One of the highlights of my work is that I get to interact in depth with small groups of students for longer periods of time than may be available to a classroom teacher. I view these opportunities as a window into understanding challenges and misconceptions that other students may be experiencing. This week I worked with three Grade 3 girls who were experiencing some difficulties with place value. I started my session by asking them to build 25 with base ten blocks. They immediately took out two ten blocks and 5 ones. I asked them to represent it in another way and they immediately showed me 25 one blocks. When I asked them to do it a third way there was a long pause. I practised my wait time and then one of the girls came up with 1 ten block and 15 ones. The other girls followed her lead. My wondering is: How many times do we make assumptions that students have an understanding of place value if they can tell us quickly that 25 is 2 tens and 5 ones? Is this enough evidence of their understanding? Students who are flexible thinkers will understand that numbers can be represented in a variety of ways: 125 can be 1 hundred, 2 tens and 5 ones but also 12 tens and 5 ones or 11 tens and 15 ones etc. Students who understand that numbers can be regrouped in a variety of ways, but still maintain their same quantity have a much better understanding of the concept of regrouping when it comes to addition, multiplication, subtraction and division. They are able to come up with alternate ways to work with numbers when problem solving because of their fluency and flexibility with numbers. They can move fluidly between the different units (ones, tens, hundreds, thousands etc.) After questioning, listening and observing I noticed that one of the students was able to identify 10 ones but she called a ten block “ten tens” when it is only 1 ten. I could tell that she was fragile in her thinking about this so I made a decision to work with her alone for the next little while. I played the following game with her: (I want her to understand that 10 ones is equal to 1 ten; that we can unitize 10 ones into 1 group of 10; sometimes we count the ones and sometimes we count the groups.) This sounds like a simple concept, but it can be very confusing for some students. This confusion becomes apparent when you listen to the language they are using.)
Race to One Hundred
Materials: 1 hundred block, about 20 tens, about 20 ones, one regular number cube (dice)
Directions: Player A rolls the number cube and takes that many ones. Player B does the same. The game continues this way and once someone gets 10 ones they can be traded in for 1 ten. Once one player gets 10 tens, these can be traded in for the hundred block.
As we played this game together I observed how this student counted the number cube. Did she subitize (recognize the quantity without counting the individual dots)? I also observed how she counted the ones that she was combining to determine whether she had enough to trade. Did she count all or did she count on? When I noticed that she counted all, I modelled for her how to count on when it was my turn. I often asked before she counted whether she thought she had enough to trade and once she counted I would sometimes ask how many more she would need. As both of our numbers grew we would count the quantities and I would model the language about 2 tens or 9 ones while pointing to the visual models which gave meaning to the math language I was using. A lot of explicit teaching based on the needs of the student was going on here.
I am happy to say that as we played the game together the student’s confidence grew and she self-corrected herself when talking about the units of ten. Next time I am thinking of building on this game and using other visuals such as ten frames, hundreds chart and number lines to help her deepen her understanding of place value and number patterns.
Think about how this will impact her fluency with numbers when working with units (cm, m etc.) within the measurement strand. Time well spent!