Tuesday, January 31, 2006
What is proof? Two-column proof versus paragraph proof
I asked you in my previous post if what I wrote was a proof (Click here to read my 'proof').
Well, yes and no.
It IS a fine proof if it was SPOKEN to someone while pointing to the various parts of the picture.
But it isn't the best proof if it was written in a book. You probably had to spend some time figuring what I meant by "this line" and "that angle".
Proof needs to COMMUNICATE clearly your thoughts. That's why we use "line segment AB" orAB in text.
Then the other is you need to CONVINCE - to be logical in your reasoning. Also it's not enough to convince a fellow student but any sufficiently educated rational person - like your parents, your math teacher, and a mathematics professor.
But, the form of the proof is not the most important thing.
Numbering your arguments is not the most important thing.
In my opinion, students don't need to write proofs in 2-column format if they want to write them as plain text (prose).
I want to show you an example comparing a proof written in two-column form or written as text. And, I will also show you MY exact thought processes when I was thinking about these. You know, I haven't done these type of problems regularly or in recent years, so I don't have the proofs memorized.
PROBLEM 1: Prove that if the two diagonals in a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
MY THOUGHT PROCESS:
Better draw a picture first of all. It's a quadrilateral with diagonals. We're supposed to prove that it is a parallelogram. I will try to draw a picture that doesn't look exactly like a parallelogram; in other words a picture that is not exact.
(Why? Because often, when looking at a picture that IS drawn exactly, we say, "Well I SEE that it's a parallelogram. No need proving it." So instead I want to draw a quadrilateral that doesn't at first sight look like a parallelogram.)
So what you have is a quadrilateral with two diagonals that bisect each other. Meaning that the intersection point is a midpoint for both of the diagonals.
Well right there it sounds like some line segments will have equal lengths. And, two lines crossing always form two pairs of vertical angles... So I will have some same angles and some same line segments. Sounds like I can easily prove that there are two congruent triangles and other two congruent triangles.
But how can one get from that to proving that the lines forming the quadrilateral are parallel?
It must be the corresponding angles stuff that will work there. I will have angles with same measure, so that makes that the lines must be parallel.
Okay, the proof is ready in my mind now. Just have to write it so others can understand.
PROOF WRITTEN IN 'PARAGRAPH' FORM:
Please look at the picture. Since the diagonals are bisecting each other, the line segments marked with one little line are equal, and similarly the line segments marked with double little lines. The two angles marked with dark blue line are equal, being vertical angles. It follows from SAS congruence theorem that the two yellow triangles are congruent.
Since they are congruent, angles A and A' have the same measure. And, angles A' and A'' are the same because they are vertical angles. So since A and A' are the same, and A' and A'' are the same, it follows that angles A and A'' are the same.
But this is equivalent to the two lines that form the top and bottom of the quadrilateral being parallel.
An identical argument using the two white triangles instead of the two yellow ones proves that the two sides of the quadrilateral are parallel.
So the quadrilateral is a parallelogram.
PROOF WRITTEN IN TWO-COLUMN FORM:
I don't know which you might like better or if either is better. I like the 'paragraph' form better myself.
Whichever form you prefer, check if you're ready to spot the error in the classic fallacy proof that 1=2!
Tags: philosophy, geometry
Well, yes and no.
It IS a fine proof if it was SPOKEN to someone while pointing to the various parts of the picture.
But it isn't the best proof if it was written in a book. You probably had to spend some time figuring what I meant by "this line" and "that angle".
Proof needs to COMMUNICATE clearly your thoughts. That's why we use "line segment AB" or
Then the other is you need to CONVINCE - to be logical in your reasoning. Also it's not enough to convince a fellow student but any sufficiently educated rational person - like your parents, your math teacher, and a mathematics professor.
But, the form of the proof is not the most important thing.
Numbering your arguments is not the most important thing.
In my opinion, students don't need to write proofs in 2-column format if they want to write them as plain text (prose).
I want to show you an example comparing a proof written in two-column form or written as text. And, I will also show you MY exact thought processes when I was thinking about these. You know, I haven't done these type of problems regularly or in recent years, so I don't have the proofs memorized.
PROBLEM 1: Prove that if the two diagonals in a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
MY THOUGHT PROCESS:
Better draw a picture first of all. It's a quadrilateral with diagonals. We're supposed to prove that it is a parallelogram. I will try to draw a picture that doesn't look exactly like a parallelogram; in other words a picture that is not exact.
(Why? Because often, when looking at a picture that IS drawn exactly, we say, "Well I SEE that it's a parallelogram. No need proving it." So instead I want to draw a quadrilateral that doesn't at first sight look like a parallelogram.)
So what you have is a quadrilateral with two diagonals that bisect each other. Meaning that the intersection point is a midpoint for both of the diagonals.
Well right there it sounds like some line segments will have equal lengths. And, two lines crossing always form two pairs of vertical angles... So I will have some same angles and some same line segments. Sounds like I can easily prove that there are two congruent triangles and other two congruent triangles.
But how can one get from that to proving that the lines forming the quadrilateral are parallel?
It must be the corresponding angles stuff that will work there. I will have angles with same measure, so that makes that the lines must be parallel.
Okay, the proof is ready in my mind now. Just have to write it so others can understand.
PROOF WRITTEN IN 'PARAGRAPH' FORM:
Please look at the picture. Since the diagonals are bisecting each other, the line segments marked with one little line are equal, and similarly the line segments marked with double little lines. The two angles marked with dark blue line are equal, being vertical angles. It follows from SAS congruence theorem that the two yellow triangles are congruent.
Since they are congruent, angles A and A' have the same measure. And, angles A' and A'' are the same because they are vertical angles. So since A and A' are the same, and A' and A'' are the same, it follows that angles A and A'' are the same.
But this is equivalent to the two lines that form the top and bottom of the quadrilateral being parallel.
An identical argument using the two white triangles instead of the two yellow ones proves that the two sides of the quadrilateral are parallel.
So the quadrilateral is a parallelogram.
PROOF WRITTEN IN TWO-COLUMN FORM:
| Argument | Reason why |
| 1. The two lines marked with one brown little line are congruent. | 1. The two diagonals bisect (given). |
| 2. The two lines marked with two brown little lines are congruent. | 2. The two diagonals bisect (given). |
| 3. The two angles marked with blue lines are congruent. | 3. They are vertical angles. |
| 4. The two yellow triangles are congruent. | 4. SAS theorem and 1, 2, and 3. |
| 5. The angles A and A' are congruent. | 5. The two yellow triangles are congruent. |
| 6. The angles A' and A'' are congruent. | 6. They are vertical angles. |
| 7. The angles A and A'' are congruent. | 7. 5 and 6 together. |
| 8. The lines that form bottom and top of the quadrilateral are parallel. | 8. 7 and the theorem that says that corresponding angles being the same is equivalent to lines being parallel. |
| 9. The lines that form the two sides of the quadrilateral are parallel. | 9. Repeat steps 1-8 using the two white triangles. |
| 10. The quadrilateral is a parallelogram. | 10. 8 and 9 together. |
I don't know which you might like better or if either is better. I like the 'paragraph' form better myself.
Whichever form you prefer, check if you're ready to spot the error in the classic fallacy proof that 1=2!
Tags: philosophy, geometry
What is proof? Two-column proof versus paragraph proof
I asked you in my previous post if what I wrote was a proof (Click here to read my 'proof').
Well, yes and no.
It IS a fine proof if it was SPOKEN to someone while pointing to the various parts of the picture.
But it isn't the best proof if it was written in a book. You probably had to spend some time figuring what I meant by "this line" and "that angle".
Proof needs to COMMUNICATE clearly your thoughts. That's why we use "line segment AB" orAB in text.
Then the other is you need to CONVINCE - to be logical in your reasoning. Also it's not enough to convince a fellow student but any sufficiently educated rational person - like your parents, your math teacher, and a mathematics professor.
But, the form of the proof is not the most important thing.
Numbering your arguments is not the most important thing.
In my opinion, students don't need to write proofs in 2-column format if they want to write them as plain text (prose).
I want to show you an example comparing a proof written in two-column form or written as text. And, I will also show you MY exact thought processes when I was thinking about these. You know, I haven't done these type of problems regularly or in recent years, so I don't have the proofs memorized.
PROBLEM 1: Prove that if the two diagonals in a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
MY THOUGHT PROCESS:
Better draw a picture first of all. It's a quadrilateral with diagonals. We're supposed to prove that it is a parallelogram. I will try to draw a picture that doesn't look exactly like a parallelogram; in other words a picture that is not exact.
(Why? Because often, when looking at a picture that IS drawn exactly, we say, "Well I SEE that it's a parallelogram. No need proving it." So instead I want to draw a quadrilateral that doesn't at first sight look like a parallelogram.)
So what you have is a quadrilateral with two diagonals that bisect each other. Meaning that the intersection point is a midpoint for both of the diagonals.
Well right there it sounds like some line segments will have equal lengths. And, two lines crossing always form two pairs of vertical angles... So I will have some same angles and some same line segments. Sounds like I can easily prove that there are two congruent triangles and other two congruent triangles.
But how can one get from that to proving that the lines forming the quadrilateral are parallel?
It must be the corresponding angles stuff that will work there. I will have angles with same measure, so that makes that the lines must be parallel.
Okay, the proof is ready in my mind now. Just have to write it so others can understand.
PROOF WRITTEN IN 'PARAGRAPH' FORM:
Please look at the picture. Since the diagonals are bisecting each other, the line segments marked with one little line are equal, and similarly the line segments marked with double little lines. The two angles marked with dark blue line are equal, being vertical angles. It follows from SAS congruence theorem that the two yellow triangles are congruent.
Since they are congruent, angles A and A' have the same measure. And, angles A' and A'' are the same because they are vertical angles. So since A and A' are the same, and A' and A'' are the same, it follows that angles A and A'' are the same.
But this is equivalent to the two lines that form the top and bottom of the quadrilateral being parallel.
An identical argument using the two white triangles instead of the two yellow ones proves that the two sides of the quadrilateral are parallel.
So the quadrilateral is a parallelogram.
PROOF WRITTEN IN TWO-COLUMN FORM:
I don't know which you might like better or if either is better. I like the 'paragraph' form better myself.
Whichever form you prefer, check if you're ready to spot the error in the classic fallacy proof that 1=2!
Tags: philosophy, geometry
Well, yes and no.
It IS a fine proof if it was SPOKEN to someone while pointing to the various parts of the picture.
But it isn't the best proof if it was written in a book. You probably had to spend some time figuring what I meant by "this line" and "that angle".
Proof needs to COMMUNICATE clearly your thoughts. That's why we use "line segment AB" or
Then the other is you need to CONVINCE - to be logical in your reasoning. Also it's not enough to convince a fellow student but any sufficiently educated rational person - like your parents, your math teacher, and a mathematics professor.
But, the form of the proof is not the most important thing.
Numbering your arguments is not the most important thing.
In my opinion, students don't need to write proofs in 2-column format if they want to write them as plain text (prose).
I want to show you an example comparing a proof written in two-column form or written as text. And, I will also show you MY exact thought processes when I was thinking about these. You know, I haven't done these type of problems regularly or in recent years, so I don't have the proofs memorized.
