The Beauty of Geometry

Every now and again, I take the opportunity to simply opine on the beauty of geometry. Math gets a bad rap because of it reputation of being cold, lifeless, functional and academic. (Some folks aren’t helping this by proclaiming that the arts are what we do to enjoy school and math is what we study to get paid later on in life.)

Don’t get me wrong, there certainly are academic ways of discussing art, music, iconography, fashion, design. There’s technique and vocab to all of these areas. Students of these disciplines are still students who must study, but their exploration isn’t saddled with the atmosphere that math is. In math, it is often believed, the box is set; the boundaries drawn. The math frontier is closed. There is no need for exploration when there is nothing to explore.

I’ve always felt like geometry has the capacity to challenge those notions. Kolams, quilts and origami help students understand the aesthetic value of straight lines, precise measurements, perfect circles and right angles. Sometimes, you have to build them to complete your understanding of them. That process alone can bring with it its own supply of feedback. When you are trying to create something visually appealing, often times, the eyeball becomes the expert in the room, not the teacher. Attention-to-detail and technique become valuable without encouragement.

At a recent professional learning opportunity, I was given some time to play with KEVA planks. So I did. The planks are all congruent rectangular prisms. So I placed one on the table. Then I placed a second on with a slight rotation (the diagonal intersection points were designed to sit right on top of each other, but the counter clockwise rotation was determined by the next block being placed so that the vertices were placed on the preceding blocks’ short-segment midpoints. It ended up being about 10 degrees.)

That all sounds pretty mathy (and probably somewhat unclear since I’ve never had to verbalize the process before). But the resulting tower is pretty cool-lookin’ (at least I think.) I simply love when objects with straight lines and right angles are arranged to look like curves. This can happen in algebra as well. As a teacher, of course I don’t know if my students will share my fascination, but fascination isn’t the goal. It’s tough to measure and, besides that, it’s fickle.

I’d encourage you to look for opportunities to change the cold, academic atmosphere surrounding the math. How can we warm this wonderful subject up? We used to take advantage of those tricky days right before a long break and do art projects. Thanksgiving Origami, or build a Christmas (or holiday… or wintertime) scene using nothing but triangles.


What ideas do you have?

Excellent Classroom Action – Art and Geometry in the Elementary Classroom

I’ve written about the connections in math and art before. The visual nature of Geometry lends itself quite nicely to this. I found that the right pairing could bring in the engagement of the visual arts while maintaining fidelity to the content.

Exhibit A: Sarah Laurens, 5th grade teacher at North Elementary in Lansing. Mrs. Laurens reached out to me excitedly while I was in her building to see a math activity that she was leading her students through. They involved quilts hand sewn by Sarah’s grandma.

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The activity went like this: Students were in groups of threes and fours gathered around one of grandma’s quilts. Each quilt was made of a series of geometric shapes. Sketch the primary “unit” shape of each quilt and identify each of the polygons that are contained within it. On the surface, it is a fairly simple activity, but listening to the students talk to each other.

[Students looking at the black and blue quilt above left]

Student 1: “Those are just a bunch of hexagons.”

Student 2: “Hexagon’s, no… no… those are octagons”

Student 1: “Yeah, yeah… same thing.”

Student 2: “They’re not the same, one’s got six sides and one’s got eight.”

Student 1: “Well… wait… one… two… three… four… five… six… six sides! See I told you!”

[Students looking at the purple and while quilt on the above left]

Student 1: “That’s an octagon with kites around the outside.”

Me: “Are they kites?”

Student 2: “They look like kites.”

Me: “They sure do. How many sides do they have?”

Student 1: “One… t, th, f… oh! five…. They’re pentagons!

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Ms. Laurens and her students were comfortably saying and hearing words like “regular”, “tessellation”, and using definitions to make sense of what these shapes are, and using the definitions to settle disagreements (the foundations of proof…)

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Interesting images like these were leading to some interesting conversations as well. I’m thinking of  a conversation I heard between two students who were trying to make sense of the shapes they were in the picture directly above to the right. They were trying to to determine if the purple section in the middle was one big shape or two smaller shapes put back-to-back.

Then after some discussion, they realized that their answer would be the same either way. (Quadrilateral was their choice for the shape name. The word “trapezoid” was getting thrown around, but the students were having to be prompted for it).

I very much enjoyed getting to see these fifth graders exploring. Ms. Laurens was excited, the students were engaged (and this was clearly not the first time they were expected to be a self-directed and collaborative).

