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!

## Cilla said,

20 May 2008 at 21:01

Andrea, you are spot on again! Terrific entry.

## Norah said,

17 May 2008 at 8:34

Andrea said:

“There can be:

Temporal issues (keeping track of time, including sequences of events, and order of operations);

Spatial orientation issues (can be related to geographic agnosia), or depth of field problems, or reading and transcription issues (reversals and/or substitutions, like what are sometimes seen in dyslexia), personal spatial awareness for gross and/or fine motor coordination including writing problems or playing an instrument or sports or dance;

Problems related to working memory, such as keeping score in a game, or not being able to do much mental math (e.g. the student can either recall a multiplication fact or think about what step is next in long division, but cannot remember that 7*8 = 56 and then what they were going to do next);

Difficulties with transferring concepts and processes from working to short term to long term memory (e.g. knowing how to do something on today’s worksheet but not the next day), or being able to understand how the smaller parts relate to the whole;

and of course, combinations of these things!”

I have all of those problems but one (the third in the list).

Makes me wish I had you as my teacher back in school. I was already falling behind in maths during primary school, and in secondary school (compare to highschool), it was pretty much over after the first year. Luckily, I managed to drop maths, physics and chemistry, or I would have never made it through school. I had so many summer schools and tutors it’s not funny, but nothing helped.

I recall explaining to friends how I used to do multiplications and divisions, the only way I could do them and not get completely lost (I ensure you my way took a very, very long time, but at least then I could do them fairly accurately), they were pretty stunned :D. I don’t think up until that point they’d really believed me when I said I couldn’t do this-or-that calculation, even with a calculator (because I just enter the numbers in the wrong order).

Maybe odd, but the only thing I have always been able to do almost faultlessly, is algebra. We only had it for the first two years in secondary school, but it used to always up my grade average for the entire maths class :D. Not that I really understood what I was doing, but I just did it.

## Igor Santos said,

15 May 2008 at 19:58

Hey there Andrea!

Thanks for the visit!

You really didn’t have to go through all that trouble just to leave me a message, English would’ve been fine. :)

I’ll link to your blog, okay?

And remember, 42 is the answer!

## Proso - pague - nó - zia « 42. said,

15 May 2008 at 10:42

[…] uma propagandazinha e ajudar todo mundo a ser lido. Quando eu a abri, uma das ligações lia “Maths * Chem = Ranting^2”, o que em bom português significa “Matemática multiplicada por Química é igual a […]

## icsouza said,

13 May 2008 at 5:08

I admired always Shakuntala Devi as a mathematical wizard. She has been helping students in fighting ‘mathephobia’. She gives all details about mathematical data. I have heard a lot of abacus and computing skills. Vedic Mathematics is drawing attention from many math-lovers, but also criticism from some quarters.

What is your opinion?

Mathematics is a beautiful subject, it is a good training for mind. Latin, mathematics and logic go to train the mind for life pursuit. It is a pity that some people do hate numbers. We have to find pleasure with numbers, patterns, symbols, problems and riddles….

## qw88nb88 said,

12 May 2008 at 22:48

Norman,

I like your idea of mathematical statements as symbolic shorthand like texting; I think i’ll borrow that one from you, in return!

andrea

## Norman Parron said,

12 May 2008 at 21:10

Great post and I like your color code idea. Which I am going to try on my students in Sept. I’ve seen the same problems with the fear of math.

Many of my students use text messaging so I show how Math is really a kind of short hand for a long sentence similar to texting. As you said there are many paths to the solutions. Getting the student past their imagined fears and defficults is the real trick.

## icsouza said,

12 May 2008 at 16:52

Andrea,

I am with you. It is an art to make mathematics easy to understand and to perform. It is not enough to understand. You have to be able to repeat it, solve problems, memorize formulae, remember the full processes, know exactly which process to follow, which solution to provide. You have to feel ease in all these operations…

## qw88nb88 said,

12 May 2008 at 14:02

cormac,

I really like the idea of the interactive instruction; I think that’s very insightful!

icsouza,

It’s not so much about making “mathematics easy” as it is about making “mathematics understandable”. Most anyone can do mathematics, but not everyone can learn it in the same way or the same speed that it is often taught.

andrea

## icsouza said,

12 May 2008 at 13:41

I found interesting that you are devising methods so as to make mathematics easy. Some people harp on the “ability to counting”. But mathematics is much more. An intelligent person should be able to progress in mathematical sciences.

Today all sciences are reduced to some form of mathematics. Reasoning is mathematical and becomes symbolic logic. I liked mathematics, though cannot say that I have a special flare for mathematics. Geometry was more difficult for me, though sometimes it is a question of reducing to a similar pattern. I always wanted to know what is calculus, integral and differential. It is good to progress gradually, instead of finishing the whole mathematics in two years, as we used to do.

I hope to get more help in remaining with mathematical skills, even when mathematics is not the subject to study or to teach.

## cormac said,

12 May 2008 at 9:52

Interesting thread – I inherited a maths class at 1st year college level a while ago, and was horrified at how they had been taught. What seems to work for me (us) in calculus, is that each student provides the next line, as Ibest they’re able, as I go through a calc problem . I think the key is that they find it so much easier to concentrate, like the difference between watching football on tv and joining inthe game….Cormacis

## bluewaveted said,

12 May 2008 at 8:24

I have to say, I’m impressed with your insight. I’ve had trouble with math back in high school, and never really had anyone to explain things to me. They seem to think that the next step is natural, while I’m stuck because they didn’t explain part of it!

