# Fixing wrong ideas about coefficients (e.g. subtract 3 from 3x to isolate x)

I tutor a student (9th grade, United States) who is in an algebra class. She consistently makes mistakes when dealing with coefficients.

• The most common one is attempting to subtract away a coefficient, e.g. if I ask how to isolate $$x$$ in $$3x = 9$$ she'll try to subtract $$3$$. This will occur if I use a vague question such as "How do we isolate $$x$$?". If I point out that the $$3$$ is multiplying the $$x$$, and ask how to reverse that, she will correctly identify division as the answer. Then, the next problem, she will try to subtract the coefficient again...
• She will also try to keep variables around when their coefficients become $$0$$, e.g. she did $$2x - 2x = x$$. When pressed she said that the 2's canceled out and left the $$x$$ behind.

This clearly represents some sort of conceptual issue with how to handle coefficients, and I'm lost as to what more I can do. Things that I have tried include:

• Solve a similar problem in front of her, explaining each step. She'll make the same mistake on the next problem, or if not the next one, the one after that.
• Repeat that we need to do the opposite of the operation to undo it. This works only if I also point out each time that the operation a coefficient represents is multiplication. Otherwise, just saying to reverse the operation has her trying to subtract coefficients.
• Get out a virtual (we meet online) set of algebra tiles. She said they were too confusing; her class doesn't use them.
• Use word problems. Even when she's just written down an equation where she knows that, e.g., $$3x$$ represents 3 chocolates of unknown cost, once the problem is turned into numbers and letters she will lose sight of the original meaning.
• Use past problems, e.g. "What did we do with the coefficient last time?". This has the best track record but it's not addressing the actual issue, it's just her blindly applying a prior method.

If I can figure out what's at the root of this issue and address it, I think it will go a long way towards her being able to do algebra independently. (Again, currently I need to prompt each step with leading questions - "The 3 is multiplying the $$x$$. What is the opposite of that? How do we reverse it?".) I get the sense sometimes that when she's stuck, she just guesses operations she knows are possible and hopes they're the correct thing to do; she doesn't appear to have a sense of what makes sense to try.

What else can I do to attack her issue with coefficients?

From very limited experience: have you tried writing $$3 \times x$$ (or using one of the various stars, dots, etc.)?

It might be that it’s not obvious that there is a multiplication here: you say that pointing that out leads immediately to division. This coefficient notation hides some things that no other core operation (addition, subtraction, division) has.

If that works, then maybe you can focus on connecting coefficient notation to operator notation.

• If you use a star or dot, I agree with this. But definitely not an x, which looks too much like the variable x. Nov 16, 2023 at 18:39
• @SueVanHattum: we always use an "x" as the multiplication sign, and always the cursive $x$ as the variable. I am not saying that this is a good idea (I strongly think it is not - as many things in our math education) but the rules are the rules.
– WoJ
Nov 17, 2023 at 12:35
• "the rules are the rules"?? We don't "always" use that x. We have five ways of denoting multiplication. x is generally used in arithmetic only. * is used when math is typed and nothing better is available. A dot or () is used when needed. And the use of no symbol can be used between variables or a variable and a co-efficient. * and the dot are best in this situation, imo. Nov 17, 2023 at 13:52
• Dot can be conflated with a decimal point, which would defeat the purpose here I think. Asterisk is probably the least confusing notation for multiplication that doesn't require parens. Nov 18, 2023 at 20:19
• Do you guys not think D. Ben Knoble's suggestion of using 3×𝑥 or any of the various stars, dots, etc, was meant as a means of breaking the mould… of breaking the student out of whatever furrow was stopping her from following the standard notation? Nov 19, 2023 at 22:21

If it were me, I'd try building on what you mentioned here:

Use word problems. Even when she's just written down an equation where she knows that, e.g., $$3x$$ represents $$3$$ chocolates of unknown cost, once the problem is turned into numbers and letters she will lose sight of the original meaning.

You mention that she loses the connection between the symbols and their meaning. This sounds like the root cause behind all her issues. Without that connection, she's just pattern matching, which means everything comes down to purely stimulus-response muscle memory.

What I would try first is having her replace the variable with an actual word. If $$3x$$ represents $$3$$ chocolates of unknown cost, try having her write $$3 \textrm{ cost}$$ (maybe even more explicitly $$3 \cdot \textrm{cost}$$) or something similar that bridges the gap between the verbal and symbolic representations of the problem. Maybe some practice doing algebraic manipulations with that sort of "mid-way" notation will help her make the connection between the symbols and their meaning.

If that doesn't work, then unfortunately you might just need to help her build muscle memory on valid manipulations. Of course that kind of approach can't carry a student through advanced math, but if she's having this much difficulty with early algebra, then future-proofing strategies for advanced math might not be too big a concern...

