Maths Assessment Books
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cimman:
Yes. I agree with you. At least my daughter always like to look for pattern recognition even before she thinks what the question is actually looking for. Sigh....Must address the root cause.rephrasing is a big problem with children. The root cause is that children are being trained to recognize problems using pattern recognition, ie. the logic, the phrasing and even the values has to be the same, before the child can recognize that it belongs to the same category of problems. What do I mean by values ? there are a number of problems where in the end, the value is zero, or the value is the same, or the value is a multiple of another value. Each of these value patterns requires a different way of processing the information.
This is an inefficient way of learning, since there are numerous ways that the problem sum can be stated and there are of course, different values in every problem. This is leads to a lot of drilling before a student can amass enough patterns to be effective.
Assessment books encourages this problem solving technique by giving multiple problems with the same value patterns, ie. zero values in the end, or same multiples of some other values, or the familiar unchanged difference, ie. Tom loses 3 units, Mary loses 3 units.
Students are not taught the basic principles of a certain class of problem and how it applies to the that specific class in all it's permutations.
The basic principles are the formulas that must be applied to that class of problem.
ie. In Before - After type of problems, the formula is
Before + Transfer (or Change) = After.
It is a very simple formula, and most students would not even recognize it as a formula, but it is. If the student is able to truly understand the application of this formula, then the student would be free from pattern recognition techniques and would be able to apply first principles to solve all problems, no matter the phrasing or the logic flow or the values.
Take for example the familiar Work Backwards heuristic. It is called Work Backwards because the information to link an unknown variable to a given value, is given right at the end of the question. Does it really matter, if that information is given at the end, at the start or even in the middle of the problem statement? Yes, it does if you're using heuristics. No, it doesn't if you're using first principles, ie. formula approach.
I've developed a technique, Table Heuristics, to solve these issues. It is a holistic framework for solving problems sums based on first principles. A large part of the framework involves problem interpretation, which involves a lot of English language comprehension. There is a visual tool for analysis, and algebra is used in the actual resolution. You'll find that primary school maths is more about English comprehension than Maths.
Why algebra ? because it is abstract enough to provide a single method to solve all problem sums for a given formula. The problem with algebra is that students find it difficult to formulate equations. The conventional way of formulating equations is to mentally process the relationships and values and then write out the final equations on paper. This is a difficult exercise for students starting on algebra.
The Table Heuristics framework has a visual aid to help students formulate the equations. Bit by bit, they build up the equations on paper, not in their minds. This leads to less careless mistakes and a clearer more visual approach to see how relationships are linked.
If you've always wondered if all these different problem sums could be solved with a single consistent approach, then yes, it can be done. Is there an alternative to massive drilling ? yes, there is a more efficient method. -
cimman:
Hi cinman,rephrasing is a big problem with children. The root cause is that children are being trained to recognize problems using pattern recognition, ie. the logic, the phrasing and even the values has to be the same, before the child can recognize that it belongs to the same category of problems. What do I mean by values ? there are a number of problems where in the end, the value is zero, or the value is the same, or the value is a multiple of another value. Each of these value patterns requires a different way of processing the information.
This is an inefficient way of learning, since there are numerous ways that the problem sum can be stated and there are of course, different values in every problem. This is leads to a lot of drilling before a student can amass enough patterns to be effective.
Assessment books encourages this problem solving technique by giving multiple problems with the same value patterns, ie. zero values in the end, or same multiples of some other values, or the familiar unchanged difference, ie. Tom loses 3 units, Mary loses 3 units.
Students are not taught the basic principles of a certain class of problem and how it applies to the that specific class in all it's permutations.
The basic principles are the formulas that must be applied to that class of problem.
ie. In Before - After type of problems, the formula is
Before + Transfer (or Change) = After.
It is a very simple formula, and most students would not even recognize it as a formula, but it is. If the student is able to truly understand the application of this formula, then the student would be free from pattern recognition techniques and would be able to apply first principles to solve all problems, no matter the phrasing or the logic flow or the values.
