Student+Misconceptions


 * Math Misconceptions **

In one of my other courses, it was recommended to think about what misconceptions a student might have regarding the subject matter prior to developing a lesson. This sounded like a good idea, and I resolved to try to put it to use. Unfortunately, theory and reality don't always meet. I learned today that it can be really hard to predict what might go through a student's head when you, yourself, already know the material very well. I am creating this page, hoping that others will share their stories of real student misconceptions. Knowing what students might be thinking will better prepare us for developing our lessons. I suggest that you give your story a title that fits with the particular Math Curriculum topic, so that others can quickly find what they are looking for on this page.
 * Student Misconceptions Page**

Please note that this is in NO WAY intended to be a "Look at the dumb things that students say" page. Please keep the stories relevant and helpful to our fellow teacher candidates. Steve R.

In the Guide for Math Makes Sense all chapters have student misconceptions listed. I have found it hard to believe that 5 to the power of 2 can be misinterpreted with 5x2. During the test, however, there were still students that had remembered the misconception rather than the correct answer. My AT jumped in at some point and said explicitly that 5 to the power of 2 is NOT 5x2 and I wonder if this was a good thing or not, as we learned that when they are told to not do something they process the action first and the NOT later (in adolescent development course, I believe)
 * Powers (Doina)**

So the students were working on the following textbook problem:
 * Ordering Integers (Steve R.)**

"Represent each situation using an integer. Show the integer on a number line: a) a loss of $6 b) a temperature of 5C below freezing c) a gain of 2kg"

This was supposed to be such a simple practice problem, that I didn't even think about how they might misinterprete it. Here are some explanations of their answers from various students:

a) "If I had 6 bucks and lost it, I'd have zero dollars" You might say "Yes, that's correct - you would have 0 dollars" Now how would you describe that loss in mathematical terms (e.g. using either + or negative signs" ? (Robin)

b) "I think that 'freezing' is something like -4" You might say, well that is close. However, water freezes at 0C - So how would you describe 5 below freezing now? (Robin) I wonder if this is a semi-forgotten memory of the fact that the water has the smallest volume at 4C? (Doina) c) "I think that gaining weight is //bad//, so this would be negative 2"

I was especially stumped by the first response because it was so simple, yet quite solid - and perfectly reasonable. I wasn't sure if I could ever really refute that one!

On a test, many students got the following question wrong: //How do you convert a decimal number into a percent?// The most common wrong answer was, "Drop the decimal and it's already percent" to which I commented on a lot of tests, "What about the decimal 0.255? Does that become 255%?" The students also had the misconception that a percent can only ever be a decimal number less than or equal to one. Bottom line: when teaching percents, make sure students understand that it is a number (//any number//) that is compared to 100.
 * Percents and Decimals (Steve H)**

Many students believe that dividing a number by another number always results in a quotient that is //less// than what you started. When given the question: "What is 70% of 35" many students will first write: 35/0.75 = ... because they assume that this will result in a smaller number. When they carry out the math and see that the number is actually bigger, they will place the decimal in an arbitrary place to make the number make sense. Note: I've yet to try (effective) ways to break students out of this habit.
 * Division with Decimals (Steve H.)**

I looked through the students' math notebooks today to gauge how much they were actually understanding. I was surprised to see that most students were not properly multiplying a fraction by a whole number: 1/4 x 4 turned into 1/4 x 4/4. We will be going over this again tomorrow and hopefully it'll make more sense to them the second time around.
 * Fractions (Hailey)**

When introducing variate algebra, a lot of students really struggle. One thing that might come at you from left field is something like:
 * Algebra (Steve R.)**

Substitute for p=2 and r=4 in the following expression: 3p + 5r Student answer: 32 + 54

Thanks to Stevie G for pointing this one out to me.

Something that surprised me a lot was that students in my class though numbers like 0.3 were less than 0. It came up a lot and every time it did, my AT and I did our best to try and explain to our students how and why that this number was still greater than 0, but it is something we really struggled with. I'm sure it is a mistake that some students in that class will make again, and it is unfortunate that I wasn't able to help them understand this.
 * Decimals being less than zero**

I used to make this mistake all the time when I was a primary school student. //Does a more negative number get bigger or small?// It's the way you word it that always mistaken the students because although going from -3 to -30 (negative numbers) may look like the number is, in fact, getting bigger, it actually is getting smaller. A better way to dictate this is by saying it is a bigger, negative number, as opposed to saying the number got smaller. I caught myself saying the number got smaller (which is the correct way of saying it) during my algebra lesson, even though I should have been more sensitive to their level because this is just one more thing they don't want to worry about - that is, on top of all the new things they are learning. All of them got confused on this simple fact, so I had to take five minutes to explain it to them until I heard a "I get it."
 * Does a more negative number become bigger or smaller? (Christe M.)**