• I think words are a great option. I often talked of it as "speaking the same language". Maybe not the best term when you get into things like 3xy. But words and the emphasis on needing to be of the same type can perhaps really help not only this basic algebra, but also things like units. Nov 18, 2023 at 12:13

This is a classical case of having learnt algebraic manipulation without meaning. The first thing to try is to build up algebra again from the foundations. You already tried using (virtual) concrete representations; I would recommend finding some such that the pupil accepts. A second is teaching flexible thinking and broad strategy use.

In any case, doing more drills is unlikely to help - the pupil, most likely, has eight grades of drills in their history, and they have not worked thus far, so they are unlikely to work in the future.

# Different levels of abstraction

1. Think of the problems that have a scale and three apples on one side, one watermelon on the other, and price or weight or whatever is given to one side, and you are asked for the price or weight or a single apple. I think the mistake you note is very unlikely to be made here. Learning to draw an equation or a word problem like this can be helpful. Other concrete representations are fine.
2. As a next step, you can write an equation with three apples, equality sign, 13 money (3 apples = 13 money). How much does a single apple cost? Maybe the pupil solves this in the head without writing anything down, but then you can ask questions and have them write down the reasoning or step, same as would be done with an equation.
3. Next step might be to write $$3 \text{apple} = 13$$. Here you treat apple as a variable.
4. The next step would be to use $$a$$ for apples.
5. And then go to using $$x$$ or other arbitrary letters.

The difficult skill the pupil needs to learn is recognizing their own uncertainty. Then they can and should move to the less abstract representation and operate there until they know what can and can not be done. This is an important metacognitive skill.

# Inflexibility in tactic use

Pupils who have difficulty with mathematics are often very inflexible in their strategy (use of different tactics). This can already manifest in very simple problems, like calculating $$5+8$$. A young child would first count from one to five, then from one to eight, and then from one to thirteen. A more advanced tactic would be to count from six to thirteen, and even more from nine to thirteen. And then one might do $$5+8 = 3 + 2 + 8 = 3 + 10 = 13$$.

You can find similar inflexibility in pupils who tend to have problems with mathematics later. They see mathematics as a collection of arbitrary rules, rather than things that one can use in several ways and play around with.

The recommended best practice in classroom use is to get several different pupils show how they have solved a given problem In one-on-one tutoring you would have to provide alternate tactics yourself. They key point is to not suggest that the pupil's own solution was bad, but rather show that there are many ways ahead.

# Varying types of exercises

The pupil is likely to meet pure calculations sans context, and also word problems where they are supposed to extract a calculation from and then proceed with the calculation.

How about giving them a solution (it could be the value of the unknown, or even an equation they should give a corresponding word problem to) and asking them to produce a problem where the solution is, well, the solution? How many different ones can they create?

Or give them an equation and ask to provide a visualization or concretization of it?

Or give some information, like the menu with prices at a restaurant, and ask them to formulate equations based on a family visiting that restaurant. A more relevant scenario is of course better.

These types of exercises require understanding of algebra and equations, but are more free and playful than just solving an equation, and do not have an algorithm one can follow to solve them. They are likely to lead to a more robust understanding of the material. They should not completely take over; variety is the goal.

• "3 apples = 13 money" - you are comparing apples to money, which is wrong. It is actually 3 apples × money per apple = money, or 3 apples × money/apple = money. Dividing both sides by apples, we get money per apple on both sides. Nov 17, 2023 at 15:41
• Yes, "apple" actually means "the price of an apple". I would not worry about this at a stage where the very idea of an equations and solving them is not grasped yet. Nov 17, 2023 at 16:48

Unfortunately I do not have experience in your (the OP’s) role, but at one time I was in the shoes of your student. When I encountered rudimentary algebra for the first time I was completely lost and none of those weird rules made sense to me. By a lucky coincidence, I took up (computer) programming a few months later and that made exactly this conceptual layer of what it means to do arithmetic operations with letters instead of numbers clear to me in very short order. I proceeded to do very well in middle and high school math with very little effort.

I think the core principle behind it is that one can very easily explore the relationship between abstractly described operations (the code) and the concrete inputs and outputs (and even the intermediate values in variables if your programming environment allows that) by running a program many times, seeing how the output changes when you vary the inputs or tinker with the intermediate steps. It is like bringing equations to life.

On paper this is much more cumbersome; since each “run” requires a lot of arithmetics I find it hard to imagine a student who would actually calculate everything out more than maybe two or three times while a computer does all the arithmetics instantly and flawlessly (and seeing stuff appear or move on screen can of course be fun and exciting).

Also, if you expect an operation (such as subtracting coefficients) to work differently than they actually do, the computer immediately shows you the difference between your expectations and reality with cold hard calculation.

