I think it is important to challenge kids to think beyond mere formula recall. This is particularly the case with gifted learners because absorbing spoon-fed material is not a challenge. Acceleration merely increases the pace, but it does little to challenge the brain. I believe that most gifted educators without formal advanced training in mathematics fail to see the difference because most have no clue what real mathematicians do.
In keeping with last week's post about challenging vs. accelerating, here is an elementary problem requiring only simple mathematical facts while going significantly above simple formula recall.
1. Draw a rectangle. Draw a triangle such that its base is one side of the rectangle and the opposite vertex is chosen arbitrarily on the opposite side of the rectangle. Find a relationship between the areas of the rectangle and the triangle.
2. If your kid has a rudimentary understanding of probability, I would phrase the problem differently.
Repeat the rectangle / triangle construction above. Pick a point at random inside the rectangle. What is the probability that you picked a point inside the triangle?
Note that the triangle is picked before the random point. Hence, we asking for the probability of picking a point interior to the this particular triangle.
The answer to this problem is simple. The area of a triangle is 1/2 of its base times its height. Since the height and base are always the same regardless of the location of the triangle's apex, the area never changes. In fact, the area is 1/2 that of the rectangle. You will pick a point interior to the triangle with a 50% probability, and this independent of where the triangle's apex is located.
If your kid is advanced enough, you can ask him or her to generalize this problem to 3 dimensions. The simpliest, 3D generalization is to think of pyramids inside prisms. The answer is 1/3 probability of picking a point inside the pyramid. This is because the volume of a pyramid is 1/3 * area of base * height of pyramid.
Saturday, August 22, 2009
Grade Skipping, Nation Deceived
This will be a very short post. Every educator and parent of gifted children should visit Nation Deceived.
Acceleration and grade skipping are not for every intelligent kid. However, when the conditions are right emotionally, intellectually, and socially, acceleration and grade skipping are extremely beneficial. In fact, for many gifted children, failing to accelerate or skip grades could be extremely detrimental.
The link above takes you to A Nation Deceived: How Schools Hold Back America's Brightest Students. Please, read the paper. Share it. Use it to advocate for gifted children.
As a side note, look in the left navigation bar at the Nation Deceived website. Download the PowerPoint. It is a good summary. You will be asked to registered before downloading it. It will ask for a data and topic of your talk. Just enter "TBD" for both. You will be able to download the PowerPooint.
Acceleration and grade skipping are not for every intelligent kid. However, when the conditions are right emotionally, intellectually, and socially, acceleration and grade skipping are extremely beneficial. In fact, for many gifted children, failing to accelerate or skip grades could be extremely detrimental.
The link above takes you to A Nation Deceived: How Schools Hold Back America's Brightest Students. Please, read the paper. Share it. Use it to advocate for gifted children.
As a side note, look in the left navigation bar at the Nation Deceived website. Download the PowerPoint. It is a good summary. You will be asked to registered before downloading it. It will ask for a data and topic of your talk. Just enter "TBD" for both. You will be able to download the PowerPooint.
Wednesday, August 19, 2009
What to Do When The Distributive Law Does Not Make Sense
My daughter has understood, for some time now, a few basic applications of the distributive law. However, it became clear yesterday that she did not really comprehend why the law is true and what it means. I have never seen a book properly tackle the concept -- for very young kids. The law is usually stated as such. Applications are introduced. An algebraic justification may be given. However, I have never seen good motivational examples. This was precisely the problem with my daughter. She saw no connection between the real world and the law.
Food is a big deal in our family. So are parties and friends. So, it occurred to me to use food and children to explain the distributive law.
Distributive Law for Division - Get a bunch of M&Ms. Get 20 in total. Put them in a pile. Tell you child that you want to divide the 20 M&Ms evenly between 2 children. Clearly, the answer is 10. Now, ask you child to split the 20 M&Ms into two piles: one with 6 and one with 14. Ask your child if you get the same number of candies per child if you divide each of the two smaller piles first. Does each child get the same number of M&Ms? Try this with varying number of children and M&Ms.
The key here is to make you kid understand that you could divide the whole pile of candies or just divide any two smaller piles and then put everything together. You may have to explain this multiple times. Do it. Repetition works wonders. Have your child do this over and over with M&Ms until it becomes natural. Have the kid choose the sizes of the two piles.
It is time to work on paper. Set up another M&M pile. Break it into two smaller piles. Divide each pile evenly among the desired number of children. Now, write the equivalent arithmetic equation. Ask the child to match the various piles of M&Ms and the correct parts of the equation. Do this several times.
It is now time to use variables to see if your kid really understood the distributive law and its interpretation in terms of candy and children. Ask him or her to solve the following problem:
(2m + 4n) / 2 = ?
