Fractions Are Part Of The Whole Algebra Thing …

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 $$\frac{\sqrt{3}}{8}t + \sqrt{\pi} = \frac{14}{9}$$


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6 thoughts on “Fractions Are Part Of The Whole Algebra Thing …

  1. jonathanavt

    Associations are a major part of the way the brain works. It’s not necessarily a bad thing if you can hold most of the problem space in your head. It’s how we can answer 3 x __ = 12 so quickly and easily. But that’s also where it ends. The power of math comes from strategies to transcend these limitations.

    Reply
  2. Gerry Varty

    Great post, Manan… as usual, your insight has uncovered and illustrated a focal point in understanding math. Here’s my take:

    Kids are taught to follow the rules, use standard algorithms, memorize basic facts so that they can be considered math-competent. We measure their proficiency on how well (and worse, fow FAST) they can repeat these mindless tasks.

    And I mean that literally. Mindless.

    3 x __ = 12 … Kids will tell you ‘4’. When you ask why, they respond ‘because it is’, proving that you can’t teach math with simple numbers. Those kids will be largely unable to solve 3.5 x __ = 8.05 , because they can’t identify the answer by inspection.

    The same thing happens with algorithms… out of every group of ‘successful’ students, some try to memorize the set of solution strategies and guess which one to use, some can succeed by remembering which basic problem type goes with which stragedy, and only a few will figure out (for themselves) how the strategies work and why we use them. Those few kids will be the only ones capable of solving non-standard problems, despite the fact that we believe their entire cohort to be ‘successful’.

    Here’s where I think you hit the nail on the head…

    Kids aren’t comfortable with fractions. They are taught fractions as localized ‘points of procedures’, rather than as partial numbers, like decimals. Because they don’t see 1 and 3/4 as ‘one and a bit’ or ‘more than 1, but not quite 2’, they don’t actually see them as numbers at all. They see them as little puzzles to be solved with discrete procedures… top x top, bottom x bottom, common denominators, improper to mixed transforms… and my personal fave, ‘ours is not to wonder why, just invert and multiply.’

    That means that EVERY algebra problem with fractions in it is a non-standard problem. That means (out of the whole cohort of kids we deemed ‘successful’), only those kids who have developed their own insights and flexible strategies will survive.

    The bad part is on us. We know that memorization and blind application of algorithms doesn’t work. It gives us this cohort of kids that WE THINK can do math, and we lament – every year – that surprisingly, they can’t. Then we go back to our rooms and teach the same way again, only louder, hoping for different results.

    Einstein said something about that, once.

    Reply
  3. Dan Springs

    It’s because the only students who LEARNED math in secondary school start go directly to calculus in college. The rest learned to pass the math tests in secondary school. I just observed 50 hours of high school math classes, from basic algebra up to statistics. There was one class for students who couldn’t keep up where the teacher’s goal was to get the students to understand a topic then move on to the next. One of his students was able to use the class to catch up and start to excel. The rest of the students had learning disabilities that weren’t being addressed in general but due to this class were making progress. Other classes had schedules dictated by the standard. X number of topics to be covered in 1 term.

    I heard a professor of physics on NPR talking about the large number of physics students who, in their senior year, had only a poor grasp of the basic laws of physics. He said that the students who had developed a full understanding of the basics of physics would have discovered them on their own. That is what I observed in the various levels of high school math but for different reasons. The problem with universities is that the professors have no teaching credentials. In middle and high schools is that the Math curriculum is a distance race and the only way to cross the finish line is to keep a fast steady pace and not stumble or even rest. The only students who can keep this pace are the ones who see beyond the pure math (the last time they saw practical applications were problems like Jimmy has 2 apples and Jenny has 3…) into the real world. The rest stumble and fall when their only option is memorize or die.

    Reply
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  6. AnnMaria

    I started teaching math at the middle school level and now teach graduate students. It never ceases to amaze me what students don’t know. I agree wholeheartedly that students need to just mess around with numbers more. “Cooking the data” so every answer is an integer gives a false picture of the real world. I don’t understand why middle school administrators think students can handle learning about sex but they need to be protected from the answer = 4/7

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