Math Misconceptions Anecdotes

License Plate Numbers

A special education student was put into the precalculus class. The advisors thought he could not possibly pass any math class so it did not matter which class he was in. I was talking to the class about their hobbies, and when I got to Willie he said his hobby was collecting license plates.

“Cool how many do you have?”

“About 10,000.”

“Where do you keep them all?”

“In my head.”

“In your head? What do you mean?”

“Whenever I see a license plate I remember it.”
”Do you know my license plate?”

“Which car?”

Before I could answer he told me the license plates of all three of my cars.

“Do you know everyone’s license plate number?”

He went around the room telling everyone’s license plate numbers.

I told him to imagine that the math information was on a license plate.

“I can do that.”

He earned a B in precalculus after never having any math except special education arithmetic. After he graduated he enrolled at Pima College.

An Unfair Test

Today prealgebra student Sara was angry at me after taking a test on fractions. She said it was unfair for me to ask what 2/3 of 12 was when I had not used 2/3 in an example in class. She said she had studied only halves, fourths, eighths and sixteenths so she could not do thirds. She said she had memorized the rules and could do the problems from the book, but she could not do such difficult problems as shading in 2/3of 12 squares (a problem from the 8^th grade TIMMS test shown below).

Shade in 2/3 of 12

Juan and PI

In a PreCalculus class, one of the brightest, highly motivated students, who also took classes at the community college, was impatient with the way class was going. He wanted to know ‘how to do it’, he hated all these experiments and quests for mental models.

During one experiment the students were discovering π by measuring the circumferences and diameters of several circular objects and then graphing the circumference of each as a function of its diameter.

Juan thought that was silly. He measured the diameters and used the formula C = πr² to calculate the circumferences. I noticed that his graph wasn’t linear so I asked what he was doing, he told me that measuring the circumferences was a waste of time.

I persuaded him, with some effort, to humor me and measure them anyway. A few minutes later he called me back to tell me that the tape measure was broken. It was showing answers that did not match his calculations.

I asked if it was possible his formula was wrong and his assured me that he was an excellent student and he was sure it was πr². I chose an object with a diameter of 1 inch. I asked him to calculate its circumference. He told me it was .785398163 inches. I asked if that was more or less than an inch. He said it was, of course, less. I asked him to consider how the circle could be shorter around than it was across.

He thought, he recalculated, he thought, he measured, he thought, he recalculated, he thought, finally, he measured the circumference, he thought some more, he began to measure all the circumferences. After he had finished, he called me over and asked what had gone wrong. I told him that he had used the formula for the area of a circle rather than the formula for circumference. The reason he had made the mistake was that he had memorized a magic charm he had no knowledge of circles.

Nurse Adding Fractions

At Good Samaritan Hospital in Phoenix a nurse was told to give ¼ gram of morphine to a patient. But the morphine tablets were ½ gram each. So she gave the patient 2 tablets. The patient died.

The First Mark on a Ruler

Edward was bewildered by trying to measure a 2 1/16 line with a ruler. The ruler he had brought to class had different sized spaces on it than the classroom rulers

He asked me if the first mark on a ruler was always a sixteenth. I explained that it depended on how many sections the ruler was divided into. I drew a 0-1 line on his paper and marked it in fourths. I asked him what the first mark after 0 would be called. He was dismayed because he thought it would be called a fourth but it did not follow the rule he had learned about sixteenths always being the first mark after the 0. I explained how the marks were named after the number of equal sections into which each unit was divided. Fourths for 4 parts, Eighths for 8 parts, etc. I drew another 0-1 line of a different length also divided in fourths and asked him what the sections represented. He asked if they were still called fourths since the sections were now a different size. He wondered what the marks would be called if they were found between 1 and 2 instead of 0 and 1.

There is a lot of potential for misconceptions when reading a ruler. I find that about one out of five students taking prealgebra do not know how to read a ruler.

Lecture Notes

If I say “dividing is a short cut for subtraction” students dutifully write the statement in their notes, anticipating the possibility that I might sometime request that answer on a test to a question like “what is dividing?” or “for what is dividing a shortcut?” It is as though the statement is transferred directly from my words to their papers without ever going through their brains. They have little interest in interpreting what the statement means or understanding my explanation. In math classes, as in many other subject areas, students have always been taught what to think and not how to think. Considering the meaning, usefulness or validity of the statement would not occur to most students who have been carefully trained not to think or question but just to memorize what the teacher seems to want them to know. Thinking and questioning might even prevent a student from passing a class if he/she happens to disagree with the teacher. This habit of mind, acquired from prior school experiences, prevents students from trying to understand math, remember it, or apply it in their lives.

Until we legitimize and teach real math, students will continue to believe that math is just a set of meaningless symbols and rules devised to confound them and impede their success.