Take for example the familiar Work Backwards heuristic. It is called Work Backwards because the information to link an unknown variable to a given value, is given right at the end of the question. Does it really matter, if that information is given at the end, at the start or even in the middle of the problem statement? Yes, it does if you're using heuristics. No, it doesn't if you're using first principles, ie. formula approach.
I've developed a technique, Table Heuristics, to solve these issues. It is a holistic framework for solving problems sums based on first principles. A large part of the framework involves problem interpretation, which involves a lot of English language comprehension. There is a visual tool for analysis, and algebra is used in the actual resolution. You'll find that primary school maths is more about English comprehension than Maths.
Why algebra ? because it is abstract enough to provide a single method to solve all problem sums for a given formula. The problem with algebra is that students find it difficult to formulate equations. The conventional way of formulating equations is to mentally process the relationships and values and then write out the final equations on paper. This is a difficult exercise for students starting on algebra.
The Table Heuristics framework has a visual aid to help students formulate the equations. Bit by bit, they build up the equations on paper, not in their minds. This leads to less careless mistakes and a clearer more visual approach to see how relationships are linked.
If you've always wondered if all these different problem sums could be solved with a single consistent approach, then yes, it can be done. Is there an alternative to massive drilling ? yes, there is a more efficient method.
May you share here what exactly is table heuristic? I find it difficult to teach for example in the topic of \"ratio\" to my girl and the best way to get ready to start a new topic is to have full understanding of why we think we need to use ratio and why it is relevant and not others.
:thankyou: -
cinman,
perhaps u can give us a diagram.
a picture speaks a thousand words.
:thankyou: -
Hi cimman,
Do you have recommendation for algebra bks?
Thank you. -
phtthp:
here are a few examples on how Table Heuristics is applied to different problems:cinman,
perhaps u can give us a diagram.
a picture speaks a thousand words.
:thankyou:
Danny had four times as much money as Peter. After Danny gave Peter $600, Peter had four times as much money as Danny. How much money did they have altogether ? Ans: 1000![](\"http://i46.tinypic.com/2wrk8ck.png\")
http://i46.tinypic.com/2wrk8ck.png\">
May and Julie had an equal amount of money at first. After May spent $18 and Julie spent $25, May had twice as much money as Julie. How much money did each girl have at first ? Ans: 32
![](\"http://i48.tinypic.com/4tq728.png\")
A big tank contained 890 liters of water. A small tank contained 170 liters of water. When an equal amount of water was added to both tanks, the big tank contained three times as much water as the small tank. How much water was added to each tank ? Ans: 190
![](\"http://i48.tinypic.com/23iv0as.png\")
A study was done on the above 3 problems and only 10% of students were able to solve the last question while 25% were successful in solving the 2nd problem (May and Jule). This is because all 3 employs different analysis techniques.
However, with the aid of the table, and using equation techniques, the analysis method is the same. When abstracted at a high enough level, the problems are the same.
Here's another 2 examples.
This one uses the Unchanged Difference heuristic. The way it has been taught is that students must 'see' that the difference is unchanged. If you view it with the aid of a table , there is no need to 'see' such a difference, the equations just pops up :
There were 2/5 as many adults as children on a bus. At Bus Stop A, 8 children and 8 adults boarded the bus. There were then 1/2 as many adults as children. How many adults were on the bus at first ? Ans: 16 adults
![](\"http://i47.tinypic.com/52l89z.png\")
This one uses the Unchanged Quantity concept, here you must 'see' that the Girls quantity remained unchanged:
In a library, 2/5 of the pupils were boys. After another 8 boys came into the library, the percentage of girls dropped to 50%. How many pupils were there in the library in the end ? Ans: 48 pupils
![](\"http://i47.tinypic.com/36cf7.png\")
there is a common thread that runs through all the problems. You will find that modelling and heuristics all uses the basic algebraic principles. However, since they have to avoid simultaneous equations, they have to fudge it, to find some other means to eliminate one of the unknown variables in a 2 unknown variables problem. The way to do that is employ heuristics or modelling techniques. However, these techniques requires the student to 'see' that there is an unchanged 'this' or unchanged 'that' and that requires a lot of drilling. With modelling, there is a lot of splitting, and shifting and aligning of boxes. All that does is to eliminate one unknown from a 2 unknown variable problem.
