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?