PROBLEM 1: Prove that if the two diagonals in a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
MY THOUGHT PROCESS:
Better draw a picture first of all. It's a quadrilateral with diagonals. We're supposed to prove that it is a parallelogram. I will try to draw a picture that doesn't look exactly like a parallelogram; in other words a picture that is not exact.
(Why? Because often, when looking at a picture that IS drawn exactly, we say, "Well I SEE that it's a parallelogram. No need proving it." So instead I want to draw a quadrilateral that doesn't at first sight look like a parallelogram.)
So what you have is a quadrilateral with two diagonals that bisect each other. Meaning that the intersection point is a midpoint for both of the diagonals.
Well right there it sounds like some line segments will have equal lengths. And, two lines crossing always form two pairs of vertical angles... So I will have some same angles and some same line segments. Sounds like I can easily prove that there are two congruent triangles and other two congruent triangles.
But how can one get from that to proving that the lines forming the quadrilateral are parallel?
It must be the corresponding angles stuff that will work there. I will have angles with same measure, so that makes that the lines must be parallel.
Okay, the proof is ready in my mind now. Just have to write it so others can understand.
PROOF WRITTEN IN 'PARAGRAPH' FORM:
Please look at the picture. Since the diagonals are bisecting each other, the line segments marked with one little line are equal, and similarly the line segments marked with double little lines. The two angles marked with dark blue line are equal, being vertical angles. It follows from SAS congruence theorem that the two yellow triangles are congruent.
Since they are congruent, angles A and A' have the same measure. And, angles A' and A'' are the same because they are vertical angles. So since A and A' are the same, and A' and A'' are the same, it follows that angles A and A'' are the same.
But this is equivalent to the two lines that form the top and bottom of the quadrilateral being parallel.
An identical argument using the two white triangles instead of the two yellow ones proves that the two sides of the quadrilateral are parallel.
So the quadrilateral is a parallelogram.
PROOF WRITTEN IN TWO-COLUMN FORM:
| Argument | Reason why |
| 1. The two lines marked with one brown little line are congruent. | 1. The two diagonals bisect (given). |
| 2. The two lines marked with two brown little lines are congruent. | 2. The two diagonals bisect (given). |
| 3. The two angles marked with blue lines are congruent. | 3. They are vertical angles. |
| 4. The two yellow triangles are congruent. | 4. SAS theorem and 1, 2, and 3. |
| 5. The angles A and A' are congruent. | 5. The two yellow triangles are congruent. |
| 6. The angles A' and A'' are congruent. | 6. They are vertical angles. |
| 7. The angles A and A'' are congruent. | 7. 5 and 6 together. |
| 8. The lines that form bottom and top of the quadrilateral are parallel. | 8. 7 and the theorem that says that corresponding angles being the same is equivalent to lines being parallel. |
| 9. The lines that form the two sides of the quadrilateral are parallel. | 9. Repeat steps 1-8 using the two white triangles. |
| 10. The quadrilateral is a parallelogram. | 10. 8 and 9 together. |
I don't know which you might like better or if either is better. I like the 'paragraph' form better myself.
Whichever form you prefer, check if you're ready to spot the error in the classic fallacy proof that 1=2!
Tags: philosophy, geometry
Saturday, January 28, 2006
What is proof?
Another obstacle in high school math are the proofs in the geometry course.
What is proof? Certainly, two-column proofs are not the only kind. In fact, they are mostly popular in high school geometry textbooks. Mathematicians, most often, just write their proofs out in sentences, and that's called "paragraph" proof (well their proofs usually take many many paragraphs worth of writing).
Keith Devlin says in his book
, "... being a proof means having the capacity to completely convince any sufficiently educated, intelligent, rational person..."
Proof is about COMMUNICATING in a CONVINCING way. Remember those things: you need to COMMUNICATE (not just write a jumbled mess of symbols and numbers) in a CONVINCING way.
Is this a proof?
PROBLEM: If E is the midpoint of BD, and AE is as long as EC, prove that the two triangles are congruent.
"Look at this picture that I drew. It's not drawn to scale or to be accurate. See, this line is as long as this line. And, since this is a midpoint, then this line is as long as this line. And now look at these two angles, here and here. They are the same, I mean have the same measure, obviously, because they are formed in the two corners when these two lines cross each other, or you can also say they are 'vertical angles'. We've studied that.
So, looking at these triangles, there's a side that's as long as this side, there's an angle taht is teh same as this angle, and there's a side that is as long as this side. Well that's SAS, or side-angle-side. I mean, by SAS congruence theorem we know these two triangles are congruent."
What do you think? Is that a proof?
If it isn't, tell me why. What is missing?
And while thinking, you can also check three other ways to write the proof of the same.
Tags: philosophy, geometry
What is proof? Certainly, two-column proofs are not the only kind. In fact, they are mostly popular in high school geometry textbooks. Mathematicians, most often, just write their proofs out in sentences, and that's called "paragraph" proof (well their proofs usually take many many paragraphs worth of writing).
Keith Devlin says in his book
, "... being a proof means having the capacity to completely convince any sufficiently educated, intelligent, rational person..." Proof is about COMMUNICATING in a CONVINCING way. Remember those things: you need to COMMUNICATE (not just write a jumbled mess of symbols and numbers) in a CONVINCING way.
Is this a proof?
PROBLEM: If E is the midpoint of BD, and AE is as long as EC, prove that the two triangles are congruent.
"Look at this picture that I drew. It's not drawn to scale or to be accurate. See, this line is as long as this line. And, since this is a midpoint, then this line is as long as this line. And now look at these two angles, here and here. They are the same, I mean have the same measure, obviously, because they are formed in the two corners when these two lines cross each other, or you can also say they are 'vertical angles'. We've studied that.
So, looking at these triangles, there's a side that's as long as this side, there's an angle taht is teh same as this angle, and there's a side that is as long as this side. Well that's SAS, or side-angle-side. I mean, by SAS congruence theorem we know these two triangles are congruent."
What do you think? Is that a proof?
If it isn't, tell me why. What is missing?
And while thinking, you can also check three other ways to write the proof of the same.
Tags: philosophy, geometry
What is proof?
Another obstacle in high school math are the proofs in the geometry course.
What is proof? Certainly, two-column proofs are not the only kind. In fact, they are mostly popular in high school geometry textbooks. Mathematicians, most often, just write their proofs out in sentences, and that's called "paragraph" proof (well their proofs usually take many many paragraphs worth of writing).
Keith Devlin says in his book, "... being a proof means having the capacity to completely convince any sufficiently educated, intelligent, rational person..."
Proof is about COMMUNICATING in a CONVINCING way. Remember those things: you need to COMMUNICATE (not just write a jumbled mess of symbols and numbers) in a CONVINCING way.
Is this a proof?
PROBLEM: If E is the midpoint of BD, and AE is as long as EC, prove that the two triangles are congruent.
"Look at this picture that I drew. It's not drawn to scale or to be accurate. See, this line is as long as this line. And, since this is a midpoint, then this line is as long as this line. And now look at these two angles, here and here. They are the same, I mean have the same measure, obviously, because they are formed in the two corners when these two lines cross each other, or you can also say they are 'vertical angles'. We've studied that.
So, looking at these triangles, there's a side that's as long as this side, there's an angle taht is teh same as this angle, and there's a side that is as long as this side. Well that's SAS, or side-angle-side. I mean, by SAS congruence theorem we know these two triangles are congruent."
What do you think? Is that a proof?
If it isn't, tell me why. What is missing?
And while thinking, you can also check three other ways to write the proof of the same.
Tags: philosophy, geometry
What is proof? Certainly, two-column proofs are not the only kind. In fact, they are mostly popular in high school geometry textbooks. Mathematicians, most often, just write their proofs out in sentences, and that's called "paragraph" proof (well their proofs usually take many many paragraphs worth of writing).
Keith Devlin says in his book
, "... being a proof means having the capacity to completely convince any sufficiently educated, intelligent, rational person..." Proof is about COMMUNICATING in a CONVINCING way. Remember those things: you need to COMMUNICATE (not just write a jumbled mess of symbols and numbers) in a CONVINCING way.
Is this a proof?
PROBLEM: If E is the midpoint of BD, and AE is as long as EC, prove that the two triangles are congruent.
"Look at this picture that I drew. It's not drawn to scale or to be accurate. See, this line is as long as this line. And, since this is a midpoint, then this line is as long as this line. And now look at these two angles, here and here. They are the same, I mean have the same measure, obviously, because they are formed in the two corners when these two lines cross each other, or you can also say they are 'vertical angles'. We've studied that.
So, looking at these triangles, there's a side that's as long as this side, there's an angle taht is teh same as this angle, and there's a side that is as long as this side. Well that's SAS, or side-angle-side. I mean, by SAS congruence theorem we know these two triangles are congruent."
What do you think? Is that a proof?
If it isn't, tell me why. What is missing?
And while thinking, you can also check three other ways to write the proof of the same.
Tags: philosophy, geometry
Friday, January 27, 2006
Can I teach my child algebra?
Homeschoolers often wonder if they are capable of teaching algebra or high school geometry to their child. You know, maybe your own knowledge on those areas is a little shaky, so you're not so sure what to do when the time comes.
I'd like to ease your worries:
First of all, maybe you don't have to do all the teaching, after all.
Quite recently, it seems, companies have popped up with products where you see an experienced teacher solving algebra or trig or geometry problems, from start to finish.
You're watching little video clips on your computer but it's like sitting in a classroom, almost. There's HomeworkTV.com and MathTV.com. These would work as a supplement to your algebra book.
And there's Math U See of course, a whole curriculum with videos.
And, there exist online animated lessons of great quality, for example our current advertiser MathFoundation.com.
Or, software that solves algebra problems for you.
Then there are lots of online tutoring services, and they can definitely be cheaper than hiring a live tutor.
And, if you are doing alright with teaching, for the most part, but once in a while get stuck with an algebra problem, there are MANY totally free message boards where you can just post your math question.
Hope this helps! Feel free to comment.
Tags: algebra
I'd like to ease your worries:
First of all, maybe you don't have to do all the teaching, after all.
Quite recently, it seems, companies have popped up with products where you see an experienced teacher solving algebra or trig or geometry problems, from start to finish.
You're watching little video clips on your computer but it's like sitting in a classroom, almost. There's HomeworkTV.com and MathTV.com. These would work as a supplement to your algebra book.
And there's Math U See of course, a whole curriculum with videos.
And, there exist online animated lessons of great quality, for example our current advertiser MathFoundation.com.
Or, software that solves algebra problems for you.
Then there are lots of online tutoring services, and they can definitely be cheaper than hiring a live tutor.
And, if you are doing alright with teaching, for the most part, but once in a while get stuck with an algebra problem, there are MANY totally free message boards where you can just post your math question.
Hope this helps! Feel free to comment.
Tags: algebra
Can I teach my child algebra?
Homeschoolers often wonder if they are capable of teaching algebra or high school geometry to their child. You know, maybe your own knowledge on those areas is a little shaky, so you're not so sure what to do when the time comes.
I'd like to ease your worries:
First of all, maybe you don't have to do all the teaching, after all.
Quite recently, it seems, companies have popped up with products where you see an experienced teacher solving algebra or trig or geometry problems, from start to finish.
You're watching little video clips on your computer but it's like sitting in a classroom, almost. There's HomeworkTV.com and MathTV.com. These would work as a supplement to your algebra book.
And there's Math U See of course, a whole curriculum with videos.
