اقدم على موقعي دراسه تقديه مفصله لكتب المنهج الجديد للصفوف الثلاث الاولى اتمنى المتابعه ومناقشتها لتعم الفائده
لقراءة الموصوع
للمناقشه
وجزاكم الله كل خير عن ابنائنا التلامبذ و زملائنا المدرسين
الأربعاء، 30 يناير 2008
الثلاثاء، 29 يناير 2008
الأحد، 27 يناير 2008
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الثلاثاء، 22 يناير 2008
Four habits of highly effective math teaching
If you were asked what were the most important principles in mathematics teaching, what would you say? I wasn't really asked, but I started thinking, and came up with these basic habits or principles that can keep your math teaching on the right track.
Habit 1: Let It Make Sense
Habit 2: Remember the Goals
Habit 3: Know Your Tools
Habit 4: Living and Loving Math
Habit 1: Let It Make Sense
Let us strive to teach for understanding of mathematical concepts and procedures, the "why" something works, and not only the "how".
This understanding, as I'm sure you realize, doesn't always come immediately. It may take even several years to grasp a concept. For example, place value is something kids understand partially at first, and then that deepens over a few years.
This is why many math curricula use spiraling: they come back to a concept the next year, and the next. And this can be very good if not done excessively (like for 5-6 years is probably excessive).
However, spiraling also has its own pitfalls: if your child doesn't get a concept, don't blindly "trust" the spiraling and think, "Well, she gets it the next year when the book comes back around to it."
The next year's schoolbook won't necessarily present the concept at the same level - the presentation might be too difficult. If a child doesn't "get it", they might need a very basic instruction for the concept again.
The "how" something works is often called procedural understanding: the child knows how to work long division, or the procedure of fraction addition or fraction division, for example. It is often possible to learn the "how" mechanically without understanding why it works. Procedures learned this way are often forgotten very easily.
The relationship between the "how" and the "why" - or between procedures and concepts - is complex. One doesn't always come totally before the other, and it also varies from child to child.
You can try alternating the instruction: teach how to add fractions, and let the student practice. Explain why it works. Go back to some practice. Back and forth. Sooner or later it should 'stick' - but it might be next year instead of this one, or after 6 months instead of in this month.
As a rule of thumb, don't totally leave a topic until the student both knows how, and understands the 'why'.
Tip: you can often test a student's understanding of a topic by asking HIM to produce an example, preferably with a picture or other illustration: "Tell me an example of multiplying fractions by whole numbers, and draw a picture." Whatever gets produced can tell the teacher a lot about what has been understood.
-------------------------------
Habit 2: Remember the Goals
What are the goals of your math teaching? Are they...
to finish the book by the end of school year
make sure the kids pass the test ...?
Or do you have goals such as:
My student can add, simplify, and multiply fractions
My student can divide by 10, 100, and 1000.
These are all just "subgoals". But what is the ultimate goal of learning school mathematics?
Consider these goals:
Students need to be able to navigate their lives in this ever-so-complex modern world.
This involves dealing with taxes, loans, credit cards, purchases, budgeting, shopping. Our youngsters need to be able to handle money wisely. All that requires good understanding of parts, proportions, and percents.
Another very important goal of mathematics education as a whole is to enable the students to understand information aroud us. In today's world, this includes quite a bit of scientific information. Being able to read through it and make sense of it requires knowing big and small numbers, statistics, probability, percents.
And then one more. We need to prepare our students for further studies in math and science. Not everyone ultimately needs algebra, but many do, and teens don't always know what profession they might choose or end up with.
I'd like to add one more broad goal of math education: teaching deductive reasoning. Of course geometry is a good example of this, but when taught properly, other areas of school math can be as well.
The more you can keep these big real goals in mind, the better you can connect your subgoals to them. And the more you can keep the goals and the subgoals in mind, the better teacher you will be.
For example, adding, simplifying, and multiplying fractions all connects with a broader goal of understanding parts or part and whole. It will soon lead to ratios, proportions, and percent. Also, all fraction operations are a needed basis for solving rational equations and doing the operations with rational expressions (during algebra studies).
Tying in with the goals, remember that the BOOK or CURRICULUM is just a tool to achieve the goals -- not a goal in itself. Don't ever be a slave of any math book.
--------------------------
Habit 3: Know Your Tools
Math teacher's tools are quite numerous nowadays.
First of all of course comes a black or white board, or paper - something to write on, pencil, compass, protractor, ruler, eraser. And the book you're using. Then we also have computer software, animations and activities online, animated lessons and such. There are workbooks, fun books, worktexts, online texts. Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on. And then there are games, games, games.
The choices are so numerous it's daunting. What's a teacher to do?
Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.
There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity to spice up your math lessons.
Basic tools
The board and/or paper to write on. Essential. Easy to use.
The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. Check my curriculum pages for some help. Two things to keep in mind:
i) Now matter what book you're using, YOU as the teacher have the control. Don't be a slave to the curriculum. You can skip pages, rearrange the order in which to teach the material, supplement it, and so on.
ii) Don't despair if the book you're using doesn't seem to be the perfect choice for your student. You can quite likely sell it on homeschool swap boards, and buy some other one.
Manipulatives. I once saw a question asked by a homeschooling parent, on the lines, "What manipulatives must I use and when?" The person was under the impression that manipulatives are a 'must' thing.