ASAP

A question from the Arizona State Assessment Program (ASAP), a test required for all high school student in order to graduate:

The article from the Rainforest Action Network says that estimates of how much rainforest is disappearing vary, but you would like to examine at least one estimate to get some idea of how fast the tropical rainforests are disappearing. One source gives the following estimate of the size of Earth’s rainforests for various years.

In 1980, there were 1,884,100,000 hectares of rainforest in the world.

In 1990, there were 1,714,800,000 hectares.

(1 hectare = 2.47 acres or 0.00386 square miles)

Using this set of data, predict how many years it would take to totally destroy all of the remaining rainforests. Explain how you made your prediction. What assumptions have to be made in order for your prediction to be valid?

70% of Arizona students solved the problem by repeated subtraction until they got to zero. They could have done it as a division problem, but few students did it that way, even though they had calculators to use on the test. Thus, it seems few Arizona students understand the relationship between division and repeated subtraction.

Whispering Dishes

While the Algebra 2 high school math students were building whispering dishes, a university professor came to visit and asked what they were doing. He said “That will not work.” The students took him out in the hall and showed that it did work. Then they showed him diagrams and math calculations to explain how it worked. He was a PhD in math and head of the undergraduate math department, and he was astonished that the high school students understood more about the math of the whispering dishes than he did.

Test Tube 1/3 Full

A friend’s 12 year old daughter was playing with her cousin’s chemistry set. She asked her mother to please fill the test tube 1/3 full of water for her.

Her mother said, “You can do that yourself.”

“Please, Mommy, do it for me, I don’t know how much is 1/3”

“You know fractions, you’ve learned them in school.”

“Don’t worry Mommy, I’ll get 1/3 right on the test at school. I always get A’s in math.”

Walking Rate

While studying distance vs. time functions, I asked if you start 5 meters from the wall and walk toward the wall at ½ meter per second, where will you be in 6 seconds? Wally began writing numbers on his paper and calculating furiously. Finally he said “I don’t know how to get the answer. Should I multiply or divide? What are the rules and steps?

I told him he could use his common sense to find the answer. I laid 5 meter sticks end to end from the wall and demonstrated walking at ½ meter every second, stopping short of the final goal. I asked him to watch and see if he could figure out where I would be in 6 seconds if I continued to walk in the same way along the sticks to the wall.

He turned back to his paper and began to write numbers on it, scratching his head. “I don’t know.”

I repeated my walk along the sticks, asking the question in different ways. I asked him to try walking along the sticks. After he walked, he found himself 2 meters from the wall. I asked him the original question. Again he said he did not know how to find the answer. He asked again for me to tell him the rules and steps.

District Superintendent

I suggested to the School District Assistant Superintendent “If you want people to do more you should increase the hourly rate for extra stuff.” He said, “They are paying people $10.00 per hour now. That is more than I make.” I wrote a program so he could enter his salary and calculate his hourly rate. He found it was about $50 per hour. (He was making $90,000 per year working 11 months and did not know that was more than $10 per hour).

800 mg Dilantin

A nurse at a Michigan hospital gave 800 mg of dilantin instead of 80 mg of dilantin to a patient. The nurse had to go around the hospital gathering 32 vials of dilantin to administer in 2 IV’s No one questioned the excessive amount. The patient died.

Meaning of Letters

I wrote the following equation on the board:

2b + 3 = 9

Immediately a student yelled out 7. How did you get that? You showed us an equation yesterday and the b was 7. The assumption was that b always would have the value of 7, no matter where it was found. His mental picture was that it was a replacement code. After you find the value of each letter you will always know them.

We use letters in many different ways in math: as yet to be determined constants e.g. a, b, and c in ax² + bx + c, or as determined constants e.g. c in e = mc², or as variables e.g. x in ax² + bx + c, to indicate units e.g. m for meters. The students do not know in what context the letters are being used.

Girls’ Heights

After making a few measurements of books and desks as a whole class activity, I assigned students to measure their own heights in centimeters using a meter stick.

Three students wrote 126.5 cm. I demonstrated how short that was for a college student (no, we don't have any midgets in the class), explaining that the measurement must have been wrong.

Immediately one girl turned to another accusingly, "I copied from you and you were wrong." Then a third girl joined in, "Yeah, I copied from you too; it's your fault we had the wrong answer."

I explained that since all three girls were different heights, they would never have the correct answers if they copied each other. There was more than one correct answer and they would need to actually measure themselves to find the correct answers.

Length, Area and Volume of a Can

Holding up a can, I asked, “How could you measure the length, area and volume of this can?”

Students answered:

Student:“You could measure the length with a ruler.”
Instructor: “Yes that would work.”
Student: “You could measure the volume by filling it with water.”
Instructor: “Good idea.”
Student: “The area is L times W, or maybe L times W times H.”

