Nine Ways of Mathematical Thinking
1. NUMBER SENSE
When we hear or read information about population growth, about the national dept, about the effect of cutting back one million dollars from the budget for Education, we need to be able to deal with the very large numbers.
A.K Dewdney in his book, 200% of Nothing, tells this wonderful story. A public radio station in Toronto was having a fund drive. An announcer asked over the air for ideas to help raise donations. A caller suggested “Why not ask people to pledge a year’s worth of pennies? On the first day, they would pledge one cent.,on the second day, two cents, on the third day, four cents,, and so on. Each day they would just double the previous day’s pledge until a year has gone by. Then they mail in a check for the total.” (156)
Those with good number sense would be aware of the results of repeated doubling. The announcer was not. He made the suggestion on the air. “Unfortunately, the amount of money that hapless donors must mail in at the end of the year would exceed all the pennies in Canada, all the money in the world. Even if a listener had $1,000. for every atom in the know universe, it would not even come close to the final amount owed.”(156)
I often asked my students what they thought an average allowance was. Then I asked if they would prefer to get an allowance of $20 a week or have their parents give them one penny on the first day of the month and double it each day. Most preferred the $20 a week rather than just pennies. If some preferred pennies, I’d try $50 a week, or even $100 a week. Nearly every student preferred that. In one month they would have $400. It’s a lot of money
Some students started doing the math. 1+2+4+8+16+32+64 = $1.27 and see that the amount is growing but very slowly. Even after doing the math for the second week they weren’t too impressed. But on the 30th day, their parents would owe them $5,368,709.12. Their total for the month would be twice that — over 10 million dollars. The numbers would of course, double again if there were 31 days in the month. I could see immediately that a few students had decided to try it with their parents. I warned them that if they tried it that they absolutely MUST stop the game as soon as the parents had given them their usual allowance. They certainly couldn’t expect their parents to hand over millions of dollars because they didn’t understand doubling problems.
In addition to having problems with large numbers, most of us have difficulty with small numbers. If a substance is a danger to your health at 54 parts per million, 54 parts per billion also seems very dangerous. But 54 parts per million = 0.000054 and 54 parts per billion = 0.000000054 or only one thousandth of the earlier number.
Number sense with percentages includes understanding that 150% of a number is not the same as 150% more than a number. 150% means one and a half. 150% of 10 = 15. 150% more than ten is 10+15 = 25. I often wonder about the statistics mentioned on television. I really think they say 150% more than when they really mean 150% of something. For example, if Joe earned $40,000 last year, and he now earns 150% OF that.
100% of 40,000 is 40,000. 50% of 40,000 is 20,000. He’d now be earning $60,000 which is, of course more than he made before. But if he really made 150% MORE THAN he made before you would take what he earned before: 40,000 and add 150% of that or 60,000 and he would now be making $100,000. There is a big difference between the two.
Number sense with fractions means being able to compare fractions. Children often believe that 1/8 is larger than 1/4 because 8 is more than 4. You should know it’s the other way around. You should be able to do a lot of problems in your head: You know that 1/2 of 8 is 4. Therefore 1/2 of 8/11 should be obvious. It is 4/11. 1/5 of 10/15 is 2/15. If this isn’t obvious, keep thinking about it.
You might cut a paper shape into 11 more or less equal parts. Mark each one 1/11. Then pick up 8 pieces. You now have 8/11. Half of 8/11 should be 4 pieces or 4/11. If you are still puzzled, get a student who really understands fractions to help you understand it.
My son showed a strong number sense when, as a first grader, he pointed to the church windows (with 5 panes across and 8 panes high) and said, “Mom, did you know that 5 times 8 and 8 time 5 have the same answer?” I asked how he knew that. He had grouped the panes vertically and then horizontally and didn’t need to count the total because he could see they were the same. That is number sense that many adults can’t grasp. And Tony, in first grade, had not begun learning about multiplication.
2. Spatial Sense
Spatial Sense helps us understand a drawing or model that is done “to scale.” In a typical dollhouse, for example, the scale is 1:12 or one inch to one foot. If a house is 32 feet high, it’s scale model would be 32 inches high. If broom is 4.5 feet long, the dollhouse broom would be 4.5 inches long.
A person with Spatial Sense can estimate distance, area, and volume fairly well. She could take a simple drawing of a whale, read the average length of that species and be able to draw a full-sized whale on the school parking lot and come fairly close to the right length.
