Mind Your Language (1)

Look at this number and say it out loud: 6.125
How did you say it?
'Six point one two five'?
'Six point one hundred and twenty five'?
'Six decimal one two five'?
'Six decimal one twenty-five'?
A few years ago I would have paid no attention to the language that I used to say decimals. In England (where I first taught) I used to say 'six point one two five' so when I first came to Canada it amused me somewhat to hear 'six decimal one two five'.
However, research by Sue Willis (First Steps in Mathematics) made me realise that I wasn't saying decimals properly and that this was not helping students understand decimals.

It is no big secret that decimals are one of the big ideas that really seem to stump some students (and adults). I've taught a lot of tricks to help learners cope with decimals but these were just papering over the cracks: they didn't help students understand the quantity of decimals.
More importantly, I couldn't fathom out what students thought decimals meant. Using a diagnostic provided in First Steps made it a lot easier for me and my colleagues to see what students were thinking. For example, what are students who make this error thinking?:

These students are saying the numbers incorrectly. in example (iii) they are saying 'three point five hundred twenty one, three point six and three point seventy five'. And since 521>75>6 then 3.521 is the biggest number of the three. For example (v) they simply ignore the leading zeroes and say 'four point nine, four point seven, and four point eight'. Fascinating, eh?

Now look how these students often answer the following questions:

I must have had so many students make this mistake without ever realising what they were thinking; instead of correcting their misconception, I tried to give a rule.
Think about how these students get this and then watch this explanation:

 

This is not a one off. I reckon there will be students in every school who make this error.
So how do we correct this? Sometimes by simply saying the number correctly:

"Six and one hundred twenty five thousandths."

This is often enough to students to rethink. In example (v) above, when students say 'four and nine hundredths, four and seven tenths, four and eight thousandths' they often realise 'Hey, seven tenths is much bigger than nine hundredths'. I can convince them of this by modelling these quantities (of which more in a future post).
Skip counting with decimals is also  a really useful activity. Students who make the above error will often continue a pattern that starts 1.2, 1.4, 1.6,  like this: 1.8, 1.10, 1.12, ...
However by saying the pattern as 'one and two tenths, one and four tenths, one and six tenths, one and eight tenths, one and ten tenths...' at this point we often see students realise 'Hey, ten tenths are one whole so it must now be two'. To reinforce this, we can use a calculator: type in 1.2 + 0.2 then keep pressing =,=,= and students will see the count continue. It is a great moment of cognitive dissonance when they see the '2' instead of the expected '1.10'!
These two diagnostics take very little time to do but have helped me so much in correcting students' misconceptions.