Manipulatives are definitely emphasized in these days. They are usually very good, but they're not the end goal of math education, and there is no need to go hog wild over them.
Manipulatives are something the student manipulates with his hands to get a better grasp of something. But the goal is to learn to do math without them.
Some very helpful manipulatives are
abacus
something to illustrate hundreds/tens/ones place value. I made my daughter ten-bags by putting marbles into little plastic bags.
some sort of fraction manipulatives. You can just make pie models out of cardboard, even.
Often, drawing pictures can take place of manipulatives, especially in the middle grades and on.
Check out also some virtual manipulatives.
Geometry and measuring tools. These are pretty essential, I'd say. For geometry however, dynamic software can these days replace compass and ruler and easily be far better.
The extras
These are, obviously, too many to even start listing.
Some game or games are good for drilling basic facts. Games are nice for about any topic. Here's one that I played with playing cards with my dd; and now she seems to have learned the sums that add to 10. And here's a game that's worth 1000 worksheets. Of course the internet is full of online math games.
I would definitely use some math software if teaching graphing, algebra, or calculus. Check MathProf for example, or Math Mechanixs. I've listed a few more here.
If you're ready to add something new to your toolbox from the online world, try The Math Forum's MathTools - a library of technology tools, lessons, activities, and support materials. Check also my pages listing interactive math activities online (there's a menu on the right).
--------------------------------
Habit 4: Living and Loving Math
You are the teacher. You show the way - also with your attitudes, your way of life.
Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day?
And then: do you like math? Love it? Are you happy to teach it? Enthusiastic?
Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your kids a way of life, and if math is a natural part of it or not.
Math is not a drudgery, nor something just confined to math lessons.
Some ideas:
Let it make sense. This alone can usually make math quite a difference and kids will stay interested.
Read through some fun math books, such as Theoni Pappas books, or puzzle-type books. Get to know some interesting math topics besides just schoolbook arithmetic. And, there are even story books to teach math concepts - see a list here.
Try including a bit about math history. This might work best in a homeschooling environment where there is no horrible rush to get through the thick book before the year is over. Julie at LivingMath.net has suggestions for math history books to buy.
When you use math in your daily life, explain how you're doing it, and include the children if possible. Figure it out together.
Miscellaneous Math Teaching tips
The child needs to know the basic addition and multiplication facts very well, or she will have difficulties with fractions, decimals, etc. These basic facts need to be known by heart.
One of the best ways to start children with math is to have them skip-count up and down from a very young age. Use a number line to show what the 'skips' or steps mean. if your child can master the skip counting by twos, threes, fours, etc., she has learned a lot about addition and later on multiplication tables will be an easy fare! See also this article How to drill multiplication tables.
Use manipulatives and pictures in your teaching. Almost all mathematical concepts can be illustrated with pictures, which can even take a place of concrete manipulatives. For example, if you can condition your child to draw lots of pie pictures when studying fractions, he/she can learn to visualize fractions as 'pies'. Then he/she won't make the addition mistake 1/2 + 1/4 = 2/6. Also certain kind of software can take place of the manipulatives.
In geometry have your child or children DRAW a lot. See examples in the Geometry ebook from HomeschoolMath.net.
When studying time, money, measuring, homeschoolers have an advantage since they can study those subjects in their natural settings. Involve your child when you measure, count money, check the time.
In middle school years, it's important to get familiar with functions, relations, and patterns - these develop algebraic thinking. Check this article about algebraic reasoning from MathCounts.org.
If you need to know the whys and wherefores of some particular math topic deeper than the textbook tells you, check Dr. Math's archives. The answers provided there are mathematically "sound doctrine", whereas math textbooks can contain all kinds of errors.
By:Maria Miller
Habit 1: Let It Make Sense
Habit 2: Remember the Goals
Habit 3: Know Your Tools
Habit 4: Living and Loving Math
Habit 1: Let It Make Sense
Let us strive to teach for understanding of mathematical concepts and procedures, the "why" something works, and not only the "how".
This understanding, as I'm sure you realize, doesn't always come immediately. It may take even several years to grasp a concept. For example, place value is something kids understand partially at first, and then that deepens over a few years.
This is why many math curricula use spiraling: they come back to a concept the next year, and the next. And this can be very good if not done excessively (like for 5-6 years is probably excessive).
However, spiraling also has its own pitfalls: if your child doesn't get a concept, don't blindly "trust" the spiraling and think, "Well, she gets it the next year when the book comes back around to it."
The next year's schoolbook won't necessarily present the concept at the same level - the presentation might be too difficult. If a child doesn't "get it", they might need a very basic instruction for the concept again.
The "how" something works is often called procedural understanding: the child knows how to work long division, or the procedure of fraction addition or fraction division, for example. It is often possible to learn the "how" mechanically without understanding why it works. Procedures learned this way are often forgotten very easily.
The relationship between the "how" and the "why" - or between procedures and concepts - is complex. One doesn't always come totally before the other, and it also varies from child to child.
You can try alternating the instruction: teach how to add fractions, and let the student practice. Explain why it works. Go back to some practice. Back and forth. Sooner or later it should 'stick' - but it might be next year instead of this one, or after 6 months instead of in this month.
As a rule of thumb, don't totally leave a topic until the student both knows how, and understands the 'why'.
Tip: you can often test a student's understanding of a topic by asking HIM to produce an example, preferably with a picture or other illustration: "Tell me an example of multiplying fractions by whole numbers, and draw a picture." Whatever gets produced can tell the teacher a lot about what has been understood.
-------------------------------
Habit 2: Remember the Goals
What are the goals of your math teaching? Are they...
to finish the book by the end of school year
make sure the kids pass the test ...?
Or do you have goals such as:
My student can add, simplify, and multiply fractions
My student can divide by 10, 100, and 1000.
These are all just "subgoals". But what is the ultimate goal of learning school mathematics?
Consider these goals:
Students need to be able to navigate their lives in this ever-so-complex modern world.
This involves dealing with taxes, loans, credit cards, purchases, budgeting, shopping. Our youngsters need to be able to handle money wisely. All that requires good understanding of parts, proportions, and percents.
Another very important goal of mathematics education as a whole is to enable the students to understand information aroud us. In today's world, this includes quite a bit of scientific information. Being able to read through it and make sense of it requires knowing big and small numbers, statistics, probability, percents.
And then one more. We need to prepare our students for further studies in math and science. Not everyone ultimately needs algebra, but many do, and teens don't always know what profession they might choose or end up with.
I'd like to add one more broad goal of math education: teaching deductive reasoning. Of course geometry is a good example of this, but when taught properly, other areas of school math can be as well.
The more you can keep these big real goals in mind, the better you can connect your subgoals to them. And the more you can keep the goals and the subgoals in mind, the better teacher you will be.
For example, adding, simplifying, and multiplying fractions all connects with a broader goal of understanding parts or part and whole. It will soon lead to ratios, proportions, and percent. Also, all fraction operations are a needed basis for solving rational equations and doing the operations with rational expressions (during algebra studies).
Tying in with the goals, remember that the BOOK or CURRICULUM is just a tool to achieve the goals -- not a goal in itself. Don't ever be a slave of any math book.
--------------------------
Habit 3: Know Your Tools
Math teacher's tools are quite numerous nowadays.
First of all of course comes a black or white board, or paper - something to write on, pencil, compass, protractor, ruler, eraser. And the book you're using. Then we also have computer software, animations and activities online, animated lessons and such. There are workbooks, fun books, worktexts, online texts. Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on. And then there are games, games, games.
The choices are so numerous it's daunting. What's a teacher to do?
Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.
There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity to spice up your math lessons.
Basic tools
The board and/or paper to write on. Essential. Easy to use.
The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. Check my curriculum pages for some help. Two things to keep in mind:
i) Now matter what book you're using, YOU as the teacher have the control. Don't be a slave to the curriculum. You can skip pages, rearrange the order in which to teach the material, supplement it, and so on.
ii) Don't despair if the book you're using doesn't seem to be the perfect choice for your student. You can quite likely sell it on homeschool swap boards, and buy some other one.
Manipulatives. I once saw a question asked by a homeschooling parent, on the lines, "What manipulatives must I use and when?" The person was under the impression that manipulatives are a 'must' thing.
Manipulatives are definitely emphasized in these days. They are usually very good, but they're not the end goal of math education, and there is no need to go hog wild over them.
Manipulatives are something the student manipulates with his hands to get a better grasp of something. But the goal is to learn to do math without them.
Some very helpful manipulatives are
abacus
something to illustrate hundreds/tens/ones place value. I made my daughter ten-bags by putting marbles into little plastic bags.
some sort of fraction manipulatives. You can just make pie models out of cardboard, even.
Often, drawing pictures can take place of manipulatives, especially in the middle grades and on.
Check out also some virtual manipulatives.
Geometry and measuring tools. These are pretty essential, I'd say. For geometry however, dynamic software can these days replace compass and ruler and easily be far better.
The extras
These are, obviously, too many to even start listing.
Some game or games are good for drilling basic facts. Games are nice for about any topic. Here's one that I played with playing cards with my dd; and now she seems to have learned the sums that add to 10. And here's a game that's worth 1000 worksheets. Of course the internet is full of online math games.
I would definitely use some math software if teaching graphing, algebra, or calculus. Check MathProf for example, or Math Mechanixs. I've listed a few more here.
If you're ready to add something new to your toolbox from the online world, try The Math Forum's MathTools - a library of technology tools, lessons, activities, and support materials. Check also my pages listing interactive math activities online (there's a menu on the right).
--------------------------------
Habit 4: Living and Loving Math
You are the teacher. You show the way - also with your attitudes, your way of life.
Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day?
And then: do you like math? Love it? Are you happy to teach it? Enthusiastic?
Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your kids a way of life, and if math is a natural part of it or not.
Math is not a drudgery, nor something just confined to math lessons.
Some ideas:
Let it make sense. This alone can usually make math quite a difference and kids will stay interested.
Read through some fun math books, such as Theoni Pappas books, or puzzle-type books. Get to know some interesting math topics besides just schoolbook arithmetic. And, there are even story books to teach math concepts - see a list here.
Try including a bit about math history. This might work best in a homeschooling environment where there is no horrible rush to get through the thick book before the year is over. Julie at LivingMath.net has suggestions for math history books to buy.
When you use math in your daily life, explain how you're doing it, and include the children if possible. Figure it out together.
Miscellaneous Math Teaching tips
The child needs to know the basic addition and multiplication facts very well, or she will have difficulties with fractions, decimals, etc. These basic facts need to be known by heart.