I’m just bummed that my schedule forced me out the door before I got to see Ms. Laurens’ closure of the activity. The students were wrapping up their discussions as I had to head out the door.

Mosaic Math

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I found some more math in Lansing, MI. Right downtown, actually. The Lansing Art Gallery has a rather artsy front entrance. It looks like this.

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See the green circle? See it? Looks like Pac Man. This one:

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Closer inspection reveals that it is a green mosaic circle partially covered by a red mosaic rectangle. This grabbed my attention while I was strolling waiting to pick up a Cottage Inn pizza. I wondered how many green mosaic pieces were “underneath” the rectangle. I also wondered what percentage of the circle was covered up by the red rectangle.

Let me know if you get a solution. I’ll let you know when I get one.

The Equity of Extra-Curricular Education

It goes without saying that the role of the educational communities extends beyond the final bell of the school day. In most communities where I’m from, schools provide a variety of extra-curricular activities that are athletic, artistic and academic in nature. In general, most of these clubs are small and get very little publicity for their efforts.

Equity isn’t an issue in many of these clubs and activities because boys and girls can participate together. Clubs and activities like robotics, art club, drama, quiz bowl, for example, typically don’t deal with equity issues (at least not based on gender) because the main factors of inclusion tend to be interest, ability, and availability.

However, there is one huge area of extra-curricular activities where gender segregation is standard operating procedure: athletics. Within this realm, there are male sports (e.g., football, wrestling, baseball) and female sports (e.g., volleyball, softball). There’s sports where both genders can play, but they each have their own teams (e.g., soccer, basketball, golf, lacrosse). There’s also sports where technically the teams compete separately, but typically schedule, practice and travel as if the teams are co-ed (e.g., cross country, track and field, bowling).

Just so we are all on the same page, I’m going to state the obvious: institutional segregation of any kind is asking for some extra scrutiny. That does not mean the segregation is wrong or harmful, but it does mean that we need to be sure that the methods, reasons, and relationships between the segregated groups are arranged deliberately and equitably.

An article written by Graham Couch appeared in my local paper on May 27, 2013, that addressed this issue of gender equality in dealing with extra-curriculars.

As I stated before, almost all extra-curricular clubs and groups are low-publicity groups whose non-participant supporters tend to be parents of the participants. In mid-Michigan, there’s a small handful of exceptions to this. Fine arts clubs (theatre, choir, art, band and orchestra) tend to draw big crowds and certain athletic events: typically basketball and football. These events are advertised in the newspapers, they have billboards, news reporters are on-site during events along with lot of spectators.

To add an additional level of complexity to this, money is involved in a few of these. Big ticket fine arts events are normally free for spectators (theatre being the exception, and if you include the marching band performances at halftime of the football games). Schools also charge admission for big ticket sporting events like football and basketball.

The issue of money isn’t a small issue. Theatre departments can be expensive clubs. Continuing the program is dependent on filling the auditorium with paying spectators for every performance. Also, most schools depend quite heavily on revenue from football and basketball games to balance their athletic budgets. So, it would make a ton of sense to schedule these events carefully to maximize their attendance. Football owns Friday nights in the fall.  Theatre performances tend to be multi-day weekend events. Thursday-Saturday in the evenings and a Sunday matinee. No gender equity issues, right?

What about basketball? Both boys and girls play it. In Michigan communities, boys basketball is typically much more popular to spectators than girls basketball. That is, it draws bigger crowds. That is, it makes more money. So, schools have worked hard to keep varsity boys basketball games on Friday evenings to maximize admissions revenue. Seems reasonable… right?

Well, there’s a civil right complaint going through the Michigan Department of Education right now that’s forcing us to deal with that issue.

From the article: “There is no perfect way to schedule girls high school basketball. There never really was and there might never be. It’s a matter of meshing equality with what’s best for student-athletes and what’s accepted by communities. And those three ideals, most places, don’t work in concert.”

