As a teacher, who teaches some middle school I am fascinated by this post by Denise at Let's Play Math. The questions here are tremendously important, and I don't think most of us don't have good answers to these. Every middle school teacher must have good answers to these questions.

ASCD's journal Educational Leadership has an article this month by Lynn Arthur Steen about math, with a significant section on fractions. In this article is a section about students having to estimate the sum of two fractions on a standardized test, and basically the results show that students have no idea in general. Steen then makes a very strong case for why we need this skill, even though few people will ever actually have to add unlike denominators in their "real" lives.

It is my belief that we owe every student the opportunity to learn how to think "mathematically". This is for the most part a birth right of most students. My most challenged students are the ones that try to dissociate their "natural" math from their school math when they are in the classroom. Many students will tell you that they are terrible at mental arithmetic, when they can do complex calculations with nickels, dimes, and quarters. I encourage you to test this with any one who tells you that they can't add in their head.

In speaking with a student this week we were speaking about 9/2 and how to calculate that number to some decimal. I asked what she pictured in here mind when I said the number nine. She replied that it was a picture of the numeral, "9". I asked her to picture again, and she drew for me a 3x3 grid of circles. This is a big improvement, but not as helpful (and this probably only my own perspective) to this problem, as being able to picture two rows, one with 5 circles and one with 4 circles. It is this lexicon of images, pictures, etc. That allows me to speak and think with fluency in mathematics. This is why the "Round-Food Model" that Denise at Let's Play Math talks about in her most recent post about fractions is important, and why it is important that we construct a powerful picture and story about the rest of the rules of fractions.

Tips to promote students' metacognition

18 hours ago

## 2 comments:

As an adult I'm trying to get to grips with fractions. I'm using the internet to help me along. Some websites are very good, others are truly awful and only serve to make life even more difficult and it seems almost a blatent attempt to push you away from fractions.

Having looked through this website and those who have said they can make fractions easy, all I can say to you is this. I truly hope you're not teaching children. Because you're doing exactly what the study suggests. You're turning children away from fractions. Do you really think you're simplifying a fraction problem with your methods? I'm an adult and I got lost halfway through your explanations. So what will your explanations do for a child? You're looking at it from a adult's perspective on how to make it easier, not a child's. You're making it overly complicated when there really is no need. What may work for you may not work for the child (or in my case idiot).

Anonymous,

I am truly sorry that you had such a strong reaction to this post and the site. There are things here that are not easy. You are right about that. I am teaching children. I am doing the things that I think will help children understand. I am sorry that you didn't find the site helpful. I don't really intend it to be a site that helps people understand fractions, but more of a site where people who teach fractions can think deeply about how to teach them. Clearly, you have been taught in ways that did not work for you, and you are looking for a way that might. You clearly feel that my methods and beliefs don't work for you, and I am in no position to argue. When/If you do, please come back and let me know what does.

It may be that, the rote systems of sayings and mantras about what do with the numbers in math problem are what works for you. The issue that I have with this, is that it can only demonstrate to students that math is capricious set of rules. That something is right because the teacher says it is. Many people appreciate that about math, but I do not. I want students to be able to move beyond calculation to the ways that math is connected to their world. This will rarely happen with such instruction.

Lastly, I am sure that you are not an idiot. I doubt that I would think of you as one, if we met in real life. You say you are searching for understanding, that is what I am doing too.

Thanks for your comment.

Matt

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