How Not to Teach Fractions

Occupy Math has complained about the quality of math education on a number of occasions. Last week Occupy Math went to the 40th meeting of the Canadian Mathematics Education Study Group at Queen’s University in Kingston, Ontario. He met lots of math education researchers and several classroom teachers. The conference was a rewarding experience and Occupy Math will be going to the CMESG conference in future years. Next year the meeting is in Montreal, but Occupy Math will bring himself to go anyway.

One of the things Occupy Math obtained at the conference, from a classroom teacher, was a Ministry of Education booklet explaining how to teach fractions for K-12 teachers. Occupy Math has linked the PDF so you can take a look if so inclined. The teacher wanted Occupy Math’s opinion on one of the points the booklet made, treating fractions as operators, which was technically a mis-use of the term “operator”. That was far from the biggest problem in the booklet.

Overall, it was one of the most awful documents I’ve ever seen on how to teach fractions. Or anything.

Occupy Math will start with a quick bullet list of the major problems with this document.

  • There was no authorship listed. This means that below “the Ministry of Education” no-one was willing to take responsibility for the document’s contents.
  • Even though the booklet went all the way through 12th grade it completely failed to mention critical mathematical skills relating to fractions. This point is unpacked later in the post.
  • The document was remarkably disrespectful of teachers – it spoke as if they were ignoramuses rather than educational professionals. More on this in a minute, too.
  • The booklet spoke of things that you should not do – without explaining why one would want to do those things or what the alternatives were.

The book completely ignores two major tools for telling if two fractions are equivalent.

Two fractions are equivalent if they have the same simplest form. For example 2/4 and 3/6 are both mildly disguised versions of 1/2. All three of 1/2, 2/4, and 3/6 are equivalent. Let’s look at two standard ways to tell if a couple of fractions are equivalent. The first is to cancel common factors and put the fractions in simplest form. If the original fractions were equivalent, the simplest forms are the same. Reducing to simplest form means you need to be able to factor the top and bottom of the fraction and notice common factors – so you can cancel them against one another. The booklet on how to teach fractions did not mention factorization techniques. They pretty much skipped one of the most critical skills. Here is the closest the booklet comes:

There are many effective strategies for comparing and ordering fractions beyond determining a common fractional unit, or common denominator. When students have a strong understanding of fractions, including the multiplicative relationship between the numerator and the denominator, they are able to use number sense and proportional reasoning to make comparisons.

Paying attention to fractions by the Ontario Ministry of Education.

The booklet is using near-gibberish to avoid mentioning factoring. Prime numbers and their use in factoring whole numbers and reducing fractions should be a central theme of a guide on how to teach fractions. Yes, this might be a bit much for third graders, but the booklet goes up to grade 12. It is like factoring is a state secret that can, at most, be alluded to.

If a Ministry of Nutrition were to write a guide to baking bread along the same lines, it might produce this:

There are many effective base materials for the creation of breads beyond using standard mixes. When a baker has a strong understanding of baked goods, including the nutritional and textural relationships caused by the choice of base materials, they are able to select among the available major ingredients to achieve a desirable and delicious result.

Sarcastic Analogies by Occupy Math.

The other standard technique for figuring out if fractions are equivalent is to cross-multiply – this means that you multiply the top of each fraction by the bottom of the other. If the resulting numbers are equal, the fractions are equivalent. Occupy Math rendered the calculations to tell if 3/4 and 39/52 are equivalent:

Since we got the same number on both sides, the two fractions are equivalent. See? Not too hard – especially as kids get access to calculators. Since cross-multiplying preserves relative size, you can also use it to tell which one of two fractions is larger (or smaller):

The reason for skipping cross-multiplication is horrible too.

The booklet does not explain why it left out a fundamental skill, but Occupy Math was sitting in a room with several math-education professionals, so he asked. One of them said “You cannot teach kids to cross-multiply because then they will do it everywhere, like when they are trying to add fractions”. Occupy Math thinks this is very close to “You cannot teach people to use screwdrivers because they will try to use them to tighten bolts or stir coffee.”

Let’s say this another way – the Ministry apparently thinks that it is too hard to teach children when a useful tool is or is not appropriate. The alternative provided in the booklet? DRAW PICTURES that embody the fractions and see if they are the same. The booklet says:

Students might also construct accurate diagrams, which can also be
useful for comparing and ordering fractions.