If your kid gets stuck, tell him or her that you have a total of 2m + 4n M&Ms. Suggest that he or she think of the big pile with 2m + 4n M&Ms as two smaller piles put together: one with 2m and one with 4n. If your child cannot do this on his own, it is time to start explaining things again. Give lots of written examples. Use simple examples. Always revert back to the interpretation using candy/children. Do not get discouraged. Keep trying because the distributive law is incredibly important.
Distributive Law for Multiplication - Use the same method as for division. For instance,
2 * 10 = 2 * (4 + 6) = 2*4 + 2*6 = 20
can be interpreted as two children, each getting 10 candies. This is the same as doubling each of the two piles of 4 and 6 M&Ms. It should not matter if you double the big pile of 10 or proceed to double the smaller piles first.
You may be tempted to explain the distributive law of multiplication by using the definition of multiplication as repeated addition. However, I advise against it because children like to relate things to concrete, familiar concepts. Candy and children are definitely concrete, so they ought to work well.
Food is a big deal in our family. So are parties and friends. So, it occurred to me to use food and children to explain the distributive law.
Distributive Law for Division - Get a bunch of M&Ms. Get 20 in total. Put them in a pile. Tell you child that you want to divide the 20 M&Ms evenly between 2 children. Clearly, the answer is 10. Now, ask you child to split the 20 M&Ms into two piles: one with 6 and one with 14. Ask your child if you get the same number of candies per child if you divide each of the two smaller piles first. Does each child get the same number of M&Ms? Try this with varying number of children and M&Ms.
The key here is to make you kid understand that you could divide the whole pile of candies or just divide any two smaller piles and then put everything together. You may have to explain this multiple times. Do it. Repetition works wonders. Have your child do this over and over with M&Ms until it becomes natural. Have the kid choose the sizes of the two piles.
It is time to work on paper. Set up another M&M pile. Break it into two smaller piles. Divide each pile evenly among the desired number of children. Now, write the equivalent arithmetic equation. Ask the child to match the various piles of M&Ms and the correct parts of the equation. Do this several times.
It is now time to use variables to see if your kid really understood the distributive law and its interpretation in terms of candy and children. Ask him or her to solve the following problem:
(2m + 4n) / 2 = ?
If your kid gets stuck, tell him or her that you have a total of 2m + 4n M&Ms. Suggest that he or she think of the big pile with 2m + 4n M&Ms as two smaller piles put together: one with 2m and one with 4n. If your child cannot do this on his own, it is time to start explaining things again. Give lots of written examples. Use simple examples. Always revert back to the interpretation using candy/children. Do not get discouraged. Keep trying because the distributive law is incredibly important.
Distributive Law for Multiplication - Use the same method as for division. For instance,
2 * 10 = 2 * (4 + 6) = 2*4 + 2*6 = 20
can be interpreted as two children, each getting 10 candies. This is the same as doubling each of the two piles of 4 and 6 M&Ms. It should not matter if you double the big pile of 10 or proceed to double the smaller piles first.
You may be tempted to explain the distributive law of multiplication by using the definition of multiplication as repeated addition. However, I advise against it because children like to relate things to concrete, familiar concepts. Candy and children are definitely concrete, so they ought to work well.
Tuesday, August 18, 2009
It's Good Enough!
Parents of highly gifted children know how hard it can be to deal with perfectionist tendencies. This is perhaps the single biggest challenge when dealing with intelligent kids. My daughter is very well adjusted. However, the perfectionist gene is a bit of an issue for her. She is not only an extreme perfectionist but also hyper-competitive. Unfortunately, the combination of these two personality traits can create serious difficulties, and dealing with them is the biggest challenge in our daily routine.
A basic strategy advocated by psychologists is the "good-enough" method. This basically means that parents tell an obsessed child that what he or she did is good enough and does not have to be perfect. This is easier said than done. While high intelligence certainly contributes to success, reasonable intelligence coupled with persistence and focus often leads to more optimal outcomes. Hence, it is extremely important for parents to learn when to encourage the perfectionist tendency and when to say "good enough."
We have been working on this for approximately one year. Our daughter is much better now when it comes to academics and the arts. She knows when to stop. She is beginning to understand when it's good enough. We cannot say the same thing about sports. Whether we play video games on the Wii, play tennis, race each other, or wrestle, she must always win. Losing is simply not an option. It does not matter to her if a kid twice her size beats her at basketball. She simply melts. She cries. She cannot help herself. This is clearly a problem. There are always going to be quite a few people better than her at sports. Hence, she will have to learn her to deal with it. There will be very few people better than her academically until she enters -- if she ever makes it -- an elite university. Hence, we need to help her deal with the reality that no matter what she tries, there is almost certainly somebody better than her.
