Arithmetic for Human Beings: An Anti-Textbook for People Who Loathe Arithmetic by Robert Froman.
I read this book in two sittings and recommend it to everyone, even those who *don't* loathe arithimetic. It was preaching to the choir to me, because it described what I did day in and day out in my CPA career, and what I do now in my everyday life with math. People say I am good with math, but what I really am good at doing is knowing how to judge how much math I need to use given an individual set of circumstances. It is this skill the author believes is not being taught to kids today.
The author has an excellent point that the way arithmetic is typically taught in school, most people do not learn how to use math unless it is modeled for them, or they figure it out on their own. The closest I've seen an author describe this is Marilyn Burns' Math: Facing an American Phobia (her "Thanksgiving dinner" example in particular), but I think this author does a better job of addressing the problem.
I found the book searching on the author, whose contributions to the Young Math series I was really impressed with. In particular, Froman's "A Game Of Functions " introduces functions to children and is really well written. He also wrote a very accessible book on topology for children. I've gleaned many ideas on how to introduce what I would have thought of as more complex math ideas to younger children through these books.
This book is called an "anti-textbook" because Froman demonstrates how blindly applying textbook arithmetic to math problems we encounter in real life will often actually produce the *wrong* answer. To demonstrate, he spends a whole chapter on applications of multiplying 979 by 743. There are many different situations wherein we might need to do this in real life - and in not one of these situation is it helpful at all to calculate the exact answer. In some situations, thinking "It's a little less than a quarter of a million" is the right answer, if, say, you are trying to rack up a million flying miles and want to know how far you have to go. A more exact answer only muddies the picture and keeps you from understanding that you are about 3/4s of the way there and have 1/4 to go.
Froman brings up another real life situation involving a business decision regarding whether or not to buy a gadget based on a buy and sell price, and sales projections. He makes the excellent point that in situations like this, exact calculations can actually do more harm than good. As he puts it, the time and effort to make exact calculations imbues in the calculations themselves an incorrect meaning. The fact is, the answer itself is an estimate, all you are trying to do in the given situation is determine whether it is good business to make the purchase. By calculating your risk of potential loss using exact dollars and cents, it implies this is an exact loss, when in fact, the entire situation is tentative.
Froman says you *should* use tentative numbers if the situation is tentative - that diving into exact calculations actually short circuits your brain and keeps you from using your intuition. In the end, he stops using arithmetic when he gets his answer - in this scenario, that the risk is worth taking, because even if the businessman cannot sell the at-risk inventory and takes a total loss on it, he will still make an overall profit. He does not calculate the exact profit because a) that wasn't the question and b)it is too tentative to bother doing the math, it won't be the right answer.
The entire time that I was reading this I was remembering my personal experience in business, and how in every business situation I was in, my ability to *not* resort to exact calculation and apply only the arithmetic that was appropriate to each situation was key to success. Employees or professionals who could not do this were let go, or sent to a department where their inability to let go of arithmetic details would not be a liability.
Froman makes the point that exact and accurate calculations are very important for higher math, and there are some (infrequent) situations in life that an exact calculation is necessary. But his problem is that young children are taught to apply math this way many years before they will need it for higher level math, and in reality many may never need it. All of us need practical math, which is highly reliant on these skills that are not in the curriculum.
The book was written in the 70's and reflects the education reform initiatives of that age, but these are not that different from today's issues. I had to laugh at some of the things he pokes fun at, he's right, it's much ado about nothing. I thought about a well-respected curriculum like Singapore. It teaches children to round and estimate numbers - but nearly always this is taught in the context of giving you a ballpark figure that you will compare your exact calculation to later, to check the reasonableness of your answer. Or it's just taught as skill to learn without context. While estimating to check an exact answer is a very important skill that I also used constantly in business and higher math, he's right that many times we are simply done when we do the estimate - this is rewarded as the "right" answer in and of itself. But how many children's books would give real life word problems and consider an estimate or rounded number a correct answer, unless the chapter was *on* rounding? Answer keys imply that every problem presented, even an estimation or rounding problem, has one exactly right answer.
The chapter on adding brought up the same kinds of issues I again ran into in business. You may have a long column of figures to add. Just attacking the data and rotely entering it into a calculator or even a spreadsheet is prone to error. Froman talks about various ways to estimate the value of the column of numbers, again with the knowledge that you may or may not even have to get more exact than that. You're at the grocery store and realize while you are heading to the register you forgot your wallet and only have a $20 - do you have enough to pay for your groceries?
Questions about whether you think you have enough in your bank account to cover a check you need to write - do you have to balance your account to the penny to answer this question? Usually, no. Froman resists the urge to call this mental math. He doesn't care if you use charts, count on fingers, write things down. The point is, do the math you need to for the given situation, and don't fall into the trap of blindly applying algorithms just because a textbook taught you do do so, if you are really looking for a different answer than what that method will produce.
There is a chapter on multiplication, addition, subtraction, division, fractions, negative numbers, estimating, and probability. Every situation is a realistic situation I've encountered or could encounter in real life, business, etc. A very worthwhile read.