And, there exist online animated lessons of great quality, for example our current advertiser MathFoundation.com.
Or, software that solves algebra problems for you.
Then there are lots of online tutoring services, and they can definitely be cheaper than hiring a live tutor.
And, if you are doing alright with teaching, for the most part, but once in a while get stuck with an algebra problem, there are MANY totally free message boards where you can just post your math question.
Hope this helps! Feel free to comment.
Tags: algebra
I'd like to ease your worries:
First of all, maybe you don't have to do all the teaching, after all.
Quite recently, it seems, companies have popped up with products where you see an experienced teacher solving algebra or trig or geometry problems, from start to finish.
You're watching little video clips on your computer but it's like sitting in a classroom, almost. There's HomeworkTV.com and MathTV.com. These would work as a supplement to your algebra book.
And there's Math U See of course, a whole curriculum with videos.
And, there exist online animated lessons of great quality, for example our current advertiser MathFoundation.com.
Or, software that solves algebra problems for you.
Then there are lots of online tutoring services, and they can definitely be cheaper than hiring a live tutor.
And, if you are doing alright with teaching, for the most part, but once in a while get stuck with an algebra problem, there are MANY totally free message boards where you can just post your math question.
Hope this helps! Feel free to comment.
Tags: algebra
Thursday, January 26, 2006
Scope and sequence
I hope everyone's teaching is going well; I guess this is a slower week for blogging for me. Maybe I'll share some of the questions asked on the site.
Someone recently asked me about MY suggested scope and sequence for teaching math, based on the way I have organized my online resources list.
Well I have never made any personal or suggested scope and sequence... Those pages are just categorized as elementary, middle school, and high school.
The thing is, you can find as many different scopes and sequences as there are textbooks. It varies from country to country, from state to state (in standards), from book to book.
People seem to have vary different ideas when it comes to when to teach which math topics.
But in my mind, you could try get your child to start algebra on 8th grade or thereabouts.
If you set that as a goal, then one should study pre-algebra topics such as integers, percent, ratio, proportion, square root, exponents on 7th grade.
One could try set a goal of going thru whole number arithmetic in the first four years, thereby leaving 5th and 6th grade for fractions, decimals, and percent.
Geometry and measuring are good topics to study on each grade. Typical state standards also include data analysis topics on all grades.
So maybe that can serve as an outline for those of you who are trying to organize your curriculum.
Someone recently asked me about MY suggested scope and sequence for teaching math, based on the way I have organized my online resources list.
Well I have never made any personal or suggested scope and sequence... Those pages are just categorized as elementary, middle school, and high school.
The thing is, you can find as many different scopes and sequences as there are textbooks. It varies from country to country, from state to state (in standards), from book to book.
People seem to have vary different ideas when it comes to when to teach which math topics.
But in my mind, you could try get your child to start algebra on 8th grade or thereabouts.
If you set that as a goal, then one should study pre-algebra topics such as integers, percent, ratio, proportion, square root, exponents on 7th grade.
One could try set a goal of going thru whole number arithmetic in the first four years, thereby leaving 5th and 6th grade for fractions, decimals, and percent.
Geometry and measuring are good topics to study on each grade. Typical state standards also include data analysis topics on all grades.
So maybe that can serve as an outline for those of you who are trying to organize your curriculum.
Scope and sequence
I hope everyone's teaching is going well; I guess this is a slower week for blogging for me. Maybe I'll share some of the questions asked on the site.
Someone recently asked me about MY suggested scope and sequence for teaching math, based on the way I have organized my online resources list.
Well I have never made any personal or suggested scope and sequence... Those pages are just categorized as elementary, middle school, and high school.
The thing is, you can find as many different scopes and sequences as there are textbooks. It varies from country to country, from state to state (in standards), from book to book.
People seem to have vary different ideas when it comes to when to teach which math topics.
But in my mind, you could try get your child to start algebra on 8th grade or thereabouts.
If you set that as a goal, then one should study pre-algebra topics such as integers, percent, ratio, proportion, square root, exponents on 7th grade.
One could try set a goal of going thru whole number arithmetic in the first four years, thereby leaving 5th and 6th grade for fractions, decimals, and percent.
Geometry and measuring are good topics to study on each grade. Typical state standards also include data analysis topics on all grades.
So maybe that can serve as an outline for those of you who are trying to organize your curriculum.
Someone recently asked me about MY suggested scope and sequence for teaching math, based on the way I have organized my online resources list.
Well I have never made any personal or suggested scope and sequence... Those pages are just categorized as elementary, middle school, and high school.
The thing is, you can find as many different scopes and sequences as there are textbooks. It varies from country to country, from state to state (in standards), from book to book.
People seem to have vary different ideas when it comes to when to teach which math topics.
But in my mind, you could try get your child to start algebra on 8th grade or thereabouts.
If you set that as a goal, then one should study pre-algebra topics such as integers, percent, ratio, proportion, square root, exponents on 7th grade.
One could try set a goal of going thru whole number arithmetic in the first four years, thereby leaving 5th and 6th grade for fractions, decimals, and percent.
Geometry and measuring are good topics to study on each grade. Typical state standards also include data analysis topics on all grades.
So maybe that can serve as an outline for those of you who are trying to organize your curriculum.
Tuesday, January 24, 2006
In which careers do you need math? And what kind?
Have you ever had your youngster ask, "Why do I need to study math? Where do I need this stuff?"
Well if not yet, you can expect the question when they get into algebra. But even before your students ask, take them into certain websites and show them!
First, Math Careers Database at Xpmath.com.
- it's not actually about careers in math, but instead a list of occupations/jobs and what kind of math topics are needed in that particular job.
I've also written an article (earlier) touching on this topic and collected a few similar links. See Where do you need square roots or algebra? Why study math? - scroll down to the bottom to see the links.
Categories: math
Well if not yet, you can expect the question when they get into algebra. But even before your students ask, take them into certain websites and show them!
First, Math Careers Database at Xpmath.com.
- it's not actually about careers in math, but instead a list of occupations/jobs and what kind of math topics are needed in that particular job.
I've also written an article (earlier) touching on this topic and collected a few similar links. See Where do you need square roots or algebra? Why study math? - scroll down to the bottom to see the links.
Categories: math
In which careers do you need math? And what kind?
Have you ever had your youngster ask, "Why do I need to study math? Where do I need this stuff?"
Well if not yet, you can expect the question when they get into algebra. But even before your students ask, take them into certain websites and show them!
First, Math Careers Database at Xpmath.com.
- it's not actually about careers in math, but instead a list of occupations/jobs and what kind of math topics are needed in that particular job.
I've also written an article (earlier) touching on this topic and collected a few similar links. See Where do you need square roots or algebra? Why study math? - scroll down to the bottom to see the links.
Categories: math
Well if not yet, you can expect the question when they get into algebra. But even before your students ask, take them into certain websites and show them!
First, Math Careers Database at Xpmath.com.
- it's not actually about careers in math, but instead a list of occupations/jobs and what kind of math topics are needed in that particular job.
I've also written an article (earlier) touching on this topic and collected a few similar links. See Where do you need square roots or algebra? Why study math? - scroll down to the bottom to see the links.
Categories: math
Saturday, January 21, 2006
Categories
You know that Blogger doesn't support categories. But I found a way to add them by using del.icio.us bookmarking service. So I'm trying them out now on my blog.
Categories
You know that Blogger doesn't support categories. But I found a way to add them by using del.icio.us bookmarking service. So I'm trying them out now on my blog.
Friday, January 20, 2006
How to teach equation solving
I got this question in my mailbox recently:
I am having a problem with show my son how to work out the proportion problem and solving equation by division, like 7.y=105 64=y.8 and find the missing term for the proportion of 3/4=x/16.
Can you help me out?
Always remember this problem solving strategy: when a problem is too difficult, make another, similar, but in some way easier problem, and observe for a strategy to solve that one.
If 7 ⋅ y = 105 and 64 = y ⋅ 8 are difficult, use easier examples first. Have him solve these ones:
2 ⋅ __ = 4
5 ⋅ x = 10
2 ⋅ x = 6
and
8 = 2 ⋅ __
12 = 3 ⋅ x
9 = 3 ⋅ x
If 'x' intimidates him, you can use an empty line.
Then ask him, HOW did he solve these ones? Well, chances are, of course, that he just 'sees' the answer, or remembers his multiplication tables and gets the answer from those.
But then SHOW him how division, in each case, gives us the answer too:
5 ⋅ x = 10. (We already know the answer is 2)
10 ÷ 5 gives the answer.
After going thru this, the initial problems should not be difficult. If they still do, then your son might have problems in understanding division concept and might need review in that area first.
One can solve the proportion 3/4 = x/16 in several different ways. This one is easy to solve thinking via fractions.
Essentially, you have two equivalent fractions: 3/4, and x/16. IF your son has difficulty solving this problem when it's written as fractions, then he should study again equivalent fractions.
Another way to solve this is to see it as an equation:
(number) = x/(number)
Can he solve easier equations such as 5 = x/2 or 2 = x/4 ?
Again, the opposite operation will work: x is divided by a number, so when solving, you need to multiply.
Hope this helps.
I am having a problem with show my son how to work out the proportion problem and solving equation by division, like 7.y=105 64=y.8 and find the missing term for the proportion of 3/4=x/16.
Can you help me out?
Always remember this problem solving strategy: when a problem is too difficult, make another, similar, but in some way easier problem, and observe for a strategy to solve that one.
If 7 ⋅ y = 105 and 64 = y ⋅ 8 are difficult, use easier examples first. Have him solve these ones:
2 ⋅ __ = 4
5 ⋅ x = 10
2 ⋅ x = 6
and
8 = 2 ⋅ __
12 = 3 ⋅ x
9 = 3 ⋅ x
If 'x' intimidates him, you can use an empty line.
Then ask him, HOW did he solve these ones? Well, chances are, of course, that he just 'sees' the answer, or remembers his multiplication tables and gets the answer from those.
But then SHOW him how division, in each case, gives us the answer too:
5 ⋅ x = 10. (We already know the answer is 2)
10 ÷ 5 gives the answer.
After going thru this, the initial problems should not be difficult. If they still do, then your son might have problems in understanding division concept and might need review in that area first.
Solving a proportion problem
One can solve the proportion 3/4 = x/16 in several different ways. This one is easy to solve thinking via fractions.
Essentially, you have two equivalent fractions: 3/4, and x/16. IF your son has difficulty solving this problem when it's written as fractions, then he should study again equivalent fractions.
Another way to solve this is to see it as an equation:
(number) = x/(number)
Can he solve easier equations such as 5 = x/2 or 2 = x/4 ?
Again, the opposite operation will work: x is divided by a number, so when solving, you need to multiply.
Hope this helps.
How to teach equation solving
I got this question in my mailbox recently:
I am having a problem with show my son how to work out the proportion problem and solving equation by division, like 7.y=105 64=y.8 and find the missing term for the proportion of 3/4=x/16.
Can you help me out?
Always remember this problem solving strategy: when a problem is too difficult, make another, similar, but in some way easier problem, and observe for a strategy to solve that one.
If 7 ⋅ y = 105 and 64 = y ⋅ 8 are difficult, use easier examples first. Have him solve these ones:
2 ⋅ __ = 4
5 ⋅ x = 10
2 ⋅ x = 6
and
8 = 2 ⋅ __
12 = 3 ⋅ x
9 = 3 ⋅ x
If 'x' intimidates him, you can use an empty line.