It occurred to me during our latest summer vacation that golf is probably a good game to teach patience, hard work, and humility. My daughter likes the game. I plan to play with her a few times a month. I have explained to her that golf is possibly the most frustrating game in the world. She says she understands. We will see. Perhaps, the lessons of the game will carry into other aspects of her life. Plus, it is very nice to enjoy quality, father-daughter time every few weeks!
A basic strategy advocated by psychologists is the "good-enough" method. This basically means that parents tell an obsessed child that what he or she did is good enough and does not have to be perfect. This is easier said than done. While high intelligence certainly contributes to success, reasonable intelligence coupled with persistence and focus often leads to more optimal outcomes. Hence, it is extremely important for parents to learn when to encourage the perfectionist tendency and when to say "good enough."
We have been working on this for approximately one year. Our daughter is much better now when it comes to academics and the arts. She knows when to stop. She is beginning to understand when it's good enough. We cannot say the same thing about sports. Whether we play video games on the Wii, play tennis, race each other, or wrestle, she must always win. Losing is simply not an option. It does not matter to her if a kid twice her size beats her at basketball. She simply melts. She cries. She cannot help herself. This is clearly a problem. There are always going to be quite a few people better than her at sports. Hence, she will have to learn her to deal with it. There will be very few people better than her academically until she enters -- if she ever makes it -- an elite university. Hence, we need to help her deal with the reality that no matter what she tries, there is almost certainly somebody better than her.
It occurred to me during our latest summer vacation that golf is probably a good game to teach patience, hard work, and humility. My daughter likes the game. I plan to play with her a few times a month. I have explained to her that golf is possibly the most frustrating game in the world. She says she understands. We will see. Perhaps, the lessons of the game will carry into other aspects of her life. Plus, it is very nice to enjoy quality, father-daughter time every few weeks!
Accelerating vs. Challenging
I have been thinking a lot about the difference between accelerating and challenging in mathematics. My prior blog posts clearly show that I am a big believer in acceleration. However, there is a big difference between doing things fast and challenging the brain to think deeply. I don't mean to say that gifted children should be challenged instead of accelerated. I mean that the process of acceleration does not seem to include enough hard problems, and this may become an issue later in the child's educational career.
How should I balance acceleration and depth of thinking? My daughter is only six. Hence, I am very careful about the amount of time we spent working on traditional coursework. We do about one hour of English and math every weekday. That is plenty for her. She moves so fast that I worry quite a bit about retention. The key here is that I want her to learn the fundamentals -- math and language arts -- while having enough time for independent exploration. She likes to do art, write stories, read about the universe and the human body, etc. Hence, I need to think carefully about how to challenge her without imposing much formal study time beyond what we do with EPGY.
Here is one strategy I have been testing. Assign one tough problem every week. Come up with a challenging problem that can be solved using what the kid already knows. The problem does not have to have an exact answer. That's not the point. The point is to teach him or her to work on something hard over an extended period of time. This means that you give the child the problem, make sure it is understood, and then leave him or her alone to think. Every few days, you ask if he or she wants to talk about the problem or need a hint.
Here are a few examples:
Alas, you got a rectangle. Divide by 2 the area of the rectangle. You want the kid to come up with a formula, algorithm, etc. Don't reveal the answer. Give only small hints if they get stuck.
Ans to 2: Set the triangle on one of the sides and call that side the base. Drop a perpendicular from the base to the opposite vertex. Alas, you you two right triangles now. Use the answer to question 1.
How should I balance acceleration and depth of thinking? My daughter is only six. Hence, I am very careful about the amount of time we spent working on traditional coursework. We do about one hour of English and math every weekday. That is plenty for her. She moves so fast that I worry quite a bit about retention. The key here is that I want her to learn the fundamentals -- math and language arts -- while having enough time for independent exploration. She likes to do art, write stories, read about the universe and the human body, etc. Hence, I need to think carefully about how to challenge her without imposing much formal study time beyond what we do with EPGY.
Here is one strategy I have been testing. Assign one tough problem every week. Come up with a challenging problem that can be solved using what the kid already knows. The problem does not have to have an exact answer. That's not the point. The point is to teach him or her to work on something hard over an extended period of time. This means that you give the child the problem, make sure it is understood, and then leave him or her alone to think. Every few days, you ask if he or she wants to talk about the problem or need a hint.
Here are a few examples:
- Compute the area of right triangle once it is understood how to compute the area of a rectangle.
- Compute the area of an arbitrary triangle once you know how to compute the area of a right triangle.
