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###### by Kate Minear, Math Specialist for Grades 3 and 4

Mathematics is about so much more than just calculation.  Mathematicians wonder, ask questions, estimate, investigate, and so much more.  We work to provide opportunities for our lower school mathematicians to do all these types of math thinking.  One structure we use is called a “Three Act Task.”

##### ACT I: Noticing and Wondering

In the first act, students look at a photo or video.  There is no question to think about yet.  Students observe carefully and brainstorm everything they notice, and then begin to pose their own “mathy questions.”  Third graders looked at a photo of a place they know well– the roof. They began by making a list of things they noticed– everything from the fact that the picture must be from last year because the new water fountain isn’t there to the fact that the first block of square windows has fewer than the rest of them because there are only 8 in each row instead of 9.

Then they began to pose their own questions.  This is the list that 3C brainstormed: It is always interesting to see how one question inspires the next.  We emphasized how mathematically interesting all the questions are, and explained that we were going to focus on how tall the low roof is (which has a practical interest to kids because balls often get stuck up there.)

##### ACT II: Estimating and Information Gathering

Now that we knew what question we were focusing on, we were ready to start making some estimates.  We asked: “What is a guess that you know is too low? What is a guess you know is too high?”

• “I know it must be more that 53 and a half inches because that’s how tall I am and it’s taller than me.”
• “I think it’s more than 6 feet because my teacher is almost 6 feet and she isn’t that tall.”
• “I think it’s less than 20 feet because most kids are less than 5 feet and I think if you stacked 4 kids on top of each other they would be taller than that.”

We hope that students will begin to do this estimation thinking with every math problem they solve, assessing how reasonable their answers are.  It is often easier to begin forming meaningful estimates when a problem has a context that allows them to make connections and draw on what they know about the world.

The next question we asked was: “What information could help you solve this problem?”  Often in math textbooks, all the information needed to find a solution is neatly provided.  When presented with a story problem, students sometimes start just adding together all the numbers without pausing to think what each number represents.  When mathematicians and researchers are using numbers to model real world events, they have to make decisions about which variables to take into account.  Our young mathematicians also need to make sense of problems and decide which information is relevant.

• How tall is the pipe?
• Could a teacher stand in front of the wall and we could see how high up they go?
• There’s a ladder off to the side.  If we know how many rungs and the space between each rung, we could count.
• I think it’s around the same height as that line on the classroom wall.  I could measure that wall.
• How tall are the mats?
##### ACT III: Solving and Comparing to the Real World Students now had a choice of what numbers to work with.  They could use 34 1/2 inches or they could choose to use around 3 feet.  They began making important decisions about how precise to be.  Will an estimate do?  Can we measure exactly enough that using half inches makes sense?

Many third graders approached the problem by figuring out how many times taller the roof is than the mat.  This is a key understanding of multiplication: that one mat represents three feet.  So if the roof is around three times as tall as the mat, it is 3 x 3 feet (or 3 x 34 inches).  Plus, this being the real world, the roof is not exactly three times as tall so they had to do some thoughtful adjusting of numbers. They shared out answers and the reasoning behind them, and then we revealed the exact height of the roof. No one got the exact answer, but they were impressively close.  After some time for cheering, we discussed why our mathematical models didn’t give us the exact real world answer and what worked well for us.

Over the course of the class, so many voices have been heard, and children have been recognized for the excellent mathematical thinking they have been doing, whether they noticed surprising details, posed creative questions, reasoned thoughtfully as they estimated, or found a particularly precise solution method.