Over the past few weeks we have been dealing with the areas volume of different shapes. After doing a multiple classroom activities, one thing that I noticed along with my classmates was that my spatial reasoning is very poor. One activity where I particularly noticed this was when we were comparing the volumes of pyramids to that of a rectangular prisms. When I was calculating the area of these two volumes with the same base shape and height, I realized that the area was calculated almost the same, except that the area of the pyramid was 1/3 that of the rectangular prism of the same dimensions. I was simply baffled that this was true. In my head I could not see how three of the pyramids that I just measured would ever fit into the rectangular prism right next to it.
This made me think about the education that I got and how little the math related concepts were hands on activities with three dimensional objects. So often in our classes we are taught that the best way for students to learn is through inquiry in the sciences, because it is through this process that the students are able to break apart the system at which they are doing things and figure out why the system works. Why have we not adapted this into the math education as well?
This was the reason that I particularly liked the activity where we calculated the volumes of several three dimensional shapes with water. After being given the hollow shapes, a measuring cup of water and a graduated cylinder, we were told to find out how the volume of each of the shapes in milliliters. After noticing that there were a few different ways that these volumes could be measure we decided that the best way for us to measure the volume would be to fill the shapes, then pour out the water that fit in the shape and measure this amount. Unfortunately, we were disappointed to realize that although this was likely the easiest way to measure the volume, it was not the most accurate as there was a lot of space that was missed in the plastic that was used to create the shape.
What I like most about this activity for elementary age students is the fact that this gets at the heart of what volume is and what it is measuring. Because of this students would definitely walk away with a concrete understanding of what volume is. Another aspect that I really think is great is that this is such a good integration activity between math and science. Measuring these volumes accurately with the tools given can be highlighted as a key skill. Also this leads for discussion of an alternative way to measure the volume by displacement.
I saw this activity, however, to be more versatile than just being able to be used in the elementary to understand volume. I think that this could be a great upper elementary or even middle school activity if you could help guide the students to make connections between the measurements that the make two dimensionally on the shapes to the three dimensional volumes. This would be good practice for students to do unit conversions. I also think that this practice would be great for students to begin to develop and refine their spatial thinking.
Over the last few weeks in my math class that focuses on Geometry in the classroom has been exploring measurement. We have had some challenging conversations to say the least, as we were given activities that asked us to measure and we quickly realized much more assumptions are made in measurement than we remembered. These activities were meant to strengthen our core measurement ideals as well as finding ways we can create a lesson that challenges and strengthens the student’s fundamental measurement skills as well.
A key question that needs to be addressed before any measuring was done is why do we need these measurements? This is a foundational thought that teachers need to address before we even start measuring so that students realize that what they are doing has purpose and is used for accurately communicating to other people. Whether that amount is time, or length is up to the specific problem, but we are always asking “How much?”
One activity that we did that I thought was beneficial was we were told to measure the length of the hallway. We were given a tape measure and told that we were to use our stride length in order to determine our length. To start off, when we were told to measure our stride, my group decided that it might be best to find the length of 10 strides, then divide by 10 in order to get the average length, since each stride length could be slightly different and we could use the average. Next we started at one end of the hallway and walked to the corner where the other hallway started. After doing this we multiplied our stride length in centimeters by the number of steps that we got in the hall to find out the length of the hallway.
After finding our hallway lengths and as we began to talk about the vast differences in we were seeing in each-others data, one thing that we realized was that we did not come up with an agreed upon starting and stopping point for how we would measure the hallway. Some of us started at the corner and measured to the next corner while others were measuring wall to wall. Another aspect of measurement that we began to discuss was what an appropriate range for error would be for each of our measurements. This was an interesting concept to begin to talk about as many in the classroom, especially those with little science background were unsure at what an appropriate error would even look like for a problem like this.
By doing this problem in this way, my classmates and I were able to be in the shoes of the students that we will someday be teaching and were able to come across some important questions that students need to ask themselves and understand in order to measure proficiently. Where is the starting and stopping point? How do I use the tool that I have to measure properly? Am I measuring the same way that it was intended to be used to measure? These are all questions that students need to ask so that they are measuring accurately. I think that this lesson did a good job at doing that and I think that it could be adapted so that students could go through the same process. By giving them the guidance of a worksheet that could help them walk though these steps and discover that these are important things to be doing the same so that we all get the same measurement. The worksheet below was created with this purpose. First off, I wanted to give the students a reason to measure. However, the goal of this worksheet is really to focus in on step five where they find their differences. In seeing these differences students will be able to see the importance of starting and stopping in the same place, walking in a straight line and using the tools they have correctly. Ultimately, by learning through inquiry in this way students could have a lasting understanding of measurement and how it can and should be done correctly.