Then ask him, HOW did he solve these ones? Well, chances are, of course, that he just 'sees' the answer, or remembers his multiplication tables and gets the answer from those.
But then SHOW him how division, in each case, gives us the answer too:
5 ⋅ x = 10. (We already know the answer is 2)
10 ÷ 5 gives the answer.
After going thru this, the initial problems should not be difficult. If they still do, then your son might have problems in understanding division concept and might need review in that area first.
One can solve the proportion 3/4 = x/16 in several different ways. This one is easy to solve thinking via fractions.
Essentially, you have two equivalent fractions: 3/4, and x/16. IF your son has difficulty solving this problem when it's written as fractions, then he should study again equivalent fractions.
Another way to solve this is to see it as an equation:
(number) = x/(number)
Can he solve easier equations such as 5 = x/2 or 2 = x/4 ?
Again, the opposite operation will work: x is divided by a number, so when solving, you need to multiply.
Hope this helps.
I am having a problem with show my son how to work out the proportion problem and solving equation by division, like 7.y=105 64=y.8 and find the missing term for the proportion of 3/4=x/16.
Can you help me out?
Always remember this problem solving strategy: when a problem is too difficult, make another, similar, but in some way easier problem, and observe for a strategy to solve that one.
If 7 ⋅ y = 105 and 64 = y ⋅ 8 are difficult, use easier examples first. Have him solve these ones:
2 ⋅ __ = 4
5 ⋅ x = 10
2 ⋅ x = 6
and
8 = 2 ⋅ __
12 = 3 ⋅ x
9 = 3 ⋅ x
If 'x' intimidates him, you can use an empty line.
Then ask him, HOW did he solve these ones? Well, chances are, of course, that he just 'sees' the answer, or remembers his multiplication tables and gets the answer from those.
But then SHOW him how division, in each case, gives us the answer too:
5 ⋅ x = 10. (We already know the answer is 2)
10 ÷ 5 gives the answer.
After going thru this, the initial problems should not be difficult. If they still do, then your son might have problems in understanding division concept and might need review in that area first.
Solving a proportion problem
One can solve the proportion 3/4 = x/16 in several different ways. This one is easy to solve thinking via fractions.
Essentially, you have two equivalent fractions: 3/4, and x/16. IF your son has difficulty solving this problem when it's written as fractions, then he should study again equivalent fractions.
Another way to solve this is to see it as an equation:
(number) = x/(number)
Can he solve easier equations such as 5 = x/2 or 2 = x/4 ?
Again, the opposite operation will work: x is divided by a number, so when solving, you need to multiply.
Hope this helps.
Visual Math Learning - math tutorials
Here's a site that was proposed to me recently, Visual Math Learning.
It has tutorials about natural numbers, basic operations, factorization, and all fraction operations, using visual models.
They include nice interactive elements where students can experiment on their own and see the visual models change.
The site has also some nice games. I spent quite some time doing the factor sliders game...
It's all free. Go ahead and check the site out!
It has tutorials about natural numbers, basic operations, factorization, and all fraction operations, using visual models.
They include nice interactive elements where students can experiment on their own and see the visual models change.
The site has also some nice games. I spent quite some time doing the factor sliders game...
It's all free. Go ahead and check the site out!
Visual Math Learning - math tutorials
Here's a site that was proposed to me recently, Visual Math Learning.
It has tutorials about natural numbers, basic operations, factorization, and all fraction operations, using visual models.
They include nice interactive elements where students can experiment on their own and see the visual models change.
The site has also some nice games. I spent quite some time doing the factor sliders game...
It's all free. Go ahead and check the site out!
It has tutorials about natural numbers, basic operations, factorization, and all fraction operations, using visual models.
They include nice interactive elements where students can experiment on their own and see the visual models change.
The site has also some nice games. I spent quite some time doing the factor sliders game...
It's all free. Go ahead and check the site out!
Thursday, January 19, 2006
Is your math curriculum coherent, focused, and logical?
Recently I wrote a blogpost about the differences between US math curricula and those of the best performing countries in international student math comparisons.
I have put those thoughts into an article form on the site. I also got permission to post two charts that show which math topics are usually studied on which grade in the US, versus in the best performing countries in the TIMSS. The differences are quite striking!
Is your math curriculum coherent?
I have put those thoughts into an article form on the site. I also got permission to post two charts that show which math topics are usually studied on which grade in the US, versus in the best performing countries in the TIMSS. The differences are quite striking!
Is your math curriculum coherent?
Is your math curriculum coherent, focused, and logical?
Recently I wrote a blogpost about the differences between US math curricula and those of the best performing countries in international student math comparisons.
I have put those thoughts into an article form on the site. I also got permission to post two charts that show which math topics are usually studied on which grade in the US, versus in the best performing countries in the TIMSS. The differences are quite striking!
Is your math curriculum coherent?
I have put those thoughts into an article form on the site. I also got permission to post two charts that show which math topics are usually studied on which grade in the US, versus in the best performing countries in the TIMSS. The differences are quite striking!
Is your math curriculum coherent?
Tuesday, January 17, 2006
Homeschooling Carnival
Carnival of Homeschooling, the 3rd week, is up and going- including an entry from my blog. Thanks!
Homeschooling Carnival
Carnival of Homeschooling, the 3rd week, is up and going- including an entry from my blog. Thanks!
Monday, January 16, 2006
Is right answer important in math?
I want to discuss this topic because of a possible confusion. Recently in a blogpost I linked to the article Math anxiety, which mentions this math-related myth:
MYTH #4: IN MATH, WHAT'S IMPORTANT IS GETTING THE RIGHT ANSWER.
Then, in another recent post I linked to an article Things not to learn in school where the author strongly convinces us that getting 70% in a test is not enough since in real life you need to get it 100% right.
So is it important to get the right answer when doing a mathematics problem?
Well, yes but it depends. This is not a simple cut-and-dried question.
It IS important if you're drilling multiplication facts.
But many times it's not your focus. For example, when a student is learning to do long division or multiplying 4-digit numbers, the emphasis should be in learning the procedure and why it works. Calculation mistakes might yield a wrong answer, but if the way the division was done is right, then some credit should be given.
And obviously all children need encouragement while they're learning - they need partial credit or acknowledgment of things they did do right, even if the final answer was wrong.
Often times, you can use a wrong answer as a springboard and delve into the WHY the calculation went wrong. There's lots of learning one can do from wrong answers.
See, in my mind kids should be taught from early on to check their answer, always be CRITICAL of their answer, "suspicion" it until they themselves can "prove" to themselves that their answer is indeed right.
They need to learn to think twice before they do things in life. Well, in mathematics we can teach them to do something similar: when the problem is done, go back and check. Think it thru again. Does the answer make sense? Can you check it? If you estimate, is your final answer in agreement with your estimation?
So a wrong answer just means your student hasn't yet perfected this "checking and criticizing your own work" process - you know, learning to think critically about your own thought processes.
Encourage your students to point it out in their exam paper, if they know they got a wrong answer but can't find their mistake. You should give more credit to that answer than to the same wrong answer without that explanation (though not full credit of course).
Categories: philosophy
MYTH #4: IN MATH, WHAT'S IMPORTANT IS GETTING THE RIGHT ANSWER.
Then, in another recent post I linked to an article Things not to learn in school where the author strongly convinces us that getting 70% in a test is not enough since in real life you need to get it 100% right.
So is it important to get the right answer when doing a mathematics problem?
Well, yes but it depends. This is not a simple cut-and-dried question.
It IS important if you're drilling multiplication facts.
But many times it's not your focus. For example, when a student is learning to do long division or multiplying 4-digit numbers, the emphasis should be in learning the procedure and why it works. Calculation mistakes might yield a wrong answer, but if the way the division was done is right, then some credit should be given.
And obviously all children need encouragement while they're learning - they need partial credit or acknowledgment of things they did do right, even if the final answer was wrong.
Often times, you can use a wrong answer as a springboard and delve into the WHY the calculation went wrong. There's lots of learning one can do from wrong answers.
See, in my mind kids should be taught from early on to check their answer, always be CRITICAL of their answer, "suspicion" it until they themselves can "prove" to themselves that their answer is indeed right.
They need to learn to think twice before they do things in life. Well, in mathematics we can teach them to do something similar: when the problem is done, go back and check. Think it thru again. Does the answer make sense? Can you check it? If you estimate, is your final answer in agreement with your estimation?
So a wrong answer just means your student hasn't yet perfected this "checking and criticizing your own work" process - you know, learning to think critically about your own thought processes.
Encourage your students to point it out in their exam paper, if they know they got a wrong answer but can't find their mistake. You should give more credit to that answer than to the same wrong answer without that explanation (though not full credit of course).
Categories: philosophy
Is right answer important in math?
I want to discuss this topic because of a possible confusion. Recently in a blogpost I linked to the article Math anxiety, which mentions this math-related myth:
MYTH #4: IN MATH, WHAT'S IMPORTANT IS GETTING THE RIGHT ANSWER.
Then, in another recent post I linked to an article Things not to learn in school where the author strongly convinces us that getting 70% in a test is not enough since in real life you need to get it 100% right.
So is it important to get the right answer when doing a mathematics problem?
Well, yes but it depends. This is not a simple cut-and-dried question.
It IS important if you're drilling multiplication facts.
But many times it's not your focus. For example, when a student is learning to do long division or multiplying 4-digit numbers, the emphasis should be in learning the procedure and why it works. Calculation mistakes might yield a wrong answer, but if the way the division was done is right, then some credit should be given.
And obviously all children need encouragement while they're learning - they need partial credit or acknowledgment of things they did do right, even if the final answer was wrong.
Often times, you can use a wrong answer as a springboard and delve into the WHY the calculation went wrong. There's lots of learning one can do from wrong answers.
See, in my mind kids should be taught from early on to check their answer, always be CRITICAL of their answer, "suspicion" it until they themselves can "prove" to themselves that their answer is indeed right.
They need to learn to think twice before they do things in life. Well, in mathematics we can teach them to do something similar: when the problem is done, go back and check. Think it thru again. Does the answer make sense? Can you check it? If you estimate, is your final answer in agreement with your estimation?
So a wrong answer just means your student hasn't yet perfected this "checking and criticizing your own work" process - you know, learning to think critically about your own thought processes.
Encourage your students to point it out in their exam paper, if they know they got a wrong answer but can't find their mistake. You should give more credit to that answer than to the same wrong answer without that explanation (though not full credit of course).
Categories: philosophy
MYTH #4: IN MATH, WHAT'S IMPORTANT IS GETTING THE RIGHT ANSWER.
Then, in another recent post I linked to an article Things not to learn in school where the author strongly convinces us that getting 70% in a test is not enough since in real life you need to get it 100% right.
So is it important to get the right answer when doing a mathematics problem?
Well, yes but it depends. This is not a simple cut-and-dried question.
It IS important if you're drilling multiplication facts.
But many times it's not your focus. For example, when a student is learning to do long division or multiplying 4-digit numbers, the emphasis should be in learning the procedure and why it works. Calculation mistakes might yield a wrong answer, but if the way the division was done is right, then some credit should be given.
And obviously all children need encouragement while they're learning - they need partial credit or acknowledgment of things they did do right, even if the final answer was wrong.
