Yesterday I blogged (
part 1) about visiting a fourth grade classroom and their lesson on estimation. After her lesson and her students left for lunch, the teacher and I debriefed. I really hope I complimented her enough. I was inspired. She asked me for some feedback. To remind you, she started the lesson by picking up a small cup, about half full of cubes, somewhat concealing it and asked students how many cubes were inside. She wanted to demonstrate that their initial answer would be a "guess" and not an "estimate". However, she had many observant students who already saw the size of the cup, the fact that it wasn't full of cubes, and even from my seat in the back I could tell that the cubes were small enough to fit inside the small cup. I told her I loved everything. The only additional thing I would have done was this:
Don't even show the kids anything. Don't go and pick up the cup. Simply say to the students:
Students, I will have a cup in my hands very soon. There will be cubes inside. How many cubes do you think will be in the cup?
Okay, look how simple this uninformative setup was. If I were a student, a bunch of questions would have just popped into my head: Hey teacher, what size is the cup? Is it a small paper cup? A medium coffe cup? or a large Big Gulp cup? What size are the cubes? As a fourth grader, I know what base ten cubes look like. Are they bigger like the size of snap cubes? Are they the size of ice cubes? and how about the amount of cubes in the cup, teacher? Are there just two at the bottom? Is the cup half full (or over half empty for you pessimists)? Is the cup full of cubes?
Any number the students produce would simply be a guess. It's a low-entry point, but doesn't hold much strength for long. How many cubes are in the cup? It could be two. It could be two hundred. Two thousand. You get the point. This strategy reminds me of a couple engineering classes I took in which we discussed a
black box. In other words, there's something inside this black box that serves a purpose or function. However, you have no idea what's inside or the parts that make it function.
Students are guessing blindly at this stage. However, I wouldn't want them to just guess. I want them to beg for more information. Even better, I want them to think what information would help them solve this question. Let's move to the next stage.
Let's spiffy up our description, but still refrain from showing students the cup, cubes, or content level:
Students, I have a small drinking cup in my hand that's about 8 oz. Inside are cubes that are slightly smaller than six-sided dice. The cup is a little less than half full. How many cubes are inside?
By this time, I hope students would be falling out of their seats trying to sneak a peek at the cup. They're lusting after more information to make a more accurate assessment. You're simply adding some labels to the black box. Heck, put the cup inside a brown paper bag for this part.
What stage is this? I think this is the in-between stage. Shall we call it guesstimating? I used to really loathe this term as I want my students to use estimation. I thought it was a silly verb and should be eliminated from our vocabulary. I no longer think that. A guesstimate implies we could make a better attempt. We could use better clues. It's the bridge between guessing and using sufficient clues and observations to make a reasonable estimate. In other words, we must encourage our students to demand better information. Demand facts. Demand relevance. Don't SETTLE! Okay, so what takes us to that next stage, estimation?
Reveal the cup. Take it out of the black box and put it in the display case. That's right, put it in the display case. Don't let them touch it.
Demand they use their intuition, their available senses, and rely less on the sense of touch. Show a picture of the cup, a cube, and a birds eye view of the cup. Remember, it's in a display case. An even greater challenge to your display case would be to put the cup on the students's desks, but don't allow them to touch anything. Hold onto any last shred of information you could provide them with. Guard it. Be stingy. Let them use their sense of sight and intuition to build their number sense. Help students build up a thirst for relevant information. Build a problem solving plan or strategy. Don't be mean about it or covet the information for malicious reasons. Give students time to respond to the clues provided. The display case is a prime stage for estimation. Students have many context clues from either a picture or physical model. This should be enough for students to really build a strong theory and/or problem solving plan. As we saw in yesterday's post, students came up with a couple of theories on how to more accurately estimate the cubes in the cup by counting the top layer or cubes on the sides. The teacher didn't just hand her students the cup. She helped them climb the
ladder of abstraction. I think the key to this is encouraging students to demand more information. Don't inundate them with all of the context clues immediately. Make them demand the clues.
Hands-On is the last stage of estimation. Without counting, allow the students to pick up the cup. I'm not saying students will change their estimate. However, it will give them opportunities to consider another perspective of the task. They are including another component: weight, size, etc. There's only one place to go from the hands-on stage and that's revealing the answer: the payoff.
Recap:
1-Black box: Keep that description minimal. Avoid a visual.
2-Label the black box: gradually reveal some information
3-Display case: Look but don't touch
4-Hand-On: incorporate one last sense in order to bring one last perspective to the estimate.
By the way, did you know that
Stevie Wonder played many of the instruments on his Innervisions album. I read somewhere that he virtually played all the instruments on about six of the nine songs. That's one of my favorite Stevie Wonder albums. If he can do that without the use of sight, imagine what we can do with all of our senses.
Part 3: Why should we be the gatekeeper of information for our students? How can you help your students build their number sense and demand more information to make a reasonable estimation? How do I incorporate estimation in my classroom?
Part 2,
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