On a recent visit to a school in a small city in the Midwest, Karen and I joined a class of 5th graders as they learned a game in Unit 3 called *Roll Around the Clock.* In the previous session, students used a clock to find and name fractions and equivalent fractions. For example, if the minute or hour hand moves from the 12 to 3, it has rotated 3/12 or 1/4 or 15/60 around the clock. Students would use these ideas in this lesson.

In this game, players take turns choosing which of two dice to roll — one has six fractions 1/2 or greater, the other has six fraction 1/2 or less — and move a marker around the clock in an attempt to get as close to 1 (12:00) as possible (meaning 1 full rotation on the clock). They can roll the dice up to three times, and the student closest to 1 at the end of each round receives a point.

As a math coach in Las Vegas I played this game many times with teachers in professional development sessions, and had seen it played in dozens of 5th grade classrooms. There is an elegance and mathematical power to this game, as students develop a conceptual understanding of adding fractions that have related denominators (twelfths, sixths, fourths, thirds, and halves). I am always intrigued by student thinking and after seeing the game played many times I was fairly certain I knew what students were going to say and do.* *

Here’s what happened as we watch the lesson unfold. The teacher introduced the game by playing a few rounds with the class. First, she rolled 7/12 and asked where the marker would go. 7/12 was easy to place. She asked if she should roll Cube A or B, and students chose Cube A, the one with fractions 1/2 and less. She rolled a 1/4.

**Teacher**: So where should I put the marker now?

A number of students raised their hands.

**Student 1**: Move it 3 places, to the 10.

**Student 2**: Move it 4 places, to the 11.

A majority of the class seemed to agree that the marker should move to the 10, but there was also some heated discussion that it should move to the 11. This didn’t surprise me – as students initially work with “clock fractions” moving 4 spaces for 1/4 is a common misconception; the same is true for students thinking 1/3 means 3 spaces on the clock.

**Teacher**: Well, let’s divide our clock into fourths, just to be sure. Can someone come up and do that?

A student came up and divided the clocks into fourths.

**Teacher**: So what do you think? We all agree that we’d move 3 spaces to show 1/4, so where should I put the marker?

The class still couldn’t totally agree. Most students were saying “on the 10”, but one or two were still saying “on the 11.” As a former fifth grade teacher, I’m pretty sure the student who was insisting the marker should be on the 11 was simply defending his original answer at all costs. I remember students like that!

Another student asked if he could come up to the board, and the teacher said of course.

**Student 3**: I can prove why it should move to the 10. What if you thought about rotating the lines, so you were dividing the clock into fourths this way?

He drew additional lines on the clock to show this:

I was dumbstruck, awed, and inspired. In all of my experiences with this game, I’d never heard this particular mathematical idea expressed.

Watching this lesson, I was once again struck by how powerful the mathematical work teachers and students do with *Investigations *is. If we truly believe *all* students have mathematical ideas (they do), create classrooms where *all* students feel part of a mathematical community and where *all* ideas are honored and respected, I believe that students are always going to surprise us with their curiosity and depth of mathematical understandings. I have seen this time and time again, no matter how many times we’ve taught the lesson or played the game. I’ve also seen many instances of deep mathematical insights coming from the students we least expect.

Visiting these classrooms (and huge thanks, as always, to all the teachers that welcome us into their classrooms) was a powerful reminder to me of how hard this work is, and the amazing things students do and say.

### Keith Cochran

*Investigations*Center for Curriculum and Professional Development. He co-directed the development of

*Investigations 3*, was a senior author of the 2nd edition, and has extensive experience providing professional development for teachers, schools, and districts implementing Investigations. Keith taught in a range of settings, including Native American Nations (AZ) and the Clark County School District (Las Vegas, NV), where he taught grade 5 and became a Math coach.

#### Latest posts by Keith Cochran (see all)

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ALL student have mathematical ideas

ALL students feel part of a community

ALL ideas are honored and respected

These should be the foundation of EVERY math classroom. Thank you for sharing such a powerful example of student thinking–when you give students the opportunity, their thinking will amaze you!

Thanks, Eileen! The role of the teacher and the curriculum are certainly key as well. Even if I’ve heard or seen the ideas or strategies before, the mathematical thinking of students impresses me. I hope this blog reminds people of times in their own practice that they were blown away (or possibly even puzzled) by student thinking. It would be great to hear those stories!