Often times, you can use a wrong answer as a springboard and delve into the WHY the calculation went wrong. There's lots of learning one can do from wrong answers.
See, in my mind kids should be taught from early on to check their answer, always be CRITICAL of their answer, "suspicion" it until they themselves can "prove" to themselves that their answer is indeed right.
They need to learn to think twice before they do things in life. Well, in mathematics we can teach them to do something similar: when the problem is done, go back and check. Think it thru again. Does the answer make sense? Can you check it? If you estimate, is your final answer in agreement with your estimation?
So a wrong answer just means your student hasn't yet perfected this "checking and criticizing your own work" process - you know, learning to think critically about your own thought processes.
Encourage your students to point it out in their exam paper, if they know they got a wrong answer but can't find their mistake. You should give more credit to that answer than to the same wrong answer without that explanation (though not full credit of course).
Categories: philosophy
Comments on blogposts
I would like for all of you readers to comment on my posts with your ideas on these math topics! Seriously. We can help each other.
Many of you are subscribing to my blog so it comes to your email address, and some subscribe to the RSS feed (117 people total are doing that! Thanks!)
So I am experimenting if it will work to have the "Comments" link in the main body of the post so it would show up in the RSS feed and in the emails. We will see.
Many of you are subscribing to my blog so it comes to your email address, and some subscribe to the RSS feed (117 people total are doing that! Thanks!)
So I am experimenting if it will work to have the "Comments" link in the main body of the post so it would show up in the RSS feed and in the emails. We will see.
Comments on blogposts
I would like for all of you readers to comment on my posts with your ideas on these math topics! Seriously. We can help each other.
Many of you are subscribing to my blog so it comes to your email address, and some subscribe to the RSS feed (117 people total are doing that! Thanks!)
So I am experimenting if it will work to have the "Comments" link in the main body of the post so it would show up in the RSS feed and in the emails. We will see.
Many of you are subscribing to my blog so it comes to your email address, and some subscribe to the RSS feed (117 people total are doing that! Thanks!)
So I am experimenting if it will work to have the "Comments" link in the main body of the post so it would show up in the RSS feed and in the emails. We will see.
Saturday, January 14, 2006
Can girls do math? A paper folding story.
This story is cute, and even though it happened several years ago, it was new to me, and I really want to share it with you.
Maybe you've heard that it's impossible to fold paper more than 7 or 8 times, no matter how hard you try?
Well, that turned out to be a myth. Then high-schooler, Britney Gallivan tackled the challenge of folding paper 12 times and succeeded (in 2002). She experimented and thought about it, and found out what exactly limits the folding process. She discovered an equation linking the minimum length of paper needed, thickness of paper, and number of foldings.
Then, she calculated she needed paper about 4000 feet long to do 12 foldings! Britney found one special toilet paper roll that would do, went to a mall with her parents and started folding... It took 7 hours for her and her parents to do it, mostly on hands and knees.
Don't you ever think or leave that impression that girls can't do math - even if you're a female yourself and weren't good in math. Remember, your students will be affected by your attitudes.
Read more and see a picture of Britney and the paper:
Champion paper-folder and Paper folding at Mathworld and Folding paper in half 12 times.
P.S.
If you wanted to do 13 foldings, you'd need about 6000 feet of paper (if it was 0.002 inches thick).
Maybe you've heard that it's impossible to fold paper more than 7 or 8 times, no matter how hard you try?
Well, that turned out to be a myth. Then high-schooler, Britney Gallivan tackled the challenge of folding paper 12 times and succeeded (in 2002). She experimented and thought about it, and found out what exactly limits the folding process. She discovered an equation linking the minimum length of paper needed, thickness of paper, and number of foldings.
Then, she calculated she needed paper about 4000 feet long to do 12 foldings! Britney found one special toilet paper roll that would do, went to a mall with her parents and started folding... It took 7 hours for her and her parents to do it, mostly on hands and knees.
Don't you ever think or leave that impression that girls can't do math - even if you're a female yourself and weren't good in math. Remember, your students will be affected by your attitudes.
Read more and see a picture of Britney and the paper:
Champion paper-folder and Paper folding at Mathworld and Folding paper in half 12 times.
P.S.
If you wanted to do 13 foldings, you'd need about 6000 feet of paper (if it was 0.002 inches thick).
Can girls do math? A paper folding story.
This story is cute, and even though it happened several years ago, it was new to me, and I really want to share it with you.
Maybe you've heard that it's impossible to fold paper more than 7 or 8 times, no matter how hard you try?
Well, that turned out to be a myth. Then high-schooler, Britney Gallivan tackled the challenge of folding paper 12 times and succeeded (in 2002). She experimented and thought about it, and found out what exactly limits the folding process. She discovered an equation linking the minimum length of paper needed, thickness of paper, and number of foldings.
Then, she calculated she needed paper about 4000 feet long to do 12 foldings! Britney found one special toilet paper roll that would do, went to a mall with her parents and started folding... It took 7 hours for her and her parents to do it, mostly on hands and knees.
Don't you ever think or leave that impression that girls can't do math - even if you're a female yourself and weren't good in math. Remember, your students will be affected by your attitudes.
Read more and see a picture of Britney and the paper:
Champion paper-folder and Paper folding at Mathworld and Folding paper in half 12 times.
P.S.
If you wanted to do 13 foldings, you'd need about 6000 feet of paper (if it was 0.002 inches thick).
Maybe you've heard that it's impossible to fold paper more than 7 or 8 times, no matter how hard you try?
Well, that turned out to be a myth. Then high-schooler, Britney Gallivan tackled the challenge of folding paper 12 times and succeeded (in 2002). She experimented and thought about it, and found out what exactly limits the folding process. She discovered an equation linking the minimum length of paper needed, thickness of paper, and number of foldings.
Then, she calculated she needed paper about 4000 feet long to do 12 foldings! Britney found one special toilet paper roll that would do, went to a mall with her parents and started folding... It took 7 hours for her and her parents to do it, mostly on hands and knees.
Don't you ever think or leave that impression that girls can't do math - even if you're a female yourself and weren't good in math. Remember, your students will be affected by your attitudes.
Read more and see a picture of Britney and the paper:
Champion paper-folder and Paper folding at Mathworld and Folding paper in half 12 times.
P.S.
If you wanted to do 13 foldings, you'd need about 6000 feet of paper (if it was 0.002 inches thick).
Math as a career - why not?
Would your youngster enjoy a career in math?
The article at Businessweek Math will rock you world tells us how mathematics is used more and more in advertisement world, in internet technologies etc.
Use of mathematics is on the rise because it is used in more and more places to model various things, including people's behaviour.
New math graduates can easily land a job - often with a 6-figure salary.
Says Tom Leighton, an entrepreneur and applied math professor at Massachusetts Institute of Technology: "All of my students have standing offers at Yahoo! and Google."
But math programs in universities and colleges aren't filling with enthusiastic American young people: estimated half of the 20,000 math grad students now in the U.S. are foreign-born.
So it looks like math graduates are in need. And you don't have to be a boy to study math, remember.
The article at Businessweek Math will rock you world tells us how mathematics is used more and more in advertisement world, in internet technologies etc.
Use of mathematics is on the rise because it is used in more and more places to model various things, including people's behaviour.
New math graduates can easily land a job - often with a 6-figure salary.
Says Tom Leighton, an entrepreneur and applied math professor at Massachusetts Institute of Technology: "All of my students have standing offers at Yahoo! and Google."
But math programs in universities and colleges aren't filling with enthusiastic American young people: estimated half of the 20,000 math grad students now in the U.S. are foreign-born.
So it looks like math graduates are in need. And you don't have to be a boy to study math, remember.
Math as a career - why not?
Would your youngster enjoy a career in math?
The article at Businessweek Math will rock you world tells us how mathematics is used more and more in advertisement world, in internet technologies etc.
Use of mathematics is on the rise because it is used in more and more places to model various things, including people's behaviour.
New math graduates can easily land a job - often with a 6-figure salary.
Says Tom Leighton, an entrepreneur and applied math professor at Massachusetts Institute of Technology: "All of my students have standing offers at Yahoo! and Google."
But math programs in universities and colleges aren't filling with enthusiastic American young people: estimated half of the 20,000 math grad students now in the U.S. are foreign-born.
So it looks like math graduates are in need. And you don't have to be a boy to study math, remember.
The article at Businessweek Math will rock you world tells us how mathematics is used more and more in advertisement world, in internet technologies etc.
Use of mathematics is on the rise because it is used in more and more places to model various things, including people's behaviour.
New math graduates can easily land a job - often with a 6-figure salary.
Says Tom Leighton, an entrepreneur and applied math professor at Massachusetts Institute of Technology: "All of my students have standing offers at Yahoo! and Google."
But math programs in universities and colleges aren't filling with enthusiastic American young people: estimated half of the 20,000 math grad students now in the U.S. are foreign-born.
So it looks like math graduates are in need. And you don't have to be a boy to study math, remember.
Thursday, January 12, 2006
US math curricula are incoherent
You probably know that in international comparisons, US students don't do real well in math.
Research into curricula in the best performing countries versus US is giving us one clue as to why this is:
US curricula tend to be
Is this really the best and most efficient way?
Just check your own math books (if you have them for several grades): do you find for example the topic of fractions on each of the books from 1st till 8th grade? Or, does your chosen math curricula teach the concept of perimeter or octagon on many many grades?
How about decimals? How many school years does it take to learn to add, subtract, multiply, and divide decimals?
US books tend to still be teaching kids basic arithmetic (such as fractions and decimals) on 7th and 8th grade, whereas those other countries textbooks move on to algebra topics, and leave arithmetic behind.
"By the end of eighth grade, children in these countries have mostly completed mathematics equivalent to U.S. high school courses in algebra I and geometry."
(Quote from The Role of Curriculum by William Schmidt.)
The increasing emphasis on testing is making this even worse. Teachers are forced to hurry through lots of math topics each year. The curriculum is "mile wide and inch deep".
In homeschool you have at least some freedom to organize the instruction a little better.
Please read the short article The Role of Curriculum by William Schmidt.
For those interested for more, you can read this longer report with some stunning charts about the difference in US math textbooks versus those of the top countries:
A Coherent Curriculum: The Case of Mathematics, Dr. Schmidt’s Summer 2002 article for American Educator (PDF file).
See also my article Is your math curriculum coherent?
Categories: curriculum
Research into curricula in the best performing countries versus US is giving us one clue as to why this is:
US curricula tend to be
- incoherent and a collection of arbitrary topics instead of focused and logical
- Average duration of a topic in US is almost 6 years (!) versus about 3 years in the best-performing countries. Lots of spiraling and reviewing is done
- Each year, US textbooks cover way many more topics than the books in the best-performing countries
Is this really the best and most efficient way?
Just check your own math books (if you have them for several grades): do you find for example the topic of fractions on each of the books from 1st till 8th grade? Or, does your chosen math curricula teach the concept of perimeter or octagon on many many grades?
How about decimals? How many school years does it take to learn to add, subtract, multiply, and divide decimals?
US books tend to still be teaching kids basic arithmetic (such as fractions and decimals) on 7th and 8th grade, whereas those other countries textbooks move on to algebra topics, and leave arithmetic behind.
"By the end of eighth grade, children in these countries have mostly completed mathematics equivalent to U.S. high school courses in algebra I and geometry."
