# Scepticism as the cornerstone for not making mistakes in arithmetic/algebra etc, especially for students who relentlessly make every possible error

As a maths tutor, some students I have tutored don't just make the odd mistake in arithmetic (including fractions) and algebra: they make every possible mistake and regularly.

My go-to approach for helping these students is to show them how to correctly apply the rule they have just made a mistake on, as and when they make that mistake.

However, for these students, their mistakes are relentless, and whilst the % of errors may decrease over time, once I show them a correct way of doing it, this % may only decrease from 80% of the time to 20% of the time a few months later, which still isn't very good, as for arithmetic and algebra, you need to be striving for errors basically 0% of the time.

"But 80% to 20% is good improvement"

No. This is algebra/arithmetic. This is not proof writing, which is a skill that you improve with over time. If you stick to the rules of algebra/arithmetic, then you shouldn't really be making errors at all. And yes, I do go through various checking methods with my students.

Moreover, such students will make every possible error on most future topics. I believe this is due to their, "I don't care about maths: I just want to pass my exams" mindset, which is counter-productive, as the only way to improve on maths is to spend time doing maths (exercises). Anyway, the point is that I think these students need a radical change in mindset/approach in order to succeed in maths.

I wonder if, for such students, an entirely different overall approach is more productive; namely, the following "sceptical" approach, which is the one I have always gone by:

Don't use a rule until 100% convinced:

1. The rule works
2. You know when the rule works and when it does not work

and more importantly,

If you are not 100% convinced the rule works, do not use it (other than to test it), and keep questioning it's validity.

They can even write an "investigation into the validity of rule X", up until the point they are convinced the rule works.

Then, at some point, they will be naturally more and more convinced of it's correctness, as they will have run out of questions as to why the rule might not work.

There are many obvious benefits of this approach, including the fact that it forces the student to actively engage in the material.

However, I'm doubtful of this approach because I doubt these type of students' abilities to correctly apply scepticism and questioning of the method. Another criticism is that this approach takes more time to carry out at first; however I don't buy this criticism as the approach probably saves time in the long run.

I'm interested to hear what other maths educators think. Has anyone tried this approach with relentless error-ridden students, and did this approach work?

• I'm not sure that being skeptical about rules will help such students much, but at any rate before the skeptic's approach you may want to try a formalist's one: for every operation state clearly the general rule you are about to apply and then apply it to the letter. Such mistakes as you described, IMHO, often stem not from having an incorrect rule in mind, but rather from having no rule at all and proceeding by "general feelings" alone. Commented Aug 21, 2023 at 21:08
• @fedja That's a good point that I cannot rebut and must take on board. I am thinking of making sheets that clarify exactly what the rules are (for example, BIDMAS rules). I have not seen this done in any book thus far (although admittedly, I haven't finished reading Wu's, "Understanding numbers", so maybe the rules are ultra-unambiguous in there). But it never fails to amaze me that the rules are not ultra-unambiguous in school-level textbooks like GCSE textbooks. A Level books are a bit better, but imo they leave a bit too much for you to figure out for yourself. Commented Aug 24, 2023 at 11:41

In part agreeing with @fedja's comment... but perhaps aggressively pursuing the idea wanting to have your students look critically at "feelings/impulses".

Namely, I am not a fan of math-as-rules. I would want to sell the idea that math "is about something". So no "authority" has the power to dictate what it is, beyond conventions on the order of speed limits or parking restrictions on weekends.

Conceivably, some students could be reached by suggesting that math is a narrative about (certain aspects of) reality, albeit in a specialized language.

One analogy is that, although (e.g.) English has various approximate grammar rules (changing with time, of course), most discussions of math in English are not marred by grammatical errors. Yes, sometimes accidental ambiguities dues to grammar-fails can impede the mathematical narrative...

One bottle-neck here is that many students seem to have an accumulated math culture that math is about (ineffable) rules... So it's hard to credibly argue against that. E.g., even in the grad courses I teach, it's only been in the last few years (with an ever-balder head, grayer beard, etc.) that students seem willing to believe me when I argue that there is a genuine reality behind definitions and notation... often contradicting their impression from typical undergrad (and grad) textbooks. :)

So, yes, it is tricky to talk to people who both want rules and want to not have to follow them carefully... No easy solution. :)

I'm sorry, but I find your attitude towards your students somewhat disturbing. In particular:

This is algebra/arithmetic. This is not proof writing, which is a skill that you improve with over time. If you stick to the rules of algebra/arithmetic, then you shouldn't really be making errors at all.

First, algebra and arithmetic are also "skill(s) that you improve with over time". So is just about everything else. When we are born, we can pee and we can suck and that's about it. Everything else is learned. Including arithmetic and algebra (but also tying your shoes).

Second "stick to the rules" is exactly the wrong thing. Which rule? If you look at various social media, such as Facebook, you will see that knowledge of which rule to apply when is lacking and, while it may be obvious to you and to me, it is not obvious to all. Clearly, it is not obvious to your students.

I also doubt that the problem is that your students don't think the rule is "correct". I would be surprised if a student said that, and, if one did, it would probably be a student who was very good at math, not one who is very bad.

Rather, if you have taught the rule well, so that they understand it, then the problem is likely to be which rule to apply when, and in what order.

Also, if your students' only goal is to pass exams, then you can motivate them by saying "well, the only way to pass exams is to practice".

Finally, if your students get down to 20% errors, then they will, in fact, pass their exams. Not every student does any math after what is required, and many would be pleased to get an 80. Indeed, those are precisely the students who are likely to hire a tutor. Students who hire tutors get worse grades than those who don't, because students who are doing well don't hire them.

• Having only read your first sentence... My teaching approach towards my students is not, "I should punish them or make them feel bad for not being 100% accurate at arithmetic/algebra". However, this is certainly how I feel. How is one ever meant to do well at 5 or 6-mark GCSE exam questions where "rearrange and solving an equation" is the "easy part", if they only correctly simplify $4a + 6 -2a + 7$ 80% of the time? The answer is: they don't! These questions cost them a lot of marks, and that's due to them not learning rote rules well enough which I believe is often due to laziness Commented Aug 23, 2023 at 13:33
• of practice... but also/ instead, due to lack of understanding and misconceptions of what the rules are. Commented Aug 23, 2023 at 13:35
• First, algebra and arithmetic are also "skill(s) that you improve with over time". Also, I don't expect students to go from, "never having seen algebra before" to "solving a linear equation in 10 seconds" in one month or even one year. But this is not what I am saying and is not relevant. Commented Aug 23, 2023 at 13:38
• "Students who hire tutors get worse grades than those who don't, because students who are doing well don't hire them." But students who have tutors tend to improve, and so may do better than students who don't hire tutors. Do you have any evidence to back up your claim, rather than a bold assertion? Commented Aug 23, 2023 at 13:47
• @AdamRubinson RE: "Students who hire tutors get worse grades than those who don't, because students who are doing well don't hire them." This statement isn't arguing that no student should hire a tutor but is just emphasizing that if the entire population has an average grade of 70 and the average grade of students who pursue tutoring is initially 50 then the initial average of the remaining non-tutored students will be higher. Self-selection into tutoring makes the non-tutored-student population unrepresentative of the tutored population. Commented Sep 9, 2023 at 12:18