Where There's a Will, There're Many Different Ways

In class this week we did a very interesting activity.  We were given the following pattern:

We were essentially asked to come up with an algebraic equation for this pattern. I started by writing a table of values, but it didn't help me too much right away.  I could see that the pattern wasn't linear (ie - increasing by the same amount of squares every time), but I was finding it really hard to find the equation. I looked around me and saw everyone else writing furiously, so I knew I had to do something.  I figured that since the first term had 2 squares only and that every subsequent term had 2 squares on either side of it, that I could start with a "+2" at the end.  Now I just had to figure out an equation for the middle of each figure.  I ended up using area. 

An easy way to see it is that you can see that figure 2 has an area of 1x3, figure 3 has an area of 2x4 and so on.  This works out to (n-1)(n+1).  So, the final equation is (n-1)(n+1) + 2.

The interesting thing is, not everybody saw it as an area with two squares on either side.  Some saw it very differently.  In fact, here are some examples of the different ways people saw this pattern:

Now, they all work out to the same simplified answer of n^2 + 1, but it just goes to show you how different people look at patterns in different ways.  It's another reminder that not everyone sees math the way I do!

As I keep going on this journey to becoming a math teacher, it just keeps getting confirmed to me that I'll need to approach my teaching from many different angles.  I think I really just always assumed that there is one "best" way to teach a concept.  In a way that's true...there is one best way for me.  And there will be one best way for every single person in my class.  The problem is, that best way will not be the same for everyone.  And that's something I'm going to remember.

"Guess Who? - Math Edition" and Other Fun-Filled Games

This week in class, we learned about some of the technology available to enhance math teaching. Our instructor introduced us to two absolutely amazing websites:  Solve Me (https://solveme.edc.org/) and Desmos Classroom Activities (https://teacher.desmos.com/).  I will discuss each one in turn, but let me just say that I'm very excited to use both of these websites when I get into a math classroom.

Solve Me is a free website which provides a few different kinds of puzzles including mobile-type puzzles, riddle puzzles, and sudoku-type puzzles. In class, we focused on  the one called Mobiles.  Mobiles is a place to practice solving equations using balanced diagrams.  Here is an example of one of the puzzles:

Some students find it much easier to solve equations visually than through the typical algebraic equation, which in this case would be x + y = 3y.  We know that y = 2, so we get x + 2 = 3(2), which works out to x = 4.  Many students struggle with this type of equation, but would be able to solve the equation logically through the above diagram by reasoning that if the heart is 2, then the right side equals 6.  The left side has 2 from the heart, so therefore since the scale is equal, the trapezoid must be 4.  It allows students to work out the problem whichever way they feel more comfortable doing!

We were also introduced to another website called Desmos.  This website is typically known as a graphing calculator, but it also has an amazing teacher section where there are many different activities to use with the class, but not only that, it allows you to actually build your own activities!  Teachers do have to pay for these capabilities, but it seems well worth it!  In class we played an activity called Polygraph, which was basically "Guess Who?" for math.  

Students navigate to the Desmos website and put in a code provided by the teacher.  This connects the student to a unique session with other students in their class.  The website pairs people up randomly and they play a game of Polygraph.  One student chooses a parabola, and the other is shown 16 parabolas (including the correct one) and has to ask "yes or no questions" in order to figure out which parabola is the one their partner chose.  This allows the class to learn proper terminology for parabolas while having fun doing it.  I loved this activity!  Here is an example of the game being played:

I honestly never realized there were such creative ways to teach math!  When I learned math in high school, it was pretty much all lecture format (the internet was around, but wasn't used to anywhere near the capacity it is now) and, while I actually enjoyed that way of learning, if these activities were available it would have truly enhanced my learning.  I also realize that for some students, these activities may be the reason they continue on in math - the activities help them understand!

In the past, technology in the math classroom was generally limited to calculators.  But now, there are just so many choices.  It's a wonderful way to help students learn and practice math without making it feel like they're learning and practicing math.

"Math People" - Do They Exist?

This week in class we watched a video by Jo Boaler, a math professor at Stanford University. (https://www.youtube.com/watch?v=pxru8H6XbR4)  In the video, she spoke of the plasticity of the human brain; that is, the brain's ability to make new connections and stretch and change to learn new things.  She argues that there's no such thing as someone "not being a math person".  She says that we all can be math people, because the brain can learn anything if we put our minds to it.

While I agree with much of what she says, I disagree with the idea that there are not "math people".  Why do I disagree?  Because I'm a math person.  I'm definitely a math person.  And what I mean by that is not just that I can do math, but that I actually like it.  I believe that in order to learn something, you have to be interested in it, otherwise your brain is just not going to be motivated to make the new connections.

Now, it might be a "chicken or the egg" situation in the sense of why non-math people don't like math.  Could it be that they initially found it very difficult and, thus, didn't like it?  Or was it that they didn't like it and, thus, didn't want to try very hard to learn it?  Hmmm, could be either...

Also, Howard Gardner's theory of multiple intelligences (https://www.learning-theories.com/gardners-multiple-intelligences-theory.html) does suggest that there is a difference between having a logical-mathematical intelligence versus linguistic, visual-spatial, etc.  If someone is not strong in the logical-mathematical intelligence, would this not hinder their learning of math?

The other reason I feel this way is that I have experienced this with music.  I love musical instruments.  I've played the alto sax since I was about 12 or 13 years old.  When I picked up the saxophone, it felt easy; I learned quickly and within a few months I was invited to play in my high school's jazz band.  Others in my school who were learning sax didn't have quite the same easy experience.

Now, you might think that I maybe just have the ability to pick up any instrument quickly, but this is not so.  My husband is very good at electric guitar - he picked that up as a teen with a similar ease to how I picked up saxophone.  Here's the thing, I have tried to learn guitar. My husband has taught me the chords, and I've tried to remember them and play them, but I just find guitar really difficult.  I get frustrated with it.  Funny thing is, I've tried to teach him to play saxophone and he feels the same way!  Now, could I learn guitar?  Sure I could, but I don't want to because it doesn't come naturally.  It would take a lot of work to learn it and to me it's just not worth it.  I'm not a "guitar person".

 I think that people feel the same way about math.  Could they learn it?  Sure, but maybe they simply feel it's not worth it and don't make the connections.  After all, there's more to life than math (isn't there?) and people who don't like math tend to have other strengths, such as writing, art, or history to name a few.  Maybe it's not such a bad thing to dislike math and not be "good at it".  After all, I'm not good at art and I'm alright with that.  I can appreciate it and I do enjoy drawing and painting - it just doesn't come easily and I'm not very adept.

So maybe the goal should not be to make everyone "math people", but rather just to make everyone "people who aren't scared of math".  And that's what I want to do as a teacher.  I just want to show students that math is not scary, and even if you don't like it, or find it difficult, you can get through it and maybe, just maybe, develop a slight appreciation for it.  I want to tell my "non-math person" students, "This too shall pass....and so shall you."