(Quote from The Role of Curriculum by William Schmidt.)
The increasing emphasis on testing is making this even worse. Teachers are forced to hurry through lots of math topics each year. The curriculum is "mile wide and inch deep".
In homeschool you have at least some freedom to organize the instruction a little better.
Please read the short article The Role of Curriculum by William Schmidt.
For those interested for more, you can read this longer report with some stunning charts about the difference in US math textbooks versus those of the top countries:
A Coherent Curriculum: The Case of Mathematics, Dr. Schmidt’s Summer 2002 article for American Educator (PDF file).
See also my article Is your math curriculum coherent?
Categories: curriculum
US math curricula are incoherent
You probably know that in international comparisons, US students don't do real well in math.
Research into curricula in the best performing countries versus US is giving us one clue as to why this is:
US curricula tend to be
Is this really the best and most efficient way?
Just check your own math books (if you have them for several grades): do you find for example the topic of fractions on each of the books from 1st till 8th grade? Or, does your chosen math curricula teach the concept of perimeter or octagon on many many grades?
How about decimals? How many school years does it take to learn to add, subtract, multiply, and divide decimals?
US books tend to still be teaching kids basic arithmetic (such as fractions and decimals) on 7th and 8th grade, whereas those other countries textbooks move on to algebra topics, and leave arithmetic behind.
"By the end of eighth grade, children in these countries have mostly completed mathematics equivalent to U.S. high school courses in algebra I and geometry."
(Quote from The Role of Curriculum by William Schmidt.)
The increasing emphasis on testing is making this even worse. Teachers are forced to hurry through lots of math topics each year. The curriculum is "mile wide and inch deep".
In homeschool you have at least some freedom to organize the instruction a little better.
Please read the short article The Role of Curriculum by William Schmidt.
For those interested for more, you can read this longer report with some stunning charts about the difference in US math textbooks versus those of the top countries:
A Coherent Curriculum: The Case of Mathematics, Dr. Schmidt’s Summer 2002 article for American Educator (PDF file).
See also my article Is your math curriculum coherent?
Categories: curriculum
Research into curricula in the best performing countries versus US is giving us one clue as to why this is:
US curricula tend to be
- incoherent and a collection of arbitrary topics instead of focused and logical
- Average duration of a topic in US is almost 6 years (!) versus about 3 years in the best-performing countries. Lots of spiraling and reviewing is done
- Each year, US textbooks cover way many more topics than the books in the best-performing countries
Is this really the best and most efficient way?
Just check your own math books (if you have them for several grades): do you find for example the topic of fractions on each of the books from 1st till 8th grade? Or, does your chosen math curricula teach the concept of perimeter or octagon on many many grades?
How about decimals? How many school years does it take to learn to add, subtract, multiply, and divide decimals?
US books tend to still be teaching kids basic arithmetic (such as fractions and decimals) on 7th and 8th grade, whereas those other countries textbooks move on to algebra topics, and leave arithmetic behind.
"By the end of eighth grade, children in these countries have mostly completed mathematics equivalent to U.S. high school courses in algebra I and geometry."
(Quote from The Role of Curriculum by William Schmidt.)
The increasing emphasis on testing is making this even worse. Teachers are forced to hurry through lots of math topics each year. The curriculum is "mile wide and inch deep".
In homeschool you have at least some freedom to organize the instruction a little better.
Please read the short article The Role of Curriculum by William Schmidt.
For those interested for more, you can read this longer report with some stunning charts about the difference in US math textbooks versus those of the top countries:
A Coherent Curriculum: The Case of Mathematics, Dr. Schmidt’s Summer 2002 article for American Educator (PDF file).
See also my article Is your math curriculum coherent?
Categories: curriculum
Solving a system of equations vs. doing dishes
I was getting ready to face the 'challenge' of a kitchen counter filled with dirty dishes, when I started thinking about the task ahead and how it compares to problem solving in math.
Then I found a funny little comparison between washing dishes and solving a system of linear equations.
You see, when you tackle that counter, you first need to organize things and move dishes around so you can see your kitchen sink again.
Just like if you have, say for example, these equations:
Just looks messy and almost discouraging! x's and y's and z's all over the place!
But then you start moving things around (moving x's, y's, and z's all to the left side), and making piles of plates, putting pots in one corner, containers in another pile (combining x's, y's, and z's), making the other side of the counter empty (zero) - and voila - it looks much simpler now. I feel I can manage the task now!
Then it's time to really start working (multiplying and dividing). Actually there are several methods, did you know? Some people do glasses first, some people start with silverware (you can use elimination method, or substitution method - or even Cramer's rule). But they all produce the same end result.
Over time, the task becomes routine (even boring) and you can just about do it without much thinking.
And, if you really just feel lazy, you can always leave dishes for tomorrow, or train a child in your family to do them (this system of equations is left for homework for the reader :).
Disclaimer: I am not guaranteeing there is a solution. Some dishes just resist washing.
Then I found a funny little comparison between washing dishes and solving a system of linear equations.
You see, when you tackle that counter, you first need to organize things and move dishes around so you can see your kitchen sink again.
Just like if you have, say for example, these equations:
| 2x - 4y + 3x | = | 2z - 4/5x + 67y |
| 4y - 0.3x - 0.98x + 34z | = | 5z + 90x - 0.2y |
| z - 2/3z + 3y -0.3x | = | 7y - 5/6z + 90x - 45x |
But then you start moving things around (moving x's, y's, and z's all to the left side), and making piles of plates, putting pots in one corner, containers in another pile (combining x's, y's, and z's), making the other side of the counter empty (zero) - and voila - it looks much simpler now. I feel I can manage the task now!
| 5 4/5x - 71y - 2z | = | 0 |
| - 91.28x + 4.2y + 29z | = | 0 |
| -45.3x - 4y + 1 1/6z | = | 0 |
Over time, the task becomes routine (even boring) and you can just about do it without much thinking.
And, if you really just feel lazy, you can always leave dishes for tomorrow, or train a child in your family to do them (this system of equations is left for homework for the reader :).
Disclaimer: I am not guaranteeing there is a solution. Some dishes just resist washing.
Solving a system of equations vs. doing dishes
I was getting ready to face the 'challenge' of a kitchen counter filled with dirty dishes, when I started thinking about the task ahead and how it compares to problem solving in math.
Then I found a funny little comparison between washing dishes and solving a system of linear equations.
You see, when you tackle that counter, you first need to organize things and move dishes around so you can see your kitchen sink again.
Just like if you have, say for example, these equations:
Just looks messy and almost discouraging! x's and y's and z's all over the place!
But then you start moving things around (moving x's, y's, and z's all to the left side), and making piles of plates, putting pots in one corner, containers in another pile (combining x's, y's, and z's), making the other side of the counter empty (zero) - and voila - it looks much simpler now. I feel I can manage the task now!
Then it's time to really start working (multiplying and dividing). Actually there are several methods, did you know? Some people do glasses first, some people start with silverware (you can use elimination method, or substitution method - or even Cramer's rule). But they all produce the same end result.
Over time, the task becomes routine (even boring) and you can just about do it without much thinking.
And, if you really just feel lazy, you can always leave dishes for tomorrow, or train a child in your family to do them (this system of equations is left for homework for the reader :).
Disclaimer: I am not guaranteeing there is a solution. Some dishes just resist washing.
Then I found a funny little comparison between washing dishes and solving a system of linear equations.
You see, when you tackle that counter, you first need to organize things and move dishes around so you can see your kitchen sink again.
Just like if you have, say for example, these equations:
| 2x - 4y + 3x | = | 2z - 4/5x + 67y |
| 4y - 0.3x - 0.98x + 34z | = | 5z + 90x - 0.2y |
| z - 2/3z + 3y -0.3x | = | 7y - 5/6z + 90x - 45x |
But then you start moving things around (moving x's, y's, and z's all to the left side), and making piles of plates, putting pots in one corner, containers in another pile (combining x's, y's, and z's), making the other side of the counter empty (zero) - and voila - it looks much simpler now. I feel I can manage the task now!
| 5 4/5x - 71y - 2z | = | 0 |
| - 91.28x + 4.2y + 29z | = | 0 |
| -45.3x - 4y + 1 1/6z | = | 0 |
Over time, the task becomes routine (even boring) and you can just about do it without much thinking.
And, if you really just feel lazy, you can always leave dishes for tomorrow, or train a child in your family to do them (this system of equations is left for homework for the reader :).
Disclaimer: I am not guaranteeing there is a solution. Some dishes just resist washing.
Wednesday, January 11, 2006
Square root of 2 and Pythagoreans' shock
Last week I asked you two questions pertaining to the image below:
1) How does this connect with irrational numbers?
Well, the two sides and the diagonal form a right triangle. You can use Pythagorean theorem to find the length of the diagonal. My picture doesn't have any lengths but I was thinking about having the side to be 1 (that's the simplest way).
If both sides are 1 and diagonal is d, then Pythagorean theorem says:
d2 = 12 + 12
Solving that, you get d = √2
And, √2 is an irrational number.
So if the sides of the square are 1, then the diagonal is an irrational number (square root of 2).
2) How does this connect with history of mathematics?
Pythagoras was a philosopher in the 6th century B.C. who founded a philosophical school or 'cult' in southern Italy. They had some mystical beliefs, such as reality is mathematical in nature, certain symbols have mystical significance, that the whole cosmos is a scale and a number. Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. They believed the whole universe was ruled by whole numbers.
These Pythagoreans apparently proved the Pythagorean theorem at some point (though they weren't the first ones to find it), plus various other mathematical results.
Then, they also found that some numbers are irrational... And that was a "biggie".
See, before that time, everyone had believed that ALL numbers are rational (all numbers are whole numbers or fractions). But Pythagoreans were able to prove that the diagonal of square and its side are "incommensurable" - that you can't find a fraction so that the diagonal would be this fraction times the side. This discovery is usually attributed to Hippas.
The LEGEND says that Pythagoreans were at sea when Hippatus discovered this, and when he told his comrades about his finding, they became so upset that they threw him in the sea!
You see, to Pythagoreans, the whole universe was ruled by whole numbers - so finding a number that was not a ratio of two whole numbers was a disastrous blow to their philosophical beliefs. Went against the dogma, so to speak.
See more:
P.S. Remember what I said: history can help get kids interested in math. It provides further insight into the topic and can help us remember better, and even understand better.
Categories: geometry, history
1) How does this connect with irrational numbers?
Well, the two sides and the diagonal form a right triangle. You can use Pythagorean theorem to find the length of the diagonal. My picture doesn't have any lengths but I was thinking about having the side to be 1 (that's the simplest way).
If both sides are 1 and diagonal is d, then Pythagorean theorem says:
d2 = 12 + 12
Solving that, you get d = √2
And, √2 is an irrational number.
So if the sides of the square are 1, then the diagonal is an irrational number (square root of 2).
2) How does this connect with history of mathematics?
Pythagoras was a philosopher in the 6th century B.C. who founded a philosophical school or 'cult' in southern Italy. They had some mystical beliefs, such as reality is mathematical in nature, certain symbols have mystical significance, that the whole cosmos is a scale and a number. Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. They believed the whole universe was ruled by whole numbers.
These Pythagoreans apparently proved the Pythagorean theorem at some point (though they weren't the first ones to find it), plus various other mathematical results.
Then, they also found that some numbers are irrational... And that was a "biggie".