Alas, you got a rectangle. Divide by 2 the area of the rectangle. You want the kid to come up with a formula, algorithm, etc. Don't reveal the answer. Give only small hints if they get stuck.
Ans to 2: Set the triangle on one of the sides and call that side the base. Drop a perpendicular from the base to the opposite vertex. Alas, you you two right triangles now. Use the answer to question 1.
Yes to Acceleration
Why is it that it is okay for gifted athletes to go straight from high school into the NBA? Forget the NBA. Isn't it a NORM for gifted, young tennis players to go pro while in their teens? The same is true for soccer. How come we accept this form of acceleration, but we don't think it is okay for kids to skip grade levels? This is complete nonsense. Kids should be accelerated whenever they are ready to handle it.
I believe that acceleration in math and science is even more critical for girls than boys. I believe that profoundly, exceptionally, and highly gifted girls (145+ IQ on the Stanford Binet) are at a disadvantage over boys because of their tendency to "dumb themselves down." Yes, when boys come into the picture and being accepted by the crowd becomes important, girls tend to dumb themselves down. While I have little data to prove it, I believe it is quite hard to act intellectually average when working three or more levels above chronological peers. Moreover, I believe that working above grade level has an ego boosting effect that may help teenage girls to deal with the pressure to be normal.
Trying to get a school to allow grade skipping can be a major undertaking. It really does not matter whether the school is public or private. Administrators will use every strategy imaginable to keep your child with kids in the same age group. The reasons could include:
None of the above reasons against acceleration is based on fact. They are myths. Despite what the professionals may say, acceleration and grade skipping are the right way to go. In fact, outcomes are far more favorable for highly gifted children who are radically accelerated (3+ years) than for those who are not. Furthermore, research shows that radically acceleration leads to a more fulfilling and successful adult life.
Here is some research in the subject of radical acceleration:
I believe that acceleration in math and science is even more critical for girls than boys. I believe that profoundly, exceptionally, and highly gifted girls (145+ IQ on the Stanford Binet) are at a disadvantage over boys because of their tendency to "dumb themselves down." Yes, when boys come into the picture and being accepted by the crowd becomes important, girls tend to dumb themselves down. While I have little data to prove it, I believe it is quite hard to act intellectually average when working three or more levels above chronological peers. Moreover, I believe that working above grade level has an ego boosting effect that may help teenage girls to deal with the pressure to be normal.
Trying to get a school to allow grade skipping can be a major undertaking. It really does not matter whether the school is public or private. Administrators will use every strategy imaginable to keep your child with kids in the same age group. The reasons could include:
- Your child is not emotionally ready to be with older kids
- Your kid will miss crucial material and then fall behind
- We have experience with gifted kids, and grade skipping is dangerous for the kid and his or her peers
- What is the rush, let him or her be a "kid"
None of the above reasons against acceleration is based on fact. They are myths. Despite what the professionals may say, acceleration and grade skipping are the right way to go. In fact, outcomes are far more favorable for highly gifted children who are radically accelerated (3+ years) than for those who are not. Furthermore, research shows that radically acceleration leads to a more fulfilling and successful adult life.
Here is some research in the subject of radical acceleration:
- Short Research Summary: http://www.davidsongifted.org/db/Articles_id_10349.aspx
- Long Review of Research: http://www.templeton.org/pdfs/funding_areas/10112_Final_Rpt_Bibliography.pdf
Friday, August 14, 2009
Strategies for Teaching Multi-digit Multiplication
Multi-digit multiplication can be challenging at first. It is quite easy to teach that multiplication means repeated addition of the same number. However, kids typically never understand the traditional method of multi-digit multiplication because their teachers themselves don't understand it. In fact, ask adults why the way they were taught to multiply actually works, and the vast majority won't know.
I first taught my daughter how to multiply using the traditional method. Unfortunately, it soon became clear that she did not understand why the method works despite the fact that she could carry out multiplications with an arbitrary number of digits.
Let me preface the following discussion by saying that multiplication with an arbitrary number of digits requires knowing the following:
I used several strategies to teach my daughter the reasons why multi-digit multiplication works.
1. Multiplication as Repeated Addition
I gave her examples of the distributive law --without telling her what I was doing -- using numbers, variables, and a combination of variables and numbers. For example:
I explained that 5 (x + y) = 5x + 5y because 5(x+y) means to add x+y to itself 5 times. Hence, you must have 5 x's and 5 y's. I actually wrote out
5(x+y) = (x+y) + (x+y) + (x+y) + (x+y) + (x+y) = 5x + 5y.
I then explained that 23 * 5 is the same thing as 23 * 5 = (20 + 3) * 5 = 20*5 + 3*5 because the definition of multiplication means I must have five 20's and five 3's. Hence, we can do the following:
23 * 5 = (20+ 3) * 5 = 20*5 + 3*5 = 100 + 15 = 115.
