Suppose I have decided to construct the Riemann Integral along the lines of tagged partitions and Riemann sums, and not along the lines of using the Darboux Integral (and then showing its equivalence).

In a standard construction of the Riemann Integral along the lines of "tagged partitions," it seems to me that we often make more definitions than are really necessary. Usually, we define partitions (and mesh norm), then we define tagged partitions, then we define Riemann sums with respect to a tagged partition, and finally, we define Riemann Integrability and the integral itself.

Is there a pedagogy reason for having to make all these definitions? Why not proceed as follows:

Definition 1: Let $a,b\in\mathbb{R}$ with $a<b$. Then, a finite sequence of nonempty closed intervals, $([x_{j-1},x_j])_{j=1}^{n}$, which satisfies the condition that $[a,b]=\bigcup_{j=1}^{n}[x_{j-1},x_{j}]$, is said to be a subdivision of $[a,b]$.

Definition 2 A function, $f(x):[a,b]\rightarrow\mathbb{R}$, which satisfies the condition that:

  • $\exists$ $L\in\mathbb{R}$ with the property that $\forall$ $\epsilon>0$, $\exists$ $\delta_{\epsilon}>0$ such that $\forall$ $([x_{j-1},x_j])_{j=1}^{n}$ with $\max\{(x_{j}-x_{j-1})\in\mathbb{R} \mid j=1,\ldots,n\}<\delta_{\epsilon}$, we have that $\forall$ $(t_{j})_{j=1}^{n}$ with $t_{j}\in[x_{j-1},x_{j}]$ for each $j=1,\ldots,n$, the statement $|\sum_{j=1}^{n}f(t_{j})(x_{j}-x_{j-1}) - L|<\epsilon$ holds.

is said to be Riemann Integrable on $[a,b]$ with Riemann Integral $\displaystyle{ \int_{a}^{b}f(x)dx}:=L$.

  • $\begingroup$ How is this different from the usual tagged partition definition? You are simply using the same constructions without giving them a name. In terms of pedagogy, I think breaking concepts down into simpler chunks is more efficient, rather than having a single complicated definition (such as your definition 2) which ultimately amounts to the same thing. $\endgroup$ – EuYu Mar 9 '17 at 6:53
  • $\begingroup$ It should be equivalent to the usual definition. It's exactly that I'm using the same constructions without given them a name. I am just wondering what peoples' opinions are about breaking these concepts down into smaller chunks vs presenting them as one big definition. $\endgroup$ – JWP_HTX Mar 9 '17 at 7:17
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    $\begingroup$ If that's the case, I think this question might be more appropriate for math educators. $\endgroup$ – EuYu Mar 9 '17 at 12:39
  • $\begingroup$ Fair enough. I have changed the tags on this question (pun intended) to more reflect that. $\endgroup$ – JWP_HTX Mar 9 '17 at 13:58
  • $\begingroup$ who are the target audience? $\endgroup$ – Alvin Lepik Mar 9 '17 at 14:31

I would say that the reason behind all these definitions (partions, tagged partitions, etc.) is that anyone who wants to understand Definition 2, has to unpack it anyway. The question then becomes, how do you structure this process.

  1. The classical approach is to do the structuring upfront. You introduce the bitesize pieces and then show step-by-step, how they fit together. Thus one defines partitions, then tagged partitions, and so on. After the student has gained some familiarity with them they are packaged together in the definition of the Riemann integral.

  2. The other approach is to through the definition at students upfront and then guide them in understanding it. If you do this at the same level of detail, you will end up with the same explanations (this part of the definition describes the partition, etc.)

In my experience students react better to concepts being presented in small chunks rather than the unpack-this-complicated-definition-on-your-own method. A very good student might be able to understand Definition 2 directly, in that case you are right, all these other definitions can be dispensed with, but most will need the step-by-step approach.

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  • $\begingroup$ Thanks for the response! I completely agree with what you've written. The reason I can see for advocating the "other" approach is from my own experience - I think I personally find it easier to just unpack one big definition than try to keep track of many smaller ones. $\endgroup$ – JWP_HTX Mar 9 '17 at 15:28
  • $\begingroup$ What I have learned teaching at my university is that the biggest difference between strong and weak students is the threshold for perseverance. Weaker students may have the capability to understand things eventually, but they give up much faster. Hence, the bite-size approach. The other thing I learned is that my personal experience of learning mathematics is very different from that of my students. For me a difficult problem was a challenge. For many of my students a difficult problem is a stumbling block. $\endgroup$ – Martins Bruveris Mar 9 '17 at 23:44
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    $\begingroup$ Yes, I agree with this also. I don't think weaker students are any less talented, and in particular, I think anyone with an interest in math has the requisite talent to do well in it - the key, as you say, is hard work. There is no substitute for that. $\endgroup$ – JWP_HTX Mar 10 '17 at 2:21

I studied Mathematics under the French system and they taught us Riemann integrals using the way you described it, i.e. by defining a Riemann Integrable function in the way you describe it in definition 2.

Well, it is difficult to change this way in France or French speaking countries because Cauchy who introduced the modern formulation for analysis that allows your Definition 2 above was himself French. Other countries, especially English speaking constraints, could teach Mathematics in a more relaxed and less rigorous way to make it more comfortable and useful for their students.

A striking example is that in all English speaking countries you can write \begin{equation} \mathit{i} = \sqrt{-1} \end{equation} while it is prohibited in the French system because the root square function's domain of definition is the positive real numbers only. As a former student of Maths, I wish I had studied in a more relaxed way and be allowed to use \begin{equation} \mathit{i} = \sqrt{-1} \end{equation} since that's what it is. For Riemann Integral for example, I don't see at all why I would prefer a rigorous definition like Definition 2 that is mostly due to Cauchy in the 19th century and is difficult to visualize or understand intuitively instead of just calculating the integral like Newton and Leibniz and all other mathematicians before Cauchy used to do and then once the student is comfortable with Integrals, we can then teach them that there are functions that are not Riemann Integrable and that there is a criterion to test for integrability which is the formulation in Definition 2 above.

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