A Streaming Epiphany

I had an epiphany this week.  To be honest, it's probably not a completely novel concept or something that hasn't already been considered, but it was an epiphany to me.

In one of my classes this week, we discussed the 4Mat learning styles.  This is a concept by which people are divided into one of four quadrants of a circle.  Each quadrant represents a certain way of learning.  You rate some statements about how you like to learn, and based on the results, you plot the points on two axes.  Once you have your four points, you join them into a quadrilateral.  The amount of the quadrilateral that's in each quadrant tells you how much you prefer that particular learning style.  Here is a photo of the four quadrants with an explanation of learning styles, courtesy of my "teaching Biology" instructor Ray Bowers:

As you can see, this particular diagram also shows what teaching style you'd prefer if you have a particular learning style.  Have a look at the diagram and notice the different ways that Why, What, How, and If people prefer to learn.  We were discussing that your learning style affects your teaching style, and that to be a well-rounded teacher, you should try to teach to all four quadrants, and not just the one you like the best.

Anyway, having said all that, I had this idea about streaming, particularly math in grades 9 and 10.  Right now, we have Applied and Academic, which, to be honest, is very similar to what it was when I was at school when the designations were General and Advanced.  The names sound better, but in essence, Applied students don't learn quite as much in the curriculum as Academic students do, and Applied students are not eligible to go into university level courses in grades 11 and 12. 

The ministry is thinking of doing away with streaming altogether, but, if you've read my previous post on this, you'll know that I'm not a fan of the idea of destreaming grades 9 and 10 math.  See, applied courses are taught with more of an emphasis on application, and Academic courses are more abstract in nature.  In an Applied math course, generally a real life example of a problem is presented and the students are encouraged to discover the concept in order to solve the problem.  Then the abstract concept is presented.  In Academic, the abstract concept is presented first, then more and more complex problems are presented, which leads to the application of the concept to real-life examples.  So, as you can see, Applied and Academic courses generally work in opposite directions.  Putting these two types of learning into one class would be incredible difficult for teachers!

This is where my epiphany comes in.  What if, instead of destreaming, we keep the streams, but both courses have exactly the same curriculum, take exactly the same exam, and earn exactly the same credit?  What if it's the learning style that makes the most difference with an Applied vs. Academic student?  What if Applied students are more interested in the If and Why and Academic students prefer the What and How?

What if...instead of looking at achievement to stream students in grades 9 and 10, we give them tests to find out their best learning style, then stream them accordingly?  We could change Applied Math to Discovery Math; the students work to discover concepts by seeing how they work in real life.  And we could change Academic Math to Directed Math, meaning that the abstract concept is presented to the students, they are directed to practice the concept, then given real life examples to apply what they've learned.

CC0 Licence - No attribution required. Retrieved from https://pixabay.com/p-1289871/?no_redirect 

This would entirely remove the stigma of streaming and give students the ability to be in a class which gives them the best opportunity to learn the curriculum.  It also helps teachers to know how come at any given concept; how best to teach it to the students.  And the best part?  Since all students get the same credit, they would be able to go into any grade 11 math course out of grade 10.

Anyway, that's about it.  I would love to hear any thoughts on this idea!

Trig Identities - Fun or Flustering?

This week in class, one of my colleagues taught a Grade 11 lesson on Trig Identities.  Those are the problems you get where you have to show that the left side equals the right side.  It's all about manipulating the trigonometric equations until they match.  I loved trig identities in high school.  For me, each one was a puzzle just waiting to be solved!  But, this is not so for many students.  Many students are scared of trig identities and dread the day they are taught in school.

Retrieved from: https://blog.enotes.com/2015/04/30/10-extra-cheesy-math-jokes-explained/

My colleague introduced a great way to practice these identities.  She said that in a prior lesson, she would have introduced the identities themselves and how they are derived.  Students, apparently, are generally alright with learning the identities, but when it comes to actually working with them to solve left side/right side problems, that's where they get worried.  Perhaps it's because it can take a few tries to solve one of these problems correctly?  The students get discouraged and don't want to finish the problems.  This is where the practice method my colleague used comes in.  She provided us with the following sheets which we had to cut into individual pieces:

Retrieved from: https://meangreenmath.files.wordpress.com/2016/05/inversetrig.png

With this activity, all the answers are there, you just need to put everything into the correct order.  The squiggly lined boxes are starting points, and all the others are steps in the problems.  In the end, you have 4 fully worked out trig problems. Honesty, I think we might have had too much fun with this activity!  But even for students who are hesitant to try to solve trig identity problems, I can see how this activity would seem more "doable".  As my instructor said, it feels much better to rearrange than it does to erase.

I think this is a great starting point for students who "hate" trig identities.  I do believe it's important for them to also practice solving these problems from scratch, but it's quite possible that activities such as this one will help them gain the confidence they need in order to stick with it.  I would be excited to use an activity like this in my classroom.

Practicing Math Can Be Fun...If You Want it to Be

These next few weeks in class, we will be presenting 20-30 minute lessons to our classmates.  Last week was grade 9; this week was grade 10.  Two of the presenters this week tackled the Pythagorean Theorem for grade 10 applied Math, and they approached it from two different ways.

