When I first started tutoring students, I often noticed that they struggled to add fractions. The addition of fractions just did not make sense to them. Part of this is caused by students having a weak understanding of fractions, and part of this is caused by them not understanding why the typical algorithms used to add fractions make sense.

Here is one model that I developed for myself, so that I could understand why addition algorithms for fractions make sense, and then use this model to help students make sense of adding fractions.

**Step 1:**

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The definition of what effective mathematics teaching looks like very much depends on what purpose we assign to teaching mathematics. A classroom where the primary objective is to teach students a specific set of mathematical skills for them to use later will look much different than a classroom where the primary objective is to teach students how to think mathematically, although there is obviously overlap between those two classrooms.

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This diagram represents a problem in education which is, by no means, the ONLY problem in education.

How do we change this paradigm?

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This is part two of a three part series on formative assessment. This post deals with some things you can do between individual lessons based on formative assessment and during a lesson. You can **read part one here**.

**Introduction**

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Part of my current role is to help teachers use formative assessment in their teaching. This has turned out to have some interesting challenges, and has helped me grow tremendously as a teacher.

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Here are seven questions my son asked today.

- Who invented buildings?
- Why don't we slip on salt?
- When you hold your eyes closed does more water get on the eyeballs than just blinking?
- Why do hummingbirds move so fast?
- Why are butterflies so pretty?
- How did we get the name "people"?
- Why do bees hum?

Kids are scientists. My job with my son is to teach how to answer his own questions.

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A few days ago, my wife told my son that he should do some mathematics from a 2nd grade workbook we had, and told him he could choose what he worked on. My son opened up the book to near the end of the workbook and decided to try some 2 digit subtraction exercises.

Here is an example of his work.

Topic:

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Teaching is a **learned activity**. As such, the act of teaching requires that the teacher have a mental model of what it means to teach. When teachers teach in ways which appear to an outside observer to be ineffective or poorly thought-out, it is because they are using a **flawed model** for understanding teaching and learning. Blaming teachers for having flawed models is like blaming students for not knowing things; it doesn't solve the problem, it may even exacerbate it.

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The following are studies which were all featured in the media in 2013. I am posting them here in the hope that they will be read more widely than they are, and that educators will examine the research themselves, and think about how this may affect their current practice.

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Like many math teachers, I have been following **Dan Meyer's discussion** on "fake world" math tasks versus "real world" math tasks with interest, especially since **one of my early blog posts was on this subject** and **one of the presentations I do for teachers** is on this very topic.

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- How many hours do teachers work?
- Powerful ideas in math
- Using my iPhone as a wireless microphone for my computer
- Questions about the flipped model of instruction
- We are homeschooling our son
- Tutorial: Converting between different media formats with WinFF
- A discussion with our Education Minister George Abbott
- Comparison of different learning theories
- I tried the Khan Academy
- The Death of the Amateur Mathematician
- Testing new Captcha system
- Take nothing for granted
- Educational chats on Twitter
- Do iPads improve mathematics instruction? Maybe
- New Math equals trouble, education expert says
- Mathematics and Multimedia blog carnival
- Can educational technology save money for schools?
- Profile of a phishing attempt
- The Case Against 21st Century Schools
- Math apps
- Open-ended problems in elementary school mathematics
- Why we need to change schools
- Children are not railroad trains
- Definition of Cyber-culture
- The competition is fierce

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