Why are so many math books poorly written? Even many of the physical sciences books seem to have this terrible dichotomy between the text explaining the concepts, and the text explaining the calculations. I suspect it’s partly because one person is writing the conceptual text, and another person is writing the calculations text. I also suspect it is because both are written by people who are naturally good at the subject, just like most maths, chem, and physics teachers are naturally good at the subject.
Well, you do want people teaching who are good at the subject. But as many of us have noticed, being naturally good at something frequently results in people who cannot understand why others aren’t equally good at it. Once in a while those adepts become snobbish, because obviously the rest of the world just isn’t smart enough to get the stuff like they are. Many of the others simply have little patience with students who “must be stupid because they can’t figure out easy things” and can’t understand the material from having the previous explanation repeated again.
Duh! If it didn’t make sense the first time around, why would repeating the same explanation make any more sense the second or third time around? What we really need is a different kind of explanation. Or, several different kinds of explanations, because what makes sense to one person doesn’t always make sense to another.
I tutor students in secondary and tertiary maths and introductory chemistry. My students have various kinds of learning difficulties, and those standard textbook or math-teacher explanations often make very little sense to them. I have to create visual or life analogies for things to make sense. (Doing this often makes me understand the maths better; there’s nothing like teaching something to really understand it.)
My students, who are in their teen years all the way through middle age, are often rather bright, but they all have various learning difficulties, which means they do not easily learn things the way they are (traditionally) taught. For most of us, “the easy things are difficult, and the difficult things are easy” — once we can get past the simple calculations, we often find the more abstract stuff, like the concepts of calculus to be rather simple. This makes absolutely NO sense to those adepts out there, but if we followed their ideas about learning, we would never get to the more advanced stuff to even be able to show what we could do.
Frequently my students understand most of a process or a concept, but there’s one little piece of it, some basic concept that they do not understand clearly, and that little speed bump makes everything downstream very confusing. We break down the processes into small bits so I can see where their “speedbumps” are. I don’t want my students to just learn “tricks”; they have to understand what they are doing, and why. Otherwise they won’t be able to take the information and apply it to new situations or to different kinds of problems.
One of the things I run into in science is that the teachers will introduce new terms right in the middle of the descriptions of new concepts. This means that the students are not understanding a word, so they can’t get a grip on the concept, and by taking that detour to briefly define the word, they lose track of the concept. This means that they often end up understanding neither the term nor the concept very well. We really need to define all the terms first before diving into explanations of the concepts. We also need to start from something that the students already understand, preferably something from real life that they understand intuitively. That way we begin from a common place of understanding, and everyone is comfortable and thus open to learning and sharing new things.
A couple of things really annoy me about some textbooks. One is that they will only give just one example of how to solve something. This is not right, because we don’t want to know how to solve that particular problem, we want to know how to solve these particular sorts of problems. To do that, we need to be able to generalise, and you cannot generalise from a sample of one. We need at least two very different examples. My other complaint is when maths texts will give you an example of how to do one of the simple problems, but do not give you an example of how to do one of the more complex problems. Damnit, those are the ones where we are getting stuck! We need those multiple examples of different kinds of problems to be able to generalise.
There are several important points that I repeatedly stress to my maths students. I never heard these points until I was in Calculus, which I think is a real shame, because I found them to be positively liberating.
- Mathematics is not really about number-crunching. Arithmetic is about number-crunching. Mathematics is about the patterns and relationships of (countable or measurable) things, and how we can figure things out. This is one of the reasons why I so enjoy the television show Numb3rs; they describe a lot of the really cool things about mathematics that the ordinary person would have never known existed. They also use a lot of really cool visual analogies. You should watch this show!
- If you don’t like the way something is, then change it! One of the reasons why we learn so many ways of solving problems in the various levels of algebra is to give us big tool boxes of different tools. By having all those tools, we are able to keep re-arranging the statement until it is in a format that is easier for us to solve. With many problems, there is more than one way to solve it (many roads lead to Rome), so you can use whichever steps are easiest for you. What, you hate fractions? Then multiply both sides of the problem by the denominator(s) and poof! the fraction disappears. Or, all fractions are really another way of writing a division problem, so divide and turn the fraction into a decimal number.
- You have a natural mathematical ability. That is different from not being able to learn mathematics the way it is often taught.
Every student argues with me about having “natural mathematical ability”; everything in their personal experience has proven to them that they have grave problems with the maths. But I just remind them that Mathematics is not really about number-crunching; mathematics is about the patterns and relationships of (countable) things, and how we can figure things out. I will re-iterate that the human brain just loves to find patterns in things (and will even create patterns when they don’t exist, like seeing shapes in clouds or faces in objects). A lot of mathematics is about the patterns of things, and using those patterns to figure things out.
In factoring, we are looking for the evidence in the equation, those clues to what happened when someone multiplied previous pieces together. “Oh, I’m no good at that,” protests my student. I grin and ask, “What if you came home and saw cookie crumbs and Sharpie marker caps. What do you think happened?” Of course parents will leap to immediate conclusions, and so can other adults and teens. You are creating likely scenarios from the evidence given. We can do the same thing in factoring, but we have to learn what kinds of evidence is there, and what it means. People who are adept at the maths do this intuitively; my students need to have those things pointed out one piece at a time, with several examples.
They need to have them listed for them on a piece of paper, so they can instead focus on listening, looking, and understanding, rather than on taking notes, and so they will have it available for them later on. I do this note-taking form them because I am organising what information they need to know later on.
I also use a lot of color to help group categories of information, or to mark which elements are the same from one step to another. For example, in algebra when we are learning to group similar elements, I will color-code things. (I first check to see if the student has any color-blindness issues; there’s no point in using both blue and green if they cannot tell the two apart!) All of the X-cubed things are one color, all of the X-squared are another color, all of the X are a third color, and all of the constants (numbers without Xs) are yet a different color. Then it is plain to see that we have different “breeds” of X and non-X items, and which of those can be added together, and which cannot. You could even say that when we are counting how many apples we have, we are not including the numbers of bananas or oranges in with them.
I am going to put some of my explanations for various math and chem concepts on a permanent page, and add to it as I remember them, meaning, it will be periodically expanded. If you have particular things that you would like to have explained, or different ways of solving particular kinds of problems, then please ask!