Is mathematics invented or discovered?

We've been solving some quadratic equations in Algebra II currently, and I've had an ulterior motive this whole unit.

Quadratic equations seem to lend themselves particularly well to math history lessons (or "math commercials" as my principal calls them--love that).  For example, when we talked about the Square Root Principle, I gave a mini-lesson on Christoph Rudolf (names that rhyme are the best, aren't they?) and the introduction of the square root symbol and how it's supposed to resemble a lowercase r, etc.

I asked them innocently here, "So, do you think mathematics is invented or discovered?"

If they said, "invented," I said something like, "So, the square root of two didn't exist until Rudolf came up with a name for it?"

If they said, "discovered," I said something like, "So, you're just going to ignore the contributions people like Rudolf made to mathematics?"

I let them hash it out a little, playing devil's advocate all the way.  And then I ended with, "Well, interesting conversation, guys," and proceeded to my next slide.  Which, inevitably had the effect of "WAIT!  Aren't you going to tell us?"

"No."

Which then had the effect of, "I'm going to Google it!"

"Go for it."[1]

The next day, Day 2, we continued with the square root principle, but now we tried to solve equations like x^2=-1.  I let them try to convince each other that there is no real solution to this equation (though their multiplication skills are still lacking, so...sigh).  Here's where we talked about imaginary numbers and a mini-lesson on Euler ensued.  I told them Euler couldn't stand not having an answer to this problem, as it--along with other problems like it--had been appearing in mathematics for nearly two thousand years.  So, Euler made his own solution, and called one of the solutions i.

"Now do you think mathematics is invented or discovered?"  We took a poll[2]:

3rd Hour

5th Hour

At the end of class, I had them write a letter to me defending their answer.  They were instructed to choose only one (invented or discovered).  Here are two really great letters, one from each point of view:

Dear Mrs. Peterson,
Mathematics was discovered, because just because a human didn't know the answer to something doesn't mean it doesn't exist.  Before the Pythagorean Theorem was invented, a right triangle still had an area.  Humans simply put words/letters/numbers and theorems to help us find the answer, and explain math, but the problem they solve, and answers they find, were always there.  Some species of animals haven't been discovered yet, but when someone finds them, they didn't invent the animal, they discovered it.

Dear Mrs. Peterson,
I believe mathematics is invented.  I believe this because invented means to create or design something that has not existed before, or make up an idea, name, story, etc.  You have to create a name for mathematics to exist.  One apple is not one apple unless you give a name to the number or quantity of the apple.

The next day, something happened that I think will go down as one of my favorite teaching moments of all time.  A student, who has said from Day 1 that she's not good at math, came up to me before class and looked at me with her precious, sincere, huge brown eyes:

"Mrs. Peterson?  I really need to ask you something."

"Go for it.  What's up?"

"Can you PLEASE tell me--is mathematics invented or discovered?  I can't stop thinking about it."

Cue burst of emotion and huge cheesy grin on my face.  Why was this so wonderful?  Because she just experienced what makes mathematics so addictive:  the deep longing to solve or to prove, and the pleasure that follows the accomplishment.

Luckily (or maybe unluckily) for her, on this day, Day 3 of our discussion, I wrote a letter to my classes, defending my point of view.  Now, I didn't give myself the same restrictions I gave them, and I typed this up the night before (I know, bad Rebecka), so it's pretty rough around the edges (hey, that's the great thing about teaching--now I have a whole year to make it better).  But, here's what I wrote:
Discovered or Invented

What was so, so cool about this whole discussion is that it really appealed to most of my students.  They were hooked.  They kept asking about it.  They wouldn't let it go.

What more could I ask for?

So, what do you say:  Is mathematics invented or discovered?  I'd really like to know your input...so I can make my letter better for next year.


[1]  When I checked in on these students, they seemed more confused than when they started.  Let's hear it for UnGoogleable Problems!

[2] Don't let the total numbers fool you.  Only 1/2-2/3 of my classes participated (don't want anyone thinking I have a class of 17!).  Not sure if the rest didn't want to commit to a single answer, if they didn't have access to a phone, or if I just didn't quite hook 'em...

