Tag Archives: student choice in learning

Circles Tasks

We looked at two diagrams as we finished up our unit on Circles. I knew that we didn’t have time for everyone to do both tasks. I wondered whether I could pull off giving students some choice in what they investigated.

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Both tasks are from the Mathematics Assessment Project.

The first can be found as Circles in Triangles in the tasks section and Inscribing and Circumscribing Right Triangles in the formative assessment lessons section.

The second is Temple Geometry from the tasks section.

What mathematical question could we explore?

Some of you saw my tweet.

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Yes. We are doing that “wonder” thing.

Most of the wonderings were about areas of various regions, and most of the questions were about the second diagram. I let each team decide what to explore. Only one team chose the first diagram.

I had copied a handout with questions about both diagrams, but I hesitated to give it to students, as it gave so much away. Instead, I gave the students a copy of the diagram only on which to work.

As has become our practice, students started working by themselves.

They practiced look for and make use of structure.

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Some practiced use appropriate tools strategically to make sense of the problem.

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Then they talked with their teams. Those working the second task almost immediately concluded that the radius of the smallest circle was half the radius of the largest circle. One student had used paper to measure the distance. Another “eyeballed” it correctly. Others constructed the diagram using dynamic geometry software and used the measuring tool to verify their conjecture. But no one was able to prove that the radius of the smallest circle was half the radius of the largest circle.

Time was ticking quickly. I still hadn’t even talked with the team who had chosen to work on the Circles in Triangles. What should I do to move student learning forward?

We moved into whole class mode. I selected a few students to share their thoughts about the length of the radius of the smallest circle. I made CS the Live Presenter so that he could show how his team translated the smallest circle so that its center lay on the point that partitioned the diameter of the medium circle into a 1:3 ratio.

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So if the radius of the smallest circle is one-half the radius of the largest circle, then what is the area of the shaded region? Every team set to work, ensuring that all of members of their team could calculate the area of the shaded region, and they did so successfully. (The evidence is on my computer at school … we are at home today for a “snow” day.)

What would have happened if I had given the teams the handout from the beginning?

From Circles in Triangles:

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From Temple Geometry:

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Would they have learned more mathematics during the lesson? Would they have practiced look for and make use of structure or use appropriate tools strategically during the lesson? Would they have engaged in productive struggle?

What would have happened if I had given the teams the handout that provided the structure for them once they got to a certain point on their own … even though they wouldn’t have had time to complete the investigation together?

I am learning that what works in our classrooms has so much to do with the students we have. Productive struggle isn’t just for students: We can plan great ideas collaboratively, but even so, we must be attentive to meeting the students in our room where they are and moving those students’ learning forward. And so, thankfully, the journey continues …


Posted by on February 26, 2015 in Circles, Geometry


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Problem Solving Points

A few years ago, I began requiring my students to earn 100 problem solving points (PSPs) each quarter, instead of requiring each student to complete a project of my choosing. PSPs are a way for students to show me what they have learned or solved outside of class. These have gone through several iterations over the past few years, but here is what I have now that seems to work.

How students earn their PSPs is their choice. Sometimes a student will ask a question in class for which a) we don’t have time to get into the answer or b) I don’t actually know the answer. When that happens, I throw the question back to the class as an opportunity to earn PSPs. Do some research to find out the significance of Euler’s line, or the shape of the faces of a dodecahedron (we were meeting in the cafeteria, and I didn’t have a model with me), or who first start using the radical symbol. Students then email me what they find, along with their source.

What do you do with the calendar insert from the Mathematics Teacher? Those problems provide another way for students to earn PSP. Some students enjoy working calendar problems. I post the current calendar in my room – and have previous calendars in a binder for students to “check out” to work.

On our class Canvas site, I post enrichment opportunities for earning PSPs in each unit – many times they are from articles I’ve read in the Mathematics Teacher – or activities that I read about on blogs and tweets. I link to some Problems with a Point. And I link to some TI-Nspire labs that would be beneficial for students but that we aren’t going through explicitly in class.

I also post general enrichment opportunities for earning PSPs. Someone recently shared about a TED Talk on Fibonacci, so I posted a link to that for students with a note “Provide evidence to your instructor that you have watched this short TED Talk for up to 5 PSP”. Students will send me an email with their reflections on watching the video.

For the Five Triangles Blog, I have a note “Send a solution to a problem on this site to your instructor for PSP”.

We have been paying more attention to Mindset this year in class, so I have an opportunity for students to explore GRIT: Angela Duckworth says that the key to success is GRIT. Watch her TED Talk here. Then determine how much GRIT you have here. Then email your instructor a reflection with a response to at least one of the following prompts:

I like …

I wish …

I wonder …

I will …

After reading Fawn’s post on Mathmunch, I posted the following for my students: is a site with plenty of opportunities for PSP!

You can DO:

-work on a puzzle

-solve a problem

-struggle with a problem

Turn in your work on the puzzle/problem.

You can MAKE:

-recreate a piece of math art

-create your own artwork inspired by the original work

Turn in your artwork.

You can WATCH:

-watch a video & leave a comment on the post about what you learned and/or questions you have

Turn in a screenshot of your comment on the post.

You can READ:

-read about a mathematician and compose a few questions you’d like to ask this person.

Turn in your two questions.

You can PLAY:

-play a math video game, then write a critique of it (likes, dislikes, suggestions, etc.)

Turn in your critique of the game.

Idea from Fawn Nguyen.

The possibilities for PSPs are endless. Some students read an article on mathematics or technology and share what they learned. Some students share a website that they have found which is helpful for learning more about mathematics. Some students do constructions using TI-Nspire. Some students write journal reflections on using a Standard for Mathematical Practice. Some students work cryptograms and problems of the week or do math history research from my school website.

In an effort for encouraging students to take the PSAT, I do give PSP for the math section of the PSAT during the second quarter (not a 1:1 ratio of math score:PSP earned). And for seniors, I give PSP one time for the math section on either the ACT or SAT. Another teacher had this idea, and I have continued it. All students get some choice about what they find interesting enough to explore for PSPs.

Student can earn some of their points in class by answering bellringer questions and Quick Polls correctly. At the end of the quarter, I total up all of their points and multiply by some scale factor that gives each student around 25-50 PSPs, depending on which quarter it is. I usually start out the year with letting them earn up to 50 from class – and decrease that number as the year goes on. I like that they can earn some of these points through class questions, because that gives them some incentive to not only be in class, but also be active participants in class. Using these points for PSPs instead of entering each assignment in the gradebook separately takes off the pressure of having to get every question correct. We are in the process of learning, and we don’t already know it all – there is plenty of leeway for students to earn PSPs and make mistakes.

I’ve wondered from time to time whether I should stop requiring PSPs, but each time I ask students to reflect on their experience with PSP, they insist that I should keep them going. And so I do, as the journey continues ….


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