Paying attention to fractions by the Ontario Ministry of Education.

Here is one of their pictures:

This might work to introduce the subject of equivalence, but figuring out if 12/37 and 72/201 are equivalent would require one heck of a picture. In essence, this document urges the teaching of techniques that will make arithmetic on fractions much too hard for students to do – as soon as the fractions are bigger than toddler-sized!

Another quote from the booklet:

Some students become over-reliant on the process of “doubling both the numerator and the denominator”. This strategy may not be an effective way to compare 7/9 and 3/5 since the doubling would continue on and on.

Paying attention to fractions by the Ontario Ministry of Education.

The booklet won’t give teachers standard, effective techniques for checking equivalence, but it helps them to avoid a technique that is often ineffective. Why? Was this technique taught under Ministry guidance in the past?

Occupy Math thinks that it is impossible to teach math by enumerating all the ways not to solve problems. Returning to the reason we cannot teach students cross-multiplication, an earlier post by Occupy Math on a related topic is Math is not a Form of Ritual Magic. In the opinion of a senior math-education professional we cannot teach cross multiplication because students will just use it indiscriminately.

This means that teaching math as rituals is so ingrained in current practice that it stops us from teaching students useful techniques – because it is believed that choosing an appropriate technique is beyond them.

As a university teacher, Occupy Math expects students to be able to reduce fractions to their simplest form and to be able to rapidly and accurately compare fractions. The easiest class taught at Occupy Math’s place of employment is calculus. Fractions are supposed to be a settled skill before starting calculus, but Occupy Math keeps needing to do remedial instruction. This document from the Ministry gives him some perspective on why this is so.

Let’s turn now to the issue of respect for teachers. Below are a series of definitions from the document. All of these statements use language suggesting deep ignorance on the part of the teachers. Definition 4 is also complete nonsense.

  1. What is an attribute? “An attribute is a quantitative or qualitative characteristic of a shape or an object. Examples include colour, size, or shape.”
  2. What is a conjecture? “A conjecture is a guess or a prediction based on limited evidence.”
  3. What is a benchmark? “A benchmark is a number or measurement that is internalized and used to help judge other numbers or measurement.”
  4. Iterating is copying or combining equal units to create a new fraction or the whole.”

Definition 3 is incorrect, incomplete, and insufficient while definition 4 misses being wrong only by failing to have a specific meaning. The entire document is written as if teachers reading it know nothing about arithmetic or fractions. The document obviously was not proof-read for content or clarity. It contains nonsense sentences. It is written in turgid jargon when clear, plain language is far more effective. Examine the following paragraph from the booklet:

In part-part relationships, the digit in the denominator indicates the number of items that are in one part of the set, and the digit in the numerator indicates the number of items that are in the other part of the set. The fractional unit, or number of equi-partitions of the whole, is determined by adding the digits in the numerator and denominator together.

Paying attention to fractions by the Ontario Ministry of Education.

This paragraph uses “number of equi-partitions of the whole” to mean “number of objects in the set”. In addition, the use of partitioning of sets to explain fractions leads students to do things with the parts of fractions that are not done elsewhere. Or useful. Reading this document from the Ministry helped Occupy Math to understand some of the bizarre things his first-year students do with their fractions. They are trying to apply what they learned in high school.

Occupy Math is appalled.

Even if the people working on math at the Ministry of Education were doing a good job, micro-managing teachers would be a bad idea. This document on teaching fractions suggests that the people working on math education at the Ministry of Education are not doing a good job. Occupy Math’s opinion is that the document on teaching fractions is a disgrace. It is remarkable that no authors are listed. This may mean that the authors know this document does not meet the standard – or it may mean that the Ministry has developed practices that would obstruct constructive feedback because it is not wanted.

In a democracy, government officials should be identifiable and responsible for their work.

Occupy Math refuses to give in to despair. While at the CMESG conference, Occupy Math and his former student Andrew McEachern designed a collection of card games, for all grade levels, to help people learn fractions. Two of the teachers at the conference gave us excellent suggestions to improve the design of the games. There are cooperative and competitive versions, as well as variations based on Texas hold-em and Magic the Gathering, to provide students with a variety of challenges. If you would like a post on how these games work, comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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2 thoughts on “How Not to Teach Fractions

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