It is much easier to use basic simultaneous equations techniques to eliminate the unknowns and avoid all these 'seeing' and moving boxes around ie. one method. Solving simultaneous equations is very mechanical. Follow certain basic rules and the child will have no problems with solving equations.
I'm planning to conduct a workshop in October to parents on this technique. Just check out the Happening forum, under http://www.kiasuparents.com/kiasu/forum/viewtopic.php?f=43&t=32278&start=120. Just drop a post in that thread and I'll notify you when the time comes.
There's no financial investment needed on your side.
Just come in and listen.
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venuschan:
Fractions, Ratios and Percentages all belong to a class of numbers, I call Relationship numbers. These numbers differ from normal absolute numbers in that they cannot stand alone. They need another number in order to make sense of its value, thus a relationship. It use the relative concept to give it a value. Normal numbers have an absolute value, without the need for a relative concept.
Hi cinman,
May you share here what exactly is table heuristic? I find it difficult to teach for example in the topic of \"ratio\" to my girl and the best way to get ready to start a new topic is to have full understanding of why we think we need to use ratio and why it is relevant and not others.
:thankyou:
The fraction has the numerator and denominator. Without the denominator, the numerator has no meaning. The denominator defines the reference point, or the baseline.
The percentage is very similar to the fraction, however, the reference point or baseline is not so obvious as it is not normally stated in an obvious way. If there is a discount of 30% discount on Gucci bags in the shop in Paragon, without knowing what is the 100% or the normal sale price, the 30% is meaningless. So for percentage problems, students are challenged to figure out what the 100% value should be linked to. This is unlike the fraction, where the 100% value is linked to the denominator, so the 100% value in a fraction is very obvious.
The ratio follows along the lines of the fraction and percentage. Without a corresponding number, the ratio is meaningless. It needs another number by it's side. However the ratio is unique with respect to the fraction and the percentage. It is the only construct that allows more than 2 relationships. ie. the ratio of the number of apples that Tom, Dick, Harry and John is given is: 2: 4: 5: 3
It is in problems that deals with proportions. A common example of proportion is found in cooking recipes. A fish sauce is made up of 3 parts of soya sauce, 5 parts of water, 1 part of sugar, 2 parts of cooking wine, 4 parts of oyster sauce. The whole, or 100% is now made up of 5 components. The interesting thing about proportions, is that it is scalable. ie. If I make a cup of fish sauce, compared to a vat of fish sauce, it will taste the exactly the same, simply because the proportions are the same, relative to each other. The proportions for making a cup of fish sauce, a bottle of fish sauce or a vat of fish sauce remains the same. If I take a spoonful of fish sauce from a vat of fish sauce, you'll find that the proportions remains the same.
A typical problem that deals with proportions have the keywords \"for every\". Not every proportion problem sums have those keywords, but those that do, it's obvious that ratio is involved.
Here's a not so obvious proportion problem. It uses the same concept as the fish sauce example that was given above:
Mr Smith had 1.5 litre of white paint in Tin A and 1.25 litre of red paint in Tin B. He poured 750ml of red paint from Tin B into Tin A. Then he poured some of the mixture in Tin A back into Tin B. If 7/11 of the final mixture in Tin B was made up of red paint, what was the volume of the mixture in Tin A that Mr Smith had poured into Tin B ? Ans: 600ml -
SAHM_TAN:
books dealing with simultaneous equations are at the secondary level. There are no books at the primary level. Tuition centers that teaches algebra at the primary level produces their own materials.Hi cimman,
Do you have recommendation for algebra bks?
Thank you.
You can create your own materials. I normally just create ad hoc equations for my child solve.
The 1st concept to teach is the equal sign. That forms the basis of an equation.