2. Alternate Method Using Distributive Property
We proceed similarly to the above method, but this may be easier to understand at first. Let's repeat the above example. 23 * 5 is the same as
(20 + 3) * 5 = 20 * 5 + 3 * 5 = 15 + 100 = 115
Let's expand this the product of two 2-digit numbers and introduce some useful record keeping method. Consider 23 * 25. This is the same as
The above method is much less confusing for young kids. The only drawback is that it uses many more rows that the traditional, grade school algorithm.
3. Alternate Method With Counting Followed by Regrouping
This is my preferred method. It is the closest to the method taught in school. Let's give an example. Suppose that you want to multiply 23 and 45. The first number is 2 tens and 3 ones. The second number is 4 tens and 5 ones. This means that the product will contain:
The final step above is to regroup into standard numerical notation. For instance, the 15 ones is equal to 5 ones and 1 ten. Hence, the 22 tens plus the 1 ten from the 15 ones becomes 23 tens. This is equal to 3 tens and 2 hundreds. The, 8 hundreds plus the two hundreds from the tens add up to 0 hundreds and 1 thousand.
One advantage of this method is that it spells out every step done in the traditional method, while coming as close to it as possible without carrying and multiplying simultaneously. It is an easy jump to the traditional method once this method is understood.
I hope that the above strategies help your kid to learn multi-digit multiplication. At the end of the day, it is of the utmost importance to understand the process. The actual method is irrelevant.
Patience: This is the key to teaching highly gifted children. Do not assume that they need no repetition. They do. They just need fewer drills than normal kids. Do not assume they will get every concept right away. They will not. These kids learn amazingly fast, but they tend to forget things unless they are absolutely passionate about them. Hence, you may have to explain things a few different ways.
Consistency: This is extremely important. You need to set a routine. Kids like structure and routine. Do not do too much in any one day. If you spend 4 to 5 days a week doing 30 mins per day, your highly gifted child could finish elementary school math in less than two years. Don't push it. Take a week off here and there. Don't study during the weekend if they are very young. Do a little every day and don't let too many days go by without doing a little bit of mathematics. Consistency is the name of the game.
Praise Them When They Work Hard. Don't Praise Them for Being Smart: Kids need to be praised when they work hard or accomplish something that is difficult for them. Kids should not be praised for being smart. There is nothing they can do about the way they were born. You job is to teach them how to set goals and work to accomplish them. Being smart is not an accomplishment.
I first taught my daughter how to multiply using the traditional method. Unfortunately, it soon became clear that she did not understand why the method works despite the fact that she could carry out multiplications with an arbitrary number of digits.
Let me preface the following discussion by saying that multiplication with an arbitrary number of digits requires knowing the following:
- Regrouping into powers of 10: into ones, tens, hundreds, thousands, etc. For instance, 15 tens = 1 hundred plus 5 tens.
- Multiplication by powers of 10: multiplying by 10 adds a zero to the right. Multiplication by 100 adds to zeroes to the right. And so on, and so forth...
- The distributive law: should be understood at a basic level. It is far more important for kids to understand this than to blindly multiply an arbitrary number of digits.
- put all the candy together and then double the number
- separately double the number in each box and then put them together
I used several strategies to teach my daughter the reasons why multi-digit multiplication works.
1. Multiplication as Repeated Addition
I gave her examples of the distributive law --without telling her what I was doing -- using numbers, variables, and a combination of variables and numbers. For example:
c (a + b) = ca + cb
5 (4 + 6) = 5 * 4 + 5 * 6
5 (x + 2) = 5x + 5 * 2
5 (4 + 6) = 5 * 4 + 5 * 6
5 (x + 2) = 5x + 5 * 2
I explained that 5 (x + y) = 5x + 5y because 5(x+y) means to add x+y to itself 5 times. Hence, you must have 5 x's and 5 y's. I actually wrote out
5(x+y) = (x+y) + (x+y) + (x+y) + (x+y) + (x+y) = 5x + 5y.
I then explained that 23 * 5 is the same thing as 23 * 5 = (20 + 3) * 5 = 20*5 + 3*5 because the definition of multiplication means I must have five 20's and five 3's. Hence, we can do the following:
23 * 5 = (20+ 3) * 5 = 20*5 + 3*5 = 100 + 15 = 115.
