Velvet's Division Homework Help
Mar. 18th, 2010 10:25 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
velvetpageAttention, parents of school-age children! Are you frustrated by the fact that your child's math homework doesn't bear even a passing resemblance to the method you were so painstakingly taught? Is your child frustrated at getting one method at school and another at home?
Never fear, help is here!
First, the big idea. Division is not a topic on its own. It's one of the four main operations, and it's connected to all the others; that is, you can use any one of the four (addition, subtraction, multiplication, and division) to solve a division question. Some methods will be more or less efficient than others, but all of them will work. The goal in math class (probably, if the teacher is up-to-date) is to get your child to build on the math they already know so they can figure out, not just the answer, but what that answer means.
Let's take a pretty simple example: 56 divided by 4. It will usually be seated in a context; for example, you might have 56 candies to be divided amongst four kids.
If the kid is totally stymied by the math, don't start with the numbers; start with raisins or some other small food item or small block. Count out 56 of them on the table, and ask your kid to divide them into four groups, one for each kid in the question. Then they can count how many raisins are in each group. When they write it down for their teacher, they should explain how they got the answer, either by drawing it (four groups of fourteen candies) or explaining in words.
If your kid is proficient at addition and subtraction, but has trouble with multiplication, you can try repeated addition or repeated subtraction. The question for repeated addition is: "Can you figure out how many times you'd have to add four in order to get 56?" If they need a starting point, you can point out that four plus four is eight; that's two groups of four. What's three groups of four? What's four groups of four? Some kids will need to keep adding groups of four until they get to 56. That's fine for the first time through, and I wouldn't push a more efficient answer until they're comfortable with that one, but it will get tedious after a while and is prone to error, so it's not an end strategy. Others will recognize that they can add small groups of groups - for example, five groups of four is 20, and 20 + 20 = 40, and then it's only four more groups of four to get to 56; so five groups, plus five more groups, plus four more groups, is fourteen. This is a really good intermediate strategy because it prepares them for the long-division method that you, their parent, learned in school.
If your kid is really focused on the number 56, you can try repeated subtraction. The question there is, "How many groups of four would you have to take out, to get to zero?" They'll subtract four, then subtract four more, then again, until they get down further. This strategy often develops after repeated addition is well-established, because adding up is usually easier for kids than subtracting. (As a side note, repeated subtraction is the concept behind the long division algorithm that we learned in school; you find the multiple closest to the digit(s) on the left, then multiply to get the actual multiple with no remainder, then subtract to get the remainder that you still need to figure out groups for. Once kids learn repeated subtraction, you can show them how that leads to long division, and they'll probably get it.)
If your kid is pretty decent at multiplication, or if they have a concept of multiplication but are not yet proficient at it, you can take it from that angle. The question then is, "Think about your four times table. What's the closest you can get to 56 in your four times table?" (The answer will probably be either 4x10, which is 40, or 4x12, which is 48.) Then you say, "Okay, so 4x10 is 40. That's ten groups of four, right? How many more groups of four would it take to get to 56?" Then you're back to repeated addition, but it's now connected to multiplication as well as addition. In fact, this is the intermediate strategy, the one that comes after repeated addition but before simple division.
If they aren't getting it with straight numbers on the page (and for that matter, even if they are, if this is their first shot at division) please, for the love of all that's holy, break out the raisins. Or the peanuts. Or the cheerios. Or the blocks. It doesn't matter what you use, and it doesn't matter if they eat it afterwards - what matters is that they have something to count that they can hold in their hands. At school, they're probably using manipulatives - blocks specially designed for supporting math concepts - but anything they can count will do the trick. Kids need to associate the numbers on the page with real-life situations and objects. When they were little, you taught them to count their fingers, their toes, the number of apple slices on their plate, and anything else they could see. The need hasn't changed - only the size of the numbers.
Once your kid gets to the point where they need a visual but not necessarily something concrete to count, you can try using an area model. The easiest way to do this is with graph paper. Make a rectangle four squares across by one square down. Ask your child how many groups of four you've drawn (1) and how many squares are in it in total (4). Now extend your rectangle to a 4x2 rectangle and ask the same questions (two groups of four and eight squares total.) Now do it again, and again, until you get to a total of 56 squares; if you count up the number of rows, you should get 14. It's easy to point out smaller divisions with that; for example, you can draw a dotted line at 4x5 and another at 4x10 and wait for your kid to figure out that they're the same amount - 4x5x2=4x10, and 40/10=40/5/2. Work with your kid to write as many different math facts for the same information as possible; fact families are valuable connections and lead directly into fractions, decimals, and algebra.