One of the best ways to start children with math is to have them skip-count up and down from a very young age. Use a number line to show what the 'skips' or steps mean. if your child can master the skip counting by twos, threes, fours, etc., she has learned a lot about addition and later on multiplication tables will be an easy fare! See also this article How to drill multiplication tables.
Use manipulatives and pictures in your teaching. Almost all mathematical concepts can be illustrated with pictures, which can even take a place of concrete manipulatives. For example, if you can condition your child to draw lots of pie pictures when studying fractions, he/she can learn to visualize fractions as 'pies'. Then he/she won't make the addition mistake 1/2 + 1/4 = 2/6. Also certain kind of software can take place of the manipulatives.
In geometry have your child or children DRAW a lot. See examples in the Geometry ebook from HomeschoolMath.net.
When studying time, money, measuring, homeschoolers have an advantage since they can study those subjects in their natural settings. Involve your child when you measure, count money, check the time.
In middle school years, it's important to get familiar with functions, relations, and patterns - these develop algebraic thinking. Check this article about algebraic reasoning from MathCounts.org.
If you need to know the whys and wherefores of some particular math topic deeper than the textbook tells you, check Dr. Math's archives. The answers provided there are mathematically "sound doctrine", whereas math textbooks can contain all kinds of errors.
By:Maria Miller
الاثنين، 14 يناير 2008
Maths misconceptions
Everyone knows the feeling of struggling with a task that other people seem to breeze through. It might be programming the DVD player or even just reading maps.
Well, this is how some kids feel with maths, and their difficulties are often rooted in misunderstandings of concepts that we, as teachers, don't give a second thought to. How much could we help them make progress if we were more aware of these misconceptions, and were able to tackle them head-on? We all know, after all, that understanding our mistakes can be a powerful learning experience.
With the help of Tim Coulson, who leads the National Numeracy Strategy, we've put together a list of some of the most common, and potentially most obstructive, of these misconceptions, and suggest some approaches that might put things right.
1. A number with three digits is always bigger than one with two
Some children will swear blind that 3.24 is bigger than 4.6 because it's got more digits. Why? Because for the first few years of learning, they only came across whole numbers, where the 'digits' rule does work.
2. When you multiply two numbers together, the answer is always bigger than both the original numbers
Another seductive 'rule' that works for whole numbers, but falls to pieces when one or both of the numbers is less than one. Remember that, instead of the word 'times' we can always substitute the word 'of.' So, 1/2 times 1/4 is the same as a half of a quarter. That immediately demolishes the expectation that the product is going to be bigger than both original numbers.
3. Which fraction is bigger: 1/3 or 1/6?
How many pupils will say 1/6 because they know that 6 is bigger than 3? This reveals a gap in knowledge about what the bottom number, the denominator, of a fraction does. It divides the top number, the numerator, of course. Practical work, such as cutting pre-divided circles into thirds and sixths, and comparing the shapes, helps cement understanding of fractions.
4. Common regular shapes aren't recognised for what they are unless they're upright
Teachers can, inadvertently, feed this misconception if they always draw a square, right-angled or isosceles triangle in the 'usual' position. Why not draw them occasionally upside down, facing a different direction, or just tilted over, to force pupils to look at the essential properties? And, by the way, in maths, there's no such thing as a diamond! It's either a square or a rhombus.
5. The diagonal of a square is the same length as the side?
Not true, but tempting for many young minds. So, how about challenging the class to investigate this by drawing and measuring. Once the top table have mastered this, why not ask them to estimate the dimensions of a square whose diagonal is exactly 5cm. Then draw it and see how close their guess was.
6. To multiply by 10, just add a zero
Not always! What about 23.7 x 10, 0.35 x 10, or 2/3 x 10? Try to spot, and unpick, the 'just add zero' rule wherever it rears its head.
7. Proportion: three red sweets and two blue
Asked what proportion of the sweets is blue, how many kids will say 2/3 rather than 2/5? Why? Because they're comparing blue to red, not blue to all the sweets. Always stress that proportion is 'part to whole'.
8. Perimeter and area confuse many kids
A common mistake, when measuring the perimeter of a rectangle, is to count the squares surrounding the shape, in the same way as counting those inside for area. Now you can see why some would give the perimeter of a two-by-three rectangle as 14 units rather than 10.
9. Misreading scales
Still identified as a weakness in Key Stage test papers. The most common misunderstanding is that any interval on a scale must correspond to one unit. (Think of 30 to 40 split into five intervals.) Frequent handling of different scales, divided up into twos, fives, 10s, tenths etc. will help to banish this idea.
Words: Steve McCormack Pictures: David Moore
from
http://www.teachernet.gov.uk/teachers/issue42/primary/features/Mathsmisconceptions/
Well, this is how some kids feel with maths, and their difficulties are often rooted in misunderstandings of concepts that we, as teachers, don't give a second thought to. How much could we help them make progress if we were more aware of these misconceptions, and were able to tackle them head-on? We all know, after all, that understanding our mistakes can be a powerful learning experience.
With the help of Tim Coulson, who leads the National Numeracy Strategy, we've put together a list of some of the most common, and potentially most obstructive, of these misconceptions, and suggest some approaches that might put things right.
1. A number with three digits is always bigger than one with two
Some children will swear blind that 3.24 is bigger than 4.6 because it's got more digits. Why? Because for the first few years of learning, they only came across whole numbers, where the 'digits' rule does work.
