Sunday, August 28, 2011
Mr. W. the numbers man: The life of a PROPORTION
Mr. W. the numbers man: The life of a PROPORTION: Every year I would stand in front of my Algebra I class with a problem behind me that looked like the following: X ÷ 3 = (X + 1) ÷ 4 I wo...
Mr. W. the numbers man: Can you make 10
Mr. W. the numbers man: Can you make 10: The first operation a child learns is to ADD . There is much time spent in adding this number to that. I strongly believe learning how t...
The life of a PROPORTION
Every year I would stand in front of my Algebra I class with a problem behind me that looked like the following:
"WHERE DO I BEGIN?". My answer was always the same. "WHAT DOES IT LOOK LIKE?"
There it was, 2 equal fractions, as big as life, and they had not a clue on how to begin. This has always bothered me. That something that you learn in the 3rd grade, and continue to learn every year would confound my students. However, I have been mistaken. This concept, that I thought went back to 3rd grade was not even being mentioned until the 7th in many schools. When I asked one 3rd grade teacher about the concept, she said "Oh really, I had no idea". She has been teaching for 13 years.
So, what am I talking about? Well, in the 3rd grade we start the idea of reducing fractions.
This is only the process that gets us there, because we have removed a value of one, 2/2 = 1. However, the only reason that we have the right to place an = sign between 6/8 and 3/4 is that 8 x 3 = 6 x 4.
The cross products are always = if it is a true PROPORTION. This enables a student to always know if they have reduced the fraction correctly.
I have had some teachers ask me not to share this concept with the class as it would be to difficult for them to understand. If they have just learned to multiply, what a great place to apply it. Seams reasonable to me. After all, in the 4th and 5th grade they are adding and subtracting fractions, and are asked to create PROPORTIONS so they can add or subtract.
Example: 3/5 = x/20 Now x = 12
+ 1/4 = y/20 y = 5
solution 17/20
How do we get the values of x and y?
To get x we divide 5 into 20 and we get 4, then we multiply 4 x 3 = 12.
To get y we divide 4 into 20 and we get 5, then we multiply 5 x 1 = 5.
This process of dividing around and then multiplying on the diagonal is of great importance and should not be forgotten as we grow in our understanding of Math.
We have just given our students a great way to solve proportions. You are dealing with a concept that is truly big. Its fingers extend into all parts of Math, and the earlier their awareness the better.
These are just a few of the places where PROPORTIONS are used . By knowing how to think in proportions it makes these types of problems easier. Thus the concepts I talk about above will take your students a long way. I have many more examples to explain these uses on my page called Proportions. Simply click on the word on my home blog page.
X ÷ 3 = (X + 1) ÷ 4
I would then ask them to solve for X. Many if not most would be staring at me like deer in the headlights. A few of them would start, but eventually the hands would rise to ask."WHERE DO I BEGIN?". My answer was always the same. "WHAT DOES IT LOOK LIKE?"
("If it looks like a duck, walks like a duck, and quacks like a duck, it is probably a duck")
There it was, 2 equal fractions, as big as life, and they had not a clue on how to begin. This has always bothered me. That something that you learn in the 3rd grade, and continue to learn every year would confound my students. However, I have been mistaken. This concept, that I thought went back to 3rd grade was not even being mentioned until the 7th in many schools. When I asked one 3rd grade teacher about the concept, she said "Oh really, I had no idea". She has been teaching for 13 years.
So, what am I talking about? Well, in the 3rd grade we start the idea of reducing fractions.
6/8 = 3/4 Why is that true?
Most will tell you that 2 goes into 6, 3 times, and into 8, 4 times. This is true, but it is not why.This is only the process that gets us there, because we have removed a value of one, 2/2 = 1. However, the only reason that we have the right to place an = sign between 6/8 and 3/4 is that 8 x 3 = 6 x 4.
The cross products are always = if it is a true PROPORTION. This enables a student to always know if they have reduced the fraction correctly.
I have had some teachers ask me not to share this concept with the class as it would be to difficult for them to understand. If they have just learned to multiply, what a great place to apply it. Seams reasonable to me. After all, in the 4th and 5th grade they are adding and subtracting fractions, and are asked to create PROPORTIONS so they can add or subtract.
Example: 3/5 = x/20 Now x = 12
+ 1/4 = y/20 y = 5
solution 17/20
How do we get the values of x and y?
To get x we divide 5 into 20 and we get 4, then we multiply 4 x 3 = 12.
To get y we divide 4 into 20 and we get 5, then we multiply 5 x 1 = 5.
This process of dividing around and then multiplying on the diagonal is of great importance and should not be forgotten as we grow in our understanding of Math.
We have just given our students a great way to solve proportions. You are dealing with a concept that is truly big. Its fingers extend into all parts of Math, and the earlier their awareness the better.