The goal is to move them through the concrete stage (things they can hold) and the representative stage (things they can draw) to the symbolic stage (number sentences) to the abstract stage (manipulating factors by substituting, for example, 5x2 for 10.) The problem with the old algorithm that we learned is that it went straight to the symbolic stage, bypassed the concrete and representative stages, and never got as far as the abstract stage at all. It's a good way for doing division after you understand division, but it's not a good way to learn to understand division.
If you're having trouble with this explanation, I suggest you get out the raisins and try it yourself. You wouldn't be the first person to figure out something important about how grade four math works, when you're supposed to be teaching grade four math. After all, the reason teachers are changing it is that this way works better; it stands to reason that there are people out there who thought they got it, but really didn't, or whose teachers thought they got it, when they knew they didn't.
If you had no trouble with this explanation, think about when it came clear to you. I'd be prepared to bet money on the fact that for four out of five of you, this did not come clear to you while you were learning long division. It came clear either before, because your parents showed you "tricks" for doing it fast, or because you had really good multiplication sense, or it came clear much later, when someone did a division question in front of you without using the algorithm and suddenly you realized why the algorithm worked. The key is that the algorithm is not really how most people learned to divide. It was how they learned to show division. They learned to manipulate the numbers some other way to make them make sense, and then fit them into the algorithm.
Never fear, help is here!
First, the big idea. Division is not a topic on its own. It's one of the four main operations, and it's connected to all the others; that is, you can use any one of the four (addition, subtraction, multiplication, and division) to solve a division question. Some methods will be more or less efficient than others, but all of them will work. The goal in math class (probably, if the teacher is up-to-date) is to get your child to build on the math they already know so they can figure out, not just the answer, but what that answer means.
Let's take a pretty simple example: 56 divided by 4. It will usually be seated in a context; for example, you might have 56 candies to be divided amongst four kids.
If the kid is totally stymied by the math, don't start with the numbers; start with raisins or some other small food item or small block. Count out 56 of them on the table, and ask your kid to divide them into four groups, one for each kid in the question. Then they can count how many raisins are in each group. When they write it down for their teacher, they should explain how they got the answer, either by drawing it (four groups of fourteen candies) or explaining in words.
If your kid is proficient at addition and subtraction, but has trouble with multiplication, you can try repeated addition or repeated subtraction. The question for repeated addition is: "Can you figure out how many times you'd have to add four in order to get 56?" If they need a starting point, you can point out that four plus four is eight; that's two groups of four. What's three groups of four? What's four groups of four? Some kids will need to keep adding groups of four until they get to 56. That's fine for the first time through, and I wouldn't push a more efficient answer until they're comfortable with that one, but it will get tedious after a while and is prone to error, so it's not an end strategy. Others will recognize that they can add small groups of groups - for example, five groups of four is 20, and 20 + 20 = 40, and then it's only four more groups of four to get to 56; so five groups, plus five more groups, plus four more groups, is fourteen. This is a really good intermediate strategy because it prepares them for the long-division method that you, their parent, learned in school.
If your kid is really focused on the number 56, you can try repeated subtraction. The question there is, "How many groups of four would you have to take out, to get to zero?" They'll subtract four, then subtract four more, then again, until they get down further. This strategy often develops after repeated addition is well-established, because adding up is usually easier for kids than subtracting. (As a side note, repeated subtraction is the concept behind the long division algorithm that we learned in school; you find the multiple closest to the digit(s) on the left, then multiply to get the actual multiple with no remainder, then subtract to get the remainder that you still need to figure out groups for. Once kids learn repeated subtraction, you can show them how that leads to long division, and they'll probably get it.)
If your kid is pretty decent at multiplication, or if they have a concept of multiplication but are not yet proficient at it, you can take it from that angle. The question then is, "Think about your four times table. What's the closest you can get to 56 in your four times table?" (The answer will probably be either 4x10, which is 40, or 4x12, which is 48.) Then you say, "Okay, so 4x10 is 40. That's ten groups of four, right? How many more groups of four would it take to get to 56?" Then you're back to repeated addition, but it's now connected to multiplication as well as addition. In fact, this is the intermediate strategy, the one that comes after repeated addition but before simple division.