When thing move across the equal sign, the sign changes. You'll have to teach the concept of the opposite sign, ie. + , -, x, /
Once the child is comfortable in moving numbers with their respective signs across the equal sign, you can move on to simultaneous equations, ie. 2 equations. Introduce the concept of the negative number. There are the standard rules here: (-) x (-) is (+), (-) x (+) is (-), and so on.
Just get your child to memorise them.
After that teach the concept of adding 2 equations, or subtracting 2 equations.
I am in the process of creating some materials for teaching algebra. If you're interested, let me know. -
just to add on to the ratio concept. The ratio is also unique in that the 100% is based on all it’s components. The fraction’s 100% is based on the denominator, a single value. The percentage 100% is also based on a single value. The ratio’s 100% is based on the summation of all it’s components. Ask your child to think of situations where 100% depends on 2 or more components and that’s where ratio is used.
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cimman:
Could you arrange one in November or December? After the exams. Thank you in advance.
here are a few examples on how Table Heuristics is applied to different problems:phtthp:
cinman,
perhaps u can give us a diagram.
a picture speaks a thousand words.
:thankyou:
Danny had four times as much money as Peter. After Danny gave Peter $600, Peter had four times as much money as Danny. How much money did they have altogether ? Ans: 1000
May and Julie had an equal amount of money at first. After May spent $18 and Julie spent $25, May had twice as much money as Julie. How much money did each girl have at first ? Ans: 32
A big tank contained 890 liters of water. A small tank contained 170 liters of water. When an equal amount of water was added to both tanks, the big tank contained three times as much water as the small tank. How much water was added to each tank ? Ans: 190
A study was done on the above 3 problems and only 10% of students were able to solve the last question while 25% were successful in solving the 2nd problem (May and Jule). This is because all 3 employs different analysis techniques.
However, with the aid of the table, and using equation techniques, the analysis method is the same. When abstracted at a high enough level, the problems are the same.
Here's another 2 examples.
This one uses the Unchanged Difference heuristic. The way it has been taught is that students must 'see' that the difference is unchanged. If you view it with the aid of a table , there is no need to 'see' such a difference, the equations just pops up :
There were 2/5 as many adults as children on a bus. At Bus Stop A, 8 children and 8 adults boarded the bus. There were then 1/2 as many adults as children. How many adults were on the bus at first ? Ans: 16 adults
This one uses the Unchanged Quantity concept, here you must 'see' that the Girls quantity remained unchanged:
In a library, 2/5 of the pupils were boys. After another 8 boys came into the library, the percentage of girls dropped to 50%. How many pupils were there in the library in the end ? Ans: 48 pupils
there is a common thread that runs through all the problems. You will find that modelling and heuristics all uses the basic algebraic principles. However, since they have to avoid simultaneous equations, they have to fudge it, to find some other means to eliminate one of the unknown variables in a 2 unknown variables problem. The way to do that is employ heuristics or modelling techniques. However, these techniques requires the student to 'see' that there is an unchanged 'this' or unchanged 'that' and that requires a lot of drilling. With modelling, there is a lot of splitting, and shifting and aligning of boxes. All that does is to eliminate one unknown from a 2 unknown variable problem.
It is much easier to use basic simultaneous equations techniques to eliminate the unknowns and avoid all these 'seeing' and moving boxes around ie. one method. Solving simultaneous equations is very mechanical. Follow certain basic rules and the child will have no problems with solving equations.
I'm planning to conduct a workshop in October to parents on this technique. Just check out the Happening forum, under http://www.kiasuparents.com/kiasu/forum/viewtopic.php?f=43&t=32278&start=120. Just drop a post in that thread and I'll notify you when the time comes.
There's no financial investment needed on your side.
Just come in and listen. -
hi cimman,
Thank you for the very clear, colorful vivid diagrams you'd illustrated.
Very much appreciated ! :thankyou:
The diagrams you'd drawn above are pitched more at Upper primary P5 / P6 level. Your coming workshop October next month also pitched at Upper primary level.
is it possible to conduct another workshop pitched at Mid primary level, ie. at P3 / P4 ? Thank you.
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