- CHEMISTRY
- SIMILAR GEOMETRIC FIGURES
- UNIT PRICING
- CHANGING MILES PER HOUR TO FEET PER SECOND
- PERCENT PROBLEMS
- SCALE WORD PROBLEMS
These are just a few of the places where PROPORTIONS are used . By knowing how to think in proportions it makes these types of problems easier. Thus the concepts I talk about above will take your students a long way. I have many more examples to explain these uses on my page called Proportions. Simply click on the word on my home blog page.
Thursday, August 25, 2011
Multiplication that answers why
I know that all of us in my generation learned to multiply the same way. 5 x 7 = 35, Mrs. Cross would say, then she would tell us, "put down the 5 and carry the 3". We all followed her directions. Then she would continue "5 x 3 = 15 + 3 = 18". Then we all put down the 18. "See class", she would say "5 x 37 is 185".
There were some of us who were thinking, of course, that's the way it has to be. Another group was thinking, alright, if that's the process I'll follow it. And then there was the group that was thinking "WHAT IN THE HECK ARE YOU DOING".
Many, if not all of those, last two groups grew up thinking, "I can't do math". If Mrs Cross had taken a different look at the same problem, maybe we would have more people in the first group.
The problem is 37 x 5 = ?, however that is (30 + 7) x 5 = ? Now what we need to teach is the distributive law of multiplication over addition. The 5 is multiplied by the 30 and the 7.
5 x 30 = 150 + 5 x 7 = 35 and 150 + 35 = 185
Now we know why we did what we did, and how we got to 185. What is more important is that the distributive law is very important in Algebra. So by learning it here, we have a natural transfer to Algebra latter on. From this point we will move to multiplying a 2–digit number by a 2–digit number.
Lets try 48 x 63 = ? We can look at it like Mrs. Cross and say, " 3 x 8 = 24, put down the 4 and carry the 2", and so on.
Or we can look at it as (40 + 8) (60 + 3) where each number in the first parenthesis needs to be multiplied by each in the second. So this generates:
40 x 60 + 40 x 3 + 8 x 60 + 8 x 3 =
2400 + 120 + 480 + 24 = 2400 + 600 + 24 = 3024
This also teaches us another important pattern that is easy to follow and will be a major part of Algebra latter on. It also allows us to work with multiplying with 0's which is an important concept to understand. Like any process it takes practice, but it also teaches us WHY something is happening. And by understanding the why the student has more power over the process and more confidence in solving the problem.
There were some of us who were thinking, of course, that's the way it has to be. Another group was thinking, alright, if that's the process I'll follow it. And then there was the group that was thinking "WHAT IN THE HECK ARE YOU DOING".
Many, if not all of those, last two groups grew up thinking, "I can't do math". If Mrs Cross had taken a different look at the same problem, maybe we would have more people in the first group.
The problem is 37 x 5 = ?, however that is (30 + 7) x 5 = ? Now what we need to teach is the distributive law of multiplication over addition. The 5 is multiplied by the 30 and the 7.
5 x 30 = 150 + 5 x 7 = 35 and 150 + 35 = 185
Now we know why we did what we did, and how we got to 185. What is more important is that the distributive law is very important in Algebra. So by learning it here, we have a natural transfer to Algebra latter on. From this point we will move to multiplying a 2–digit number by a 2–digit number.
Lets try 48 x 63 = ? We can look at it like Mrs. Cross and say, " 3 x 8 = 24, put down the 4 and carry the 2", and so on.
Or we can look at it as (40 + 8) (60 + 3) where each number in the first parenthesis needs to be multiplied by each in the second. So this generates:
40 x 60 + 40 x 3 + 8 x 60 + 8 x 3 =
2400 + 120 + 480 + 24 = 2400 + 600 + 24 = 3024
This also teaches us another important pattern that is easy to follow and will be a major part of Algebra latter on. It also allows us to work with multiplying with 0's which is an important concept to understand. Like any process it takes practice, but it also teaches us WHY something is happening. And by understanding the why the student has more power over the process and more confidence in solving the problem.
Wednesday, August 24, 2011
Mr. W. the numbers man: How many DECREASING NUMBERS are there?
Mr. W. the numbers man: How many DECREASING NUMBERS are there?: I remember doing a lot of Math problems over the years, but not one that could span many class years. Hear is one, that if teachers all too...
How many DECREASING NUMBERS are there?
I remember doing a lot of Math problems over the years, but not one that could span many class years. Hear is one, that if teachers all took some time, could be used from 2nd grade on. As you can see the question asks how many decreasing numbers are there? First, what is a decreasing number? Second, where do I start? Third, why undertake a project of this magnitude?
1. A decreasing number is any number where all digits contained in the number are less than the previous digit. With that as a definition, the numbers 22, 326, and 567 are not decreasing. The 2 and 2 are of course equal, although 2 is less than 3, the 6 is greater than the 2, and in 567 the digits are increasing. We have two new important terms of Math in our bank.
These numbers are in the family of decreasing numbers, 74, 310, 54321. As we see in each example, the digits decrease from left to right.
