No more couch historians!

[HoreTore]

Math problems in school exams are generally tailored for calculator use, as otherwise they would be impossible to solve except for people like Stephen Hawkings. Calculators are a prime example of expanding what a person could possibly do.

You're close, but allow me to expand:

Whether you use calculators or not will decide what kind of mathematics you are able to show. I've given my students tests both without and with calculators. Whether or not I let them use a calculator has nothing to do with "going easy" on them, or whatever. It will determine what ability my test can uncover.

If they do not use a calculator, I must keep the numbers low, or otherwise they'll spend ages on the algorithms instead of the actual maths I intended them to learn. And that's a problem, since your ability to solve problems with high numbers is the difference between "good" understanding and "excellent" understanding. Solve a simple 11x+7y=61 equation, fine. Ramp up the numbers, and the probability increases heavily that you will make a mistake if you do not know the procedure well enough. To do the latter you need calculators, because otherwise you're spending all the time solving problems irrelevant to the problem I wanted to check if you were able to solve.

A test without calculators will be a test to check your understanding of basic algorithms. A test with calculators will check your understanding of mathematical problems of a higher level than simple division/multiplication.


I often hear adults whine about how calculators ruined maths, and how thy got lower grades when they got calculators. They often lament how their grades went down after calculators were introduced, and blame it on the calcs. The reality is that their understanding of maths was low to begin with, and they were kept up only by their ability to perform algorithms, not their understanding of maths.


Finally, a quick question to the OP to check whether he has any knowledge, or is just another hypocrite:

The problem 101 multiplied by 99 should be solved by anyone wit good knowledge of maths in under 3 seconds, without the use on any help outside your mind. Can you do that?

[Papewaio]

A better way to show understanding of maths is solve the sum of 1 to a million in one minute.

I can do it easily, after all I'm just an average in a sum of all parts.

[Pannonian]

A better way to show understanding of maths is solve the sum of 1 to a million in one minute.

I can do it easily, after all I'm just an average in a sum of all parts.

I didn't quite get Horetore's point until reading this. I can write down the working out, and I can explain the reasoning. But without a calculator, I'll need a pen and paper to get the answer.

[Tellos Athenaios]

I didn't quite get Horetore's point until reading this. I can write down the working out, and I can explain the reasoning. But without a calculator, I'll need a pen and paper to get the answer.

Yes: pen and paper: 5 * 10^5 (10^6 + 1) = 5 * 10^11 + 5^ 10^5. Which is: 5 00 000 500 000 = 500000500000.

[Tellos Athenaios]

You're close, but allow me to expand:

Whether you use calculators or not will decide what kind of mathematics you are able to show. I've given my students tests both without and with calculators. Whether or not I let them use a calculator has nothing to do with "going easy" on them, or whatever. It will determine what ability my test can uncover.

If they do not use a calculator, I must keep the numbers low, or otherwise they'll spend ages on the algorithms instead of the actual maths I intended them to learn. And that's a problem, since your ability to solve problems with high numbers is the difference between "good" understanding and "excellent" understanding. Solve a simple 11x+7y=61 equation, fine. Ramp up the numbers, and the probability increases heavily that you will make a mistake if you do not know the procedure well enough. To do the latter you need calculators, because otherwise you're spending all the time solving problems irrelevant to the problem I wanted to check if you were able to solve.

A test without calculators will be a test to check your understanding of basic algorithms. A test with calculators will check your understanding of mathematical problems of a higher level than simple division/multiplication.

It depends heavily on the type of math but it has little to do with the "size" of the numbers. Three examples, all are equally difficult:

• 4999 * 5001
• 4*6
• -1/ ((x + 1)(1 - x))

Which is why you only get a minimal reward for providing the result and the real points are awarded if you can demonstrate how you solved it without the calculator.

I often hear adults whine about how calculators ruined maths, and how thy got lower grades when they got calculators.

I never heard that one, I heard the reverse (they got lower grades once the calculators were removed). The typical complaint is that because people can use the calculator they do, and because they do they lose out on the practice needed to use the brain thing so they are stuck without a calculator.

Now there are problems which simply cannot be "solved" or computed without the use of the calculator in any reasonable time frame but then generally you'd simply provide the formula as answer and leave entering the formula in a calculator as an exercise for the reader.

[HoreTore]

It depends heavily on the type of math but it has little to do with the "size" of the numbers. Three examples, all are equally difficult:

• 4999 * 5001
• 4*6
• -1/ ((x + 1)(1 - x))

Which is why you only get a minimal reward for providing the result and the real points are awarded if you can demonstrate how you solved it without the calculator.

