Math in Art. Art in Math.

When I first discovered that there were connections between Math and Art, I was fascinated.

So, I started integrating some art into math lessons.  Originally, my purpose was two-fold: 1. Make math class fun. 2. Show students this new world of Mathematics.



Fail.  This was definitely not a good teaching strategy.  Students mostly enjoyed the tasks, no doubt.  But the way I presented the tasks... it was more show.  The students did not think they were actually doing math, but were mostly humouring me since it was definitely more fun than math.





Yet, it was not total failure.  Surprising 'aha' moments popped out of integrating some art into math class.  Take a look at the polynomial function in the picture above represented in a pie.  Suddenly the 'zeroes' of the function mean something more than that curvy line crossing the x-axis.  Zero means don't draw any circles.

 

Students get a deeper understanding of fractals, similarity, patterns, infinity.... when they actually need to physical use a pencil or paintbrush to draw out designs.  Take a look at the series of pictures above. The painting on the left is a 2' x 3' painting, and each consecutive picture is just zoomed in to the next level of detail.

This hexagon of Pascal's Triangles is about 2 meters across.  Surprisingly large!  How does it work?  We first set up a colour code: 0-green, 1-blue, 2-orange, 3-yellow, 4 purple.  Then each student painted one of the triangles in either Mod 3, 4, or 5.  So, what skills are they using here?  Nothing too intense.  The numbers they were adding together were less than 5.  A bit of modular arithmetic.  Yet, the aha moments that DID come out were surprising.  Grade 12 Math students making statements like... 'Hey, these are all zeroes, so if we keep adding zeroes it will still be zero.' Very true.  Great observation.  Funny thing is, they never needed to add zeroes together before. Even though they knew that loads of nothing is still nothing, it felt more powerful when a big patches of green zero-ness appeared.

If students were given an assignment to graph that extensive list of equations above, they would scream with the dullness of it.  Lines.  Circles. Parabolas.  What's the purpose of it all.  Give them the opportunity to design a face.... and students almost beg me to show them (ie teach them) about inequalities, restricted domains, transformations, etc.  Why?  Because they want to fill the eyes in with color, keep the smile within the face, or move that nose just a tad up so it doesn't overlap with the mouth!

I attempt to put a little more thought into integrating art into my math classrooms now.  Students need to feel that what they are doing is truly mathematics to appreciate the deeper levels of mathematics involved.  Hopefully, I can take this to the next level in my classroom next year!

The three pictured below were actually an art class assignment from my current students... but I still think they are pretty awesome!

Freedom to Choose

Pushing my students to do math has always been a battle.  Try to make it interesting.  Keep them entertained.

This has gotten easier over the past few months.  Instead of painting up the math with exciting colours, I have stripped it bare.  I stopped forcing them to work on an assigned list of problems that is due the following day.

Originally, my students had less freedom.
Term 1: Traditional.  Assigned seating.  Students quietly face 'front', listening, taking notes.  Practice problems build in difficulty. Homework checks (when I remembered).
Term 2: Non-traditional.  Random groups every day.  Work standing at whiteboards.  Big problems.

In the last little while, I have been allowing them some more freedom.
1. Work however you want.  (Paper or Whiteboard. Standing or sitting.)
2. Work with whomever you want.
3. Work on whatever you want.

Strangely enough, they push themselves more!  Instead of resisting and complaining about a list of assigned practice problems that are too hard... they are choosing to work on the problems that are hard! Why?  Because those are the ones that they know they need to practice.

Here's an example what happened in a review lesson.

Students walked in to 4 questions on the board, from 4 completely different units.  On a scale of 1 to difficult, most of them pushed difficult.

On another board I wrote:
1 - No idea what to do ( ~ 40%)
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2 - Understand what to do after checking textbook or talking to somebody. (~60%)
3 - Understand what to do, but made a mistake. (~80%)
4 - Understand what to do, completely correct. (~100%)

I told them to try each of these problems to see what level of understanding they have so they know what needs work.  This worked as a great 'quiz' replacement.  Students were 'willing' themselves to get a 3 or a 4.  They were sitting elbow to elbow.  Perfect environment for cheating on a quiz.  But if its not for mark, who are you cheating?  The class was quiet for about 10 minutes as they sweated over the problems individually.  Slowly a few students pulled out their textbook (their resource) for some ideas.  A few whispered to each other.  They studied their textbooks in pairs.

Suddenly, it was no longer about right or wrong.  How many points can the student earn.  It was about 'How well do I actually understand this?'  For me to assess my class, I did not need a quiz average.  I just needed to watch and listen.

At the end, students did not ask me what the right answer is.  They asked me good questions.  How do I multiply terms with exponents?  When might I need to use the tan inverse function?  Why is any number to the power of 0 equal to 1?  They frowned at me when I gave them a little exponent rule 'x^0 = 1' because they wanted to understand the rule.  Check. Check. Check!!

Following this 'quiz', I told students to flip to the cumulative review of the whole year and work on whichever problems they want, however they want, with whoever they want.  Here's a few pictures.

You will see students working on paper, portable whiteboards, and vertical whiteboards.
True, the classroom was fairly buzzing with noise and students regularly go into off-topic conversations.  I haven't figured out how to totally prevent the off-topic-ness, but I believe that quality working time trumps quantity working time.

A few things that really stand out lately in this approach are:
1. Hope.  Instead of writing themselves off as a failure when they make mistakes... they have a sense of hope that they can master difficult concepts.

2. Fun.  Instead of complaining about work, students enjoy the casual, stress-free atmosphere where they have freedom to choose.

3. Challenge.  Nobody truly likes working on tasks that are 'brainless' or 'useless' when they are working for themselves.  So with the freedom to choose what they want to work on, students challenge themselves to questions that look difficult.

As a teacher, this environment of students working independently and collaboratively gives me the time to informally assess the needs of individual students, coach where needed, and to teach small groups when they require it.