2. Alternate Method Using Distributive Property
We proceed similarly to the above method, but this may be easier to understand at first. Let's repeat the above example. 23 * 5 is the same as
(20 + 3) * 5 = 20 * 5 + 3 * 5 = 15 + 100 = 115
Let's expand this the product of two 2-digit numbers and introduce some useful record keeping method. Consider 23 * 25. This is the same as
The above method is much less confusing for young kids. The only drawback is that it uses many more rows that the traditional, grade school algorithm.3. Alternate Method With Counting Followed by Regrouping
This is my preferred method. It is the closest to the method taught in school. Let's give an example. Suppose that you want to multiply 23 and 45. The first number is 2 tens and 3 ones. The second number is 4 tens and 5 ones. This means that the product will contain:
- 3 ones, five times = 15 ones
- 2 tens, five times = 10 tens
- 3 ones, 4 ten times = 12 tens
- 2 tens, 4 ten times = 8 hundreds
The final step above is to regroup into standard numerical notation. For instance, the 15 ones is equal to 5 ones and 1 ten. Hence, the 22 tens plus the 1 ten from the 15 ones becomes 23 tens. This is equal to 3 tens and 2 hundreds. The, 8 hundreds plus the two hundreds from the tens add up to 0 hundreds and 1 thousand.One advantage of this method is that it spells out every step done in the traditional method, while coming as close to it as possible without carrying and multiplying simultaneously. It is an easy jump to the traditional method once this method is understood.
I hope that the above strategies help your kid to learn multi-digit multiplication. At the end of the day, it is of the utmost importance to understand the process. The actual method is irrelevant.
Patience: This is the key to teaching highly gifted children. Do not assume that they need no repetition. They do. They just need fewer drills than normal kids. Do not assume they will get every concept right away. They will not. These kids learn amazingly fast, but they tend to forget things unless they are absolutely passionate about them. Hence, you may have to explain things a few different ways.
Consistency: This is extremely important. You need to set a routine. Kids like structure and routine. Do not do too much in any one day. If you spend 4 to 5 days a week doing 30 mins per day, your highly gifted child could finish elementary school math in less than two years. Don't push it. Take a week off here and there. Don't study during the weekend if they are very young. Do a little every day and don't let too many days go by without doing a little bit of mathematics. Consistency is the name of the game.
Praise Them When They Work Hard. Don't Praise Them for Being Smart: Kids need to be praised when they work hard or accomplish something that is difficult for them. Kids should not be praised for being smart. There is nothing they can do about the way they were born. You job is to teach them how to set goals and work to accomplish them. Being smart is not an accomplishment.
Thursday, August 13, 2009
When should variables be introduced?
When should variables be introduced? I debated this for many days when I learned that my daughter was highly gifted. I asked myself if I should concentrate on basic arithmetic, reasoning, etc. I went back and forth between first teaching her to add two and three digit numbers and introducing variables simultaneously with those concepts. The key here is that my daughter could already add single digit numbers and some two digit numbers.
I tried various strategies to teach the concept and use of variables. First, I explained the idea of an equation. I made it simple. I simply said that an equation is when you have two things that are equal. I taught her how we write an equation using standard notation. I used simple things like 5 = 5, 1 + 2 = 3, etc. I then showed her examples of things that are not equations. I used inequalities like 4 <> 3, 4 <> 1 + 2, etc. The key was to present a ton of examples. The next step was a fun one. I taught my daughter how to solve simple linear equations using M&Ms. Yes, you read that correctly. I used candy. We typically give our daughter some sort of dessert after dinner. So, I figured it may be fun to use dessert to teach elementary math. I got a bag of M&Ms and a small bowl. I set 1 M&M on the table and 2 under the bowl -- making sure she did not know how many were under the bowl. I then put 3 M&Ms the other side of the table. I asked her to tell me how many M&Ms would have to be under the bowl so both sides of the table had the same number. It should not shock you to learn how quickly she figure it out since I told her she could only eat the M&Ms under the bowl if she got the right answer. M&M algebra became a daily favorite of my daughter after dinner. I just made sure she never had too many! We played this game for about a week. Once she was getting the answers quickly, I took out some paper. I set up an M&M problem and then wrote down the corresponding equation on a piece of paper. I put an empty box to represent the bowl. I set up a number of examples asking her to tell me which of the M&M piles corresponded to what number and what corresponded to the empty box. I then asked her to tell me the number needed in the box to make the equation true. The next step was easy. We replaced the empty box with letters like n, x, etc. I used different letters and moved back and forth between empty boxes and letters always asking her to set up an equivalent M&M algebra problem. A few days later, variables became second nature to her. She understood they were just meant to represent the number we don't know. Clearly, I let her eat the M&Ms when she got the right answers.
I know. Some parents are going to argue that I shouldn't have used candy. However, you can be careful with the amount of sugar and still make the learning process fun. After all, we all know kids love candy. This may not work for your kid for medical or other reasons, but there is always something you can use in place of candy. Just make sure your kid loves it.
Let me get back to the original question of this post. When should variables be introduced? I now believe they should be taught as early as possible. The concept is only marginally more abstract than that of a number. Why we wait until pre-algebra is beyond me. In fact, Stanford's Education Program for Gifted Youth first teaches the idea in K and 1st grade. It uses many different examples in many different situations. The upshot of this early introduction is that 2nd and 3rd grade EPGY students can solve an impressive array of equations and understand how to translate word problems into equations and vice versa. For instance, one of the standard questions in EPGY's 2nd grade final exam is to solve a system of equations like the following:
m + n = 5
m - n = 1
This is asking to find two numbers 1 apart that add to 5, and this is precisely how they are taught to solve the system. They do not look at it the same way that algebra students do. They learn the role of variables and what the problem means.
I hope this post gives you some creative ideas to teach about variables and equations. I will write about how to teach other concepts in future posts.
I tried various strategies to teach the concept and use of variables. First, I explained the idea of an equation. I made it simple. I simply said that an equation is when you have two things that are equal. I taught her how we write an equation using standard notation. I used simple things like 5 = 5, 1 + 2 = 3, etc. I then showed her examples of things that are not equations. I used inequalities like 4 <> 3, 4 <> 1 + 2, etc. The key was to present a ton of examples. The next step was a fun one. I taught my daughter how to solve simple linear equations using M&Ms. Yes, you read that correctly. I used candy. We typically give our daughter some sort of dessert after dinner. So, I figured it may be fun to use dessert to teach elementary math. I got a bag of M&Ms and a small bowl. I set 1 M&M on the table and 2 under the bowl -- making sure she did not know how many were under the bowl. I then put 3 M&Ms the other side of the table. I asked her to tell me how many M&Ms would have to be under the bowl so both sides of the table had the same number. It should not shock you to learn how quickly she figure it out since I told her she could only eat the M&Ms under the bowl if she got the right answer. M&M algebra became a daily favorite of my daughter after dinner. I just made sure she never had too many! We played this game for about a week. Once she was getting the answers quickly, I took out some paper. I set up an M&M problem and then wrote down the corresponding equation on a piece of paper. I put an empty box to represent the bowl. I set up a number of examples asking her to tell me which of the M&M piles corresponded to what number and what corresponded to the empty box. I then asked her to tell me the number needed in the box to make the equation true. The next step was easy. We replaced the empty box with letters like n, x, etc. I used different letters and moved back and forth between empty boxes and letters always asking her to set up an equivalent M&M algebra problem. A few days later, variables became second nature to her. She understood they were just meant to represent the number we don't know. Clearly, I let her eat the M&Ms when she got the right answers.
I know. Some parents are going to argue that I shouldn't have used candy. However, you can be careful with the amount of sugar and still make the learning process fun. After all, we all know kids love candy. This may not work for your kid for medical or other reasons, but there is always something you can use in place of candy. Just make sure your kid loves it.
Let me get back to the original question of this post. When should variables be introduced? I now believe they should be taught as early as possible. The concept is only marginally more abstract than that of a number. Why we wait until pre-algebra is beyond me. In fact, Stanford's Education Program for Gifted Youth first teaches the idea in K and 1st grade. It uses many different examples in many different situations. The upshot of this early introduction is that 2nd and 3rd grade EPGY students can solve an impressive array of equations and understand how to translate word problems into equations and vice versa. For instance, one of the standard questions in EPGY's 2nd grade final exam is to solve a system of equations like the following:
m + n = 5
m - n = 1
This is asking to find two numbers 1 apart that add to 5, and this is precisely how they are taught to solve the system. They do not look at it the same way that algebra students do. They learn the role of variables and what the problem means.
I hope this post gives you some creative ideas to teach about variables and equations. I will write about how to teach other concepts in future posts.
Can Your Kid Get a Truly Individualized Education?
Can your kid get a truly individualized, primary education? I believe this is next to impossible. Public and, to a larger extend, private schools argue that they educate the "whole" child, catering to individual needs. This philosophy sounds wonderful, but the reality of day-to-day instruction is quite different. This is not the result of apathy. It is a consequence of basic economics and simple arithmetic.
Teachers can and do attempt to individualize instruction. Unfortunately, this is typically limited to remedial help or additional worksheets. Gifted programs such as California's GATE use additional curriculum and ability-based grouping, but they do not really allow students to move at their own pace. Good private schools market themselves as capable of handling accelerated learners. However, a few well posed questions quickly reveal that the vast majority of schools are incapable of handling students two or more standard deviations from mean IQ. Simply put, your child will NOT get what he or she needs when faced with special needs or radical acceleration.