Wrong.

You need to be more confident in the multiplication algorithm to solve 4999*5001 than to solve 4*6 or 49*51. The procedure is the same, but the evidence clearly states that more people do more mistakes once lower numbers are substituted for higher numbers in the same mathematical problem. That's why a lecturer will scale up thenumbers if he is unsure on whether to give you an A or a B on an oral test, for example.

But please feel free to argue against commonly accepted academic fact. That's always fun to watch.

[Tellos Athenaios]

Wrong.

You need to be more confident in the multiplication algorithm to solve 4999*5001 than to solve 4*6 or 49*51. The procedure is the same, but the evidence clearly states that more people do more mistakes once lower numbers are substituted for higher numbers in the same mathematical problem. That's why a lecturer will scale up thenumbers if he is unsure on whether to give you an A or a B on an oral test, for example.

But please feel free to argue against commonly accepted academic fact. That's always fun to watch.

Has nothing whatsoever to do with math. The algorithm to compute 4999*5001 or 4*6 or 49*51 is exactly the same, but the important insight you can get from math has nothing to do with generic multiplication. It is a special case, and if you recognise that pattern you can do it much easier. It is in fact 5000 * 5000 - 1, 5 *5 -1, 50 * 50 -1. And 5000 * 5000 is easy to do in the decimal system, because it's 5 * 5 * 1000 * 1000 - 1 which reduces to 5 * 5 shifted left by six, minus 1.

That is math. Taking the long route and writing out multiplication as a big sum indicates the victim does not understand the math. Likewise if you spot a kid working out the sum of 1 to 1 million by hand instead of taking the shortcut, the kid doesn't understand the math behind it. There's no shame in not understanding the math as long as we don't pretend that being able to perform computations (algorithms) equates to understanding math.

[HoreTore]

Has nothing whatsoever to do with math. The algorithm to compute 4999*5001 or 4*6 or 49*51 is exactly the same, but the important insight you can get from math has nothing to do with generic multiplication. It is a special case, and if you recognise that pattern you can do it much easier. It is in fact 5000 * 5000 - 1, 5 *5 -1, 50 * 50 -1. And 5000 * 5000 is easy to do in the decimal system, because it's 5 * 5 * 1000 * 1000 - 1 which reduces to 5 * 5 shifted left by six, minus 1.

That is math. Taking the long route and writing out multiplication as a big sum indicates the victim does not understand the math. Likewise if you spot a kid working out the sum of 1 to 1 million by hand instead of taking the shortcut, the kid doesn't understand the math behind it. There's no shame in not understanding the math as long as we don't pretend that being able to perform computations (algorithms) equates to understanding math.

(I already gave an example of the quadrat sentence in an earlier post)

So... You are of the opinion that the difficulty does increase when the numbers used are higher?

[Tellos Athenaios]

(I already gave an example of the quadrat sentence in an earlier post)

So... Your argument isn't that higher numbers does not increase the difficulty level, then? You are of the opinion that the difficulty does increase when the numbers used are higher?

My opinion is that once you take the step from "here's an algorithm which lets you compute this kind of problem" to "here's how you can deduce what algorithm will compute your problem" the numbers switch from being the things that make life difficult to fairly arbitrary constants. In other words when you take the step from computations to math. If you the question is about math the numbers don't matter at all.

[HoreTore]

My opinion is that once you take the step from "here's an algorithm which lets you compute this kind of problem" to "here's how you can deduce what algorithm will compute your problem" the numbers switch from being the things that make life difficult to fairly arbitrary constants. In other words when you take the step from computations to math. If you the question is about math the numbers don't matter at all.

I have no objections to this, other than the remaining problem of how it's still easier to find algorithms hen higher numbers are present. It's not really a talking point, it's a simple fact one must be aware of. Higher numbers are not more difficult because they're harder, they're more difficult because our brains tends to mess up more. For example, adding an extra zero when writing "100000" is a lot easier to do than when writing "10".

Increasing the numbers is a nifty way to test an algorithm(or your ability to produce an algorithm) without actually adding any new problems to the original problem. It's comparable to the satanic practice of adding an irrelevant piece of information to a text problem("calculate the m2 of a room with sides of 2m and 4m" is easier than "calculate the m2 of a room with sides of 2m and 4m and a height of 3m").

[Ironside]

It depends heavily on the type of math but it has little to do with the "size" of the numbers. Three examples, all are equally difficult:

• 4999 * 5001
• 4*6
• -1/ ((x + 1)(1 - x))

Mathwise, 19*31 is also as simple. The formula is the same. A change in the size of the numbers though.