**A word problem from the Khan Academy**

Last year, I was asked by my American colleagues to give my assessment of mathematical material on the Khan Academy website. Among other things I looked for the so-called “word problems” and clicked on a link leading to what was called there an “average word problems” but happened to be a “word problem about averages”. It appears that the problem there changed since last year. I reproduce here the old one, it is more of interest for discussion of “word problems”.

Gulnar has an average score of \( 87\) after \( 6\) tests. What does Gulnar need to get on the next test to finish with an average of \( 78\) on all \( 7\) tests?

**Hints given by the Khan Academy**

What follows are hints as were given, one after another, by the Khan Academy website:

Hint 1.Since the average scores of the first \( 6\) tests is \( 87\), the sum of the scores of the first \( 6\) tests is \( 6\times 87=522\).

Hint 2. If Gulnar gets a score of \(x\) on the \(7\)th test, then the average score on all \(7\) tests will be:\[

\frac{522+x}{7}

\]

Hint 3.This average needs to be equal to \(78\) so:

\[

\frac{522+x}{7} = 78

\]

Hint 4.\(x=24\)

**“Questions” method for word problems**

And here is the same problem solved by “steps” or “questions” method. (As a child, I learnt it in a direct face-to-face communication with a live teacher and with my peers.)

Question 1.How many points in total did Gulnar get in \(6\) tests?

Answer:\(6 \times 87=522\)

Question 2.How many points in total does Gulnar need to get in \(7\) tests?

Answer:\(7\times 78 = 546\)

Question 3.How many points does Gulnar need to get in the \(7\)th test?

Answer:\(546 – 522 = 24\)

Crucially, the whole point of the “questions” method is that students have to formulate these questions *themselves*.

**Self-guiding questions**

Actually, a student has to be taught to start his/her ”questions method” attempt at a word problems asking* himself or herself* appropriate *self-guiding questions*. In this case, these self-guiding questions are likely to be something like

Question A.“Gulnar has an average score of \(87\) after \(6\) tests.”What questions can be asked about these data?

Question B. “Gulnar needs to get an average of \(78\) on all \(7\) tests”.What questions can be asked about these data?

Therefore the use of “questions” method in mathematics education involves gently nudging a child towards *reflection* and analysis of his/her own thought process – at the level, needs to be emphasised, actually accessible to the child, and it was confirmed by mathematics education practice of dozens of countries around the world. This is why I prefer the term “questions method” to the more commonly used, in British education literature, words “steps method”; the word “questions” emphasises the pro-active and reflective components of thinking, while the word “steps” might inadvertently imply a passive algorithmic approach.

**A comparison between the two approaches**

However, it is instructive to compare the two approaches in more detail.

The Khan Academy solution is longer than the “questions method” solution because Hints 2 and 3 are already equivalent to a solution of a 3-step problem (you need to express algebraically the new total, compute the new mean and compare with the desired result).

The “step” or “questions” method of solving the problem directly encourages students to look at the core of the concept of average, it invites them to re-visit previous material and solve the problem of finding the total from the average. It encourages them to *talk* about mathematics, with themselves and with others.

In more technical terms, the “questions” method triggers the all-important dynamics of encapsulation / de-encapsulation of the concept of “average” (and we have to remember that this is a pretty abstract concept – this is understood even in the mass culture, in expressions such as “*2point4 children*“). The terms “encapsulation” and “de-encapsulation” are less frequently used, and perhaps a few words of clarification may be useful; I quote Weller et al.* Intimations of infinity*, Notices AMS 51 no. 7 (2004) 741-750:

The encapsulation and de-encapsulation of processes in order to perform actions is a common experience in mathematical thinking. For example, one might wish to add two functions \(f\) and \(g\) to obtain a new function \(f+g\). Thinking about doing this requires that the two original functions and the resulting function are conceived as objects. The transformation is imagined by de-encapsulating back to the two underlying processes and coordinating them by thinking about all of the elements \(x\) of the domain and all of the individual transformations \(f(x)\) and \(g(x)\) at one time so as to obtain, by adding, the new process, which consists of transforming each \(x\) to \(f(x)+g(x)\). This new process is then encapsulated to obtain the new function \(f+g\).

Mathematical concepts are shaped and developed in a child’s mind in a recurrent process of encapsulation and de-encapsulation. Returning to solution of Gulnar’s problem by “questions” method, we see this process in action.

And what is even more important, self-guiding questions are *meta-questions*, that is, questions aimed at finding the optimal way of reasoning.

From a basic pedagogical point of view, if the didactic aim of the problem is to reinforce the understanding of averages, then the “questions” method appears to be more useful; it gives a student a joint and cohesive vision of mathematics. And the “questions” method is more difficult to implement in a computer based medium – by its nature, it requires live interaction between teacher and students, and between students.

But I claim much more: the “questions” method can act as a trigger of the fundamental recurrent cycle of mathematical thinking: *intentional and purposeful* encapsulation / de-encapsulation of abstract entities. Like breathing and heartbeat in a living body, the encapsulation / de-encapsulation cycle makes the difference between live mathematics and zombie mathematics. And, the last but not least, it is the very essence of thinking in computer science and computer programming.

**Some general comments**

The flaw in the Khan Academy hints is something that can be observed in the current English education practice as well: instead of giving to students proper didactic support, hints are frequently just pieces cut off a ready solution of the problem.

But, in my opinion, mathematics cannot be learned as a mass of disjoint bits and pieces. Mathematics is not a sum of facts; learning mathematics does not mean memorising facts, it means gaining understanding of connections between mathematical facts, and, moreover, of connections between connections and analogies between analogies. Word problems and “questions” method allow to do that in natural human language exploiting its in-built powerful logical facilities (the latter otherwise manifest themselves, in a child’s study of mother tongue itself, as its *grammar*).

Giving a student a ready piece of a solution instead of gently nudging him in the direction of a solution is, of course, a very precise form of communication. But it proves only that the teacher can solve the problem, it does not teach a student to solve unfamiliar problems independently.

An effective mathematics teacher should be a *diagnostician* and *communicator*: he or she should *see* student’s error or difficulty, *understand* its underlying causes, *talk* to the student using an accessible mathematical language. Moreover, the teacher should *relate*, both via empathy so important for a diagnostician, and via non-verbal hints (an approving gesture can worth a hundred words when a student needs a lightest possible hint that he/she is on right track to a solution of the problem).