# Short Reflection: Decimals, Division and Place Value

I’ve shared a couple of lessons in recent blog posts that went well.  The following is a lesson which I’m much less sure about.

With my S2 class who, historically, would be likely borderline National 5 candidates in S5 I’ve been trying to develop their “sense” and “feel” for mathematics.  Much of their appreciation of the subject seems to be in the form of procedures and algorithms.  They are not fluent with many of these procedures, but more, I have the sense that there is a weak conceptual foundation on many ideas.

I know these young people have the capacity to reason, but in the first couple of weeks working with them I’ve witnessed many errors in lesson starters due to a lack of fluency with procedures they’ve tried to memorise.  I argue that if the learners had a stronger conceptual understanding of this previous work then they would have the capacity to make sense of their answers and identify that they make sense.

I normally begin lessons with a mix of content from previous lessons and/or touch on knowledge required for today.  Occasionally, I break loose of this mould.  I started the lesson today with this prompt:

As I walked around the class I was disappointed with what was being produced, although not surprised.  I think that leaving the world calculation in the task was a mistake.  I prompted pupils to try and make their argument in more than one way.

I considered that perhaps, if I shared some ideas of how to represent the situation they would be more confident.  I spent some time chatting at the board, trying to get some responses from the room.  This resulted in the board looking like this…

I now asked the class to attempt the prompt again this time with 4 ÷ 10 = 0.4 as the statement they had to illustrate.

I found it hard to decide if this was simply them mimicking without engaging with the concepts.  This was exactly what I was trying to avoid.  I didn’t want to do a lesson where they imitated me and then promptly forgot everything – as they’d clearly done when taught this by a procedure focused approach.  I was aiming for deeper learning.
Next, I asked them to consider 3 ÷ 100 = 0.03.  I had expected them to make the natural leap and continue to use diagrams and number lines. This, generally, did not materialise.  The vast majority related the question to either money (which I suppose is an improvement upon their attempts at the initial task).  Only one pupil began to draw a diagram.

A chat on the idea of ÷ 10 and then ÷ 10 being the same as ÷ 100 didn’t go particularly well.  I muddled on with the following diagrams.

I posed a couple of questions on the board, which pupils did in their jotters, most of them could demonstrate at least procedural competency.  I’d made up a sheet, which was about pattern spotting which I then asked them to do the first half of.  The majority of the class did fine on this, with only a couple unable to get full marks.  I know this wasn’t the case last week as in a couple of starters this topic was done very badly.

Going forward, I have a few points to contemplate
• It is difficult to illicit evidence of conceptual understanding at the time of acquisition.  I will attempt a similar entry task in a few days to try to gauge how much learning happened from this episode.
• Most of the “meaning” almost entirely came from me – I am still unsure of what sense of these calculations they have in their own minds.  Surely there is more to their understanding than what I’ve been able to draw out of them?
• I will work more on the number line model.  Placing pairs such 0.3 and 0.03 at appropriate points on the number line to emphasise their relative values.
• The illusion of performance – fluency in procedures can obscure what is or is not going on in our student’s minds.

This lesson today was messy and not obviously successful in measurable ways.  Progress was slow.  I know with example problem pairs and skipping the initial tasks and whole class discussion I could have had the kids churning out near perfect work on problems such as 6.52 ÷ 2000.   But sometimes this race through performing tasks is not sufficient, if it is built on memorisation and a lack of understanding.  None of this material was new content for these learners, but if felt like it!  Having been previously taught this material in a direct method, I once again find myself considering the shortcomings of direct instruction as a sole pedagogy.

So – I ask the questions, based upon what you have read: