The whole may be greater than the sum of the parts, but the whole still needs those parts. A reflective teacher will think carefully about when to concentrate on the whole, and when on the parts.

If you were teaching someone golf, you wouldn’t spend days on a driving range, never going out on a course. Your student would not get the idea of what the game is, or why they need to be able to drive straight and to a desired length. Nor would it be much fun! Similarly if the person only played games of golf it would be difficult for them to develop their game. Practice driving and putting is needed. A serious student of golf would also read and watch experts at golf.

Learning music is similar. Anyone who is serious about developing as a musician will spend a considerable amount of time developing their technique and their knowledge by practicing scales, chords and drills. But at the same time they need to be playing full pieces of music so that they feel the joy of what they are doing. As they play music, as opposed to drill, they will see how their less-interesting practice has helped them to develop their skills. However, as they practice a whole piece, they may well find a small part that is tripping them up, and focus for a while on that. If they play only the piece as a whole, it is not efficient use of time. A serious student of music will also listen to and watch great musicians, in order to develop their own understanding and knowledge.

In each of these examples we can see that there are aspects of working with the whole, and aspects of working with the parts. **Study of the whole** contributes perspective and meaning to study, and helps to tie things together. It helps to see where they have made progress. **Study of the parts** isolates areas of weakness, develops skills and saves time in practice, thus being more efficient.

It is very important for students to get an idea of the purpose of their study, and where they are going. For this reason I have written earlier about the need to see the end when starting out in a long procedure such as a regression or linear programming model.

It is also important to develop “statistical muscle memory” by repeating small parts of the exercise over and over until it is mastered. Practice helps people to learn what is general and what is specific in the different examples.

We are currently developing a section on probability as part of our learning materials. A fundamental understanding of probability and uncertainty are essential to a full understanding of inference. When we look at statistical evidence from data, we are holding it up against what we could reasonably expect to happen by chance, which involves a probability model. Probability lies in the more mathematical area of the study of statistics, and has some fun problem-solving aspects to it.

A popular exam question involves conditional probability. We like to use a table approach to this as it avoids many of the complications of terminology. I still remember my initial confusion over the counter-intuitive expression P(A|B) which means the probability that an object from subset B has the property of A. There are several places where students can come unstuck in Bayesian review, and the problems can take a long time. We can liken solving a conditional probability problem to a round of golf, or a long piece of music. So what we do in teaching is that first we take the students step by step through the whole problem. This includes working out what the words are saying, putting the known values into a table, calculating the unknown values in the table, and the using the table to answer the questions involving conditional probability.

Then we work on the individual steps, isolating them so that students can get sufficient practice to find out what is general and what is specific to different examples. As we do this we endeavour to provide variety such that students do not work out some heuristic based on the wording of the question, that actually stops them from understanding. An example of this is that if we use the same template each time, students will work out that the first number stated will go in a certain place in the table, and the second in another place etc. This is a short-term strategy that we need to protect them from in careful generation of questions.

As it turns out students should already have some of the necessary skills. When we review probability at the start of the unit, we get students to calculate probabilities from tables of values, including conditional probabilities. Then when they meet them again as part of the greater whole, there is a familiar ring.

Once the parts are mastered, the students can move on to a set of full questions, using each of the steps they have learned, and putting them back into the whole. Because they are fluent in the steps, it becomes more intuitive to put the whole back together, and when they meet something unusual they are better able to deal with it.

It is interesting to contemplate what “the whole” is, with regard to any subject. In operations research we used to begin our first class, like many first classes, talking about what management science/operations research is. It was a pretty passive sort of class, and I felt it didn’t help as first-year university students had little relevant knowledge to pin the ideas on. So we changed to an approach that put them straight into the action and taught several weeks of techniques first. We started with project management and taught critical path. Then we taught identifying fixed and variable costs and break-even analysis. The next week was discounting and analysis of financial projects. Then for a softer example we looked at multi-criteria decision-making, using MCDM. It tied back to the previous week by taking a different approach to a decision regarding a landfill. Then we introduced OR/MS, and the concept of mathematical modelling. By then we could give real examples of how mathematical models could be used to inform real world problems. It was helpful to go from the concrete to the abstract. This was a much more satisfactory approach.

So the point is not that you should always start with the whole and then do the parts and then go back to the whole. The point is that a teacher needs to think carefully about the relationship between the parts and the whole, and teach in a way that is most helpful.

## 4 Comments

[…] Parts and whole […]

There’s a subtlety here I may have missed. In sports coaching*, we distinguish ‘shaping’ and ‘chaining’ in the same kind of way that you speak of the whole/part dichotomy (and we often talk of whole-part-whole teaching almost synonymously). However, an important feature in motor skills training is that the initial ‘whole’ – the ‘shaping’ phase(s) – is often an intentionally crude and simplified approximation to a complete skilled performance. We then train elements of the skill or movement to provide increased detail or refinement. That is then incorporated into whole-performance exercises.

This also needs to do more than provide perspective; it incorporates transitions (hence, perhaps, ‘chaining’) from one detailed action to the next. In a sports context – and perhaps even more in a musical context – transitions are often critical and may even need to be practised in isolation themselves.

There is a question in this ramble: were you intending to provide a crude shaping overview with your ‘whole problem’ walkthrough or do you mean a complete, but fully detailed, first walkthrough? And (given that this is maths) is it even possible to provide an approximate ‘shaping’ framework that can be progressively refined, or do we have to move from a complete simple approximation to a _different_ complete rigorous solution?

*Among other things, I coach a sport when I’m not on the day job.

Hi

Thanks for your thoughtful comments. You can tell I’ve never coached a sport – know more about the piano. You brought up a couple of great insights. In the whole problem walk-through I would use a straightforward problem with the information given in the order you need to use it, with no extra info – at least for this particular example.

And the idea of transitions is interesting too. I’ll need to think some more about what that would mean in a probability/statistics lesson. You might like to read about liminality (see wikipedia!). I find the implcations to education interesting and one day hope to read enough to write a post about them.

Agreed. I’m trying to incorporate more of the “whole” early in my stats classes this year, starting with an experiment and simulation that asks the class to make a conclusion already on the first day (see simulation post by @gwaddellnvhs), and incorporating a decent sized project early in the year. Then we’ll hit specific skills hard in the context of earlier projects, and finally a big project at the end to put it all together.