Written from the perspectie of an AS Maths/Further Maths student.

]]>Yes, it is. If you learn math by memorizing formulas and applying them over and over, you’re not ever going to understand anything. You’re not going to have a sudden epiphany one day. Read your textbook. Although most high school textbooks are absolu…

]]>Saving this one for later. ]]>

Or perhaps when the examiners stop testing a concept, the students are not eager to learn the concept, and eventually the teachers stop teaching the concept.

The cycle gets more vicious when the generation of students become the generation of teachers …

… until the examiners start testing the concept again.

]]>I totally agree with your statement

“So I guess my point is that while it is possible that your young friend has been badly taught, it is more likely that the teacher did what she had to do in order to get the majority of ordinary students through the exam with decent grades”.

Teaching proofs/first principles and assesing students on these aspects would mean fewer good grades at the end of the day and our society or at least the powers that be decided that we should not go along such paths.

]]>For example, have students measure their displacement and their velocity for a variety of fairly simple motions, and then look for comparisons and patterns between the two sets of data (perhaps by graphing them), and then use these observations to build a inductive understanding of the relationship between distance functions and velocity functions, and then extend this understanding to a variety of different functions AND then talk about secant lines, etc…

Even if you skip the notation, the idea of the secant line converging to a tangent line is completely accessible to students, especially if you use numerical examples.

As for students who do not understand graphs at all, I developed this tool to help jump start classroom discussions: http://davidwees.com/graphgame/

]]>Many insightful points on A-level Math. In some Junior Colleges there are close to 100% students scoring distinction in A-level Math just to show that either the exams papers is too easy or students are too smart to spot questions from past-year papers and drill to score perfect marks. Do the students learn the real math to prepare them for university education? ]]>

Thanks for visiting my blog. ]]>

Nevertheless, thank you for your contributions to this discussion, and your blog looks nice.

]]>You are welcome to visit my blog tomcircle.wordpress.com, I would be honored if you find any idea useful to you.

Best regards,

Cornelius Goh

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]]>Another observation of A-level Calculus teaching, it never teaches students to be rigorous. Before applying all kinds of techniques to integrate, first check the domain of definition in which whether the function is defined? continuous? If the function is not continuous, then integration stops because it is not integratable over that domain (or interval).

In French lycÃ©es Baccalaureat, this rigorous practice is very much emphasized. When I was a student then in France, the professor yelled at me because I started straight to integrate without the prior check of integratability over the domain of definition and continuity.

These are interesting questions, indeed. Are you aware of the “Ask NRICH” bulletin board at nrich.maths.org? That is a forum on which questions like this can be discussed – I do recommend that you post your question there, and you’ll get a series of good answers.

Julian

]]>d/dx (e^x) = lim h-> 0 (e^x.e^h – e^x)/(h)

This can be factored into

e^x lim h-> (e^h -1)/(h) as you have mentioned.

Now this is then equal to 1.

My question is this ; suppose you replace e with say a number , a. Then, why does

lim h->0 (a^x -1)/(h) not equal to 1. What makes it so different?

I would also like to add that I do NOT want an answer that says that this is experimentally proven. That would be a disgusting and frankly speaking dishonest answer.

Also what does a number, b, raised to the number of an irrational mean? For instance,

a^ root 2 =???

The way I would do it is as follows;

Let a^ root 2 = b

Then raising again to the power of root 2,

a^2 = b^ root 2

But this is (in hindsight) plain stupid.

I have therefore accepted the fact that we can only approximate a^ root 2. ( in passing , I am tempted to ask, how can we prove that a^root 2 is an irrational?)

Consider also the proof that root 2 is an irrational number.

Suppose root 2 is rational.

Then by default, root 2 = p/q where p and q are elements of the integer set and q not equal to 0. Let p/q be a fraction in its simplest form. Then,

2q^2 = p^2

Then p can be replaced by 2k

Therefore,

q^2 = 2k^2

Then q must be even.

But p/q is a fraction in its simplest form.

Therefore, this is a contradiction and root 2 is an irrational ( Q.E.D)

Now I want to generalize this proof and prove that root a is irrational where a is an element of positive real numbers and a not equivalent to n^2 where n is an element of rational numbers.

This proof however is still not good enough. What about the generalization nth root of a is an irrational where a is an element of real numbers ( for odd nth) and a not equal to m^n

for m an element of rational number?

I would also like to ask what root 2 means and how we can draw out root 2 if it cannot be measured fully? (The drawing is easy; simply draw a right angle with sides 1 and 1 and the hypotenuse will be root 2) my question is what is the meaning of such a number physically? This also applies to the number pi. The circumference of a circle is given by the formula C = 2 x PI x R.

Consider r to be a rational number .

Then C is an irrational.

What does that mean physically ? How can we draw something that cannot be measured?

Thanks

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