Of course, it doesn’t seem very likely for your student to be particularly interested in programming, so this answer may very well be for the exclusive benefit of hypothetical future readers. But perhaps there is a chance you could still make use of this idea: there are quite a few graphical programming environments out there – Wikipedia lists a whole bunch (maybe Scratch seems plausible? I have no experience with any of them) – that are supposed to be extremely accessible and easy to use. You could make a program relating to one of the practice problems (e.g. a program that calculates the price of various fruits or chocolates) and study together what happens with the results when the inputs change or when you mess with the intermediate steps.

• $[\ldots]$ one can very easily explore $[\ldots]$ by running a program many times, seeing how the output changes when you vary the inputs or tinker with the intermediate steps. It is like bringing equations to life. On paper this is much more cumbersome; --- Regarding this, you might find my Paleozoic era computer experiences interesting. See my several comments to this Mathematics Educators answer. Nov 17, 2023 at 11:48
• This is a big problem with modern mathematical pedagogy. Many many many mathematics teachers think they know best how to teach mathematics, but actually they do not. They do not realize that precision is so extremely crucial, and it takes other teachers (e.g. programming teachers or even compilers) to actually teach students precision, but it only works if they are motivated since it is "not in their syllabus". Horrible. 'Cold hard' computation is really good, and it is ironic that even the adjectives "cold" and "hard" are used stereotypically and further the pedagogical misconception. Nov 17, 2023 at 15:38

tl;dr You might just work out whatever algebraic manipulations the student suggests, showing them the results of their ideas until they get a hang of it.

### Show them how the math would work out.

It might help to show the results of the student's ideas.

For example:

1. The most common one is attempting to subtract away a coefficient, e.g. if I ask how to isolate $$x$$ in $$3x = 9$$ she'll try to subtract $$3$$.

Okay, so starting with the equation and then subtracting $$3 ,$$ \begin{align*} 3x&=9 \,, \\ 3x \color{red}{-3} & = 9 \color{red}{-3} \,, \\ 3x - 3 & = 6 \,, \end{align*} so what next?

2. She will also try to keep variables around when their coefficients become $$0$$, e.g. she did $$2x - 2x = x$$. When pressed she said that the 2's canceled out and left the $$x$$ behind.

Okay, so let's cancel those $$2's ,$$ $$2x - 2x \,, \\ \left(2 - 2\right) x \,, \\ \left( 0 \right) x \,, \\ 0x \,,$$ so what next?

Or, did you want to cancel the $$2's$$ by division? If so: $$2x - 2x \,, \\ \frac{2x - 2x}{2} \,, \\ \frac{2x}{2} - \frac{2x}{2} \,, \\ x - x \,,$$ and then what next?

Basically:

2. Ask the student what to do next.

3. Interpret the student's idea as something mathematically valid.

• If there're multiple reasonable interpretations, maybe choose a few good ones to demonstrate.
4. Show how that mathematically-valid idea works out.

5. Loop back to Step-(2).

They might end up bumbling around a bit, suggesting weird ideas that might make some busy-work for you if the results of their ideas don't reduce so well. Still, if you walk them through the steps and help them to understand their options, even seeing how ugly the math can in a wrong direction might help them understand why such a direction wouldn't be one to go in.

### Don't do fake math.

They might suggest something that wouldn't make sense. For example, from

The most common one is attempting to subtract away a coefficient, e.g. if I ask how to isolate $$x$$ in $$3x = 9$$ she'll try to subtract $$3$$.

, they might've been thinking something like \begin{align*} 3x&=9 \,, \\ 3x \color{red}{-3} & = 9 \color{red}{-3} \,, \\ x & \neq 6 \,. \end{align*} It might be best to ignore such interpretations entirely.

If the student presses for such an interpretation, you might just tell them that you don't understand what they mean. You only understand math, and you just need them to tell you what they want to do in terms of math.

• The last interpretation (3x - 3 -> x) is definitely what she was thinking; when I let her get further by herself she will confidently do 4x + 10 -> 14x (example from last session) Nov 16, 2023 at 16:11

A thing that I do in my low-level college courses is to spend a beat with each new example/exercise to inspect the statement and make sure we all agree with what we're looking at. "What operations are here?" is among the most important questions; even pointing at each connection as needed.

I notice in the OP's question they lean on the word "coefficient" many times (10), and "multiply" not so much (3). I'd suggest, with this student, that they lean more on the latter word to make sure they're reading the statements correctly.

For the next several equations they look at, ask the student, "What operation connects the $$3$$ to the $$x$$?", and so on, and see if the expected decoding gets more natural.

Following up on Daniel R. Collins' suggestion, teach the student a step-by-step way to solve and check problems. For problems that are given to her in the terse algebraic format (like "3x = 9", which omits the multiplication signs), the first step needs to be:

• Always translate the problem into a format she understands. And write out the translated problem. Specifically, show the multiplication signs. For example, "3 ˟ x = 9" or "3 · x = 9".
• In all subsequent steps, always write out the multiplication signs.