See, before that time, everyone had believed that ALL numbers are rational (all numbers are whole numbers or fractions). But Pythagoreans were able to prove that the diagonal of square and its side are "incommensurable" - that you can't find a fraction so that the diagonal would be this fraction times the side. This discovery is usually attributed to Hippas.
The LEGEND says that Pythagoreans were at sea when Hippatus discovered this, and when he told his comrades about his finding, they became so upset that they threw him in the sea!
You see, to Pythagoreans, the whole universe was ruled by whole numbers - so finding a number that was not a ratio of two whole numbers was a disastrous blow to their philosophical beliefs. Went against the dogma, so to speak.
See more:
P.S. Remember what I said: history can help get kids interested in math. It provides further insight into the topic and can help us remember better, and even understand better.
Categories: geometry, history
Square root of 2 and Pythagoreans' shock
Last week I asked you two questions pertaining to the image below:
1) How does this connect with irrational numbers?
Well, the two sides and the diagonal form a right triangle. You can use Pythagorean theorem to find the length of the diagonal. My picture doesn't have any lengths but I was thinking about having the side to be 1 (that's the simplest way).
If both sides are 1 and diagonal is d, then Pythagorean theorem says:
d2 = 12 + 12
Solving that, you get d = √2
And, √2 is an irrational number.
So if the sides of the square are 1, then the diagonal is an irrational number (square root of 2).
2) How does this connect with history of mathematics?
Pythagoras was a philosopher in the 6th century B.C. who founded a philosophical school or 'cult' in southern Italy. They had some mystical beliefs, such as reality is mathematical in nature, certain symbols have mystical significance, that the whole cosmos is a scale and a number. Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. They believed the whole universe was ruled by whole numbers.
These Pythagoreans apparently proved the Pythagorean theorem at some point (though they weren't the first ones to find it), plus various other mathematical results.
Then, they also found that some numbers are irrational... And that was a "biggie".
See, before that time, everyone had believed that ALL numbers are rational (all numbers are whole numbers or fractions). But Pythagoreans were able to prove that the diagonal of square and its side are "incommensurable" - that you can't find a fraction so that the diagonal would be this fraction times the side. This discovery is usually attributed to Hippas.
The LEGEND says that Pythagoreans were at sea when Hippatus discovered this, and when he told his comrades about his finding, they became so upset that they threw him in the sea!
You see, to Pythagoreans, the whole universe was ruled by whole numbers - so finding a number that was not a ratio of two whole numbers was a disastrous blow to their philosophical beliefs. Went against the dogma, so to speak.
See more:
P.S. Remember what I said: history can help get kids interested in math. It provides further insight into the topic and can help us remember better, and even understand better.
Categories: geometry, history
1) How does this connect with irrational numbers?
Well, the two sides and the diagonal form a right triangle. You can use Pythagorean theorem to find the length of the diagonal. My picture doesn't have any lengths but I was thinking about having the side to be 1 (that's the simplest way).
If both sides are 1 and diagonal is d, then Pythagorean theorem says:
d2 = 12 + 12
Solving that, you get d = √2
And, √2 is an irrational number.
So if the sides of the square are 1, then the diagonal is an irrational number (square root of 2).
2) How does this connect with history of mathematics?
Pythagoras was a philosopher in the 6th century B.C. who founded a philosophical school or 'cult' in southern Italy. They had some mystical beliefs, such as reality is mathematical in nature, certain symbols have mystical significance, that the whole cosmos is a scale and a number. Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. They believed the whole universe was ruled by whole numbers.
These Pythagoreans apparently proved the Pythagorean theorem at some point (though they weren't the first ones to find it), plus various other mathematical results.
Then, they also found that some numbers are irrational... And that was a "biggie".
See, before that time, everyone had believed that ALL numbers are rational (all numbers are whole numbers or fractions). But Pythagoreans were able to prove that the diagonal of square and its side are "incommensurable" - that you can't find a fraction so that the diagonal would be this fraction times the side. This discovery is usually attributed to Hippas.
The LEGEND says that Pythagoreans were at sea when Hippatus discovered this, and when he told his comrades about his finding, they became so upset that they threw him in the sea!
You see, to Pythagoreans, the whole universe was ruled by whole numbers - so finding a number that was not a ratio of two whole numbers was a disastrous blow to their philosophical beliefs. Went against the dogma, so to speak.
See more:
P.S. Remember what I said: history can help get kids interested in math. It provides further insight into the topic and can help us remember better, and even understand better.
Categories: geometry, history
Sunday, January 8, 2006
Things not to learn in school
Sorry this is kind of off topic. Just too good to pass... It's comparing real LIFE and school tests:
Things not to learn in school
Things not to learn in school
Things not to learn in school
Sorry this is kind of off topic. Just too good to pass... It's comparing real LIFE and school tests:
Things not to learn in school
Things not to learn in school
Saturday, January 7, 2006
Square and its diagonal
Two questions for you to think about:
1) How does the above image of a square with one diagonal relate to irrational numbers?
2) How does this image connect with history of mathematics?
Answers will be here next week... : )
1) How does the above image of a square with one diagonal relate to irrational numbers?
2) How does this image connect with history of mathematics?
Answers will be here next week... : )
Square and its diagonal
Two questions for you to think about:
1) How does the above image of a square with one diagonal relate to irrational numbers?
2) How does this image connect with history of mathematics?
Answers will be here next week... : )
1) How does the above image of a square with one diagonal relate to irrational numbers?
2) How does this image connect with history of mathematics?
Answers will be here next week... : )
Thursday, January 5, 2006
Learn to like math as the teacher
Good teacher loves his/her subject matter. That's old wisdom.
Obviously not all people who teach math actually like or love it - I've had homeschooling moms admit that, for example.
The sad thing about that is that you're probably transferring your attitudes to your students: your students will end up not liking math either.
What to do?
Well, try change your attitude:
--> Think back and try find the reason as to WHY you don't like math. Was it some bad experience in school? See if you can overcome or ignore that factor, whatever it is.
Maybe you had not-so-enthusiastic math teachers yourself. Maybe you don't like math because you don't understand it. Or, you feel "I'm just not a math person - I'm no good at math."
The last one is probably not true... it's largely a myth. Any normal person with normal intelligence can learn basic math.
Maybe you as the homeschooling parent suffered from math anxiety yourself? If so, I encourage you to read this excellent article that explores the reasons for math anxiety and some very common myths about math, such as:
MYTH #1: APTITUDE FOR MATH IS INBORN.
MYTH #2: TO BE GOOD AT MATH YOU HAVE TO BE GOOD AT CALCULATING.
MYTH #3: MATH REQUIRES LOGIC, NOT CREATIVITY.
MYTH #4: IN MATH, WHAT'S IMPORTANT IS GETTING THE RIGHT ANSWER.
MYTH #5: MEN ARE NATURALLY BETTER THAN WOMEN AT MATHEMATICAL THINKING.
It's a long article but WELL worth reading for all math teachers!
Coping with math anxiety
Categories: philosophy, math
Obviously not all people who teach math actually like or love it - I've had homeschooling moms admit that, for example.
The sad thing about that is that you're probably transferring your attitudes to your students: your students will end up not liking math either.
What to do?
Well, try change your attitude:
--> Think back and try find the reason as to WHY you don't like math. Was it some bad experience in school? See if you can overcome or ignore that factor, whatever it is.
Maybe you had not-so-enthusiastic math teachers yourself. Maybe you don't like math because you don't understand it. Or, you feel "I'm just not a math person - I'm no good at math."
The last one is probably not true... it's largely a myth. Any normal person with normal intelligence can learn basic math.
Maybe you as the homeschooling parent suffered from math anxiety yourself? If so, I encourage you to read this excellent article that explores the reasons for math anxiety and some very common myths about math, such as:
MYTH #1: APTITUDE FOR MATH IS INBORN.
MYTH #2: TO BE GOOD AT MATH YOU HAVE TO BE GOOD AT CALCULATING.
MYTH #3: MATH REQUIRES LOGIC, NOT CREATIVITY.
MYTH #4: IN MATH, WHAT'S IMPORTANT IS GETTING THE RIGHT ANSWER.
MYTH #5: MEN ARE NATURALLY BETTER THAN WOMEN AT MATHEMATICAL THINKING.
It's a long article but WELL worth reading for all math teachers!
Coping with math anxiety
Categories: philosophy, math
Learn to like math as the teacher
Good teacher loves his/her subject matter. That's old wisdom.
Obviously not all people who teach math actually like or love it - I've had homeschooling moms admit that, for example.
The sad thing about that is that you're probably transferring your attitudes to your students: your students will end up not liking math either.
What to do?
Well, try change your attitude:
--> Think back and try find the reason as to WHY you don't like math. Was it some bad experience in school? See if you can overcome or ignore that factor, whatever it is.
Maybe you had not-so-enthusiastic math teachers yourself. Maybe you don't like math because you don't understand it. Or, you feel "I'm just not a math person - I'm no good at math."
The last one is probably not true... it's largely a myth. Any normal person with normal intelligence can learn basic math.
Maybe you as the homeschooling parent suffered from math anxiety yourself? If so, I encourage you to read this excellent article that explores the reasons for math anxiety and some very common myths about math, such as:
MYTH #1: APTITUDE FOR MATH IS INBORN.
MYTH #2: TO BE GOOD AT MATH YOU HAVE TO BE GOOD AT CALCULATING.
MYTH #3: MATH REQUIRES LOGIC, NOT CREATIVITY.
MYTH #4: IN MATH, WHAT'S IMPORTANT IS GETTING THE RIGHT ANSWER.
MYTH #5: MEN ARE NATURALLY BETTER THAN WOMEN AT MATHEMATICAL THINKING.
It's a long article but WELL worth reading for all math teachers!
Coping with math anxiety
Categories: philosophy, math
Obviously not all people who teach math actually like or love it - I've had homeschooling moms admit that, for example.
The sad thing about that is that you're probably transferring your attitudes to your students: your students will end up not liking math either.
What to do?
Well, try change your attitude:
--> Think back and try find the reason as to WHY you don't like math. Was it some bad experience in school? See if you can overcome or ignore that factor, whatever it is.
Maybe you had not-so-enthusiastic math teachers yourself. Maybe you don't like math because you don't understand it. Or, you feel "I'm just not a math person - I'm no good at math."
The last one is probably not true... it's largely a myth. Any normal person with normal intelligence can learn basic math.
Maybe you as the homeschooling parent suffered from math anxiety yourself? If so, I encourage you to read this excellent article that explores the reasons for math anxiety and some very common myths about math, such as:
MYTH #1: APTITUDE FOR MATH IS INBORN.
MYTH #2: TO BE GOOD AT MATH YOU HAVE TO BE GOOD AT CALCULATING.
MYTH #3: MATH REQUIRES LOGIC, NOT CREATIVITY.
MYTH #4: IN MATH, WHAT'S IMPORTANT IS GETTING THE RIGHT ANSWER.
MYTH #5: MEN ARE NATURALLY BETTER THAN WOMEN AT MATHEMATICAL THINKING.
It's a long article but WELL worth reading for all math teachers!
Coping with math anxiety
Categories: philosophy, math
Wednesday, January 4, 2006
Growing to love or hate math?
Have you ever met anyone who feels, "I hate math", or "I don't understand math at all", or "I'm not a math person" ?