A person with Spatial Sense understands that, with several containers of different shapes, having the water level at the same height does not mean the volume is the same in each container. Young children often believe that a skinny glass filled with one cup of milk contains more than a wide glass where the one cup of milk leaves the glass only two-thirds full.
3. Problem Solving
Problem Solving here does not include difficult calculations. It involves understanding what information is being asked for, what information is given, and how to use the given information to reach the answer.
One of my favorite problems says there is a farmer who raises only pigs and chickens. He has two visitors We might call them Jack and Jill. Jack counts 18 animal heads. Jill counts 50 animal legs. (No tricks: we are not counting people.) How many pigs and how many chickens does the farmer have.
Students must first realize there is some missing information. Students who calculate without thinking take the numbers 18 nd 50 and either add, subtract, multiply, or occasionally divide. When they get an answer they can’t tell you if they think this is the number of pigs or the number of chickens. It’s just their “answer.”
Students who think before calculating are puzzled about the “number of legs” and soon realize it is important to know that pigs have four legs and chickens have two. And the number of heads? Pigs and chicken have one head each. Now you should be able to solve the problem.
With elementary students, I used smaller numbers and we arranged the numbers on a chart. We started by listing possibilities for chickens 0, 1, 2, …18 in the first column. In the next column we could list the number of chicken legs. In the third column, We could list the number of pigs (the total 18 – number of chickens). In the fourth column, we place the number of pig legs. In the last column we list total legs (chicken legs + pig legs). When it equals 50, we have solved the problem.
Older students sometimes prefer trial and error. The might guess 9 chicken and 9 pigs, count the legs and then guess again up or down until they find it.
High school students often use algebra : P + C = 18 and 4P + 2C = 50. Nowhere in our curriculum are students taught how to approach this problem. They must use Mathematical Reasoning and Problem Solving.
4. Understand Measurement
Measurement starts with a knowledge of different units of measure including 12 inches in a foot, 3 feet in a yard, 16 ounces in a pound, 4 cups in a quart and many more. It also requires an understanding of metric measurement: 1000 millimeters in a meter, 100 centimeters in a meter, 1000 meters in a kilometer, etc.
Beyond that, you must understand shich numbers are always exact (like 8 people, or 93 candy bars) and which are measured numbers (like 5000 feet, 100 yards, 4 ounces, 60 miles per hour, etc.) and that measured numbers are NEVER which are never exact. You must understand how this affects the answer.
If a bus will hold exactly 20 people, how many buses do you need for 90 people. You cannot answer 4 and a half buses. Half buses do not exist. You need 5 buses.
On the other hand 25 cakes for 20 people does equal one and one-fourth cake per person. Fractional parts of a cake do exist. We should realize, however, that your piece might be slightly larger or smaller than my piece because measurement is never absolutely precise.
5. Mathematical Communication
This involves understanding the problem clearly and being able to explain your reasoning in writing or verbally. Students today are often graded, at least in part, on how clearly they can explain how they solved a problem, including both their procedure and their reasoning. In each of the examples I have used, I try to explain my reasoning as clearly as possible.
6. Mathematical Reasoning
Some people have difficulty explaining their reasoning. If asked to estimate how many quarts of paint are needed to paint a room that is 20 feet long, 10 feet wide, and 8 feet high, you might reason that you need to calculate the area of each of the 4 walls, but that opposite walls have equal area.
Thus you have 20 x 8 (length x height)= 160 square feet each for the two long walls . I hope you weren’t planning to use length x width… that would give you the area of the floor, and usually also the ceiling. But we are painting the walls.
10 x 8 = 80 square feet for each of the two short walls.
Twice 160 =320 square feet. Twice 80 = 160 square feet Add these for the total area for 480 square feet.
The paint can says one quart covers 100 square feet. We can therefore assume we need 5 quarts. But, if there are three doors that won’t be painted and 5 large windows, we could subtract their area and possibly manage with only 4 quarts.
This is using mathematical reasoning.
7. Understanding Statistics
We have shown several examples of understanding statistics. You would consider the ad that says “Four out of five dentists prefer…” and wonder if they only spoke to five dentists, or even worse, if they spoke to thirty, but selected fives responses for purposes of this ad.
When you read statistics about the number of girls at a certain age who were sexually active, you need more information. What was the size of the sample? ( how many girls were interviewed?} How were these girls selected? Was it a random sample or were they high school girls staying for detention one day?
