I was teaching some advanced second graders an enrichment math lesson the other day when I learned something. I often like to mix and combine skills, so that kids can see how math is really used, as well as make it fun. At the end of this particular lesson, I used the data we had collected throughout our time together to summarize what had happened. As I attempted to make sense of the numbers, I found myself making conclusions or at least forming a hypothesis that could be tested. When I pointed this out to my second graders, I thought to myself, “Wow, this is a pretty good lesson I’m learning right now.” I was thankful that I hadn’t stopped at only having my students do the original lesson’s math.
The lesson involved making estimates and then measuring actual length to the nearest inch. The first thing I did was model. I took a wooden block out of a box; my “Box o’ blocks”! After standing it up on the table, I asked, “How tall do you think this is?” I received some wild guesses from my second graders. Someone thought it might be a foot. Another student said two inches.
I picked up the block and measured its length with a little mini ruler. I showed the students where the block ended on the ruler. They eventually settled on the idea that the block stopped between the five and the six. One of the students suggested that it was five and a half.
After praising this smarty, I asked them if it was okay to measure the block laying down, because I was holding it flat in my hand. We were supposed to be measuring its “height.” Their spatial reasoning skills were sound, and we all agreed that we were measuring the length of the same side, no matter which direction it was facing.
After teaching estimating and measuring and before breaking the group up into teams, I explained the directions. Each team would get some random blocks. They were to work together to build a tall tower. It had to be free standing; No holding it. Every block should be used. I would give them two minutes to construct the tower. When the timer goes off, the teams will form an estimate of how many inches tall the tower is. Once the team has decided on a number and communicated it to Mr. Weimann (me), they get a yardstick to measure the actual height.
I would be keeping track of our estimates and accurate measurements. The idea was to try to get your estimate as close as possible.
Next, it was time to form teams. I just had kids who were sitting near each other form teams to make it go faster. I dumped random blocks in front of each group and told them to get started. After the first two minute timer sounded, I stopped everyone. I had the groups come up with estimates of how tall their towers were. As each team shared their estimate, I had them provide a name for their team, as well. Clowning around, I purposely misspelled the names they gave me. That had them laughing.
After I wrote the estimate under a team’s name, I handed them the yardstick. Watching them estimate the height was fascinating. One group had a girl who used two fingers squished together to climb the tower with the members counting as she jumped. They figured her fingers constituted about an inch. They were very accurate. Another group had a student using his arm, presumably thinking it was a foot long. After a round or two I reminded groups that the medium sized block was already measured. We found it to be exactly five and a half inches tall. They could use that in their estimates. I don’t think any of them did, but we can revisit that.
As it turned out, we only had time for three rounds. After collecting all of the wooden blocks, I went over the chart that I’d made. I had second graders figure out the difference between each estimate and measurement. I wrote that data on the board in a different color. We then added all of the differences from each team together to total them on the bottom of each column. Although two teams had the same total, three inches , one of them had never supplied any data for one of the rounds; Their tower kept tumbling, and they were never able to estimate or measure it.
The “BeeKays” began rejoicing for having the largest total, presumably the winners, but other second graders squashed their victory dance with unwelcome information. Like golf, the total that is the smallest was winner. It took a little convincing, but I explained that the goal was to get the estimate as close as possible to the actual measurement. The smaller the difference, the better the estimate. “Look, the very first round had a team whose measurement ended up being exactly the same as the actual measurement! The difference between the two was zero. This zero was the winner of that round.”
Next, was the very cool part for me. Here is where I joined my students in learning, albeit through teaching. I had asked the students what data was. They eventually settled on the synonym, “information.” I added to this the word “useful” and proceeded to show them how we could use the data to draw conclusions. The team that had thought they won because the sum of all of their differences was the greatest, eleven and a half, did actually win something. They were the winners of which team improved the most.
“What do you notice when you compare the differences of each tower they built?” I asked the group. “They got smaller and smaller. The first tower was estimated to be eight inches shorter than it actually was. The group over corrected a little on the second tower, estimating it to be a little taller than it was, but only by three inches. The final tower was within half of an inch of its estimate!” I pointed to each difference on the chart as I explained its meaning. “The estimates got more and more accurate, as the BeeKay team practiced.” I let that sink in.
After pointing out that the data shows some improvement in the other two groups, it isn’t as consistent as the middle group’s. As I time ended, I taught my second graders that we just analyzed data and developed conclusions based on the information we collected during our lesson. “What might we expect to happen if we built some more towers and continued this exercise of estimating and measuring?” I asked.
Hands shot up all over the room. “They would get closer and closer,” someone shared.
Putting it into mathematical terminology, I restated, “That’s right. The differences between the estimates and the measurements should get smaller and smaller as you get better at estimating.”
Teachers use data all of the time to measure how students are doing. Do you ever show students how the data works? Give it a try.