Confusing Contexts

A few weeks back during one of our previous classes, we had a brief discussion on what contexts we would choose to use in teaching fractions. Many of us mentioned food, baking, measuring time and distances, money. These we believed were things students knew about and had back ground knowledge on, and potentially things our students would find interesting or care about - who doesn't like food?

This past week during an observation, I realized more so just how important the context is that we choose to use. I was in a seventh grade classroom. The students were reviewing for their upcoming test the next day over percentages (comparing percents, fractions, and decimals, finding percentage change, what's the percentage of a whole?). I was walking around the classroom, helping students who had questions. Most of the students in the class seemed unafraid to ask questions if they were uncertain about something - which seemed to be a result of the sort of classroom the teacher had cultivated. Anyway, as I was walking around I had a girl ask me over to see if she was solving a problem correctly. The problem gave her the price of a good, the percent of a mark up on the good, and asked for the new price of the good after the mark up. When I asked her what she was thinking, she explained to me how she had solved the previous problem (given the price of a good, and the percentage of discount, find the new price). She asked if she would then solve this one in the same way - after all she was given a price and a percentage and asked to find the new price. So to her the two problems seemed very similar, if not the same. I asked her if she knew what "mark up" meant and she shook her head no. I helped to explain to her the concept and then she was able to solve the problem on her own.

Originally, viewing the problem, it sounded like the previous one. While she may not have known or simply had forgotten, not knowing what a mark up was, she made an assumption that felt logical to her so she could keep going and solve the problem. Luckily she had asked a question, but not all students will. On a test, she might have gotten points off, even though once she understood the situation she knew how to correctly solve it.

Context is important. We want to see what students have learned and understand, instead of whether they know what the term "mark up" means. We may think students know what a mark up is (it uses the word up, so wouldn't students think increasing?), but they may not. Lots of students haven't had jobs yet and are more familiar with ideas of sales and discounts, than the marking up of prices by suppliers. When using contexts, we need to be careful and ensure they are relate-able to students and that students understand them and any vocabulary or ideas that accompany it.

Proportion Sense

Recently in class we began discussing fractions. We began by looking and discussing proportions without the use of numbers. It seemed like it would be simple, but without the use of fractions at our disposal it became difficult.

We were given a picture of two boxes of chocolates(like below) and asked which one was more nutty?

It was difficult to stop yourself from using and comparing fractions - since we already have that knowledge. Our future students, however, will be just developing how to use fractions and relate them. Instead, we had to rely on the picture and ways in which we could manipulate it. What we ended up finding most helpful was the idea of buying enough of each box so you have the same number of chocolates and then comparing how many nuts each one has. An idea  that relates back to finding a common denominator. Once you had the same number of each, making a comparison between the two was logical.

However, after accomplishing this, when faced with another proportional problem we fell into mistakes. We were given a problem comparing mixtures of blue dye with water. Given certain a number of beakers of dye and water, we had to decide which would produce a stronger blue color. It was interesting to see how different we reacted to this problem. Many of us ignored the idea of getting to the same number of beakers and began to cancel out blue dye and water combos. The result was a false belief that the two mixtures shown below would be the same color.

We failed to take into account the ratio of water to blue dye. It was startling to find ourselves making this mistake.

These two activities really helped me to come to understand the importance of proportional sense. My future students will likely be making similar mistakes, but having a discussion on these ideas and working through proportions without numbers will be helpful. Introducing students first to proportions and making sense of the relationships between them will get students asking important questions:

• What quantities are being compared?
• What other factors do I have to consider? 
• What roles do these variables play/what's their relationship?
• How is it affecting the goal(if it's nuttier or more blue)?

Through this, students can begin to develop a sense of how to compare proportions and what to take into account. Then, when dealing with the numbers and fractions they should to able to reason if the answer they arrive at is a logical one or not. They can rely on asking questions to led them to the correct calculations and hopefully dealing with fractions will not seem as daunting.