in order.)
2) Counting things one by one. As your children learn number names, give them practice counting things, helping them when necessary and praising them as they get it right. Counting things one by one helps them count and it reinforces the order of number names while they are young. You can have them count candy, such as M&M's, or poker chips, or the hearts (spades, clubs, diamonds) on the face cards in a deck of cards, or the dots on dice. If you have games like Chutes and Ladders or Monopoly, etc., it will give them lots of practice counting the dice and the squares they move past on the game board. Eventually they will even start to see groups of squares they won't even have to count one-by-one.
3) Naming number names by groups; e.g., by 2's, by 5's, and by 10's in particular. Once they have learned to name numbers in order, teach them to name numbers by two's, then by fives, and by tens. Once they understand WHAT it is you are teaching them, you can give them practice by the next step, #4.
4) Counting by groups; e.g., by 2's, by 5's, by 10's, etc. Make sure you get them to see how much faster it is to count out large quantities by groups, rather than one at a time. You have to point this out to most kids or they will tend to count things one at a time and not even think about counting by groups even though they know how to count by groups; they just don't think to do it, unless they have been told and shown at least once that it is a faster way to count.
5) My children had trouble learning what I would call the "transition" number names. They had trouble learning what comes after the 9's in the two digit numbers; e.g., after 29, 39, 49, etc., even though they could say numbers by tens: 10, 20, 30, 40, etc. So it took additional practice working with that in particular. I had to get them to see that what came after, say, 49, when they were counting by one's was the same thing that came after 40 when they were counting by 10's; that is, they needed lots of practice in seeing that when you "finished" the forties you went into the fifties, when you finished the seventies you went into the eighties, etc. So we did extra practice naming numbers starting at the 7 in each "decade"; i.e., 37, 38, 39, ? 47, 48, 49, ? 87, 88, 89, ?
6) Kids also have difficulty sometimes saying numbers in order out loud because they will accidentally jump from, something like fifty-six to seventy-seven or to sixty-seven because they get confused between changing the one's or the ten's place number. It is not a sign of any significant difficulty, but you need to watch for it so they learn not to do it.
7) Kids need to learn to read and to write numbers. This is not too difficult with single digit numbers, but it is somewhat difficult with multi-digit numbers, since the number ten, for example, written out looks like one, zero. Kids can just learn it is 10. At this stage they don't necessarily need, and might not be able to appreciate, a rationale. You can just say something like, "I know this looks like one, zero, but it is the way you write 'ten'." Similarly 11, etc. At some point, if they seem like they can follow it, you can show them that ten through nineteen all have a "1" on the left side, and that all the twenties have 2's on the left side, etc., but I wouldn't get into talking about columns or place-value. If you feel they might think it interesting, you might explain that the "teen" in each of the -teen number names is like "ten" and that the teens are like three-teen, four-teen, five-teen, and that twelve is like two-teen and so the numbers look like a ten except for the numeral that replaces the "0" in the ten. Once you get to twenty, this is easier, and you may even want to start with it -- twenty one is written like twenty but with a one at the end; twenty two is like twenty with a two at the end, etc. (I will get to place-value later.)
8) When your children are very young, you can very naturally, without any fanfare, introduce them to fractions by breaking a cookie into roughly two parts in front of them and saying something like: "Here, I'll give you half a cookie and I will eat half [or I'll give your brother the other half]." Similarly with one-fourth of something when a reasonable occasion arises. Or you might give them "half a glass of milk" and identify it as such.
9) When your children start to study fractions in school, you can make it easier for them by explaining every fraction has two parts, which, when written, are a top number that is said first, and a bottom number which is said second (in the form you will have to explain to them --e.g., "fourTHS" instead of "four"). Let them know the bottom tells how many "pieces" you divide something up into, and the top part tells you how many of those pieces "you have" or "you are talking about". So if you divide a cookie into halves, and you get one piece, you have one half a cookie. If you have four people in your family and two of them are women, then two fourths of your family are females. You can ask them what fraction of their family they are, what fraction the children are, what fraction of the legs of a dog are front legs, or left legs, or left front legs. Etc. I find kids get a real kick out of telling you all kinds of bizarre fractions like these once they catch on to seeing how to name fractional parts of things. At some point you can also show them that fractions can be more than one whole thing, say, by breaking two cookies into halves and giving them three of the halves and asking them how many halves they have. And helping them see that three halves then is the same as one-and-a-half cookies, just as you probably already have shown them that two halves are the same as a whole cookie (except for some of the crumbs that fall when you break the cookie in half).
