At all grade levels, especially when introducing a new concept, I think it is important to provide experiences for students that have them create representations of that concept (place value, adding, multiplication, perimeter, etc) in different ways. Most students move through stages of understanding with a concept and many need to begin with a concrete representation (using materials such as blocks in groupings for multiplication) before moving on to pictorial or visual ways of representing. Some students are able to think and represent symbolically right away, but most do so after concrete and visual experiences with the concept.

As in all classes, in the class I am teaching in, there is a wide range of learners although they are all grade 3s. One way of helping students experience mathematics together as a whole class is to provide experiences that touch on different levels of understanding. Students who are working at a concrete level can make connections to the pictorial and symbolic representations they are exposed to, even if they are not fully ready to work at that level yet. Students who are working at a symbolic level have their understanding strengthened and stabilized by being able to represent their thinking concretely and pictorially.

We have been doing lots of work with concrete representations of multiplication and are moving towards pictorial representations. Today we played a game we called "Grid Arrays". I printed off centimetre graph paper (I googled centimetre graph paper printable). Students each had two dice and rolled the first for how many rows in their array and then the second for how many squares in each row. The students coloured in the array and wrote the equation over it.

I had noticed in our last array "game" that some students were adding an extra column. For instance, if they rolled 4 rows, they drew 4 squares or build a column of 4 cubes first. Then if they rolled a 3, they added three more to each row. So instead of making a 4x3 array, they were actually making a 4x4 array. This told me we needed more experience with concrete and visual models and connecting the equation to what we were doing. Today, when we went over the grid array game, I modeled it for them on a chart. So if I rolled a 4 (for 4 groups or rows) I just made a little mark down the first four squares in the grid. Then when I rolled a 3 (for how many square in each group or row) I made a point of counting the first one I had already marked and then marked the next two, counting up to 3 for each row. After the marking of the squares, I went back and lightly coloured in the whole array. This really seemed to help. I am going to do a check with students building arrays for me this week as well.

We have also been working with numbers up to 1000 and have been using the base ten blocks to build numbers before representing them with pictures. Thanks to my colleague next door, I had just enough for each student to work with. I think we can underestimate how many materials we really need for a whole class. It's so important that the students don't just get to see the manipulatives - they each need to hold them and work with them.

We rolled dice, spun spinners and turned over cards to find out how many 100s, 10s or 1s we needed to include. Variety is the spice of life ;)

And you couldn't imagine how giddy a class of 22 grade 3s could get when they were told after letting the dice, or me choose the numbers, that they could choose ANY number they wanted to represent. Oh my, the excitement! This boy was very pleased with how he put together 999.

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