2. When you multiply two numbers together, the answer is always bigger than both the original numbers
Another seductive 'rule' that works for whole numbers, but falls to pieces when one or both of the numbers is less than one. Remember that, instead of the word 'times' we can always substitute the word 'of.' So, 1/2 times 1/4 is the same as a half of a quarter. That immediately demolishes the expectation that the product is going to be bigger than both original numbers.
3. Which fraction is bigger: 1/3 or 1/6?
How many pupils will say 1/6 because they know that 6 is bigger than 3? This reveals a gap in knowledge about what the bottom number, the denominator, of a fraction does. It divides the top number, the numerator, of course. Practical work, such as cutting pre-divided circles into thirds and sixths, and comparing the shapes, helps cement understanding of fractions.
4. Common regular shapes aren't recognised for what they are unless they're upright
Teachers can, inadvertently, feed this misconception if they always draw a square, right-angled or isosceles triangle in the 'usual' position. Why not draw them occasionally upside down, facing a different direction, or just tilted over, to force pupils to look at the essential properties? And, by the way, in maths, there's no such thing as a diamond! It's either a square or a rhombus.
5. The diagonal of a square is the same length as the side?
Not true, but tempting for many young minds. So, how about challenging the class to investigate this by drawing and measuring. Once the top table have mastered this, why not ask them to estimate the dimensions of a square whose diagonal is exactly 5cm. Then draw it and see how close their guess was.
6. To multiply by 10, just add a zero
Not always! What about 23.7 x 10, 0.35 x 10, or 2/3 x 10? Try to spot, and unpick, the 'just add zero' rule wherever it rears its head.
7. Proportion: three red sweets and two blue
Asked what proportion of the sweets is blue, how many kids will say 2/3 rather than 2/5? Why? Because they're comparing blue to red, not blue to all the sweets. Always stress that proportion is 'part to whole'.
8. Perimeter and area confuse many kids
A common mistake, when measuring the perimeter of a rectangle, is to count the squares surrounding the shape, in the same way as counting those inside for area. Now you can see why some would give the perimeter of a two-by-three rectangle as 14 units rather than 10.
9. Misreading scales
Still identified as a weakness in Key Stage test papers. The most common misunderstanding is that any interval on a scale must correspond to one unit. (Think of 30 to 40 split into five intervals.) Frequent handling of different scales, divided up into twos, fives, 10s, tenths etc. will help to banish this idea.
Words: Steve McCormack Pictures: David Moore
from
http://www.teachernet.gov.uk/teachers/issue42/primary/features/Mathsmisconceptions/
Maths misconceptions
Everyone knows the feeling of struggling with a task that other people seem to breeze through. It might be programming the DVD player or even just reading maps.
Well, this is how some kids feel with maths, and their difficulties are often rooted in misunderstandings of concepts that we, as teachers, don't give a second thought to. How much could we help them make progress if we were more aware of these misconceptions, and were able to tackle them head-on? We all know, after all, that understanding our mistakes can be a powerful learning experience.
With the help of Tim Coulson, who leads the National Numeracy Strategy, we've put together a list of some of the most common, and potentially most obstructive, of these misconceptions, and suggest some approaches that might put things right.
1. A number with three digits is always bigger than one with two
Some children will swear blind that 3.24 is bigger than 4.6 because it's got more digits. Why? Because for the first few years of learning, they only came across whole numbers, where the 'digits' rule does work.
2. When you multiply two numbers together, the answer is always bigger than both the original numbers
Another seductive 'rule' that works for whole numbers, but falls to pieces when one or both of the numbers is less than one. Remember that, instead of the word 'times' we can always substitute the word 'of.' So, 1/2 times 1/4 is the same as a half of a quarter. That immediately demolishes the expectation that the product is going to be bigger than both original numbers.
3. Which fraction is bigger: 1/3 or 1/6?
How many pupils will say 1/6 because they know that 6 is bigger than 3? This reveals a gap in knowledge about what the bottom number, the denominator, of a fraction does. It divides the top number, the numerator, of course. Practical work, such as cutting pre-divided circles into thirds and sixths, and comparing the shapes, helps cement understanding of fractions.
4. Common regular shapes aren't recognised for what they are unless they're upright
Teachers can, inadvertently, feed this misconception if they always draw a square, right-angled or isosceles triangle in the 'usual' position. Why not draw them occasionally upside down, facing a different direction, or just tilted over, to force pupils to look at the essential properties? And, by the way, in maths, there's no such thing as a diamond! It's either a square or a rhombus.
5. The diagonal of a square is the same length as the side?
Not true, but tempting for many young minds. So, how about challenging the class to investigate this by drawing and measuring. Once the top table have mastered this, why not ask them to estimate the dimensions of a square whose diagonal is exactly 5cm. Then draw it and see how close their guess was.
6. To multiply by 10, just add a zero
Not always! What about 23.7 x 10, 0.35 x 10, or 2/3 x 10? Try to spot, and unpick, the 'just add zero' rule wherever it rears its head.
7. Proportion: three red sweets and two blue
Asked what proportion of the sweets is blue, how many kids will say 2/3 rather than 2/5? Why? Because they're comparing blue to red, not blue to all the sweets. Always stress that proportion is 'part to whole'.
8. Perimeter and area confuse many kids
A common mistake, when measuring the perimeter of a rectangle, is to count the squares surrounding the shape, in the same way as counting those inside for area. Now you can see why some would give the perimeter of a two-by-three rectangle as 14 units rather than 10.