Wonder when and where those feelings got started?
I recently read an article called Formula to bring back thrill to math.
The author argued that calculus should be reserved to college studies, not high school. His point was, while the nation is suffering from a shortage of math teachers, the good math teachers are needed, not at advanced-placement calculus classes, but instructing 'the masses' so that they would learn to love math.
He also said that besides a drastic shortage of math teachers, there is also a severe shortage of math majors. In other words, fewer and fewer kids are studying math as their major in college, and there are fewer and fewer (qualified) math teachers... Of course the result is that schools then hire just about anyone to teach math.
I just wonder, is there a connection between the lack of good math teachers and the common feelings of students of not liking math? And then, less and less youngsters choosing to study math? Is this a vicious cycle?
So what makes one love math or hate math? What makes people hate math?
Categories: philosophy
Wonder when and where those feelings got started?
I recently read an article called Formula to bring back thrill to math.
The author argued that calculus should be reserved to college studies, not high school. His point was, while the nation is suffering from a shortage of math teachers, the good math teachers are needed, not at advanced-placement calculus classes, but instructing 'the masses' so that they would learn to love math.
He also said that besides a drastic shortage of math teachers, there is also a severe shortage of math majors. In other words, fewer and fewer kids are studying math as their major in college, and there are fewer and fewer (qualified) math teachers... Of course the result is that schools then hire just about anyone to teach math.
I just wonder, is there a connection between the lack of good math teachers and the common feelings of students of not liking math? And then, less and less youngsters choosing to study math? Is this a vicious cycle?
So what makes one love math or hate math? What makes people hate math?
Categories: philosophy
Growing to love or hate math?
Have you ever met anyone who feels, "I hate math", or "I don't understand math at all", or "I'm not a math person" ?
Wonder when and where those feelings got started?
I recently read an article called Formula to bring back thrill to math.
The author argued that calculus should be reserved to college studies, not high school. His point was, while the nation is suffering from a shortage of math teachers, the good math teachers are needed, not at advanced-placement calculus classes, but instructing 'the masses' so that they would learn to love math.
He also said that besides a drastic shortage of math teachers, there is also a severe shortage of math majors. In other words, fewer and fewer kids are studying math as their major in college, and there are fewer and fewer (qualified) math teachers... Of course the result is that schools then hire just about anyone to teach math.
I just wonder, is there a connection between the lack of good math teachers and the common feelings of students of not liking math? And then, less and less youngsters choosing to study math? Is this a vicious cycle?
So what makes one love math or hate math? What makes people hate math?
Categories: philosophy
Wonder when and where those feelings got started?
I recently read an article called Formula to bring back thrill to math.
The author argued that calculus should be reserved to college studies, not high school. His point was, while the nation is suffering from a shortage of math teachers, the good math teachers are needed, not at advanced-placement calculus classes, but instructing 'the masses' so that they would learn to love math.
He also said that besides a drastic shortage of math teachers, there is also a severe shortage of math majors. In other words, fewer and fewer kids are studying math as their major in college, and there are fewer and fewer (qualified) math teachers... Of course the result is that schools then hire just about anyone to teach math.
I just wonder, is there a connection between the lack of good math teachers and the common feelings of students of not liking math? And then, less and less youngsters choosing to study math? Is this a vicious cycle?
So what makes one love math or hate math? What makes people hate math?
Categories: philosophy
Monday, January 2, 2006
Geometry and Euclid
You know, some math history can help spark interest in whatever you're teaching, and enliven the math (make it 'live math' so to speak). So today I want to educate you just a little bit about Euclid and geometry.
Did you know that a typical high school geometry course today with axioms, definitions, and theorems follows after the way Euclid presented geometry in his book Elements... and that this happened around 300 B.C. in Alexandria!
So the theorems your student is learning date back 2300 years!
Euclid was great - not because he found many great theorems back in his time (he didn't), but because he organized all then known mathematics into a logical presentation in his book Elements.
The geometry parts of Elements guided geometrical research for a long time after Euclid, and your school course of geometry is basically still based on this book.
So Euclid gives his name to Euclidean geometry - also called plane geometry.
In the beginning of his book, Euclid stated some definitions and postulates (we would call them axioms). Then he went on to prove various theorems, using only the postulates, and previously proven theorems.
A system organized this way is called an axiomatic system, and that's what your high school geometry book does too.
You might be interested in checking out a website that makes Euclid's Elements 'live' with interactive Java applets.
You might ask, "So why do we study geometry in such an ancient way?"
Because it is NOT an ancient way! Euclid started this 'business' of organizing math into axiomatic systems, and mathematicians have been doing that ever since. In other words, mathematicians organize everything in that manner, not just geometry.
Actually, in school mathematics, high school geometry is the ONLY place where you encounter an axiomatic system: axioms that are assumed as true without proving, and theorems logically proved from those.
Of course people argue whether that is good or bad. Mathematicians say it is good; it gives youngsters the only opportunity to encounter proofs in school mathematics. Some say it's not good because it's just too difficult for today's teenagers.
I feel it is VERY good and needful that school mathematics involves some proving and justification of the math facts instead of mathematics that is plain 'announced'. But I also feel there could be other ways to do this than the current high school geometry course. It would be better, in my opinion, to involve proving in other levels and other math topics, too - but not necessarily in this rigid 'two-column proof' way.
But while things stand as they do, you might be interested in reading my previous articles: Why high school geometry is difficult and What to do about it?
I wish everyone a prosperous year 2006!
Categories: history, geometry
Did you know that a typical high school geometry course today with axioms, definitions, and theorems follows after the way Euclid presented geometry in his book Elements... and that this happened around 300 B.C. in Alexandria!
So the theorems your student is learning date back 2300 years!
Euclid was great - not because he found many great theorems back in his time (he didn't), but because he organized all then known mathematics into a logical presentation in his book Elements.
The geometry parts of Elements guided geometrical research for a long time after Euclid, and your school course of geometry is basically still based on this book.
So Euclid gives his name to Euclidean geometry - also called plane geometry.
In the beginning of his book, Euclid stated some definitions and postulates (we would call them axioms). Then he went on to prove various theorems, using only the postulates, and previously proven theorems.
A system organized this way is called an axiomatic system, and that's what your high school geometry book does too.
You might be interested in checking out a website that makes Euclid's Elements 'live' with interactive Java applets.
You might ask, "So why do we study geometry in such an ancient way?"
Because it is NOT an ancient way! Euclid started this 'business' of organizing math into axiomatic systems, and mathematicians have been doing that ever since. In other words, mathematicians organize everything in that manner, not just geometry.
Actually, in school mathematics, high school geometry is the ONLY place where you encounter an axiomatic system: axioms that are assumed as true without proving, and theorems logically proved from those.
Of course people argue whether that is good or bad. Mathematicians say it is good; it gives youngsters the only opportunity to encounter proofs in school mathematics. Some say it's not good because it's just too difficult for today's teenagers.
I feel it is VERY good and needful that school mathematics involves some proving and justification of the math facts instead of mathematics that is plain 'announced'. But I also feel there could be other ways to do this than the current high school geometry course. It would be better, in my opinion, to involve proving in other levels and other math topics, too - but not necessarily in this rigid 'two-column proof' way.
But while things stand as they do, you might be interested in reading my previous articles: Why high school geometry is difficult and What to do about it?
I wish everyone a prosperous year 2006!
Categories: history, geometry
Geometry and Euclid
You know, some math history can help spark interest in whatever you're teaching, and enliven the math (make it 'live math' so to speak). So today I want to educate you just a little bit about Euclid and geometry.
Did you know that a typical high school geometry course today with axioms, definitions, and theorems follows after the way Euclid presented geometry in his book Elements... and that this happened around 300 B.C. in Alexandria!
So the theorems your student is learning date back 2300 years!
Euclid was great - not because he found many great theorems back in his time (he didn't), but because he organized all then known mathematics into a logical presentation in his book Elements.
The geometry parts of Elements guided geometrical research for a long time after Euclid, and your school course of geometry is basically still based on this book.
So Euclid gives his name to Euclidean geometry - also called plane geometry.
In the beginning of his book, Euclid stated some definitions and postulates (we would call them axioms). Then he went on to prove various theorems, using only the postulates, and previously proven theorems.
A system organized this way is called an axiomatic system, and that's what your high school geometry book does too.
You might be interested in checking out a website that makes Euclid's Elements 'live' with interactive Java applets.
You might ask, "So why do we study geometry in such an ancient way?"
Because it is NOT an ancient way! Euclid started this 'business' of organizing math into axiomatic systems, and mathematicians have been doing that ever since. In other words, mathematicians organize everything in that manner, not just geometry.
Actually, in school mathematics, high school geometry is the ONLY place where you encounter an axiomatic system: axioms that are assumed as true without proving, and theorems logically proved from those.
Of course people argue whether that is good or bad. Mathematicians say it is good; it gives youngsters the only opportunity to encounter proofs in school mathematics. Some say it's not good because it's just too difficult for today's teenagers.
I feel it is VERY good and needful that school mathematics involves some proving and justification of the math facts instead of mathematics that is plain 'announced'. But I also feel there could be other ways to do this than the current high school geometry course. It would be better, in my opinion, to involve proving in other levels and other math topics, too - but not necessarily in this rigid 'two-column proof' way.
But while things stand as they do, you might be interested in reading my previous articles: Why high school geometry is difficult and What to do about it?
I wish everyone a prosperous year 2006!
Categories: history, geometry
Did you know that a typical high school geometry course today with axioms, definitions, and theorems follows after the way Euclid presented geometry in his book Elements... and that this happened around 300 B.C. in Alexandria!
So the theorems your student is learning date back 2300 years!
Euclid was great - not because he found many great theorems back in his time (he didn't), but because he organized all then known mathematics into a logical presentation in his book Elements.
The geometry parts of Elements guided geometrical research for a long time after Euclid, and your school course of geometry is basically still based on this book.
So Euclid gives his name to Euclidean geometry - also called plane geometry.
In the beginning of his book, Euclid stated some definitions and postulates (we would call them axioms). Then he went on to prove various theorems, using only the postulates, and previously proven theorems.
A system organized this way is called an axiomatic system, and that's what your high school geometry book does too.
You might be interested in checking out a website that makes Euclid's Elements 'live' with interactive Java applets.
You might ask, "So why do we study geometry in such an ancient way?"
Because it is NOT an ancient way! Euclid started this 'business' of organizing math into axiomatic systems, and mathematicians have been doing that ever since. In other words, mathematicians organize everything in that manner, not just geometry.
Actually, in school mathematics, high school geometry is the ONLY place where you encounter an axiomatic system: axioms that are assumed as true without proving, and theorems logically proved from those.
Of course people argue whether that is good or bad. Mathematicians say it is good; it gives youngsters the only opportunity to encounter proofs in school mathematics. Some say it's not good because it's just too difficult for today's teenagers.
I feel it is VERY good and needful that school mathematics involves some proving and justification of the math facts instead of mathematics that is plain 'announced'. But I also feel there could be other ways to do this than the current high school geometry course. It would be better, in my opinion, to involve proving in other levels and other math topics, too - but not necessarily in this rigid 'two-column proof' way.
But while things stand as they do, you might be interested in reading my previous articles: Why high school geometry is difficult and What to do about it?
I wish everyone a prosperous year 2006!
Categories: history, geometry
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