Instructor: “You could find the area by wrapping it with paper and finding out how much paper you need.”

Student: “Oh, then area is the surface?”

Instructor: “You need to add the area of the top, bottom and side of the can to find the total surface area. If the top is a circle how does L times W work?

When studying geometric shapes, students often demand to have a formula for each shape, and they are very disappointed to find that there is not a formula for everything.

Ratio Problem

A question from the textbook:

If a company can make 500 cartons in .8 hours, how many cartons can it make in 2 hours?

Conrad insisted it was a unfair question since they did not tell how many cartons were made in 1 hour and it could not be solved without that information. Carmen had used 8 hours instead of .8 hours and insisted her answer of 125 cans was correct because only the decimal point was in the wrong place.

Who’s Winning

Which information is more important to a sports fan, the score for the game (the numbers) or who is winning (the sign)?

Rounding Error

While giving a ‘standardized’ test to a group of students I had never worked with before, one student called me over and asked how to ‘round off’ 65.23. I explained that this was a test and I couldn’t actually teach how to do things. She looked up plaintively and said, “Please just tell me if it’s closer to the 65 or the 23.”

Thalla and the House Numbers

During a discussion in a high school Pre-Algebra class about using number lines, I could see one of the students was completely lost.

I tried again to explain a number line with little success. So, I said, “It’s like street addresses, you know, the one hundred block, the two hundred block, the three hundred block, and so on.”

Still no sign of enlightenment.

“Where do you live.”

She gave me her address.

“So, when you go home, you’ll go north on Kino, turn left on your street, and it will be the last house on the right in the first block.”

The girl was dumbfounded. “How do you know where I live.”

“Well, it’s like the numbers in this hallway. The numbers are in order.”

Blank look again.

“When you have to go to a room, how do you find it?”

“I walk around and look at the room numbers until I find the one I want.” Her tone indicated that she thought my question was stupid. She clearly believed everyone did it that way and couldn’t see why I would question it.

After school Thalla came in with two friends. She said, “Ms. Cardell, show them the trick.” I asked one of her friends her address and explained how she would get to her house from school and she too was amazed. The girls were convinced that math teachers were magicians.

Cut Out One Square Inch

After studying the English and metric systems of measurement, I gave the students in the prealgebra class inch/cm rulers and pieces of paper. I asked them to each cut out one square inch of the paper. They cut out the following shapes:

.

I gave the same problem to my Introductory Algebra class with the following results:

	Square Centimeter	Square Inch	1 “ long rectangle
Prealgebra	20%	40%	40%
Introductory Algebra	43%	43%	14%

Measure from Toe to Toe

I demonstrated to the students how to find the length of their paces by walking forward, then stop in mid-stride to measure the distance from the front of one toe to the front of the other.

When they began to measure in centimeters, using a meter stick, I found some were placing the stick from toe to toe as instructed, but were getting the wrong readings.

Instead of placing one end of the stick at the end of a toe, they were randomly placing the stick, and randomly reading any number that happened to land near a toe. Thus they could have read almost any number from 1 to 100 cm!

I showed them they needed to measure with the zero end of the stick placed at one toe, and read the number beside the second toe.

Place Decimals on a Number Line

Only half of the students in the prealgebra class could correctly place the following decimals on a number line from 0 to 1: .6, .069, .609, .60

0 1

My computer program, Decimal Scales, gives students practice with decimal scales. You can download it from my web page at
www.cardells.net/cacav/refmat.htm

Shade in 20% of these objects:

Represent the colored squares as a fraction, decimal and percent of the whole grid.

Add and subtract fractions with spaghetti and a ruler

Give each student a ruler and two pieces of spaghetti of different sizes to add and subtract their lengths.

Area as Length Plus Width

I write on the board A = L + W. Then I give the students a lot of problems to do. E.g. the area of houses, swimming pools, their room, the desk, their notebook, a piece of paper, (real things). Never had a student argue or question the formula. Later told them that the formula was wrong. Most would just switch to the new formula with no more thought than using the old one. None of them ever understood what area was. Need to go back and learn about area. Cut out square inches, feet, etc and use them to measure. Compare that with measuring around the outside with a tape measure (perimeter= distance around)

Models for Signed Numbers

Positive and negative numbers. First shot was as vector addition because that was my model. I assumed everyone thought about it the same way. Each member of the faculty had a different mental model. Each had a picture that dominated their thinking about signed numbers. Students contributed. E.g. War, distance, canceling particles, up and down stairs, walking back and forth, piles and holes, elevator, water tank, weights and balloons, bank balance, football yardage, temperature.

Metric Lab

While students are measuring their height, arm length, finger length, etc. tell them to say the units for each measurement as they measure. E.g. “20 centimeters”. They will be using the new vocabulary in context while they measure.