PBL: Problem Based Learning


Seeing mathematics in the world around you is a trainable skill.
This presentation will focus on participants rather than the instructor. I am simply sharing the tremendous ideas of others and how I use them to improve my own teaching.


My Rough Guidelines for Incorporating Digital Media

  1. Encounter an interesting image or video.
  2. Formulate questions associated with the image or video.
  3. Reflect on your own practice. Which questions are worth student investigation?
  4. Structure tasks with you, the instructor, as facilitator. You should fade into the background. Students should be doing the thinking and the doing.
  5. Determine where in the curriculum the activity fits and align it to standards.
  6. Determine how you will craft assessment tasks associated with the activity.


Good Places to Start

Math Class Needs a Makeover
Watch Dan Meyer's talk. Mr. Meyer points out some glaring issues math teachers face in classrooms and provides a rationale for incorporating digital media in the classroom.







101 Questions
Create an account on this site. Start formulating questions for images and videos. If you do not find an image or video interesting, skip it and go on. If you find an image or video interesting, be sure to use the links in the top right to find the URL or to download (pictured above).

Deconstructing a Lesson Activity, Part 1
Deconstructing a Lesson Activity, Part 2
Read Fawn Nguyen's post on organizing classroom activities. She provides great practical tips to follow when structuring classroom activities.

3-Act Lessons
Dan Meyer's Google Doc on 3-Act Lessons. This is a powerful place to start. A great example of a low threshold high ceiling task is the Sunglasses Problem.



If you are not yet on Twitter, you are missing out on a powerful tool for streamlining professional development.



Share your effective practices and ideas with others. Sam Shah has two great posts on the matter. If you are looking for blogs to follow, Sam has a great list to get you started in his Blog Roll if you scroll down a bit.

If you do want more information on how to start a blog, now is a great time. Check out the Explore MTBoS page.



An administrator told me about a problem from the world outside school he faced while building his home last year. He had gone to the post office to confirm the location in which he wished to place his mailbox was acceptable to the U. S. Postal Service mailbox guidelines. The administrator poured the footing and ordered the stone for the mailbox. A few days later, the mail delivery person informed the administrator the mailbox placement was incorrect. As a result of a miscommunication, the Post Office instructed the administrator to move his mailbox, causing the administrator to lose his considerable fiscal investment in the mailbox. The issue has to do with the route the mail delivery person drives each day. Below is a map of the neighborhood in question.


Exhibit A: The neighborhood. Mailboxes appear as an "X." The house in question is drawn in red. Street names have been suppressed to protect the innocent. Red ovals indicate either a cul-de-sac or a potential entrance from a main road into the neighborhood.

Some background information: mail delivery personnel prefer not to get out of their vehicle. Many mail delivery vehicles have the driver's side door on the right hand side, so a person could infer the mail delivery route using the condition the mailbox must be on the right hand side of the delivery vehicle.

The administrator came to our classroom Thursday. He served as a guest speaker for twenty minutes, drawing the annotations on the Google Maps image. It reminded me of a press conference format; the students asked questions about relevant and irrelevant information in an effort to fully understand the problem.

This problem is the perfect follow-up to the "Chomp the Graph" activity on graph theory. Students can investigate routes using directed paths. Students can compute distances by coordinatizing the image and applying the scale which appears in the lower left. What the students really want to know is whether it is possible to optimize the mail route in a way that benefits the administrator. My students will work on this problem next week in the computer lab. I am eager to see where the investigation takes us.

The October 2013 NebraskaMATH Newsletter



contains published work relating to one of my earlier blog posts,
The Classroom Seating Problem.

Here is the direct link to the article regarding Andrew Bartow's solution to the classroom seating problem.



Special thanks to the University of Nebraska - Lincoln Center for Science, Mathematics, and Computer Education for providing the opportunity for my student to publish his solution.

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Our school district started a new alternative school this year. I went to the alternative school during our professional days. In retrospect, my visit was serendipitous to say the least.

