What are the major misconceptions that students have when it comes to simplifying algebraic expressions, or collecting like terms? Specifically, what happens when students are faced with squared and non-squared terms?
Using real life data and explanations from students all around the world from my Diagnostic Questions website, we can find out! :-)
Have a look at the question below.
What percentage of students do you think get it correct?
What is the most popular incorrect answer?
What explanations do students give for the incorrect answers?
Give the question a go yourself, and try to come up with an explanation of the correct answer that would make sense to students:
At the time of writing, this question has been answered 113 times, and has been answered correctly just 46% of the time. The choices of incorrect answer are pretty evenly spread across the other 3 options, with B coming out on top with 24% of students.
To see this data, read student explanations, and filter by things such as gender and age, just visit the Data Question page.
Collecting like terms, or simplifying algebraic expressions, can be a difficult area for many students. I feel it is down to the abstract nature of algebra as a topic - for many students it is seen as a mysterious set of rules that they need to remember and apply correctly. There is a real lack of deep understanding. And the problem with rules in mathematics is that there are so many of them, and so it is little surprise when they are misremembered or misapplied.
This is no more evident than in the responses students give to the question above.
Firstly, let's take a look at the wonderful ways students explain the correct answer:
Correct Explanations for C
Here are some of my favourites:
Because x+2x=3x and you add x2 which makes 3x+x2 and you add the y2 to make 3x+x2+y2
I think this because you shouldn't add the squared numbers you should just leave them. so then you haven left 2X and X which turns into 3X then you add your squared numbers
Because you have a X2 its a totally different number to 2X so you just leave it to one side then add 2X and another X = 3X so then your answer is 3X + Y2 + X2
And remember, as discussed in this video, students who answer this question and get it wrong can benefit from reading this explanations (and more) from students all around the world, hopefully helping to resolve their own algebraic misconceptions.
And what can we learn about students' thinking from their explanations of the incorrect answers?
Incorrect Explanations for A
Students answering A seem to think that x squared is the same as 2x, as the following explanations illustrate:
Because x squared is x x plus the x which makes it 3x plus the 2x which makes it 5x. Then you leave the y squared as it is because there are no other ys in the question.
i think the answer is a because if you break down all of the x squared thats x+x+x then add the 2x which is 5x. finally add y squared and the answer is a.
Because x2+x+2x = 5x all together and y2 is on its own so that would be the answer
How would you help resolve this misconception? Possibly with some number example, showing clearly that 4 squared, is not the same as 2 x 4.
Incorrect Explanations for B
Students answering B may have a related, but subtly different misconception. Here, x's and powers are seemingly being combined indiscriminately:
I think the answer is b because there is 2x and there is 1 x on its own and 1 x means 1 and there is an x squared so 2x + x = 3x and then 3x + x squared = 4x squared you add the y squared so it = 4x squared + y
x squared is X2 but if you put another 2 x's into it then it become 2x squared then add on the other 2x and you get 4x squared. then add on the y2 that was supposed to be added anyway
Because is you add x + x + x + x together you get 4 and your wondering only three well 2x stands for 2 x's also we get the squared from the first x
Again, a number example might help make this concept a little less abstract for the students.
Incorrect Explanations for D
D is an interesting, but all too common one. Students selecting this answer appear to be indicating that they are not aware that things like 4a mean 4 x a, and not 4 + a:
i think this answer because x2+x+2x=5x and = the y on to it it would be 4xy2
So there you go. What a minefield of misconceptions algebra can be! Hopefully by reading other students' explanations, students displaying these common misconceptions will benefit. And if that fails, I find using number explains often helps make the subject of algebra, and specifically simplifying algebraic expressions, a little less abstract.
Below is a video explaining how to access the pages I talk about in this blog post: