## Carl's Teaching Blog

#### Tag: Misconceptions

So I was going to start teaching simple and compound interest in my banking and investment class so I started off with this question for the do now:

If you borrow \$100 from someone and they charge you 4% interest, how much will you have to pay them after the five month loan is finished.

Before giving students time to work I asked eight people to weigh in with estimates, or guesses at what they could expect.  Each person in class had time to think and make a choice and gave an answer after considerable thought.  They also were told to estimate, they didn’t use calculators or much pencil/paper calculation.  They came up with quite a variety of answers, but (SPOILER ALERT) none of the answers were \$104.  Here are some of the answers that were interesting.

• \$102.50 – Here students seemed to think the 4% means divide by 4, and perhaps they knew the ‘slide the decimal over’ rule. So it seems did 100/4=25.00, and then moved the decimal over.
• \$120 – Here a student was very certain that this was the correct answer. This student assumed that the interest was applied for five months straight, so they used the \$4 of inters and multiplied it by 5 to get \$20.
• \$145, \$140, \$160 – It seems that these answers were blind stabs in the dark.  Students probably picked a number that was sufficiently bigger than \$100.

All these students were having trouble with guessing what would be an appropriate amount to have to pay back for this loan.  We talked about how easy it would be for them to get duped into paying large amounts of money for the different loans like the student loans and credit card offers they will see in college.  This set a clear context for the day’s lesson, which was talking about credit card APRs.

Numeracy is really important, as I saw in a recent blog post at Algebra’s Friend, there are a number of ways to help students build this skill.  I Plan to have students start focusing on doing that kind of thinking during the Do Now activity at the beginning of each class.  Students will start by working on tasks like the one below and ask them to think through it.

Which is a better deal?

 Store A Store B Store A Store B 1 Television15% off of \$300 Television250 5 Microwave20% off of 120 Microwave30% of 150 2 Stove 5% off of 500 Stove10% off of 530 6 Table60% off of 400 Table15% off of 200 3 Ipod30% off of 200 Ipod10% off of 180 7 Monitor25% off of 100 Monitor15% off of 320 4 Shake Weight50% off of 30 Shake Weight15% of 20 8 Health Master25% off of 120 Health Master10% off of 90

Each day we do it we will work on little tricks with percents.  I have taught a number of tricks before to help students do this (i.e. 50% is just half, 25% is half of half, 10% is just moving the decimal place over, etc.).  While it may be successful, I would like to have students understand the deeper concepts of number sense. (Perhaps I should take “Nix The Tricks” off of my Amazon wish list?)

What do you think?  What is the best way to teach percentages and number sense to high school students?

17/30 #MBToS30

I began my first day of my probability course today by asking each student individually students to do the following:

Introduce yourself to the class, say your feelings about math using a scale from 1 – 10, and use the word “probability” in a sentence about yourself.

The probability question led to some pretty interesting answers, so I will list a couple of them here:

There’s a proabability that I’ll pass this class

There is a probability that I will struggle with this class

There is a probability I will that I’ll start college in the fall

Aside from noticing their lack of positivity, these quotes told me that my students don’t really have an understanding of the word probability.   It seems that they are substituting the word “probability” with the word “possibility.” In all of these contexts the word “possibility” would make sense syntactically, and for 3 out of 18 students to make this mistake, it seems that this might be a misunderstanding, and not just a typo.

To say “there is a probability” is not really a phrase that one would hear normally, and it doesn’t make sense in that sentence, but substituting “probability” for “possibility” in math might be a misconception that many students might not have to challenge.  Students who only see the phrases “find the possibility”, or “what’s the possibility” may immediately realize that they are quantifying the likelihood of possible events and decide to do the appropriate procedure.  These same students maybe confused in later math classes when see phrases like  “compound probability” or “highly probable” and find themselves struggling with their conception of the word.

I tried to make a clear connection between probability and possibility while clarifying the distinction by saying “Probability is similar to possibility, because both talk about something that could happen.  We use probability to take possibility a step further, and actually come up with a number that allows us to say just how likely it is for an event to happen”.

In case you’re wondering about the kids’ feelings for math, the average was about 5.9, with some 1s along with some 10s.  Students share their feelings about math on the first day of  class because the students get to hear and see a range of values.  Some kids always love math, and others always hate it. This prompts me to give the speech where I say “I want us all to recognize that some people in here don’t like math.  It’s already an uncomfortable environment for some people when they walk in the door.  Let’s not make it more uncomfortable by making fun of people, being disrespectful, and separating yourself from the class experience by getting on your phone.”  I hope it makes the less comfortable kids feel supported, and prompts the strong kids to consider their role in the class.

7/30 (First week of blogging in the books!)  #MTBoS