This is the second part in a series of articles that describes the problem solving rules that my son and I use to help him tackle his Maths homework, and shows how these “Rules of Maths” can be generalized to grown-up sized problems.
The first article covered Rules 1-3 (and a half)!
Rule 0: Make Space
I forgot to mention this rule it in the last article, but it is important nonetheless:
I always start Jonathan’s homework sessions by creating a suitable physical space for us to work.
For us, the dining table is the best place for homework, so I try to remove anything from the table that’ll distract us from the homework. At the same time, I try to make enough space for him to spread out any materials he needs. It helps that we keep a “homework pack” full of stationary and paper for rough notes etc. in the dining room cupboard.
As well as eliminating environmental distractions, I try to help Jonathan eliminate internal distractions. He doesn’t work well if he’s restless or tired or hungry or thirsty or busting for the toilet, so I try to address these things by choosing a suitable time for him to do his work and ensuring that his physical needs are met.
The lesson for me is that, when I need to do some thinking, it will help if I can create space for my thinking. Finding a quiet, distraction-free space in a bustling workplace or a family home can be challenging, so I might have to get creative. Perhaps I could take myself off for a walk, or use headphones to block out the noise. In the past I’ve even resorted to locking myself in the bathroom to get some quiet! Perhaps one day I’ll get that study with the special “thinking chair” that I dream of…
Sometimes the distractions in my physical environment are minor compared to the ones in my head. It is often the mental clutter that I fund most difficult to shift. But I’m not going to dwell on any of this just now… I’ll mention it again under some of other rules.
Finally, it is good to be consistent about the place I choose as my thinking place. As I mentioned, we always sit at the dining table for Jonathan’s homework. As a result, Jonathan associates the place with the need to concentrate and get on with his work. In other words, the place helps create the right mood. The same is true of my thinking places.
Rule 0.5: Enlist Help
I missed this rule, too: we encourage Jonathan to Enlist Help with his homework.
One of the most important things in Jonathan’s workspace is… me. I don’t expect Jonathan to get on with his homework on his own. Rather, I’m there to scaffold his learning experience, to help him if he gets stuck or goes off on the wrong track.
Situations where you have to figure things out completely alone are less common in life than they area at school. The education system often emphases independent work over collaborative effort. Exams are almost always independent, as are other assessments. Many pieces of work are supposed to be done by children working alone. When I was at school, talking during many lessons was strongly discouraged. There are good reasons for this emphasis. But on the whole, real-life isn’t like that. In most situations, collaboration is more important than lone working.
In most situations, there are people around us who can help us figure out the answers to problems. I’m not suggesting we should get other people to do our thinking for us, but having people to bounce ideas off can be very helpful. (And, in my experience, I can always enlist God’s help, even if there is nobody else around to aid my thinking).
Rule 4: Write Out the Question
We’re back on track with the rule-numbering, as rules 1-3 (and a half) were in the previous article.
Whenever we start working on a maths problem, one of the first things I do is remind Jonathan of our 4th problem solving rule. I get him to Write Out the Question.
The first benefit of this is that it ensures that he has actually read the question.
If you’ve ever taken an exam, you were probably given this sage advice, “start by reading the question”. Whenever someone’s said that to me, I have invariably thought this was obvious and unnecessary counsel. Nevertheless, there have been times when this is exactly what I’ve failed to do! By encouraging Jonathan to take a moment to jot down the question in his answer book I can make sure he avoids making the same mistake. Hopefully, then, when Jonathan writes something like “6 x 4 =”, he’ll notice that he’s supposed to be multiplying and not adding or dividing, and proceed accordingly.
Now, school maths questions are usually quite clear cut: there are only so many ways of interpreting “6 x 4 =”. Adult questions are sometimes more complex. Indeed, I sometimes mistake the nature of problems completely. For example, at work I might try to provide a technical solution to a problem that is really a management or political issue. At home, I might try to find intellectual solutions to situations that are really emotional, practical or spiritual challenges. Obviously, then, taking time to understand a problem is an essential step to solving it. And I find that jotting down a brief description of the problem is a great way to get a handle on the nature of a problem.
The second benefit of writing out the question is that it ensures that Jonathan is able to extract the relevant question from its context, and assemble it all in one place.
Most of the exercises he’s given have a lot of addition information that isn’t relevant to the question that he’s working on, all of which is a distraction from the question itself. For example, they might include a title, the assessment criteria, a little scenario, some pretty pictures, examples, submission date for the work. Then there’s all the other questions on the sheet, the question numbers and the fact that the question is often on a different page in his book from where the answer needs to be written. When I get Jonathan jot down the question, it helps him focus on what he’s supposed to be working on.
It is the same for me, except that once again it is often the mental clutter that gets in my way. When I jot down a question, it helps me eliminate all the junk and focus on what is really important.
Rule 5: Think out Loud
Sometimes, when I’m working with Jonathan, he just sits there, staring into space. I can’t always tell if he’s just lost concentration, or if he’s trying to figure something out. Other times, he suggests an answer to a problem that I know is wrong, but I can’t tell if that’s because he’s made a mistake with an appropriate strategy, or because he’s used the wrong strategy, or because he’s broken Rule 2 and just guessed. The obvious way to overcome this problem is to remind Jonathan of Rule 5: Think Out Loud.
I’ve already mentioned the value of enlisting others in my own problem solving (Rule 0.5). Often times, it isn’t enough just to explain your problem to other people. Rather, I have to help others to help me by explaining what progress I’ve made and how I’ve got stuck. Only then can they offer appropriate insights into my problems and guide me to finding solutions that I would never have found on my own.
Curiously, however, thinking out loud can work well even when there is nobody actually listening. Many people have experienced a situation where the answer to a complex problem comes to them as they’re explaining the problem to someone else, even though their partner has said nothing. Indeed, this phenomena is so common in the programming world that it has given rise to terms like “Cardboard Programmer” and Rubber Duck Debugging.
In the absence of someone to explain things to, writing out problems or even explaining them to inanimate objects can be a valuable technique in finding solutions to my problems. (Although I’m persuaded that divine guidance is genuinely available for those who seek it, I don’t doubt that prayer works this way, too, at times).
That’s All for Now
That’s all I’ve got time for today, but there are more rules to come. Stay tuned!