I walked into an English teacher's classroom. After pausing momentarily, she asked, "Where are you going?" As I stepped out of the room, I replied, "I left my phone in the car. I need to get my camera." I returned and took a photograph of the classroom layout.Classroom_Seating_ProblemExhibit A: English classroom. Note the students in the back left corner of the background. These two students will bump into each other as they pull their chairs away from the table.

The English teacher asked me, "Is there a mathematical way I could figure out the best way to arrange my tables?" I asked her how many students does the arrangement need to seat. She said 19 students. I posed the problem to my MTPS students with only the photograph as a reference. Students began modeling the problem in several ways: drawing on graph paper, cutting out paper trapezoids and taping them to notebook paper, applying masking tape on the floor of the classroom to represent the tables, even using manipulatives.

Taylors_diagramExhibit B: A physical model of a possible table configuration, affectionately referred to as "Taylor's Diagram"

Before going to the lab to model with Geogebra, I did receive some clarification on the table and classroom dimensions. I included them in a follow-up file for students.

Seating_Problem_with_DimensionsExhibit C: The table photograph with dimensions listed.

The English teacher also added there is a tenth table available should we need to use it. I will publish some of the students' modeling work later. For now, I am interested in potential solutions readers will pose.

The "Home Depot Garden Club Photo" investigation described in an earlier post has been dubbed the "Berry Bush Problem" by my students, likely in an effort to conserve syllables. Another reason for the renaming is due to the mislabeling of a plant at the Home Depot: what we thought was a blackberry bush was actually a blueberry bush. The activity associated with the photograph will be the basis for my presentation on 9/30 in Kearney at the Nebraska Association of Teachers of Mathematics (NATM) Annual Meeting. The description in the conference listing reads:

CCSSM Practice Standards 2 and 4: Using Images to Maximize Student Engagement: This session will model how to use an image as a hook and build a lesson which engages students in the listed practice standards. Participants will formulate questions and conduct investigations based on these questions. The featured image is one the presenter took while shopping at Home Depot. Geogebra will be used in this session. Topics may include, but are not limited to, volume, area, linear functions, temperature, and modeling. Participants would benefit from bringing a laptop or tablet device with wireless capability.

For reference, here are the two Standards for Mathematical Practice featured:

CCSS.Math.Practice.MP2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

CCSS.Math.Practice.MP4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.


I learned a great deal from working alongside my students and acting as a facilitator. Here is a subset of the outcomes I saw in my classroom:

  • Transfer of mathematical authority from teacher to student.
  • Ownership and intrinsic motivation in the revision process.
  • Students developing healthy strategies for working through point of frustration.
  • Procedure viewed as process rather than the culmination of the mathematics.
  • Rich discussion regarding potential sources of error, particularly roundoff error in Geogebra.
  • Collaboration and mentoring between experienced and inexperienced math students.
  • Understanding we make sacrifices in modeling that may be revisited later.











Initial student video:
Andrew, Matt, Nik, Ryan


At the NATM talk, I will show examples of Geogebra sheets and PDFs written by other students in class and provide insight on these students' mathematical backgrounds. I will also offer information on how I assess such a project in a way that encourages revision.


Revised student video:
Andrew, Matt, Nik, Ryan




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I took this photo while shopping with my wife. Students in my MTPS class worked on mathematical investigations related to information from the photo. I will post more about the students' work as it develops. The activity ran like this:

4 minutes: Present the image to students. Formulate as many questions as you can as an individual. What does this photo make you wonder?

5 minutes: Partner up. Write down questions and observations you each had in common.

7 minutes: Groups of 3 to 4. Decide on 3 questions the group will use as a basis for mathematical investigation.

20 minutes: Limited resources (peers, textbooks, pencil, graph paper, ruler) Come up with a plan for how you will address these questions mathematically. How will you maximize on your time in the computer lab?

THEN we go to the computer lab.

My primary point of emphasis with this image is to investigate the temperature conversion. There's something going on there. I will post student work examples before Monday 9/30. Feel free to leave comments or questions.