9. Misreading scales
Still identified as a weakness in Key Stage test papers. The most common misunderstanding is that any interval on a scale must correspond to one unit. (Think of 30 to 40 split into five intervals.) Frequent handling of different scales, divided up into twos, fives, 10s, tenths etc. will help to banish this idea.
Words: Steve McCormack Pictures: David Moore
from
http://www.teachernet.gov.uk/teachers/issue42/primary/features/Mathsmisconceptions/
Well, this is how some kids feel with maths, and their difficulties are often rooted in misunderstandings of concepts that we, as teachers, don't give a second thought to. How much could we help them make progress if we were more aware of these misconceptions, and were able to tackle them head-on? We all know, after all, that understanding our mistakes can be a powerful learning experience.
With the help of Tim Coulson, who leads the National Numeracy Strategy, we've put together a list of some of the most common, and potentially most obstructive, of these misconceptions, and suggest some approaches that might put things right.
1. A number with three digits is always bigger than one with two
Some children will swear blind that 3.24 is bigger than 4.6 because it's got more digits. Why? Because for the first few years of learning, they only came across whole numbers, where the 'digits' rule does work.
2. When you multiply two numbers together, the answer is always bigger than both the original numbers
Another seductive 'rule' that works for whole numbers, but falls to pieces when one or both of the numbers is less than one. Remember that, instead of the word 'times' we can always substitute the word 'of.' So, 1/2 times 1/4 is the same as a half of a quarter. That immediately demolishes the expectation that the product is going to be bigger than both original numbers.
3. Which fraction is bigger: 1/3 or 1/6?
How many pupils will say 1/6 because they know that 6 is bigger than 3? This reveals a gap in knowledge about what the bottom number, the denominator, of a fraction does. It divides the top number, the numerator, of course. Practical work, such as cutting pre-divided circles into thirds and sixths, and comparing the shapes, helps cement understanding of fractions.
4. Common regular shapes aren't recognised for what they are unless they're upright
Teachers can, inadvertently, feed this misconception if they always draw a square, right-angled or isosceles triangle in the 'usual' position. Why not draw them occasionally upside down, facing a different direction, or just tilted over, to force pupils to look at the essential properties? And, by the way, in maths, there's no such thing as a diamond! It's either a square or a rhombus.
5. The diagonal of a square is the same length as the side?
Not true, but tempting for many young minds. So, how about challenging the class to investigate this by drawing and measuring. Once the top table have mastered this, why not ask them to estimate the dimensions of a square whose diagonal is exactly 5cm. Then draw it and see how close their guess was.
6. To multiply by 10, just add a zero
Not always! What about 23.7 x 10, 0.35 x 10, or 2/3 x 10? Try to spot, and unpick, the 'just add zero' rule wherever it rears its head.
7. Proportion: three red sweets and two blue
Asked what proportion of the sweets is blue, how many kids will say 2/3 rather than 2/5? Why? Because they're comparing blue to red, not blue to all the sweets. Always stress that proportion is 'part to whole'.
8. Perimeter and area confuse many kids
A common mistake, when measuring the perimeter of a rectangle, is to count the squares surrounding the shape, in the same way as counting those inside for area. Now you can see why some would give the perimeter of a two-by-three rectangle as 14 units rather than 10.
9. Misreading scales
Still identified as a weakness in Key Stage test papers. The most common misunderstanding is that any interval on a scale must correspond to one unit. (Think of 30 to 40 split into five intervals.) Frequent handling of different scales, divided up into twos, fives, 10s, tenths etc. will help to banish this idea.
Words: Steve McCormack Pictures: David Moore
from
http://www.teachernet.gov.uk/teachers/issue42/primary/features/Mathsmisconceptions/
الأحد، 13 يناير 2008
السبت، 12 يناير 2008
السبت، 5 يناير 2008
الجمعة، 4 يناير 2008
المشروع العربي لتحسين مستوى الرياضيات
فكرة المشروع :
اثناء عملي كمعلم رياضيات و الذي امتد ما يقرب من 12 عاما ما زالت تواجهنى نفس المشكلة كل عام و هي ضعف المستوى التحصيلي لعدد كبير من تلاميذ الفصل فعدد قليل (لا يتعدى 50 % ) هو الذي يتفاعل و يتجاوب معي اثناء الحصة الدراسية و اعتقد ان هذه حقيقة تواجه كثير من معلمي الرياضيات و ربما يعود ذلك الى عدم المام الطالب باساسيات المادة ، فمثلا ينتقل الطالب من المرحلة الابتدائية الى الاعدادية دون المامه الكافى بجدول الضرب او اثناء دراسته فى المرحلة الاعدادية لا يعرف الطالب كيفية جمع و طرح الاعداد الصحيحة او حل المعادلة البسيطة و يرجع ذلك الى اسباب عده منها على سبيل المثال كثافة الفصل الدراسي - التزام المعلم بخطة تدريس المنهج فى فترة زمنية معينة كل هذا يؤدي الى اهمال هذه الفئة من الطلبة (الطلبة المهمشين).
و لا أخفى سرا اننى كنت اصاب بالاحباط في بعض الاحيان و انهي عملي و انا غير راض عن ذلك ...