Trig Art Projects 2012

At our school, our Pre-Calc/Trig students create trig art projects every fall.  I know it's a pretty popular thing to do, but I gotta say, I'm pretty sure I got some of the most amazing trig art ever.

The math criteria:

• Use at least 3 trigonometric functions
• Each function must have at least 2 periods
• There must be at least 2 shifts (vertical or horizontal)

Once students figure out their functions, they graph each function on a transparency or a sheet of wax paper.  This acts as their blueprint.  From there, they copy their blueprint onto a quarter of a poster board and add their artistic genius.  Finally, they fill out a rubric, telling me things about each function, such as its domain and range (they must state the domain and range that they used for THEIR project; i.e., "all real numbers" does not cut it), amplitude, and any shifting.

Students working with Desmos in the Math Lab

I took a day to let them play on Desmos.  This was fantastic because they were able to see immediately what  their transformations did to each function, without having to graph each function fifty times by hand.  Also, I showed them how to restrict the domain of their functions, which taught them a teeny bit of programming.  I feel that using math to understand technology is one of the greatest things I can teach my students.  So a day in the Math Lab to play with trig functions was well worth it, in my opinion.


I let my students vote on the Most Creative Award.  Here's their winner from 1st hour:

And from 4th hour:

Each winner got a gift card to Target in the approximate amount of $3pi ($9.42).  And, no, when I asked for that amount, the cashier's face was not nearly as disconcerted as I was hoping.

But the kids laughed.  And that's what matters.

Teepees for Factoring

One thing that's been very shocking to me as I've made the transition from college instructor to high school teacher is the lack of number sense that many of my [Algebra II] students possess. Estimation skills, knowledge of times tables 0-12, and the recognition of a negative sign are so sorely lacking.  On a daily basis, I get told that a negative plus a negative is a positive, because, "two negatives make a positive, Mrs. Peterson."  Students also regularly explain to me that when we multiply a negative times a positive, the product will take the sign of the larger factor.

I haven't figured out how to break these bad habits.  But I'm working on it.

Needless to say, the thought of teaching factoring was a bit daunting.  How can I teach them to un-distribute, if they can't distribute correctly in the first place?

And then a couple colleagues of mine introduced me to the x-method, or what I call the teepee method.  This may be old news to many, but I had never seen it before, and I found that it was just the bit of organization some of my students needed in order to factor trinomials.

So, let's say we want to factor x^2-9x+20.  We create the following teepee:

Then we find two numbers that multiply to be the top number and add to be the bottom number:

And, viola!  Then we can factor the original trinomial: (x-4)(x-5).

This by no means solves all my problems.  How can we find those numbers in the first place if we don't know how to multiply?  However, it is a nice little organizer for those students who are visual learners.

I can't take any of the credit for this visual organizer.  I'm just passing along what I learned from my wonderful department.  But, in the words of LeVar Burton, "Don't take my word for it."  Here's what some of my students wrote when I asked them to choose their favorite form of factoring from the ones we had discussed so far (GCF, difference of squares, and trinomial factorization) and tell me why...

“Trinomials are my favorite because I like to make lil x’s and then put the numbers inside that would make the others true.”

“My favorite factoring exercise is trinomial factoring because it really makes you think.  The x’s really help too.”

“The trinomial factoring is my favorite because it’s easy to use the teepee.”

Review Game: TRIG BRAIN POWER!

To prepare for an upcoming test last week, my Pre-Calc/Trig students and I had a great time playing TRIG BRAIN POWER! (because I can't think of a better name, and yes, it must be in all caps).

It's quite simple.  On the SMART Board, my "home page" was a unit circle with six adorable colored brains sitting on the positive x-axis:

I split students up into six teams, one team for each colored brain.  I would show a review exercise, and when every single person on the team had a written response, the team raised their hands, which signaled me to come check their answer.  The first team with the correct answer got to move three spaces on the unit circle, the second team got to move two, and the third team got to move one.  After each exercise, I had the winning team explain their strategy, and if there were no questions we moved on to the next exercise.  The team that rotated through the greatest angle at the end of the hour won (we had to add 2pi rads a couple times!).