## qw88nb88 said,

11 May 2008 at 18:10

Marlo,

ZPM is a fictional gizmo.

## qw88nb88 said,

11 May 2008 at 18:00

Kate,

Without being able to sit down with your son and go through things together, it’s a bit difficult to make specific recommendations.

However, this previous post, “Dividing We Stand” may give you some ideas. It’s about long division instead of algebra, but many of the concepts will transfer.

Also, one thing I have found to be rather consistent is that we have little bits of confusion in our arithmetic that trip us up in algebra (just as we later have little bits of confusion in our algebra that trip us up in calculus). Being able to identify what those are and getting them ironed out also helps a great deal.

Sometimes the format of the homework or tests can make a difference as well: “Testing, 1, 2, 3 …”

EVERYONE:

It’s a long shot, but it might be that the two of us are geographically close for tutoring. You can contact me by sending a message to andreasbuzzing care of my gmail account.

andrea

## Kate said,

11 May 2008 at 16:47

Wow!

This is absolutely the best post I have read that describes my sons’ struggle with mathematics. He is taking (failing) algebra and has decided he hates it and his teacher : ( Summer school here he comes. His teacher observed that he seems to get stuck on problems that have more than five steps. If you have any recommendations to help with this, I would really appreciate it. Also, I just want to add another kudos for your observation about examples. My son is allowed to use his algebra book when taking tests. He is very frustrated because he can’t make the leap from an example in the book to the test queston.

Thanks so much! I just might send this to his algebra teacher.

## qw88nb88 said,

11 May 2008 at 15:54

Norah asked,

I have. Dyscalculia can be expressed in different aspects, so part of the process if figuring out in what ways the difficulties are presenting, and then finding ways of working around them.

There can be:

Temporal issues (keeping track of time, including sequences of events, and order of operations);

Spatial orientation issues (can be related to geographic agnosia), or depth of field problems, or reading and transcription issues (reversals and/or substitutions, like what are sometimes seen in dyslexia), personal spatial awareness for gross and/or fine motor coordination including writing problems or playing an instrument or sports or dance;

Problems related to working memory, such as keeping score in a game, or not being able to do much mental math (e.g. the student can either recall a multiplication fact or think about what step is next in long division, but cannot remember that 7*8 = 56

andthen what they were going to do next);Difficulties with transferring concepts and processes from working to short term to long term memory (e.g. knowing how to do something on today’s worksheet but not the next day), or being able to understand how the smaller parts relate to the whole;

and of course, combinations of these things!

Because of all these issues, it is difficult for the dyscalculic student to follow the usual math instruction, because it moves too fast, and they cannot understand what’s going on, much less retain enough information or keep things in order. Frequently they have a multiple gaps in knowing the terms for parts or processes, as well as incomplete or erratic recall of various arithmetic facts.

Sometimes students will be diagnosed with Non-verbal Learning Disorder (NLD or NVLD) instead of dyscalculia. By my understanding, NVLD is similar to Asperger’s, but without the math-geek aspect, and a bit less difficulty with social interactions. (There are endless debates about whether it’s a “real” or different diagnosis, and what it means, et cetera. NVLD is not in the DSM, but it’s “real” in the sense that you can create almost any set of descriptions and find some people who meet them.)

People with dyscalculia are not necessarily “stupid” and can be especially gifted in language or art (art schools are known for having numbers of students who are “math-impaired”). Dyscalculia makes life just as difficult as dyslexia (one could have both), but is less-well known. Both create such problems in school that those affected are often perceived as not being intelligent. This is damaging not only for one’s self-esteem, but also socially. Some kinds will act out (it’s both easier and more acceptable with your peers to be “bad” than “dumb”) and many will withdraw, either literally or mentally.

There are two VERY important things for working with dyscalculic students.

One of the biggest problems for dyscalculic students is the whole overwhelming and angst-ridden history with the subject; by the time they reach junior high/middle school, even the very idea of maths is enough to make one ill. We have to work past this and get comfortable with each other before we can make much headway on the subject. Sometimes we begin by going outside and chatting, just to break away from the whole gut-wrenching sitting-at-the-table-with-a-math book scenario.

The second thing is to identify what strengths they do have in the maths (those fragment skills) so we can build upon them. Because dyscalculic students feel so overwhelmed, and because all of these skills are so inter-linked, it is hard for them to know that they even

havestrengths, much less what those might be.andrea

## marlo tortosa said,

11 May 2008 at 12:49

could you explain to me what thay now abount the Zero Point Power (ZPP) or what we call the Zero Point Module. I always came across this words on some movies that, It can help alot for this time.

## Norah said,

11 May 2008 at 12:49

Do you teach any students with dyscalculia?

## Diane J Standiford said,

11 May 2008 at 11:49

Boy, do I wish I’d had YOU as my math/chemistry/algebra/geometry teacher! I got straight A’s in English, Drama…but math made me tune out, and I wanted SO badly to know it. Just as you stated, a concept or new word was suddenly introduced out of nowhere and there I went, all I heard after that was blah blah blah. Now, at 51, I see high school grads who can’t do any simple math without their calculator or computer. Seriously, don’t give up on this issue…the USA needs your help!