What High School Geometry Could Be

If you've read any of my writing on education, you already know something about how I think teaching should be done. If you haven't, here's the short of it: The testing machine that is our education system is conducive to programming robots, not educating people, but since that doesn't appear to be changing any time soon, it falls on the teacher to think of creative ways to encourage genuine problem solving while still fostering the ever-so-highly-valued exam-taking skills.

My goal this year has been to make my geometry classroom look more like my computer science classroom. A number of factors are goading me to maintain the run-of-the-mill, Regents-focused, common-core- and common-assessment-aligned Geometry syllabus, but my will is set on disrupting the norm with something hands-on, memorable, or perhaps even... (brace yourself) worthy of making a student say "I like geometry." 

A project that is successful in that regard must embody at least three principles:

  1. It must require student choice. More is better.
  2. It must resemble a real life application.
  3. It must be an end in itself. To superimpose exam questions on the project content is to dilute the meaning of the project itself and redirect the students' attention back to the very falsehood we meant to avoid in the first place: an exam is the ultimate end and indicator of success.

This year, I've come up with the "Geometric Dwelling" project.

The Specifics

In this example diagram, students can describe symmetry exiting in this unusual blueprint.

In this example diagram, students can describe symmetry exiting in this unusual blueprint.

In this project, students form teams as large as seven to build a model dwelling (apartment, condo, or house). Each student must build his or her own room and no more than one room. After planning in school and constructing at home, the rooms must be brought to school and fit together. In this way, building a dwelling requires precise measurements and planning. Furthermore, a dwelling must satisfy the following requirements (presumably set forth by an imaginary "client")

  • Every room must be connected to at least two other rooms.
  • The dwelling must exhibit symmetric beauty in every room. For instance, rotational or reflective symmetry.
  • Every dwelling must have a bathroom. The bathroom must be the smallest room in the dwelling.
  • Every dwelling must have at least one bedroom.
  • Every dwelling must have a room that is not a bedroom, kitchen, or bathroom. (Examples include: family room, sitting room, living room, swimming pool room, band practice room...)
  • Every dwelling must have exactly one kitchen/kitchenette. The kitchen must be smaller than another room (but not the bathroom.)
  • Every room must have at least one piece of furniture, built to scale.
  • The overall appearance of the house should be consistent.
 
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Finally, to support the team's design choices, each member of the team must submit a written justification of the symmetry exhibited in the dwelling. Here are some actual student responses.

The bed in my room, bed ABDC, is a reflection of my partner’s bed , bed A’B’D’C’, in her bedroom (on the left). My bed, ABDC is reflected over line XY to form bed A’B’D’C’. [XY is formed by] cnstructing the midpoints of line EF (horizontal line of the dining room) and line GH (horizontal line of the the living room), which were point X and Y, respectively. [...] Bed ABDC and A’B’D’C’ are reflections because both beds are equidistant from the line of reflection, line XY. [...] Also, any line drawn using the corresponding points of bed ABDC and A’B’D’C’ would be perpendicular bisector of the line of reflection because being so is also a property of reflection.
My kitchen exhibits rotational symmetry with my partner’s, Eric’s, living room because both the bedroom and the kitchen have congruent walls and doors and the corresponding points are equidistant from the center of rotation which is the middle of the door connecting the kitchen and living room. The angle of rotation is 180º and the angle between the corresponding points are 180º.

Scoring

For scoring, students were graded on four criteria. If you would like more detail on the rubric, please refer to the assignment website:

Along this wall, doors are completely off alignment, but walls are perfectly flush.

Along this wall, doors are completely off alignment, but walls are perfectly flush.

Connections – The doorways of every room that this individual's room is adjacent to are aligned with the doors of this room.

Walls – Every room that this individual's room is adjacent to is flush both in height and width.

Along this wall, doors are not aligned and the wall is not flush vertically.

Along this wall, doors are not aligned and the wall is not flush vertically.

Theme - Each individual's room is skillfully made. It is clear it was not rushed. It embodies the theme and a justification is precise.

Design - This part of the rubric is scored as a group. Students may score a '5' when all dwelling requirements are met and the scale of the furniture is satisfactory.

 

 

 

Results

Overall, I was immensely satisfied with the execution of this project. Students were very actively engaged while planning their dwelling and individual responsibilities. Groups were huddled together in patches throughout the classroom and many groups were using whiteboard space to draft a blueprint. Because this was the first year I implemented the project, I also collected feedback from my students on their experience. In general, feedback was positive, with students citing an appreciation for collaboration and the chance to work over the course of multiple days. I have included photos of student work an both positive and negative quotes below: 

 

 

Please rate the extent to which you agree with the following statement: "Doing projects is preferable to taking exams."

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Please explain your answer to the previous question.

With doing group projects there is more room for error. The reason being is because you have people to check after you and help you when you need it.
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I would rather do a project than an exam because when doing a project you actually have to think rather than memorize how to do a problem. When doing the project we have to be creative and use outside information which shows more of an effort than a test.
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Projects should not replace exams because the projects only test the extent of your social, cooperative, and communication skills as well as a little bit of knowledge. However, the exams fully test the knowledge/understanding/intuition of a person.
I believe that doing projects rather than exam is preferable because you get to spend a more than good amount of time working on your end of the unit “assessment”. You have the chance to look back on your notes to make sure your project is perfect. Also, many people, like me, spend a lot of time studying and getting passing grades for homework and classwork, but when there’s an exam, we still pass but with a low grade.
 
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I like taking test because it holds less requirements when displaying knowledge.
 
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When working on a project, normally the amount of time and effort that one puts in is evident. For exams however, it is more difficult to prepare for and therefore doesn’t reflect the information we actually learned. The only reason why I would keep my rating at a 4 is because tests are still the standard way to measure a student’s progress.