I visited a well-known, Los Angeles private school last year when my wife and I were searching for a challenging primary education. Our daughter has exhibited an affinity for math and science, so we approached the director of the elementary science program. After enduring a fifteen minute marketing pitch, we asked the director to explain how she would handle a first grader who already knows reading, multiplication, fractions, etc., and is years ahead of her chronological peers. Her answer shocked us. She said that every child has weaknesses and that she would simply "hold back" my daughter's mathematical development to help her on areas where she might be weaker. She also stressed that while kids enter first grade at varying developmental stages, most exit second grade at similar degrees of proficiency. My wife and I listened intently how the school would hold our baby back while we paid $20,000+ per year. This was ludicrous. There are real, easily identifiable, intellectual differences. Bear in mind that it is irrelevant for this discussion whether fat tails or a standard bell curve is the best description of the distribution of ability in a population. It is a well-established fact that the set of kids with IQs above 145+ is quite small relative to the overall population. It is also a fact that the range of subject-specific abilities follows a curve with a heavy mass density around the mean. In other words, there is a clear distinction between the rare cases of high ability and the rest of the student population at the beginning of first grade. What the science director above implied is that her school has a magical method to homogenize brains!!! Furthermore, she argued, the majority of students exit second grade on even ground. Isn't it amazing how the outliers predicted by ability distribution curves re-appear after this magical, first-through-second grade lobotomy? We visited a number of other well-known, private schools, but the same story repeated itself over and over. It was deja vu, over, and over, and over again. The bottom line is that traditional schools, however well-intentioned, are not equipped to deal with exceptional students. There are a handful such as The Mirman School for Gifted Children that offer a unique environment ideal for many highly gifted children. However, even Mirman may be wrong for your kid for reasons that have little to do with academics.
How should one educate a highly gifted child? This is a difficult question whose answer clearly depends on the child as well as the family's ability to provide a balanced, maturity-appropriate social environment. Homeschooling is clearly an option provided there is flexibility and time to embark in this serious commitment. However, I believe there are other options which I will explore in future blog posts.
Teachers can and do attempt to individualize instruction. Unfortunately, this is typically limited to remedial help or additional worksheets. Gifted programs such as California's GATE use additional curriculum and ability-based grouping, but they do not really allow students to move at their own pace. Good private schools market themselves as capable of handling accelerated learners. However, a few well posed questions quickly reveal that the vast majority of schools are incapable of handling students two or more standard deviations from mean IQ. Simply put, your child will NOT get what he or she needs when faced with special needs or radical acceleration.
I visited a well-known, Los Angeles private school last year when my wife and I were searching for a challenging primary education. Our daughter has exhibited an affinity for math and science, so we approached the director of the elementary science program. After enduring a fifteen minute marketing pitch, we asked the director to explain how she would handle a first grader who already knows reading, multiplication, fractions, etc., and is years ahead of her chronological peers. Her answer shocked us. She said that every child has weaknesses and that she would simply "hold back" my daughter's mathematical development to help her on areas where she might be weaker. She also stressed that while kids enter first grade at varying developmental stages, most exit second grade at similar degrees of proficiency. My wife and I listened intently how the school would hold our baby back while we paid $20,000+ per year. This was ludicrous. There are real, easily identifiable, intellectual differences. Bear in mind that it is irrelevant for this discussion whether fat tails or a standard bell curve is the best description of the distribution of ability in a population. It is a well-established fact that the set of kids with IQs above 145+ is quite small relative to the overall population. It is also a fact that the range of subject-specific abilities follows a curve with a heavy mass density around the mean. In other words, there is a clear distinction between the rare cases of high ability and the rest of the student population at the beginning of first grade. What the science director above implied is that her school has a magical method to homogenize brains!!! Furthermore, she argued, the majority of students exit second grade on even ground. Isn't it amazing how the outliers predicted by ability distribution curves re-appear after this magical, first-through-second grade lobotomy? We visited a number of other well-known, private schools, but the same story repeated itself over and over. It was deja vu, over, and over, and over again. The bottom line is that traditional schools, however well-intentioned, are not equipped to deal with exceptional students. There are a handful such as The Mirman School for Gifted Children that offer a unique environment ideal for many highly gifted children. However, even Mirman may be wrong for your kid for reasons that have little to do with academics.
How should one educate a highly gifted child? This is a difficult question whose answer clearly depends on the child as well as the family's ability to provide a balanced, maturity-appropriate social environment. Homeschooling is clearly an option provided there is flexibility and time to embark in this serious commitment. However, I believe there are other options which I will explore in future blog posts.
Subscribe to:
Posts (Atom)