Having a large and random or representative sample is important. It would not make sense to interview only Republicans to get unbiased opinions of the President’s performance.
You also need to understand how statistics seem to prove cause and effect when none exists. If you read that, on the average, vegetarians weigh twenty pounds less than those who eat meat, can you conclude that being vegetarian caused the weight difference? It could just as easily show that losing weight caused people to become vegetarians.
It may well be a complete coincidence. (two unrelated things that occur together) They might both be caused by some third factor. People who are better educated or more health conscious might eat less and also make the choice to be a vegetarian.
One of my favorite examples of statistics is found in Dewdney’s book, 200% of Nothing. pp. 9-10. It describes a situation where tests scores for a school went down 60% one year and then up 70% percent the next year, This seems to imply they are now 10% above where they had been two years before. It takes a great deal of work to show this is not true, but a story makes it clear.
“A man by the name of Smith … spotted a $5 bill on the pavement. He … picked it up and put it in his pocket. His other pocket already contained a $10 bill. Smith smiled. ‘My wealth has increased by 50%….'” He gets home, finds he had a hole in his pocket and has lost the $5 bill. “‘That’s not so bad,’ he said. ‘Earlier, my wealth increased by 50%, new it has decreased by only 33%. I’m still ahead by 12%.'”
If you still don’t understand, ask a math major to explain in detail.
8. Understanding Probability
Many people believe that, in a lottery, a series of numbers like 9, 17, 19, 27, 31, 37 has a far better chance of winning than 1, 2, 3, 4, 5, 6. This is nonsense, of course. The first list looks familiar perhaps, and we have never seen a series like the second. But each combination of numbers has exactly the same chance of being chosen as any other combination. The second list has the advantage of being easier to remember and is less frequently chosen because people are so sure it can’t win.
The only advice for choosing a number is to avoid the fallacy that, if there have been a lot of threes in the last several weeks that threes are “Hot” and likely to keep appearing for a while. Nonsense. These things happen by chance, not because a number is hot. But, if a lot of other people are including threes, you should avoid them. You are not more likely to win, but just slightly less likely to have to share your winnings with other people.
Another piece of information about lotteries. Lets look at one where each number is selected independently, that you really can have a number more than once. “5555” Then you know you can have 10 possibilities for the first number (0-9), ten for the second and so on. 10x10x10x10 = 10,000. Your changes are 1 in 10,000. Does it mean that if you play this game 10,000 times and spend $10,000 that you’re pretty sure to win at least once? NO. Just because there are 10,000 possibilities doesn’t mean that all numbers will eventually be picked. Each is chosen randomly. Some numbers may appear many times. Others may not appear. In other words, Don’t WASTE your money.
If the game you are looking at selects balls from the same set, then a number cannot appear twice. Then you can calculate your chances as 10 x 9 x 8 x 7 if there are only 4 balls. Now the possible combinations are only 5040. Your chances are nearly twice as good. But you’d still be wasting your money. And notice that the lotteries with the biggest winnings always have more than 4 numbers to choose from. This means your possible choices for each number are much higher than ten. The odds of winning are terrible.
For example if you could choose a number from 0-60 and you had 6 ball, your chances would be 61 x 60 x 59 x 58 x 57 x 56 You could bet the same number every day of your life and chances are that you would never win anything. You could bet the same number 100 times every day of your life you still wouldn’t win anything. Save your money.
9. Connections to Other Disciplines
Mathematical thinking is NOT just thinking in math classes. It applies in every field. Stay alert to ways you can use it in every subject you study.
Cole quotes Sir Arthur Eddinngton
“What do we really observe? …Relativity theory has returned the answer — we only observe relations. Quantum theory returns another answer — we only observe probabilities.” (8)…
She then adds:.
“Used correctly, math can expose the glitches in our perceptual apparatus that lead to common illusions — such as our inability to perceive the true difference between millions and billions — and give us relatively simple ways of protecting ourselves from our own ignorance. (9-10)
I would like to add that, if you haven’t learned something in each of these areas in high school, no matter what your major, you should study some basic statistics and probability or other general classes in math. And, as you study, reflect regularly on how what you are learning will be useful in your field of study or in your life in general. A good education should always included math.
If you did not read them earlier, you might want to read: Answers to the Quick Test
If you would like to continue to the next way of thinking: Scientific Thinking