Problems With Place Value

Place value is so much more than Base 10 blocks.
This is a common theme has emerged in a number of recent Collaborative Inquiry sessions where we focused on what students know and don't know about place value. A lot of questions from textbooks and worksheets tend be of the type shown below:

The danger with these types of questions is that I have seen students get the right answers but have no firm understanding of quantity whatsoever. They might learn a strategy such as 'The first digit goes in the first space, the second digit goes in the second space and so on...' or 'The digit on the right tells you how many little blocks, the one next to it tells you how many rods, and the one next to that tells you how many flats and so on...' If base 10 blocks are the only representation used then there is a real danger that students will develop misconceptions such as on that was highlighted by Sue Willis in First Steps in Mathematics. Grade 4 students were correctly able to identify that 4 rods and 3 smalls were 43 (below).   

However, when asked how many there would be if they were cut into individual pieces, two-thirds of Grade 4 students said “I don’t know, I would have to count them.” 
Similarly, Grade 6-7 students described a large cube as ‘the thousands cube’ but thought that if it was cut up, there would be 600 small cubes (100 each side).  This sort of error explains why some students think that 24+25 is 13 as it is simply 2+4+2+5.

With this in mind, we visited a number of classes to try some different sort of place value and quantity questions. We set out to deliberately bring out any student misconceptions by asking them to order a set of numbers such as:

547, 600 - 3 twenties, 5 tens 7 ones 4 hundreds, 5 hundreds 23 tens, 4 fifties

As the students began to order these, it quickly became clear that many were making errors based on a simplified grasp of standard partitioning. As one teacher said "Boy, we've done way too much of that...".
So in addition to the bog-standard standard partitioning questions, we realised the importance of asking questions like:

• Write 57 in at least three different ways
• 1000 take away 47 tens is the same as how many tens?
• How many 20s in 100? 500? 1000?
• How many 25s in 500? 1000?

This will give students a deeper and more flexible understanding of quantity and place value which will be necessary if they are to have good number sense.