10) As they learn to add numbers, give them plenty of practice by letting them play games where they add numbers together. They can play with two or more dice, for example. Or they can play "double war" in cards, a game where each player turns over two cards, and the player with the highest SUM wins all the cards turned over. (When a player runs out of cards to turn over, he or she picks up the cards s/he has won and uses them. Each player keeps doing this until one player has all the cards.) Or when they are old enough to start to understand the game, they can play blackjack just for fun without betting anything. They will like just trying to win each hand. As your children get better at adding and subtracting, you can show them neat "magic" tricks with numbers, such as how to add up the numbers that are on the BOTTOMS of the dice they have rolled, without having to pick up the dice to see those numbers. (The opposite sides of dice add up to 7, so if the three is rolled, a four is on the bottom; if a six is rolled, the one is on the bottom. So if you roll two dice and get a five and a three, you know that there is a two and a four on the bottom, and can sum them up to six. Also, the opposite sides of TWO dice will add up to 14, so you could add the five and the three you see and subtract that from 14 to still get 6.) Once a kid learns how to do this trick, s/he can amaze his/her friends, and get lots of practice. Especially if using three or four dice.
11) I believe it is important that children play games that give them practice adding single digit numbers up to sums of at least 18, since 18 is the largest number you ever get when you regroup or "borrow" numbers by the "standard" subtraction "method" or recipe (algorithm); e.g, if you are subtracting 9 from 38, in the standard American algorithm, you change the "thirty" to "twenty with 10 ones", and that gives you 18 ones. (If you were to get 19, you would not have had to regroup in the first place, because you could have subtracted any digit from the 9 that you began with, without having to "borrow" to do it; e.g., if you were subtracting something from 39, you would never have to "borrow" from the "thirty", since with a 9 in the one's column, you could subtract ANY number from it in the one's column.) If you are not opposed to letting them play cards, "blackjack" or 21 is an easy and excellent game for practice in developing this particular skill.
12) Children run into great difficulty learning "place-value"-- what the different columns of numbers represent, AND WHY, etc. And many learn it only by rote (they never learn the "why"), which causes problems later in a number of places. I think there is an easy and great way to teach place-value, and to teach about regrouping, borrowing, etc. using poker chips with different colors. (Stacks of poker chips can also teach about fractional relationships; e.g., if you start with a stack of 32 poker chips, half of that stack is 16, half of that is 8, half of that is 4, half of that is 2, and half of that is 1; and you can show them the relationships among the stacks: e.g., 4 is half of the stack with 8 in it, and 1/4 of the stack with 16. Etc., etc.) Plus, when they are first learning to count, and also learning to count by two's, etc., they can count poker chips and stack them into two's, five's, ten's. So I recommend that you buy a pack or two of poker chips (be sure they have stacks of at least three colors -- commonly red ones, white ones, and blue ones), which you can get for a few dollars a pack at some of the discount stores or at some drugstores. And I also recommend your buying two decks of cards, since you can give kids practice in counting and adding and subtracting with them. They can count cards or count the objects on the faces, or add and subtract the face values, in a number of different games they might play, or in a number of different tasks you might ask them to do, that they often will find fun.
13) If children have learned fractions and place-value, decimals will not be all that difficult, with some help and explanation. And once they have learned these things, percentages will be easy as well.
14) Finally, for now, you can lay some groundwork as early as kindergarten or first grade for word problems in general and for algebra later, by asking questions like "If I have a bag, and you have a bag, and we each have the same number of things in our bags, and together we have four things, how many things do we each have?" Let the child figure it out however s/he wants to; don't make there be some particular way to do it. As the child gets older or more sophisticated in arithmetic, you can make the question more sophisticated: "I have a bag and you have a bag, and I have twice as much as you, and together we have nine things in our bags...." or "If we double what you have and then add three, you will have 13...." or even harder: "I have five more than you do in my bag, but if you double what is in your bag, you will have five more than I do...." Surprisingly perhaps, kids can figure these out. Sometimes they do so by trial and error or by lucky guesses; but all of them give them more and more practice with numbers and with relationships between numbers. And they often seem to love doing these things, at least in small doses. And they also like doing "progressions", such as "if numbers start out going 1, 2, 4, 8, what number will be next, and HOW DO YOU KNOW?" You can quickly begin to make the progressions harder and they will still catch on. Or you can make two different progressions in the same problem: what should come after 3, 4, 6, 8, 12? (16) And why? (There are two progressions here: 3, 6, 12, as the first, third, and fifth numbers in the series, and 4, 8, _ as the second, fourth, etc. numbers.)
15) You may have your own areas of math that you find interesting: geometry,
trig, topology, etc. Try to devise games or puzzles using insights from
those areas that your children might find fun to play with and think about.
There are various inexpensive math puzzle, riddle, logic, or "magic" books,
and free Internet sites, available that teach many different aspects of
math in different fun ways. Simple objects can be used to teach math
elements also. Nobel physicist Richard Feynman told, for example,
about how when he was still in his high chair, his father would bring home
color tiles and would line them up in various ways for (and with) him so
that there would be patterns, such as blue-white-blue-white-blue-white,
or color patterns alternating by thirds or some other way. It was something
of just a fun game for the baby, but a game that had a deeper meaning and
point to his father. As long as it stays interesting or fun for the child,
I do not see any harm in it, and it might have much educational developmental
value for later.