Place value: Guess the Misconception
This week on Guess the Misconception we have an innocent looking question about place value. What could possibly go wrong ?
What do you think the most popular wrong answer is, and why might a student select it?
And here are the results:
So, less than one-third of students get this question correct.
But things get even more interesting (or perhaps scary is a better choice of word) when we break the results down by ages of students answering the question:
Yes, 10 and 11 year olds answer the question better than 13 and 14 year olds… who answer the question better than 15 and 16 year olds! So, it seems the best thing I could do for my Year 11s when they come to secondary school is to keep my mouth shut lest I cause them any more misconceptions :-)
Before we take a closer look at the reasons students get this question wrong, it is worth considering why the success rate on this question decreases with the age of the student. I have two theories. The first is that as students get older, they encounter more and more maths, and thus have more and more things to remember. Those 15 and 16 year olds may not have remembered what hundredths were, but they may have been able to recall the definition of an alternate angle, or how to calculate the frequency density needed to draw a histogram. Secondly — and very much related to the first point— is the question of when was the last time older students considered hundredths? I know that naming the decimal places is not on our Key Stage 4 scheme of work. For me, it is assumed knowledge, and as such I do not teach it explicitly. But, it appears that my assumptions were false.
Okay, so why do students get this question wrong? Well, let’s dive into some student explanations to try to find out:
Answer A
Students answering A are clearly muddling up hundredths and hundreds. This may be a literacy issue, it may be them not reading the question carefully enough, or — as is the case with the student below — they may simply not know the difference between the terms:
The number 200.358 is made up of 2 hundreds, 0.3, 0.05 and 0.08. The digit in the hundredths place is 2, because it has 2 hundreds
Answer B
Answer B most likely comes from a misunderstanding of how the decimal system works. Students who are familiar with the place value of units, tens and hundreds over-generalise and assume it should work the same for numbers after the decimal point. So, we get explanations like this:
I think the answer is B because there are 2 digits after the number three and for a number to be in the 100’s zone it needs to have two digits after it
Answer D
Answer D represents another form of over-generalisation. Here the student assumes there is some symmetry in the naming of positions either side of the decimal point. So, if the third number to the left of the decimal point is hundreds, then surely the third number to the right of the decimal point is hundredths? That certainly would be nice, but alas, no. Maths is flipping complicated when you think about it:
Because it goes backwards. 256, out of that number 200 is the hundred. This is the same for decimals except that it’s backwards
Because as it is hundredths it will go into the negatives by 3 place values. And the number which is negative by 3 place values is 8
So, what do we do about this?
For me, the answer lies in lots of intelligently varied practice. So, a selection of questions like the following, with students asked to identify the value of the 3, and give its name (e.g. 3 tens, or 3 hundredths) will hopefully help develop the knowledge needed to not fall into any of the traps laid above.
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
