I don’t know when it starts, but at some point, students learn about fractions. They learn to add them, subtract them, multiply them, divide them, and eventually how to work with exponentiation of fractions. The rules they learn are crazy.

Cross multiply.
$$\frac{3}{4} \pm \frac{5}{6}$$
Flip and change.
$$\frac{3}{4} \div \frac{5}{6}$$
Multiply across.
$$\frac{3}{4} \times \frac{5}{6}$$

Then at some point, students take Algebra.

Solve for \(x\).
$$x + 3 = 7$$
$$2x – 3 = 7$$

And here begins the deterioration of fraction education. Let’s just say, for sake of argument, that by the time a student gets to Algebra, they actually have the mechanics of working with fractions correct and they understand in (basic) concept of what a fraction can represent.

Now, here’s what I see in (college) Algebra (I or II) courses.

I can ask students to compute the above fraction problems and most students have no problems. Typically, even in College Algebra, students have had a high school Algebra course so they come with some basic knowledge about the symbolic manipulation. Thus, I can ask to solve for \(t\) in $$3t – 8 = 19$$ and most students correctly answer \(t = 9\) since they are capable of managing the mechanics of $$\begin{aligned}3t – 8 & = 19\\ 3t & = 27 \\ t & = 9 \end{aligned}$$

Now here is one of the most annoying, baffling, irritating, angering things I see. We just solved for \(t\) in \(3t – 8 = 19\). Great! Now, I ask: Solve for \(t\)
$$\frac{2}{3}t – \frac{4}{5} = \frac{6}{7}$$

and students are lost. I don’t just mean “how do you add fractions again?” kind of lost. I mean, more than a handful, are flummoxed as to what the next step is. Nothing has changed in form! Only that the numbers have gone from integers to fractions. At this point, I get crazy suggestions, “Do I subtract \(\frac{2}{3}\)?” “Is this the same as $$\frac{-2}{15}t = \frac{6}{7}$$ because I just ‘combined’ the left-hand side?”

Some students just change the problem altogether! They’ll get “rid of the fractions” by writing this problem as $$2t – 4 = 6$$ And why? The idea is to multiply the equation by the LCM, thereby doing away with fractions and working only with integers. There’s nothing wrong with that. But some students forget this step and just remember “get rid of the fractions” and so they do! By magic! Other times, teachers just avoid giving problems with fractions in them because “the focus is on the algebraic process” and argue that “there is no need to further complicate the understanding of the algebraic process by using more ‘complicated’ numbers”.

When first introducing Algebra to students, I can understand the pedagogical and didactic needs to give simple examples that do not lose focus of the new concept. However, once the basic algebraic process has been taught, it is essential to ‘complicate’ the problems with fractions (among other things).

What compounds fraction phobia is not just that problems with integer coefficients and constants, but also the fact that the solutions to these problems work out as integers! This leads to a cascade of neglect with fractions.

It is as simple as asking that students solve not just \(3t – 8 = 19\), but also solve for \(t\) in $$\begin{aligned} 3t – 8 & = 20 \\ 3t – 8 & = \frac{20}{3} \\ 3t – 8 & = \frac{19}{4} \\ 3t – \frac{2}{5} & = \frac{4}{7} \\ \frac{3}{4}t – \frac{5}{6} & = \frac{7}{8} \\ \frac{a}{b}\cdot t – \frac{m}{n} & = \frac{p}{q}\end{aligned}$$

And of course play with the signs.

Why am I harping about this? Because the sooner students see fractions and the more regularly they are required to use them in all problems, the higher the likelihood that students will successfully progress further with math coursework. Right now, many students are bureaucratically progressing further.

Without fail, in every class I have taught, regardless of college or university, fractions are the single largest stumbling block for students. And rather than address the issue anywhere, there is this strange (subconscious) philosophy to pander to the weakness by constructing problems that only require working with integers.

For example, consider Factoring Quadratics. Students are taught how to factor $$x^{2} + 4x – 12$$ by “finding two integers that multiply to -12 and add to 4” — and in this case the factoring is \((x + 6)(x – 2)\). But, when asked to factor $$x^{2} + 4x – 13$$ students are dead in the water.

Worse yet, the fraction avoidance is not the only problem, there’s also the “small integer” bias that is also culprit. I asked students to factor $$x^{2} + 17x + 60$$ Many were unable to “see the factoring” because they couldn’t think of two numbers that multiply to 60 but added to 17. What’s amazing is that nowhere in my class have I emphasized this method of factoring. In fact, I have done nothing but harp on completing the square as a general method. They just have remembered this “see the factoring” method from high school or elsewhere.

The general indifference to students’ fractions woes begins in precalculus. But if for the bulk of a student’s math career, fractions are ignored because “they are difficult and are not material to the concepts”, then all that’s been done is disillusion students about their abilities and about what is necessary. But the material in precalculus is, well, a prerequisite for Calculus. And Calculus is pretty much a mainstay for many of the STEM fields. No fractions? No STEM. Find a different career path. Poof.

The solution then isn’t to further hide fractions away. The solution is to incorporate them early and often into Algebra. Part of content mastery, is not just conceptual understanding — the concepts, especially at the basic math and Algebra level are simple — but also mechanical understanding. If a student can solve $$3x – 8 = 19$$ but is unable to solve $$3x – 8 = 20$$ we don’t have mastery. We have the beginnings of an understanding of how to work with some of the symbolic and numeric manipulation, but not in adequate depth.

I see a lot of things like $$\frac{2z + 4}{z + 2} = 3$$ because “the \(z\)s cancel and we have six divided by two” or $$\frac{2z + 4}{z + 2} = 4$$ because “the \(z\)s and the twos cancel and we have just four left over”, etc. Yet, students are completely capable of reducing $$\frac{18}{4}$$ to \(\frac{9}{2}\).

So whatever the system is in place that moves students through, there are too many knowledge gaps with working with fractions.

There is no need to ignore fractions in an Algebra class. For anyone teaching Algebra, it is a simple matter of replacing all integers in problems with fractions that when fully-reduced do not resolve to an integer.

By using fractions, it also becomes immediately evident to see which students are just guessing their way into solutions because they know to expect integer solutions and how many actually understand the mechanics.

I hear too many students say “I got this weird number. Is that ok?” where that “weird number” is a fraction! This is a great disservice that’s been done to students into letting them think that if the solution isn’t an integer, they must’ve done something wrong!

Fractions are part of the whole Algebra thing! Don’t ignore them! Don’t shy away from them! And don’t stick with integer-based problems because they’re easier to grade! And while you’re at it, ask students to solve for \(t\) in $$\frac{\sqrt{3}}{8}t + \sqrt{\pi} = \frac{14}{9}$$