نعم ..ما زلنا نسير فى دائرة مفرغة أو كما يقولون :"اننا نحرث في البحر"
و لكن ما العلاج ؟
.الى ان مررت بهذه التجربة :
من المعروف ان كل معلم له اسلوبه فى توصيل المعلومة و كان اسلوبي يعتمد على تحويل الدرس الى حوار بينى و بين الطلبة يمثل خطوات التفكير و كان لهذا الاسلوب نتيجة طيبة على الاقل يقيس مدى استيعاب الطالب للدرس .
و اثناء اهتمامي بالبرمجة فى الفترة الاخيرة و تطوير برنامج تعليمي لحل المعادلات ، رأيت أن اقوم بعمل قسم للمهارات الاساسية يحتوى على جمع و طرح الاعداد الصحيحة حيث ان كثير من الطلبة يعجزون عن جمع و طرح الاعداد الصحيحة واتتنى فكرة ان أقوم بتحويل الاسلوب الذي اتبعه فى الشرح و برمجته باستخدام ادوات برمجة بسيطة و جعل البرنامج يوجه عدة اسئلة الى الطالب مع وجود اختيارات للاجابة ليختار منها الطالب الاجابة الصحيحة يتم ذلك بترتيب منطقى للخطوات فعند اجابته للسؤال ينتقل للسؤال التالي ... و هكذا .
مع وجود اداة لتقييم الطالب فى نهاية المحاولة و تحديد مستواه : ممتاز - جيد جدا - جيد - متوسط - ضعيف.
و لا مانع من وجود مؤثرات صوتية لتشجيع الطالب مثل: احسنت - انى فخور بك.
تم ايضا عمل اعدادات تتيح للمعلم التعديل فى الفترات الزمنية لكل محاولة و التعديل فى الاجابات ما يتناسب و مستوى كل طالب .
تنفيذ الفكرة :
قمت بتجربة البرنامج على احد الطلبة ضعاف المستوى و كانت مفاجأة كبيرة ان تكون النتيجة رائعة بهذا الشكل ففي خلال ساعة استطاع الطالب ان يتقن مهارة جمع و طرح الاعداد الصحيحة ، ليس هذا فقط بل بدأ يهتم المادة و يتجاوب معي فى الفصل ...
و اطلقت على هذا الاسلوب المحاولة و الخطأ .
و اعتبرت هذا الاسلوب جزء من العلاج .
و فكرت ان اقوم بنشره من خلال مشروع يعتمد على استخدام البرمجة التفاعلية من خلال موقع على الشبكة ليستفيد منه الطلبة العرب.
ارجو ممن له اهتمام بمجال الرياضيات ان يتكاتف معي لبناء و استمرار هذا المشروع.
فما زال العلم هو اخر اسلحتنا في المعركة...................
أحمد فكري علي صاوي
الخميس، 3 يناير 2008
الأربعاء، 2 يناير 2008
الرياضيات العقليه
بسم الله الرحمن الرحيم
اود هنا ان اقدم نقد لطريقة عرض مهاره كانت تدرس في الرياضيات بدون ان تكون موجوده بكتاب التلميذ اما اليوم فهي موجوده بكتاب التلميذ ك هذه المهاره هي الرياضيات العقليه. لابدا كلامي اقدم هذا المثال البسيط للرياضيات العقليه
9 + 6 = 10 + 6 - 1 = 10+ 5 = 15
ماحدث هنا اننا بدلا من ان نحسب 9+6 التي تبدو صعبه حسبنا 10 + 6 ثم طرحنا 1 الى هنا والكلام رائع المشكله تكمن في كيفية تدريب التلمبذ على ذلك فالكتاب يطلب من التلميذ ان يكمل عبارات مثل
7+8 =7 + …..-…..=5+ …..= 15
وعند تقييم التلميذ مطلوب منه اكمال مثل هذه العبارات .................... تخيل لقد تحولت الطريقه التي تدرس لزيادة مهارة التلميذ في العمليات الحسابيه الى طريقه للحفظ وادت الى بطء حل العمليات الحسابيه على عكس الهدف المرجو من تدريس الرياضيات العقليه . اعتقد ان الحل هو ان تعقد امتحانات شفويه للرياضيات العقليه تحل باختيار الجواب الصحيح بعد سماع السؤال وان يحدد وقت (15 ثانيه مثلا) لكل فقره. هذا ما اظنه صحيح كما اتمنى حذف هذا الجزء من كتاب التلميذ مع بقائه في دليل المعلم وفي المنهج. ارجو افادتي بارائكم الكريمه
اود هنا ان اقدم نقد لطريقة عرض مهاره كانت تدرس في الرياضيات بدون ان تكون موجوده بكتاب التلميذ اما اليوم فهي موجوده بكتاب التلميذ ك هذه المهاره هي الرياضيات العقليه. لابدا كلامي اقدم هذا المثال البسيط للرياضيات العقليه
9 + 6 = 10 + 6 - 1 = 10+ 5 = 15
ماحدث هنا اننا بدلا من ان نحسب 9+6 التي تبدو صعبه حسبنا 10 + 6 ثم طرحنا 1 الى هنا والكلام رائع المشكله تكمن في كيفية تدريب التلمبذ على ذلك فالكتاب يطلب من التلميذ ان يكمل عبارات مثل
7+8 =7 + …..-…..=5+ …..= 15
وعند تقييم التلميذ مطلوب منه اكمال مثل هذه العبارات .................... تخيل لقد تحولت الطريقه التي تدرس لزيادة مهارة التلميذ في العمليات الحسابيه الى طريقه للحفظ وادت الى بطء حل العمليات الحسابيه على عكس الهدف المرجو من تدريس الرياضيات العقليه . اعتقد ان الحل هو ان تعقد امتحانات شفويه للرياضيات العقليه تحل باختيار الجواب الصحيح بعد سماع السؤال وان يحدد وقت (15 ثانيه مثلا) لكل فقره. هذا ما اظنه صحيح كما اتمنى حذف هذا الجزء من كتاب التلميذ مع بقائه في دليل المعلم وفي المنهج. ارجو افادتي بارائكم الكريمه
Primary Mathematics Syllabus 2007
I'll get some information about the Math syllabus in singapore.