2. George Polya, the great American Mathematician of the early 20th Century always said, "When a problem is to big to solve, start small, and build up through a pattern". This is what I propose you do. In the 2nd grade we figure out how many decreasing numbers there are below 100. That is, we start small, and we begin to build successes on a difficult problem. So, what is the first decreasing number? It is 10 of course. But the next one is 20, and 21 then 30, 31, and 32. The last decreasing 2–digit number is 98.
Your 2nd grade class's job is to figure out exactly how many there are.
3. I know many of you are wondering why do it. For two reasons, why not, and to discover a true beauty in numbers. There is not a infinite number of decreasing numbers. After all, we only have 10 digits, thus we have a largest decreasing number 9,876,543,210. Also, there is only one of this size.
So, as your students move from 2nd to 3rd grade we simply ask for the number of decreasing numbers to a higher number, lets say 500. Like our 2–digit numbers, with our 3–digit numbers we need to develop a pattern, and that is the power we are placing in the hands of our youngsters. The first decreasing 3–digit number is 210, and the next is 310. We need not gasp at the amount of them, because there are not that many of them. However, as we develop the pattern it starts to become fun.
Enjoy the process of this problem, and let them enjoy the creation of the pattern.
1. A decreasing number is any number where all digits contained in the number are less than the previous digit. With that as a definition, the numbers 22, 326, and 567 are not decreasing. The 2 and 2 are of course equal, although 2 is less than 3, the 6 is greater than the 2, and in 567 the digits are increasing. We have two new important terms of Math in our bank.
These numbers are in the family of decreasing numbers, 74, 310, 54321. As we see in each example, the digits decrease from left to right.
2. George Polya, the great American Mathematician of the early 20th Century always said, "When a problem is to big to solve, start small, and build up through a pattern". This is what I propose you do. In the 2nd grade we figure out how many decreasing numbers there are below 100. That is, we start small, and we begin to build successes on a difficult problem. So, what is the first decreasing number? It is 10 of course. But the next one is 20, and 21 then 30, 31, and 32. The last decreasing 2–digit number is 98.
Your 2nd grade class's job is to figure out exactly how many there are.
3. I know many of you are wondering why do it. For two reasons, why not, and to discover a true beauty in numbers. There is not a infinite number of decreasing numbers. After all, we only have 10 digits, thus we have a largest decreasing number 9,876,543,210. Also, there is only one of this size.
So, as your students move from 2nd to 3rd grade we simply ask for the number of decreasing numbers to a higher number, lets say 500. Like our 2–digit numbers, with our 3–digit numbers we need to develop a pattern, and that is the power we are placing in the hands of our youngsters. The first decreasing 3–digit number is 210, and the next is 310. We need not gasp at the amount of them, because there are not that many of them. However, as we develop the pattern it starts to become fun.
Enjoy the process of this problem, and let them enjoy the creation of the pattern.
Tuesday, August 23, 2011
The ways to make 15
We are starting 1st grade. Math expands this year. We really begin to add, and by the end of the year we are doing some subtracting, or adding in the opposite direction. We will talk more about that later, but today I want to talk about 15. 15 you ask, what is important about 15? Well, if you are learning to play the card game Cribbage, every way you can make 15, using 2, 3, 4, or 5 cards is important. More importantly, it gives us an insight to the adding of single digits into the teens. Like adding to 10 in Kindergarten we start by looking at the combinations that make 15.
Have them tell you all the ways they can think of for 2 numbers to make 15. 14 + 1, 13 + 2, 8 + 7, 10 + 5, and all of the others. Then ask them to put them in order so they display a pattern:
1 + 14 = 15
2 + 13 = 15
3 + 12 = 15
4 + 11 = 15
5 + 10 = 15
6 + 9 = 15
7 + 8 = 15
Now look at the pattern. As numbers get bigger on the left the numbers decrease on the right. This gives the student a look toward subtraction as they get the idea of decrease and increase at the same time. The big step is to go one better. This may be a challenge for your 1st graders, but with your encouragement I believe they can do it. How many ways can I use three digits, the same or different,
to sum to 15? For most 5 + 5 + 5 might be the first thought. Great! Have them write it down. However, as they will see there are many more, like 4 + 5 + 6, and 1 + 1 + 13. There are two major goals in this exercise. The first is of course is knowing your combinations of adding that will duplicate as the numbers get bigger. By the way, using a large number line that is probably on your wall is a great visual for them. The second is recognizing that by starting small and using a PATTERN, we can form all the combinations possible.
This is not a small problem, I would contend, if you gave them this to work on it would be the largest problem they ever had. Like rearranging letters to make words, they are rearranging numbers they know to make an equal total. They are creating things that are equal: 4 + 5 + 6 = 1 + 3 + 11.