Things we’ve tried in Michigan:

  • Girls’ basketball season to be played in the fall to avoid direct conflict. Courts ruled in several years ago that fall scheduling was an unnecessary detriment to female athletes getting recruited to play at the college level as most of the recruiting work was done in the winter.
  • Girls’ being played on different days. But this added bus trips and made scheduling within conferences more complex. Especially considering that the gyms in most schools are also being used in the winter for volleyball and wrestling. Plus each basketball program has at least a junior varsity team, and many have freshman teams.
  • Replacing the typically slot for JV boys (right before the varsity boys) with varsity girls on Friday nights. But, as Couch puts it: “The argument against that schedule, however, is the girls game feels like a JV game.”
  • So, we’ve tried alternating the schedules, putting boys varsity games first followed by girls’ games. But, as Couch puts it: “When the girls play second… many fans leave as they’re warming up or beginning play. And the athletes notice.” He continues: “Is it equitable to have your fan base walk out on your girls game on a Friday night?” Fulton AD and CMAC president Chad Podolak said. “Maybe it is, because you’re playing on a Friday night, but the girls I’ve talked to just feel terrible about it.”
  • How about both varsity teams play the same nights, but at opposite locations. So, the same two schools are competing just one school hosts the girls games and one school hosts the boys game. Well, as Couch puts it: “And, if the teams are instead separated — one on the road, one at home — the fan base is split, but not equally.”

So, it there seems an equity issue. But, I am concerned that this might be bigger than high school sports.

Consider that Michigan is a fantastic sports state. The fans are enthusiastic and loyal. Michigan supports six major programs (Red Wings, Lions, Tigers, Pistons, Spartans, and Wolverines). Not all of these teams are good all the time. In fact, the Lions are almost always frustratingly poor. The Pistons have been one of the worst teams in the NBA the last three or four years. The Tigers have gone through several stretches where they have struggled mightily (although right now, that is not the case). There has never been any talk of these teams leaving Detroit. You never hear that. Ever.

The Detroit Shock were a WNBA team that started in 1998. In the first twelve seasons, all they did was make the playoffs 8 times, the finals 4 times and won three WNBA championships. (If the Lions supported those winning percentages over their existence, they’d have won 11 super bowls by now.) In 2009, The Shock franchise moved their team to Tulsa, hoping for more enthusiastic support from its local fan base.

From Couch: “The issue, and often proverbial elephant in the room, comes down to this: Basketball is the only spectator sport in which girls and boys go head-to-head, in high school or college. And it’s a sport built around height and athleticism, one where the physical differences in genders are pronounced.”

So, here’s my question: To what degree is it the individual high school athletic departments responsibility to help solve this social preference? The athletic departments, conferences and schedule-makers are contorting themselves to no end to try to balance out a social imbalance that seems to exist beyond the scope of the school. Do we really think that schools are responsible for causing the issue?

And before we explore the issue of schools having the responsibility to solve this problem, let’s first discuss is local school districts have the ability to solve this problem?

Let’s get back to what extra-curricular activities of all sorts are designed to do: give young people an additional way to be engaged by the school curriculum to support the overall educational experience for their 13-run from kindergarten to graduation.

Does forcing girls’ basketball to bear the civil rights burden in a way that the girls bowling team doesn’t make it a better educationally-supportive activity for the students? Are the schools (whose lack of adequate funding is always used as a political tool) at fault for trying to maximize their revenue? At what point are the schools simply called upon to create equality of opportunity and the community is on it’s own to respond? Do these problems all go away if we outlaw charging admission at games? Might the loss of revenue make the sports go away, too?

I don’t know. And it seems that I’m not the only one asking these questions.

From Couch: “The question is, at what age or level do these realities matter? When does it stop being strictly about educational opportunity and begin to morph into spectator-driven sport?”

Isn’t Geometry an Art?

The things I do to get the students to learn geometry...

The things I do to get the students to learn geometry…

By the time most students arrive in my classroom to take geometry, they have seen Math 7 and Math 8 (which are two sides of the same pre-Algebra coin) and Algebra I. The stated goals of this sequencing is to “prepare the students with the necessary algebra skills to be successful in Algebra I (and later on in Algebra II) and also to be successful on the state tests (currently the Michigan Merit Exam, soon to be replaced by the Smarter Balanced Assessment).

And then there’s geometry…

And our textbook decides that it wants to teach this class right in line with the sequencing pattern: right between Algebra I and Algebra II comes “Algebra G” or Geometric Algebra. I’ll admit it makes some sense. We have spent several years training our students to “do algebra.” Typically that includes plotting points, graphing lines, solving equations and systems, and manipulating functions. If that has been the environment for the last three math classes, then, for the sake of the students, it would make sense to keep the model the same, right? Consistency and predictability breed success, right?