If they aren't getting it with straight numbers on the page (and for that matter, even if they are, if this is their first shot at division) please, for the love of all that's holy, break out the raisins. Or the peanuts. Or the cheerios. Or the blocks. It doesn't matter what you use, and it doesn't matter if they eat it afterwards - what matters is that they have something to count that they can hold in their hands. At school, they're probably using manipulatives - blocks specially designed for supporting math concepts - but anything they can count will do the trick. Kids need to associate the numbers on the page with real-life situations and objects. When they were little, you taught them to count their fingers, their toes, the number of apple slices on their plate, and anything else they could see. The need hasn't changed - only the size of the numbers.
Once your kid gets to the point where they need a visual but not necessarily something concrete to count, you can try using an area model. The easiest way to do this is with graph paper. Make a rectangle four squares across by one square down. Ask your child how many groups of four you've drawn (1) and how many squares are in it in total (4). Now extend your rectangle to a 4x2 rectangle and ask the same questions (two groups of four and eight squares total.) Now do it again, and again, until you get to a total of 56 squares; if you count up the number of rows, you should get 14. It's easy to point out smaller divisions with that; for example, you can draw a dotted line at 4x5 and another at 4x10 and wait for your kid to figure out that they're the same amount - 4x5x2=4x10, and 40/10=40/5/2. Work with your kid to write as many different math facts for the same information as possible; fact families are valuable connections and lead directly into fractions, decimals, and algebra.
The goal is to move them through the concrete stage (things they can hold) and the representative stage (things they can draw) to the symbolic stage (number sentences) to the abstract stage (manipulating factors by substituting, for example, 5x2 for 10.) The problem with the old algorithm that we learned is that it went straight to the symbolic stage, bypassed the concrete and representative stages, and never got as far as the abstract stage at all. It's a good way for doing division after you understand division, but it's not a good way to learn to understand division.
If you're having trouble with this explanation, I suggest you get out the raisins and try it yourself. You wouldn't be the first person to figure out something important about how grade four math works, when you're supposed to be teaching grade four math. After all, the reason teachers are changing it is that this way works better; it stands to reason that there are people out there who thought they got it, but really didn't, or whose teachers thought they got it, when they knew they didn't.
If you had no trouble with this explanation, think about when it came clear to you. I'd be prepared to bet money on the fact that for four out of five of you, this did not come clear to you while you were learning long division. It came clear either before, because your parents showed you "tricks" for doing it fast, or because you had really good multiplication sense, or it came clear much later, when someone did a division question in front of you without using the algorithm and suddenly you realized why the algorithm worked. The key is that the algorithm is not really how most people learned to divide. It was how they learned to show division. They learned to manipulate the numbers some other way to make them make sense, and then fit them into the algorithm.
(no subject)
Date: 2010-03-18 04:18 pm (UTC)Yes, I used to (quickly) solve divcision problems by going "Aha, I need to divide 56 by 4. 4*10 is 40, too low. 4*20 is twice that, so too high. What about 4*15? 60, that's close, but still high, let's try 4*14 scribble ah, right, 14 it is, then."
On the other hand, I was playing around with integration and derivation at the age of 12, so I suspect I am a bit atypical.
I do remember it took me a while after figuring out long division before I understood why 22 / 0.5 was higher than 22, but that (too) clicked eventually.
(no subject)
Date: 2010-03-18 05:43 pm (UTC)I'm going to get a high school teacher friend of mine to borrow a few textbooks for me and brush up over the summer. In the meantime, I can wax eloquent about how unconscionable it is that a kid like me, with number sense and a high level of interest and generally a teacher's dream student, could have gotten all the way through high school math with marks in the eighties and no clue why she was doing any of it.
(no subject)
Date: 2010-03-18 06:16 pm (UTC)But that's a digression. I just wanted to say, yeah, that's how it's taught. It's just a shame that most kids (or teachers/schools) rely so heavily on the algorithm that they forget the other techniques and so can't back up a little to an earlier stage if they need to.
(no subject)
Date: 2010-03-18 06:28 pm (UTC)I also think it has a lot to do with inconsistencies of teaching methodology. I teach using a problem-solving method. I know full well that the grade six teachers at the middle school down the road probably aren't doing the same, or not as completely as I am, because this is my specialty subject and I've been developing my skills in it for two years now; only about one in fifteen teachers has any additional certification in math. I know for a fact that the grade four teacher isn't using a complete problem-solving methodology, because I teach in an open-concept school and I can hear his lessons. Which leaves me the only teacher in three years who is using a consistent, cross-strand, cross-curricular approach to mathematics. Of course their skills will be weak when they aren't consistently reinforced.