1 + 1 + 13 = 15
1 + 2 + 12 = 15
1 + 3 + 11 = 15
1 + 4 + 10 = 15
1 + 5 + 9 = 15
1 + 6 + 8 = 15
1 + 7 + 7 = 15
By starting with these your class has practiced their sums to 14 as well. I believe that by there need to make 15 accurately they solidly place the values that sum to 14 in their minds. Now we have run out of what we can do starting with 1 so:
2 + 2 + 11 = 15 we cannot use 2 + 1 it is in our above list
2 + 3 + 10 = 15
2 + 4 + 9 = 15
2 + 5 + 8 = 15
2 + 6 + 7 = 15
We are now finished with the 2's, and have practiced our sums to 13. I will let you finish the list on you own. I ends with 5 + 5 + 5.
Remember, it is the pattern that is significant, the concepts will be obvious from there. Have a good time with your numbers.
Have them tell you all the ways they can think of for 2 numbers to make 15. 14 + 1, 13 + 2, 8 + 7, 10 + 5, and all of the others. Then ask them to put them in order so they display a pattern:
1 + 14 = 15
2 + 13 = 15
3 + 12 = 15
4 + 11 = 15
5 + 10 = 15
6 + 9 = 15
7 + 8 = 15
Now look at the pattern. As numbers get bigger on the left the numbers decrease on the right. This gives the student a look toward subtraction as they get the idea of decrease and increase at the same time. The big step is to go one better. This may be a challenge for your 1st graders, but with your encouragement I believe they can do it. How many ways can I use three digits, the same or different,
to sum to 15? For most 5 + 5 + 5 might be the first thought. Great! Have them write it down. However, as they will see there are many more, like 4 + 5 + 6, and 1 + 1 + 13. There are two major goals in this exercise. The first is of course is knowing your combinations of adding that will duplicate as the numbers get bigger. By the way, using a large number line that is probably on your wall is a great visual for them. The second is recognizing that by starting small and using a PATTERN, we can form all the combinations possible.
This is not a small problem, I would contend, if you gave them this to work on it would be the largest problem they ever had. Like rearranging letters to make words, they are rearranging numbers they know to make an equal total. They are creating things that are equal: 4 + 5 + 6 = 1 + 3 + 11.
1 + 1 + 13 = 15
1 + 2 + 12 = 15
1 + 3 + 11 = 15
1 + 4 + 10 = 15
1 + 5 + 9 = 15
1 + 6 + 8 = 15
1 + 7 + 7 = 15
By starting with these your class has practiced their sums to 14 as well. I believe that by there need to make 15 accurately they solidly place the values that sum to 14 in their minds. Now we have run out of what we can do starting with 1 so:
2 + 2 + 11 = 15 we cannot use 2 + 1 it is in our above list
2 + 3 + 10 = 15
2 + 4 + 9 = 15
2 + 5 + 8 = 15
2 + 6 + 7 = 15
We are now finished with the 2's, and have practiced our sums to 13. I will let you finish the list on you own. I ends with 5 + 5 + 5.
Remember, it is the pattern that is significant, the concepts will be obvious from there. Have a good time with your numbers.
Sunday, August 21, 2011
THE USE OF 10'S IN ADDING
I can remember being 11 or 12 years old and my dad taking me to the bowling alley to keep score for his bowling team. At first I was a bit intimidated. I had been bowling for a few years already and could keep score quickly and accurately. There were 5 men on each team, they bowled fast, and they expected an accurate accounting of their scores. I had no calculator to help me, but as long as I was accurate they were happy, and they paid me accordingly. At the end of an evening I would have about $5.00 in change in my pockets, and was the richest kid on the block in 1958 or 1959.
The reason I tell the story, is not because of the money I made, but the speed addition that I had learned that made it possible. Today, when we go to the bowling alley, a computer keeps score for us. Many people do not even know why they have the score they have, and would not know how to figure out their own score, based on the number of pins they have knocked down each frame. However, even if we never enter a bowling alley, we can learn to add quickly by tens, and to use tens to add quickly.
For the first thought ADDING QUICKLY BY TENS we can look at a few examples.
1. 24 + 18 = ? In our mind we count 24, 34, 44, because 44 is 20 more than 24
Now 18 is 2 less than 20, so we need 2 less than 44, or 42,
Thus we can say from 24; 34 then 42.
2. 59 + 35 = ? Again, in our head we think 59, 69, 79, 89 + 5 = 94
Or we can think 69, 79, 89, 99, less 5 = 94
Either one is fine, but by using 10's we can get there quickly and accurately
3. 137 + 27 = ? Nothing changes after 100, so think 137; 147, 157, 167
but 27 is 3 less than 30 so we get 164.
Or 147, 157, + 7 = 164
Our first and second graders can become quite good at this with a lot of practice, and with it they will develop a clearer feeling for subtraction. Now what about USING 10'S TO ADD QUICKLY.
When I visited a Kindergarten class last year, I was given small groups ( 4 or 5) and I gave them a test.
