Algebraic Expressions: Guess the Misconception
All you algebra lovers out there will have been wondering when the language of maths will be featuring in this Guess the Misconception series? Well, we all know students don’t have any miscocneptions whatsoever with algebra…
If only.
Yes, this week on Guess the Misconception we have an algebraic classic. What do you think the most common wrong answer given by students to this question is:
And here are the results:
Yes, C is the most common wrong answer, and only 39% of students attempting this question get it correct. Pretty worrying, right? So, before we propose a solution, let’s take a look at some student explanations to try to diagnose the cause of the problems.
Answer A
The main attraction for students selecting this answer was the apparent lack of a multiplication sign:
i think this because this has a division on its own and has nothing else and all the others have multiplication involved
This may require a return to the basics of algebraic notation.
But it also unveiled a whole host of other fascinating misconceptions:
Because it could be a top heavy fraction so the biggest number could be at the top and the smallest number at the bottoms where as all the rest are normal fractions
This is because all the others say that the fraction needs to be times to m and k, but on this, it doesn’t say this.
Answer B
Here a key issue was the fact that only one of the letters appears to be being divided by 2:
A, C and D all have expressions of mk being halved, in whatever format, while B does not.
This is a specific misconception related to the order you carry out divisions and multiplications and the effect it has on the overall answer. Failure to recognise that the order does not matter may come from a misunderstanding of the order of operations (BIDMAS), and may be rectified by means of numeric example. Having students calculate 3 × 4 ÷ 2, and demonstrate that that answer is exactly the same as 3 ÷ 2 × 4 may help.
But once again, other misconceptions were unearthed:
Because all the expressions have m and k multiplied together whereas this expression is multiplying k with m and a number.
I think this because the letter have been separated
Answer C
Finally we have the most common choice of wrong answer. Students choosing this failed to recognise the equivalence between dividing by 2 and multiplying by a half:
because the other 3 have m times k and the divided by 2 but C has 1/2 divided by m times k which isn’t the same
every other equation had either a 2 in the equation or at least equal 2 but this one shows half
For me, it is both fascinating and worrying that these students do not recognise that answers A and C are the same. Before continuing on with algebraic manipulation, these students would probably benefit from being convinced that multiplying by a half and dividing by 2 are the same. Some simple examples with numbers and a calculator may suffice. Or, depending on how capable students are with fraction multiplication, they could be convinced by doing something like 5 × ½ by first changing the 5 into 5/1.
Correct answer
Interestingly, many of the reasons given for the correct answer did not rely on slick algebraic manipulation, but instead made use of subsitution:
Let’s say m = 2 and k = 5 and lets do A (MK / 2). The answer is 5. Then you do B (M/2 x K). The answer to this is 5. Then you do C (1/2 x MK). The answer to that is 5. Now we do D (M/2 x K/2) which leaves us an answer of 2.5 therefore this answer is wrong
Indeed, those answers that correctly used algebraic manipulation were few and far between:
because all of them were = to ‘mk / 2’ and the m/2 k/2 = mk/4 which is not the same to other mk/2
So, how do we help these students? Well, once we have dealt with the specific misconceptions suggested by each of the incorrect answers, more practice dealing with algebraic terms and expressions is needed, especially when we consider the host of other misconceptions with algebra revealed by the student explanations.
The substitution route favoured by many students who got the question correct is a good one — indeed the beauty of many algebraic topics, including solving equations and rearranging formulae is that answers can always be checked at the end by substitution. But relying on substitution alone that is not enough.
One of my favourite activities to develop a deeper understanding of algebraic equivalence is “equivalence” from Don Steward’s incredible Median blog. Students are encouraged to substitute numbers into expressions, and then reflect on why certain expressions result in the same answer. It is in addressing the question of why that key algebraic insights can be made and discussed.
Why not try this diagnostic question out on your students, either in class or as part of a homework, and see how they get on? Talk about the correct answer, and also the wrong ones. Better still, you can ensure students receive a regular diet of quality questions like this — together with all the teacher insights you can ever want-by setting up our free schemes of work. We have free maths schemes from Year 1 to GCSE, with all the awarding bodies represented. Just click here to get started.
Have a great week
Craig
Subtle advertisement alert: my book How I wish I’d taught maths, which contains an entire chapter dedicated to the practicalities, benefits and considerations when using diagnostic questions in the classroom, is available to buy from Amazon and John Catt Educational Ltd.
