While researching yesterday's topic, I stumbled across a list of mathematical misconceptions some of which I've had students happily share.
I'm sure you'll recognize some or all of the misconceptions listed below. I'm also sure some will make you smile at the memory of a teacher telling you that exact thing in elementary school.
I know, I heard them myself. So here is the list.
1. Three digit numbers are always bigger than two digit numbers. This rule comes about because when they first learn numbers, they are only exposed to whole numbers. In that case, this rule is correct but once decimals are thrown into the learning, it no longer applies. 3.24 is not bigger than 6.2.
2. When you multiply two numbers together, the result is always larger than either of the original but that is only true with whole numbers. Once students begin using fractions or decimals, this may not be true. one example is 1/2 times 1/6. The result, 1/12, is smaller than either one.
3. Often students think the fraction with the larger number in the denominator means its larger such as in 1/4 and 1/8. They sometimes think 1/8 is larger than 1/4 because 8 is larger than 4. I think this has to do with 8 is larger than 4 normally with what they've been taught so when the context changes their understanding does not.
4. Most students see two dimensional shapes in only one orientation such as a triangle with the base always at the bottom part of the shape rather than placing it at the top with the vertex pointing downward or off to the side. Teachers need to change the orientation so students do not get in the habit of seeing it one way.
5. In squares the diagonal appears to be almost the same length as the sides and students may assume they are the same.
6. When multiplying by 10, simply add a zero. This works for a whole number but not for a decimal number. You could add a zero but it does not help you to remember to change the position of the decimal.
7. Ratios where students get used to comparing one object to another such as two carrots to three peppers rather than looking at two carrots to five vegetables. When the situation comes up where they need to set up a part to a whole, they often have trouble.
8. Students often confuse perimeter to area because they count squares for both of them without understanding the whole square inside the shape is counted for area while they are only counting one side of the square for the perimeter.
9. Students often have difficulty determining the scale used by the measuring item. Not all scares are divided into 10's. Many students do not count the markings to figure that out, they assume its always going to be 10.
I understand why students are taught many of these rules when they are in elementary school but it does a disservice teaching these are "rules". Students need to to quit learning "rules" which only apply to a narrow population of numbers. Hopefully, teachers will quit doing these so students are more open to learning new situations.
Let me know what you think. I'd love to hear.