I had 9 squares with the numbers 1 to 9 on them. I spread them out on a table and ask them to put them in pairs that would add to 10. All of the groups had no trouble finding the pairs that made 10. They even asked me for another 5 so that the left out 5 they had would have a partner to make 10. Then we lined them up: 1 – 2 – 3 – 4 – 5 9 – 8 – 7 – 6 – 5
I then said when adding always look for your 10 pairs it makes it easier. There is a pattern here to be observed and remembered, and with practice it makes larger groups of adding easier. I believe that we can take this further by looking for 3 digits to make 10 or even 4 digits to make ten. With these patterns we build confidence in math, eliminate the the anxiety that we see in so many children, and doing problems can become enjoyable, not stressful.
I finish with this story. When I was studying for my Math degree, I gave my dad a sum to do. It was five, 4-digit numbers. He placed his fingers on the top number and slowly moved his hand down. He then proceeded to write down the correct answer that I had already tabulated. I said, "Dad, how did you do that so fast?" He said, "I just captured all the 10's and the rest was easy, doesn't everybody do it that way?" "No", I said. His comment was, "you see son, that is the way I was taught in the 1920's". Have we gone the wrong direction since then. Arithmetic has not changed. For all those youngsters who are today just learning to count to 10, and those who are adding, subtracting, and multiplying their way through the basics. Give them the advantages of the patterns of Math, and they will fly through the concepts.
The reason I tell the story, is not because of the money I made, but the speed addition that I had learned that made it possible. Today, when we go to the bowling alley, a computer keeps score for us. Many people do not even know why they have the score they have, and would not know how to figure out their own score, based on the number of pins they have knocked down each frame. However, even if we never enter a bowling alley, we can learn to add quickly by tens, and to use tens to add quickly.
For the first thought ADDING QUICKLY BY TENS we can look at a few examples.
1. 24 + 18 = ? In our mind we count 24, 34, 44, because 44 is 20 more than 24
Now 18 is 2 less than 20, so we need 2 less than 44, or 42,
Thus we can say from 24; 34 then 42.
2. 59 + 35 = ? Again, in our head we think 59, 69, 79, 89 + 5 = 94
Or we can think 69, 79, 89, 99, less 5 = 94
Either one is fine, but by using 10's we can get there quickly and accurately
3. 137 + 27 = ? Nothing changes after 100, so think 137; 147, 157, 167
but 27 is 3 less than 30 so we get 164.
Or 147, 157, + 7 = 164
Our first and second graders can become quite good at this with a lot of practice, and with it they will develop a clearer feeling for subtraction. Now what about USING 10'S TO ADD QUICKLY.
When I visited a Kindergarten class last year, I was given small groups ( 4 or 5) and I gave them a test.
I had 9 squares with the numbers 1 to 9 on them. I spread them out on a table and ask them to put them in pairs that would add to 10. All of the groups had no trouble finding the pairs that made 10. They even asked me for another 5 so that the left out 5 they had would have a partner to make 10. Then we lined them up: 1 – 2 – 3 – 4 – 5 9 – 8 – 7 – 6 – 5
I then said when adding always look for your 10 pairs it makes it easier. There is a pattern here to be observed and remembered, and with practice it makes larger groups of adding easier. I believe that we can take this further by looking for 3 digits to make 10 or even 4 digits to make ten. With these patterns we build confidence in math, eliminate the the anxiety that we see in so many children, and doing problems can become enjoyable, not stressful.
I finish with this story. When I was studying for my Math degree, I gave my dad a sum to do. It was five, 4-digit numbers. He placed his fingers on the top number and slowly moved his hand down. He then proceeded to write down the correct answer that I had already tabulated. I said, "Dad, how did you do that so fast?" He said, "I just captured all the 10's and the rest was easy, doesn't everybody do it that way?" "No", I said. His comment was, "you see son, that is the way I was taught in the 1920's". Have we gone the wrong direction since then. Arithmetic has not changed. For all those youngsters who are today just learning to count to 10, and those who are adding, subtracting, and multiplying their way through the basics. Give them the advantages of the patterns of Math, and they will fly through the concepts.
Wednesday, August 17, 2011
What's in the ROOM makes a difference
In my 29 years of secondary teaching I have visited many high school classrooms of Math teachers. You always know when you have walked into a Math class. You generally see a number line that starts at –20 and goes to at least 100.There are Geometric figures, pictures of Mathematicians, charts, and graphs all around the room. I used to have my version of the TEN COMMANDMENTS OF ALGEBRA on the wall. If a student was not sure if he could do something, those rules would keep him out of trouble.
In Elementary school I do not expect to see this same amount of Math as each class teaches Math, History, English, Science, Social Studies, etc. However, I seldom see more than a simple square with the integers 1 – 100. You never see 0, and the negative integers are almost never there. This concerns me. The list of integers –20 to 100 should be on the wall, starting in Kindergarten, and the counting numbers should be on tiles on the floor. These could be used to have the students stand on before recess. The teacher could say, "find a multiple of 3 and stand on it before we go to lunch", "David go stand on an even number", or "Maria, find the number that is two bigger than 13". Imagine a teacher saying, "lets go walk and sing our 2's, or 3's. And as they do it they have a visual that corresponds.
