This was the first question:
We were expecting some answers of 15cm but in actual fact, there were very few of these. The most common answer was 9cm. They weren't measuring, they were counting. But they started the count "1" at the '7' mark, then "2" at the 8 mark and so on. Now, both myself and the class teacher were pretty sure that if we gave the students a proper ruler, they would have been more successful, but is this because they are actually measuring or merely reading a number?
The second question revealed the extent of this misconception though:
The majority of the students circled the correct answer, c. We then asked the students to explain how they got their answer. Some gave this:
It was the same mistake seen in the first question! These students were counting dots, not measuring length. It was clear that they knew how to calculate area of the two shapes, but were using the wrong values.
Others gave this answer:
These students were again counting to get the length/base and width/height, but this time were counting the dots inside the shape to do so!
It is perhaps not surprising why students should have this misconception. All their measurement experience from primary years is based on using a count to get a measure. So they have developed this misconception that measurement is discrete not continuous.
So how do we change this? Well, we started by using the response to the first question to create a contradiction:
Starting at the tail of the caterpillar, I asked who thought it was 9cm. I then gradually moved my finger towards the head, and then asked how long the head was. They said 2cm which I wrote above the head. I then put my marker on the 7cm mark and told them that I was about to draw a line and that they had to tell me when I reached 1cm. (By drawing the line, I was hoping that they would see the length as a continuous measure and not a discrete count). Sure enough, they told me to stop when I had drawn 1cm and I recorded this on the diagram. I then compared the 1cm to the 2cm (see above) and asked "Can it be both?" At this point, faced with the contradiction, the students had to agree that their 2cm answer (and hence their 9cm one) was wrong. By then drawing further line segments, they could see that the actual length was 8cm. We immediately followed this with some practice whereby they used a broken ruler to measure the perimeter of various shapes and they did this with great success.
What I learned from this is that when I see students use a ruler to get the right numeric value, I assume that they have measured this. Now I know that they might have counted. These quick questions helped us uncover and confront this potentially problematic misconception.