Negative decimals: Guess the Misconception
This week on Guess the Misconception we have an interesting question that has been causing thousands of students all around the world (including my Year 11s!) no end of troubles.
What do you think the most common incorrect answer is?
How did you get on? Well, here come the results…
Almost two-thirds of students got the question incorrect. Scary, hey? And both C and D are vying for the title of the most popular incorrect answer. Why might this be? Well, let’s take a look at some actual student explanations to find out more.
Answer B
Only 8% of students selected answer B, but this could represent a student in your class, so it is worth investigating further. Students selecting B seem to be inventing some interesting rules for dealing with this kind of question:
Because you do 3 take a way 2.5 and then add on the minus sign. So the answer is -0.5
Answer C
Almost one-quarter of students went for option C. Their explanations reveal a combination of a misinterpretation of the word more in the question (which, I concede, is a little ambiguous), together with a misunderstanding about how to add positive numbers to negative numbers. These two student explanations capture this clearly:
Answer C, because if you add 3 to 2 it equals 5 is you put the number together it will equal -5.5
In my opinion, I add 3 more to -2 equals -5, then i put the 5 back on so its -5.5
Answer D
Here we have the most popular wrong answer. Students selecting this answer have moved on the number line in the right direction, but then something has gone wrong. Explanations suggested this was either due to a common mistake when counting to start the count on the initial number instead of the jump to the next number, or an issue with what happens when they have to cross zero:
I used a number line to get my answer i went up in 0.5 because it started with -2.5
Because the 3 more is a whole number you can ignore the 0.5 and just add on the 3 to the 2 remembering that the “2.5” was a negative 2.5(-2.5)
As I mentioned at the start, this question caught out my Year 11 class. I learned from my previous mistake, and so before teaching Vectors I made sure I asked them this question so I could identify any issues with this concept that could come back to haunt students when dealing with the algebraic manipulation that often comes into play at the end of the more challenging vector questions. And sure enough, when I asked the question at the start of the lesson there were issues.
So, having had the usual class discussion based on my students’ answers, I went through the following exercise with students:
This is another example of my dabbling with the principles of variation. I want my students to notice what happens as the pattern continues so that the answers to questions like the one above do not appear magical but are instead part of a logical sequence of questions that start with things that they are already know. However, at the same time I do not want this to just become a pattern-filling exercise, and hence my prompt for them to cross the “gap of understanding” (I really need to work on a better term for this), and use what they have learned above and apply it to these new problems.
This was all reinforced by constant reference to a number line, my current favorite of which is this from the Math Learning Centre:
This combination of practice and visualisation seemed to help, and Vectors went a lot smoother this year than 2 years ago. But I was also sure to include questions like the one above in my daily low-stakes quizzes and non-topic homeworks over the next few weeks and months.
Why not try the 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 hereto get started.
And if the intelligent sequencing of examples is of interest to you, then I discuss it further in Chapter 7 of my book How I wish I’d taught maths. The book also 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.
Have a great week
Craig