The visuals in the class would give the teacher a chance to see another form of learning, and it would give her a clear understanding of the child's ability to follow directions. Some students may want to know what the negative integers are even though they are not part of the Kindergarten curriculum. At this point, telling them that they are a reflection of the positive integers with 0 as the mirror should a good explanation. Or you can tell them how they help to take things away rather than add things on.
The environment of the classroom says a lot about what is important to a teacher. You may see a room covered with pictures of astronauts and the shuttle, pictures of race cars, football teams, baseball teams,
or whales. These do tell us about the teacher, but remember these pictures will become part of hundreds of youngsters minds. What we have or do not have there will make a difference. To a young mind it is saying this is important or not important.
As I sat in my own 4th grade class, above my teachers desk, dead center of the room was a statement in big letters. I read it every day. I had no choice, it was directly in front of my desk. It seemed to be saying, "read me first before you do anything else". I am 64 years old and I have never forgotten those words; "ASK A QUESTION AND YOU ARE A FOOL FOR FIVE MINUTES, DON'T ASK AND YOU ARE A FOOL THE REST OF YOUR LIFE". To this day I am not ashamed to ask for directions, or get help from a worker at a store. Anything that I feel will make my life easier. Moreover, as a teacher I realized there were no poor questions.
As teachers we need to consider all the things we teach. As I have said before, you are the warriors of education, and these are just suggestions that may help you, and give your students a better idea of true beauty of Mathematics and all its patterns.
In Elementary school I do not expect to see this same amount of Math as each class teaches Math, History, English, Science, Social Studies, etc. However, I seldom see more than a simple square with the integers 1 – 100. You never see 0, and the negative integers are almost never there. This concerns me. The list of integers –20 to 100 should be on the wall, starting in Kindergarten, and the counting numbers should be on tiles on the floor. These could be used to have the students stand on before recess. The teacher could say, "find a multiple of 3 and stand on it before we go to lunch", "David go stand on an even number", or "Maria, find the number that is two bigger than 13". Imagine a teacher saying, "lets go walk and sing our 2's, or 3's. And as they do it they have a visual that corresponds.
The visuals in the class would give the teacher a chance to see another form of learning, and it would give her a clear understanding of the child's ability to follow directions. Some students may want to know what the negative integers are even though they are not part of the Kindergarten curriculum. At this point, telling them that they are a reflection of the positive integers with 0 as the mirror should a good explanation. Or you can tell them how they help to take things away rather than add things on.
The environment of the classroom says a lot about what is important to a teacher. You may see a room covered with pictures of astronauts and the shuttle, pictures of race cars, football teams, baseball teams,
or whales. These do tell us about the teacher, but remember these pictures will become part of hundreds of youngsters minds. What we have or do not have there will make a difference. To a young mind it is saying this is important or not important.
As I sat in my own 4th grade class, above my teachers desk, dead center of the room was a statement in big letters. I read it every day. I had no choice, it was directly in front of my desk. It seemed to be saying, "read me first before you do anything else". I am 64 years old and I have never forgotten those words; "ASK A QUESTION AND YOU ARE A FOOL FOR FIVE MINUTES, DON'T ASK AND YOU ARE A FOOL THE REST OF YOUR LIFE". To this day I am not ashamed to ask for directions, or get help from a worker at a store. Anything that I feel will make my life easier. Moreover, as a teacher I realized there were no poor questions.
As teachers we need to consider all the things we teach. As I have said before, you are the warriors of education, and these are just suggestions that may help you, and give your students a better idea of true beauty of Mathematics and all its patterns.
Saturday, August 13, 2011
Mr. W. the numbers man: Math and the Home Front
Mr. W. the numbers man: Math and the Home Front: "Many years ago a colleague of mine asked me, 'if you had all the money you could spend on education, where would you put it'. I said, 'pare..."
Friday, August 12, 2011
Mr. W. the numbers man: Math and the Home Front
Mr. W. the numbers man: Math and the Home Front: "Many years ago a colleague of mine asked me, 'if you had all the money you could spend on education, where would you put it'. I said, 'pare..."
Math and the Home Front
Many years ago a colleague of mine asked me, "if you had all the money you could spend on education, where would you put it". I said, "parental involvement". He said that no one else had said that and he had asked over one hundred educators that same question. Today I have not changed my mind. It all starts at home. It is ease to blame the teacher, but they only see the child 7% of their lives. Send them off to school with curiosity, and the love of learning.
The new year is about to start for our students. They will be one grade higher or starting anew. Most will be coming home with questions about their Math. What will we say? How will we approach their questions? Will we say, as some parents do, "I was never good at math, I can't help you". Some may even go as far as saying "I wasn't any good at Math, and you probably won't be either".
If your teenager is taking Trigonometry, and your last Math class was Algebra, you might have trouble helping him out. However, that does not make you bad in Math, it only means that you are unaware of the concepts that he is asking about. If your child is in the 6th grade or below, you should be able to help him. The last thing we need to do is place more anxiety in their minds. At times their own teachers do that for them. After all you would not tell them, sorry I cannot read.
Just because you were not that pop-bottled glasses youngster in your 3rd to 6th grade class that always earned the highest grade in Math class, does not mean you can not handle their Math. The four basic operations are the same, and by sitting down together, writing out what you know, and showing them how you see the problem solved will go a long way in building their confidence. You might even show them problems that are similar that you solve on a daily basis. This shows them the importance of what they are learning.
We want them to be life learners. We want them to get up in the morning with the excitement of learning something new. While I am thinking about it, school is not FUN. Going to a theme park on the weekend, playing video games, or building a tree house, these are fun. I do not believe that telling our kids "have fun" every morning, as they leave for school, is the message we want to convey. Perhaps saying "go learn something new", and tell me about it later, might be a better message. Perhaps I am being picky, and I am sure my own family would tell me so.
As a teacher, I wanted to make Math interesting, and easy to understand through the use of patterns. Often times my students would say that they had fun learning the concept, however my intent would be to have them wanting to know what comes next. It is like leading them to a beautiful meadow, and then watch them discover the wild flowers.
The new year is about to start for our students. They will be one grade higher or starting anew. Most will be coming home with questions about their Math. What will we say? How will we approach their questions? Will we say, as some parents do, "I was never good at math, I can't help you". Some may even go as far as saying "I wasn't any good at Math, and you probably won't be either".
If your teenager is taking Trigonometry, and your last Math class was Algebra, you might have trouble helping him out. However, that does not make you bad in Math, it only means that you are unaware of the concepts that he is asking about. If your child is in the 6th grade or below, you should be able to help him. The last thing we need to do is place more anxiety in their minds. At times their own teachers do that for them. After all you would not tell them, sorry I cannot read.
Just because you were not that pop-bottled glasses youngster in your 3rd to 6th grade class that always earned the highest grade in Math class, does not mean you can not handle their Math. The four basic operations are the same, and by sitting down together, writing out what you know, and showing them how you see the problem solved will go a long way in building their confidence. You might even show them problems that are similar that you solve on a daily basis. This shows them the importance of what they are learning.
We want them to be life learners. We want them to get up in the morning with the excitement of learning something new. While I am thinking about it, school is not FUN. Going to a theme park on the weekend, playing video games, or building a tree house, these are fun. I do not believe that telling our kids "have fun" every morning, as they leave for school, is the message we want to convey. Perhaps saying "go learn something new", and tell me about it later, might be a better message. Perhaps I am being picky, and I am sure my own family would tell me so.
As a teacher, I wanted to make Math interesting, and easy to understand through the use of patterns. Often times my students would say that they had fun learning the concept, however my intent would be to have them wanting to know what comes next. It is like leading them to a beautiful meadow, and then watch them discover the wild flowers.
Wednesday, August 10, 2011
Can you make 10
The first operation a child learns is to ADD. There is much time spent in adding this number to that. I strongly believe learning how to get to 10 should come first . If we look at the counting integers 1 to 9 we should ask, what unique set of 2 of them add to 10?
Before calculators fill their hands in 6th or 7th grade this needs to be pairs to look for when adding. Then as they sum by 10's their speed and confidence will increase.
This is just the start. Now the next question makes further learning easier. And like the pattern above it is important they see this as a pattern. Which UNIQUE SETS of three numbers make 10?
1 + 9, 2 + 8, 3 + 7, 4 + 6, 5 + 5 (ALL ADD TO 10)
Before calculators fill their hands in 6th or 7th grade this needs to be pairs to look for when adding. Then as they sum by 10's their speed and confidence will increase.
This is just the start. Now the next question makes further learning easier. And like the pattern above it is important they see this as a pattern. Which UNIQUE SETS of three numbers make 10?
1 + 1 + 8 = 10
1 + 2 + 7 = 10
1 + 3 + 6 = 10
1 + 4 + 5 = 10
2 + 2 + 6 = 10
2 + 3 + 5 = 10
2 + 4 + 4 = 10
3 + 3 + 4 = 10
In this pattern if we tried 1 + 5 + ? it would be 4, and we already have it.
So we need to go to 2, but we can not do 2 + 1, as we already did that.
As the children create these it is up to us to lead them to the pattern.
Have them just think of numbers that fit first. By doing this they do
all the other combinations they practice up to 10; 2 + 3 = 5, 3 + 4 = 7,
and they are writing them down over and over.
Which unique set of 4 numbers make 10?
Monday, August 8, 2011
Mr. W. the numbers man: Zero, the forgotten digit
Mr. W. the numbers man: Zero, the forgotten digit: "As all our youngsters begin their schooling in math they are asked to be able to count and write the numbers to 100, and count backwards from 1..."
