Every day, I notice and wonder things about both the students thinking and the mathematics in my classroom. Over the past couple of weeks, however, there are a couple of things I have noticed that still have me wondering….
We do Number Puzzles in our first Investigations unit, serving as a review of properties of numbers that students have worked with over the years. For those who have not done number puzzles, this is an example:
At the end of this activity (14 puzzles), I have the students design their own number puzzle, trade with another pair of students and talk about the solutions. It is a great formative assessment of their learning and it is fun to see some students creating puzzles that are impossible or have more than one possible answer. The conversations are amazing. However, this year, a student completely stumped me. I wish I had saved his puzzle, but it was two of his four clues that got me wondering. First clue: My number is odd. Second clue: My number becomes even when you make it a decimal. Hmmmm. Thankfully I do not have to recap the conversation here because Christopher Storified our conversation here: https://storify.com/Trianglemancsd/it-s-even-when-i-make-it-a-decimal
My first thought, after I got done reeling over his thinking, was, when was the last time the students have revisited even/odd numbers? I know in second grade, students investigate odd/even numbers in terms of being able to break a number (positive integers) into two equal groups or share between two people. The exact math focus points are…
So, they establish that a number is even if it can be broken into two equal groups and if there is a leftover, it is odd. But, at this point, they have only dealt in whole numbers. Do we ever revisit that when they begin working with rational numbers or negative integers? Do we just assume that students keep the understanding with positive integers and don’t try to apply it to other numbers? What an interesting thing I have never thought about before! It blows my mind that after 19 years of teaching (12 of those years in 5th grade) that I have never had this conversation before with a student. This whole conversation has me digging back into the math introduced in earlier years to see if there are other things that we never revisit in our work as we move into decimals, fractions, negatives…etc. Just for some added fun, Christopher said he thinks he could master “Billy’s” even/odd quiz. Billy made a quiz here: pic.twitter.com/p24gMG4Uwd and you can check your answers here: https://www.dropbox.com/s/2chzj2n3534stjo/billy.parity.answer.key.pdf?dl=0
My second wondering stems from my students’ work yesterday with Volume. Up to this point, we did a lot of building with unit cubes and in Minecraft of rectangular prisms in route to formalizing our strategies for finding volume as l x w x h or area of b x h. Students were very comfortable with finding the volume of rectangular prisms, with or without the cubes drawn in the diagrams. They could talk about the arrays in each layer they saw, and how many layers they saw “stacked” in finding the volume. I was feeling so confident that these understandings would follow us right into our work with two rectangular prisms combined to form a solid figure. It was looking great as we worked with the gridded diagrams. Here is an example of one student’s work:
Then hit the unmarked models…
Even anticipating there would be some confusion, I was honestly blown away by the number of students struggling to break apart the picture and think about the lengths, widths, and heights they needed to reason about layers of cubes. I found that some of the students felt the need to use every number in the picture in their solution while others were making no connection to their reasoning about arrays and layers. They could describe how they broke the shape apart into two pieces, however when I asked how many cubes were in the bottom layer of the prism to the right, they could not see the 3 x 2 as 6 cubes in that layer, and if they did (and had broken it horizontally) would do 9 layers instead of the 5.
I was struggling with how to help them go back to their amazing work with cubes. Why was this visualization disappearing? In hopes of having them have a better visualization, I asked them to build the above picture in Minecraft. Done. Every single struggling student automatically laid down a 2 x 3 array, made 5 stacks of them before moving into four 12 x 2 arrays on top of that. What?!? This has me completely perplexed. How can they see it to build it without a second thought, but cannot communicate about it otherwise? More importantly, where do I go from here? They obviously cannot build every figure they need simply because when the numbers are larger, it is completely time consuming and I want them to connect their concrete building to a more abstract reasoning. I am thinking from here, I will have some continue to build until they don’t need it, pushing them to abstract with questions such as, “If you were to build this right now, describe it to me.” Pausing occasionally to ask how many cubes that would be.
These two wonderings, although so obvious at times, have seemed so complex to me over the past weeks. I love every moment of looking into my students’ reasonings’ because it challenges me to be a better teacher and look deeper into my own understandings about how children learn. Great stuff.