I essentially adapted a game called BrainSavvy that I found on SMART Exchange.  If you want the Notebook file for this particular edition of TRIG BRAIN POWER!, just shoot me an email.

I liked this because it involved minimal prep, the students had to work with each other, they worked hard, and they had fun.

Function Transformations/Domain and Range: Day 2 (and a bit more)

Day 2:  Domain and Range of Parent Functions

I started the year with domain and range of a finite set of points, because, it's an easy concept and I wanted my students to know--Algebra II is totally conquerable.

Fast forward a month, and I think they might just ready to handle a continuous case.  We broke it up sloooooowly and built towards finding the domain and range of the four functions they found the day before.

Domain and Range

Obstacle #1:  Closed circle v. Open Circle
I really enjoyed this part of the lesson because the students were totally in to the closed circle v. open circle, which is awesome...but not before we jumped over a few hurdles.  When I asked them what an open circle denotes in mathematics, a few people proudly reported, "Parenthesis!"

No, sweethearts, an open circle does not mean parenthesis, last time I checked the dictionary.

So, we got to talk about my favorite branch of all--analysis.[1]  How could we write all the numbers between -5 and 4, but not including -5 (see fourth slide above)?  Of course, [-4.9, 4] was suggested.  But then poor -4.99 (just to mention one) gets left out!  Any number they suggested that was close to -5, but bigger, I could always find a number that was even closer (thanks, density!).

Hmmm...

And then a kid suggested something that was absolutely brilliant.

Incorrect, but brilliant nonetheless.

"Could we use -4.999...?"

Cue look of How far do I dare take this subject with these kids?

I wasn't quite sure.  And still am not so sure.  I decided to write this on the board:

-4.999... = -5

"This is what we know, and can prove, mathematically:  that -4.9 repeating is equal to -5.  So, sadly, writing [-4.999..., 4] does not help our quest because that's the same as [-5, 4], which is what we were trying to avoid in the first place."  BRILLIANT thought though, loquacious kid in the front row.

So, new notation is all that we could come up with in order to fix this dilemma.  Parenthesis.  As was wildly suggested before.

Now if only they could remember a few of the deeper ideas as opposed to just "open circle=parenthesis."

Obstacle #2:  Domain of Deceiving Functions
Here's what I mean by a deceiving function.  I took y=x^3 and showed a graph like this:

Most students were convinced that the graph would never pass x=-3 on the left, and x=3 on the right; hence the domain must be something like [-3,3].  And who can blame them?  They haven't developed a good sense for what the graph of a function is yet.

But that's ok.  Because we have Desmos.

So, to abettercalculator.com we went to graph the function, along with the line x=-3:

The kids' case was looking good.  Until we started scrolling down:

And down...

Aw, isn't that a lovely linearization...

And changing the y-axis view even more...

Gah!

And this convinced many (though not all)...the domain is indeed all real numbers.

Days 3-5 were spent focusing on vertical and horizontal translations.  We tried vertical stretching/shrinking, too, but I started to lose several of them, so I decided parent functions, domain/range, and translations were plenty for now.  We can come back for the rest later.  That's the beauty of 180 days, as opposed to the 48 that I'm used to.

Before the unit test, we played Kate Nowak's Speed Dating Game, but I adjusted it for these topics.  You can find the game cards I created here.

[1]  I claim analysis/advanced calculus as my emphasis in grad school...mostly because that's what I took the most classes in and that's what I took my written comprehensive exams in.  In any case, while I can't say I've read that many texts on analysis, I can say, that Understanding Analysis by Stephen Abbott is, by far, the best text I've ever seen on introductory analysis.  All calculus teachers should be required to read it.  Truly!  It's the best.  Get the whole thing for free here.  Go.  Read.  ENJOY.  He's a master teacher.