Zero, the forgotten digit
As all our youngsters begin their schooling in math they are asked to be able to count and write the numbers to 100, and count backwards from 10 – 0. Funny, we are never asked to know the alphabet z to a, but I digress. I am willing to bet that most of you fine Kindergarten teachers count 1 to 10, but you are asked to have your children learn these same numbers backwards ending with 0. A bit confusing in my mind. I think we could look at it with a small change, and it may make the concept of the larger numbers easier.
Rather than starting with 1, lets start with 0.
These are our digits: 0, 1, 2, 3, 4, 5, 6, 7, 8. 9. There are ten of them, and that is why our system is called BASE TEN. We write our numbers based on these ten digits, and no more. More over, this list of digits is our pattern, 0 to 9. It tells us size, order, magnitude. So lets use that to our advantage. Now that we have listed our digits, 0 – 9, what is next, and why?
You students will probably know that it is 10, but now you can point out something as you move from 10 through 19. You repeated your 0 – 9, and placed a 1 in front of all of them to get 10 – 19. As you year progresses, and you stretch toward 100, our 0 – 9 keeps repeating, and our 1 goes to 2 the to 3 and
4 all the way to 9. This gives us our last 2–digit number 99. But, by this time, hopefully they realize that again we have run out of digits in our 2 places, and we need to add another place. This makes the creation of the number 100 more real to the students, as it follows the pattern. It also gives the digit 0 more strength in the minds of your students. It may mean, having no value, however, its importance in Math is undeniable. Try this method if you are not using it already , or let me know why it may bother you. Remember, it is just an idea, and because I believe in pattern first, concept second, we need to begin with a pattern.
Rather than starting with 1, lets start with 0.
These are our digits: 0, 1, 2, 3, 4, 5, 6, 7, 8. 9. There are ten of them, and that is why our system is called BASE TEN. We write our numbers based on these ten digits, and no more. More over, this list of digits is our pattern, 0 to 9. It tells us size, order, magnitude. So lets use that to our advantage. Now that we have listed our digits, 0 – 9, what is next, and why?
You students will probably know that it is 10, but now you can point out something as you move from 10 through 19. You repeated your 0 – 9, and placed a 1 in front of all of them to get 10 – 19. As you year progresses, and you stretch toward 100, our 0 – 9 keeps repeating, and our 1 goes to 2 the to 3 and
4 all the way to 9. This gives us our last 2–digit number 99. But, by this time, hopefully they realize that again we have run out of digits in our 2 places, and we need to add another place. This makes the creation of the number 100 more real to the students, as it follows the pattern. It also gives the digit 0 more strength in the minds of your students. It may mean, having no value, however, its importance in Math is undeniable. Try this method if you are not using it already , or let me know why it may bother you. Remember, it is just an idea, and because I believe in pattern first, concept second, we need to begin with a pattern.
Mr. W. the numbers man: So who am I
Mr. W. the numbers man: So who am I: "Hi, My name is Albert Woicicki, and my students called me Mr. W. while I was teaching high school. During those 29 years, I found, that as..."
So who am I
Hi, My name is Albert Woicicki, and my students called me Mr. W. while I was teaching high school. During those 29 years, I found, that as I had learned so well with patterns, so did my students. However, many of my students came to me without having much of a background in patterns. I also, was surprised to have students come to me with unusual concept beliefs, and that many were afraid of math.
I promised myself that upon retirement I would write a book on the concepts of arithmetic, as I see them being taught. I have done that, and it is now being edited and necessary drawings are being added. I have also written the skeleton of 65 lessons for the internet that follows the book. Because of the comments of a friend, I also added a section to the book for parents and guardians of our children. I did this, because as she said it is hard to understand the new texts. My notes are there to reassure them that Math has not changed, but some of the words are different and there are some new terms.
I do not believe that elementary school teachers teach the wrong concepts. However, I do believe what they are asked to do is a mile wide and an inch deep. That is, they are forced to teach a ton of tiny concepts that the students find difficult to tie together. Then they are asked to spend so much time testing they do not have enough time to connect the dots.
I have tremendous respect for these warriors or education that are asked to teach so much, about so many different things, much of which they have spent little time studying. Yet, they do a great job, day after day, year after year. I only hope I can be of help to them.
I promised myself that upon retirement I would write a book on the concepts of arithmetic, as I see them being taught. I have done that, and it is now being edited and necessary drawings are being added. I have also written the skeleton of 65 lessons for the internet that follows the book. Because of the comments of a friend, I also added a section to the book for parents and guardians of our children. I did this, because as she said it is hard to understand the new texts. My notes are there to reassure them that Math has not changed, but some of the words are different and there are some new terms.
I do not believe that elementary school teachers teach the wrong concepts. However, I do believe what they are asked to do is a mile wide and an inch deep. That is, they are forced to teach a ton of tiny concepts that the students find difficult to tie together. Then they are asked to spend so much time testing they do not have enough time to connect the dots.
I have tremendous respect for these warriors or education that are asked to teach so much, about so many different things, much of which they have spent little time studying. Yet, they do a great job, day after day, year after year. I only hope I can be of help to them.
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