This will cause her to always be slower than her classmates. But it is better to be slow and right, than fast and very wrong.

I'm not a teacher, so maybe I don't get your advanced teaching techniques, but why isn't anyone explaining what 3x means in reality? When I have 3 apples, I can show 3 apples. Why aren't you breaking down the x? Instead of 3 dot x or 3 star x, why not show them X + X + X = 6 then point out they need to divide the 6 into 3 pieces to match the Xs, so you get X+X+X = 2+2+2.

I think you need to work on muscle memory, more than concepts. You have given her the concepts before, but clearly it wasn't sufficient. No problem with repeating the "why" also, while doing the drill. But as a "side", not the main course. There's a decent chance that the concepts start to sink in AFTER she masters the monkey skills, not before.

Do imitation and practice. Including very small step (within a problem) and immediate imitation by her. And very gradual removing of scaffolding (say unimitated problems, but still step by step. Consider also, a very easy problem type and only gradual increase in difficulty (more gradual than you think reasonable).

When she makes a mistake and gets it wrong, make her repeat the same problem. Not just the step she missed, but the whole thing (line by line writing it out). I did this to myself in AP Calculus and it helped immensely. It's much to easy to assume that a failure ("dumb mistake") was corrected, otherwise, but often it's not.

The tiles were a decent try, but since you have low contact hours, I would not push something that is a distraction from what she's doing in school. Word problems are even more contraindicated (1. Word problems are often harder in terms of cognitive load. 2. you tried it and it didn't help.)

I don't know everything about how you are teaching her, but there were a few things that made my spidey sense tingle. Bullet one you're actually getting partial success. Lean into that and do things that build towards longer cycles efore she forgets. (Probably keeping the problems more similar). Bullet two of the things tried reads too abstract to me (and I'm good at math). And then on bullet five, you have something good happening and want to dismiss it since it's "blind procedure following". Would you rather succeed the wrong way or fail the right way!?

Good luck and stick with it. Half a loaf is better than none. This is a big transition for the kid and harder than many people realize.

P.s. Take a look at this essay. (Ignore the discovery based learning...that's not the key issue here...but other things like cognitive load and prealgebra difficulty should resonate. If they do, I would read more of what he has to say in favor of smaller increment instruction.)

https://fillingthepail.substack.com/p/peter-liljedahl-wants-to-make-kids

• The blog post seems to be behind a subscription, but a readable copy is saved on the Wayback Machine. Nov 17, 2023 at 16:51
• Thank you, The Amplitwist. Nov 17, 2023 at 17:10
1. Without math, you can ask what is left 8f you take one apple from 2 apples. Write down algebraic expressions for one apple and two apples, and abbreviate apples by a. So she may know how to compute 2a-a.

2. With math, it simply distributive rule. It tells you why 30-10=20, why 2 mill-1 mill=1 mill. And why 2a-a=a.

3. More rigorously, American kids need to write down, at least once in their whole life, that 2x-x=2•x-1•x(also read out 2 copies of x minus one copy of x)=(2-1)•x=x.

After all these, never be mad at kids if they keep making similar mistakes. They may just want to tease you.You just repeat the above for them, say, 10 times, like how we train kids to learn Chinese characters. Good luck with your journey and good luck with your students. In particular, if she starts to learn Algebra I in 9th grade.

Have you tried to use the balance metaphor to solve the equation?

To solve the equation $$3x=9$$ using the balance metaphor, you can imagine a scale with two sides in balance. On one side of the scale, there are three bags (each containing a quantity 'x'), and on the other side, there are nine coins.

To maintain the balance while finding out how much is in one bag (one 'x'), you would take away the same number of coins from each side. So, you would remove one bag (containing 'x') from one side and three coins from the other side, doing this three times.

Each time you remove a bag and three coins, the balance is maintained. After three repetitions, you end up with one bag on one side and three coins on the other side, showing that the contents of one bag (one 'x') is equal to three coins, hence $$x=3$$.

There are some interactive applets that implements this metaphor on the web. See, for instance: https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Pan-Balance----Numbers/

Maybe a bit of a lateral solution compared to the other answer here, but given you're meeting online, have you considered using emojis to represent your unknowns? This can help keep things tangible.

$$3x$$ doesn't look like 3 apples, but $$3🍎$$ seems much easier to intuitively read as "three apples".

Some people are good at abstraction. You could use virtually anything to represent $$x$$ and they'll follow along. Others get caught up in the symbology even though the specific symbol is irrelevant (i.e. only the consistent usages between instances of the same unknown value is relevant).

Given her lack of intuitive understanding of the concepts you've listed, it seems highly likely that she's not seeing the wood for the trees and instead fixating on the symbols you're using. A more intuitive syntax might help lower the entry threshold for her.