But what’s the risk? Well, as I’ve talked about before on this blog, the Algebra taught in most places is significantly lacking in it’s ability to engage students. So, while the theory is that our students come in well-prepared and well-trained, the reality is that most students come in with sensitive pressure points and calluses similar to someone who has walked for three years in the same pair of shoes. While the shoes are certainly familiar, they might also be smelly and worn out. The ankle support might be gone and the students have learned how to walk funny in them to avoid the blisters and shin splints that have plagued them in the past.

Is this any less geometry then having the student solve equations?

Is this any less geometry then having the student solve equations?

Perhaps what some of them need is a new pair of shoes.

Today’s learning target is getting students to be able to visualize cross-sections and 3D shapes made when 2D shapes are rotated around specific lines.

To approach that, I started with this activity and I noticed very quickly that the students started “walking funny.” I had asked them to put on their old Algebra shoes and the predictable disengagement started setting in very quickly. I wasn’t going well. Following the Dan Meyer Model, I could find at least three indicators that I was doing math education wrong.

Then something dawned on me: Why do we insist that students do algebra all the time? Geometry is the measuring of the earth, a true down-to-earth, visual math. Why can’t it be art?

So, we changed things up. I had them all get a piece of paper and I acted like an art teacher. I put stuff on a stool and had them draw what they saw.

Engagement hit 100% fairly quickly, especially once I struck a pose and asked them to draw me (see the pic at the top).

If 100% are willing to do this, and 40% are willing to graph lines, which is a better math activity?

If 100% are willing to do this, and 40% are willing to graph lines, which is a better math activity?

Apparently, the new shoes I gave them to wear were a lot more comfortable and many more of them were willing to follow along the path. Content-based discussion and collaboration began to happen without me telling them to do so. As a result, when we had to discuss how the activity supported the learning targets, the connections were much better.

Then I gave them back the original handout (which, for the record, I still think is a good handout), and the students did much better with it the second time.

So, I ask again: why do we insist on turning Geometry into an Algebra course? It doesn’t seem like it has to be. Our current textbook says it has to be, but the textbook only has the authority we choose to give it. (See The Blessed Textbook Conflict if you want my take on the authority of textbooks.)

Today’s activity would suggest that if we are willing to consider what a class would look like if we really wanted the students to simply learn geometry, perhaps it would look more like an art class instead of a math class.

Increasing Engagement Through Art

A half-hour of drawing might make the difference to whether or not the data table gets filled out...

A half-hour of drawing might make the difference to whether or not the data table gets filled out…

I have a teaching colleague to thank for this idea. He currently teaches all of our Algebra I students, mostly freshmen. We have noticed a struggle-disengagement cycles that is self-repeating and driven by inertia. The student begins to struggle, which leads to disengagement, which leads to more struggling, which leads to… you get the idea. However, my colleague created an art activity that might be a game-changer for some of these students.

It goes like this. First, create an image on a piece of graph paper. Whatever you want. Could be your name or a picture or whatever. But there are conditions. It must include at least 10 line segments. Endpoints must have integer coordinates. It must have one pair of parallel lines. It must have at least one pair of perpendicular lines. After the pictures were drawn and colored, then the students picked 10 line segments and found coordinates of the endpoints, slope, function rules in standard and slope-intercept form.

Doing all of those things ten times over is as good as a practice worksheet, except better! Better because, you activated the right side of the brain for those who need it (which is a healthy portion of our students). There is a greatly reduced temptation to cheat or shortcut because the products are unique (and the more competitive students will see to it that their picture isn’t copied). Plus, like a good billiards player, you are always thinking two shots ahead. By tossing in the parallel and perpendicular pieces, you are running the risk that the student might begin to make some conjectures about the look of lines and their slopes.

How much more Algebra is this student learning now?

How much more Algebra could this student be learning now?

The Art of Geometry: My students’ first tries at Kolams

ImageI recently turned to Indian Folk Art to help my students make sense of rotational symmetry. For some, it was a time for them to get to show-off their artistic abilities. For other this was a fantastic struggle for some as they built their understanding. ImageIt was awesome to see the different levels of understanding. First the students were able to recognize the definition and make sense of the written words. Then they were asked to recognize and interpret online images for symmetric properties. Finally, they were asked to create an original rotationally symmetric piece. ImageAnd while not all of the art work captured the essence of the art of India, much of it demonstrated that the students were truly building authentic understanding of a spatial reasoning concept.ImageBottom line: I just built my own understanding. I’ve been told for years how the deepest understanding comes from having the students CREATE. It was awesome to see the process and it seems to be working.Image