this time I present P1........Wnat to read your comments
Primary 1:
this time I present P1........Wnat to read your comments
Primary 1:
- Whole Numbers
Numbers up to 100
counting to tell the number of objects in a given set
comparing the number of objects in two or more sets
use of ordinal numbers (first, second, up to tenth) and symbols (1st, 2nd, 3rd, etc.)
number notation and place values (tens, ones)
reading and writing numbers in numerals and in words
comparing and ordering numbers
number patterns
Exclude:
use of the terms ‘cardinal number’ and ‘ordinal number’
use of the symbols > and <
Addition and subtraction
concepts of addition and subtraction
use of the addition symbol (+) or subtraction symbol (-) to write a mathematical statement for a given situation
comparing two numbers within 20 to tell how much one number is greater (or smaller) than the other
recognising the relationship between addition and subtraction
building up the addition bonds up to 9 + 9 and committing to memory
solving 1-step word problems involving addition and subtraction within 20
addition of more than two 1-digit numbers
addition and subtraction within 100 involving
a 2-digit number and ones
a 2-digit number and tens
two 2-digit numbers
addition and subtraction using formal algorithms
Mental calculation
addition and subtraction within 20
addition and subtraction involving
a 2-digit number and ones without renaming
a 2-digit number and tens
Multiplication and division
multiplication as repeated addition (within 40)
use of the multiplication symbol (×) to write a mathematical statement for a given situation
division of a quantity (not greater than 20) into equal sets:
given the number of objects in each set
given the number of sets
solving 1-step word problems with pictorial representation
Exclude:
use of multiplication tables
use of the division symbol (÷) - Measurement
Length And Mass
measurement and comparison of the lengths / masses of two or more objects in non-standard units
use of the following terms:
long, longer, longest
short, shorter, shortest
tall, taller, tallest
high, higher, highest
heavy, heavier, heaviest
light, lighter, lightest
Exclude:
finding the difference in length / mass
Time
telling and writing time to the hour / half hour
Exclude:
24-hour clock
Money
identifying coins and notes of different denomination
matching a coin / note of one denomination to an equivalent set of coins / notes of another denomination
telling the amount of money
in cents up to $1
in dollars up to $100
use of the symbols $ and ¢
solving word problems involving addition and subtraction of money in dollars only (or in cents only)
Exclude:
combinations of dollars and cents - Geometry
Basic Shapes (rectangle, square, circle, triangle)
identifying and naming the 4 basic shapes from 2-D and 3-D objects
describing and classifying shapes
Patterns
making / completing patterns with 2-D cut-outs according to one or two of the following attributes
shape
size
colour
making / completing patterns with 3-D models
cube
cuboid (rectangular block)
cone
cylinder - Data Analysis
Picture graphs
collecting and organising data
making picture graphs
use of a symbol/picture to represent one object
reading and interpreting picture graphs in both horizontal and vertical forms
Exclude:
picture graphs with scales
TIMSS Schedule of Events 2008
TIMSS 2007 is the fourth in a cycle of internationally comparative assessments dedicated to improving teaching and learning in mathematics and science for students around the world. Carried out every four years at the fourth and eighth grades, TIMSS provides data about trends in mathematics and science achievement over time.
To inform educational policy in the participating countries, this world-wide assessment and research project also routinely collects extensive background information that addresses concerns about the quantity, quality, and content of instruction. For example, TIMSS 2007 will continue collecting detailed information about mathematics and science curriculum coverage and implementation, as well as teacher preparation, resource availability, and the use of technology.
TIMSS Schedule of Events 2008
Sampling adjudication January
Fourth Science and Math Item Review Committee Meeting to conduct scale anchoring of achievement data May
Eighth National Research Coordinator Meeting to review draft International Report – text, graphics, and tables June
ISC/IEA conduct international press conference to release International Report; ISC posts TIMSS 2007 International Report and Technical Report on web December
To inform educational policy in the participating countries, this world-wide assessment and research project also routinely collects extensive background information that addresses concerns about the quantity, quality, and content of instruction. For example, TIMSS 2007 will continue collecting detailed information about mathematics and science curriculum coverage and implementation, as well as teacher preparation, resource availability, and the use of technology.
TIMSS Schedule of Events 2008
Sampling adjudication January
Fourth Science and Math Item Review Committee Meeting to conduct scale anchoring of achievement data May
Eighth National Research Coordinator Meeting to review draft International Report – text, graphics, and tables June
ISC/IEA conduct international press conference to release International Report; ISC posts TIMSS 2007 International Report and Technical Report on web December
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