The first presenter had a math doodle sheet.  It was a Pythagorean Theorem handout with blanks to fill in, places to colour, and generally looked conducive to doodling.  I thought this was very unique!  Many students love to colour and doodle and this would give them an outlet for that while letting them learn the lesson at the same time.

The second presenter had a very cool Pythagorean Theorem board game for us to play!  It was a way for students to practice using the Pythagorean Theorem without getting bored.   It looked like this:

Students would need to show their work on a separate piece of paper as they go through the game.  The only question I have is whether their completed sheet would be handed in for marks or whether it's just for practice?

The reason I ask is that I once volunteered in a grade 9 applied math class.  I was going over the geometry of angles of triangles and I had thought that they might like to do a fun worksheet to practice.  I made up this worksheet:

The thing is, after lessons were given in that class, homework was always assigned from the textbook.  I completely agree with that - I believe that students should always have assigned questions from the textbook so they can practice a variety of questions and word problems to prepare for future tests.  Also, one of the questions from the textbook was to be handed in as part of their portfolio which was to be marked later in the year.  The students did not want to do the worksheet; they wanted to work on their textbook homework.  They felt that the questions in the textbook were more important and that my worksheet was just extra work; and I can completely see where they were coming from.  So I would just worry that a board game in class may not be as well received as hoped!  The students may see it as extraneous and may just want to get to work on textbook questions.  

Now, a teacher could theoretically use the board game instead of the textbook work, but the textbook questions are always right there for reference, and the answers are in the back so the students can check later.  What I'm saying is, the textbook is nice and organized, it's a great way for students to practice questions and they can reference back to those exact questions later, whereas the board game, while super fun, may not provide that same stability.

Maybe I'm coming from more of an "old school" way of teaching math, but I can definitely say that the "old school" way of teaching is still alive and well, and not only that, students are thriving on it!  How do I know?  Because the teacher I volunteered with is an "old school" teacher and gets incredibly high student reviews.  

Having said that, I totally realize that there are many different styles of teaching and that students can thrive on all of them!  But I wonder if the board game would be best used as a review?  Perhaps after the students have done the textbook questions and feel good about them, they could use the board game in class during a review period before a test?  Then again, that's probably just my "old school" style speaking again. 😁

To Stream or Not to Stream...That is the Question

Happy New Year everyone!  In class this week we talked a little bit about streaming.  That refers to how schools have different grade 9 and 10 classes for different students.  In Ontario, the two most common streams are Applied and Academic.  But it doesn't start there...

When I was in high school back in the 90s, we had Basic, General, and Advanced classes.  To be honest, these made sense!  Students who were strong in a given subject went into the advanced class, students who weren't so strong did general, and students who weren't ready even for general were placed in basic.  Now, it didn't mean that if you were in placed in general for Math that you had to be in general for English.  It depended on the individual class.  You may be more advanced in some areas than others.

For a short period of time after I left school, classes were destreamed, then the government introduced a new streaming program in the form of Applied and Academic classes.

There are downsides to streaming of course.  One is that if a student takes an applied Math course in grade 9 and does well, they may wish to take an academic course in grade 10. Unfortunately it doesn't work that way; to take grade 10 academic Math, you need grade 9 academic Math. However, if a person who has taken grade 9 applied wants to take grade 10 academic, they could take the grade 9 academic credit over the summer and be ready to go - it's not a closed door. 

Some would say that it makes students feel bad to be streamed.  But I don't believe this is a reason not to do it. We are supposed to be training students for life in the real world.  The truth is, not everyone is treated equal in the working world!  And employers don't generally take feelings into account when choosing who to hire.  People who are better at certain jobs will absolutely be chosen over people who aren't. Furthermore, saying that everyone is at the same level as everyone else in every subject just doesn't jive with reality.

Destreaming I think would be a nightmare for teachers.  Imagine having students in a Math class - some who are exceptionally good at the subject, and others who have barely squeaked by the grade before.  How can you make all students happy?  You're either going to overwhelm the students who are having trouble, or underwhelm the ones who are proficient.  This happens in any course, yes, but the spectrum would be far shorter in a streamed class than a destreamed class.

Streaming is also a good way to individualize curriculum for students.  If a student is very strong in Math, but not at all strong in English, put them in a more academic stream in Math and a less rigorous course for English - it makes total sense!  High school is a perfect time to discover strengths and weaknesses to help students make post-secondary choices.  Some may wish to go on to Math or Science in university, others may wish to go to culinary school for example.

Lastly, I think destreaming has the potential to result in a high amount of failure for students.  Putting them in a higher academic level than they're ready for will likely result in them having to take the course over again. Now, in an ideal situation, students could take courses over and over again until they're proficient enough to move on to the next level (same with elementary grades), but in reality, people don't want to be in school for longer than they need to be and school boards just don't have the resources to make that happen.

Of course, I could be wrong.  More data is needed.  If Ontario decides to destream, then it would be interesting to see the outcome!  Maybe it would turn out better than I think!  We'll just have to wait and see what happens.

I'd love to hear input on this!  If anyone has any comments, feel free to post!

Who Are We to Question Why, Just Invert and Multiply!

At the end of last week's class, our instructor posed an interesting question...  She was talking about dividing fractions and, of course, we all know the rule to divide fractions is "invert and multiply".  That is, you keep the first fraction the same, invert (or flip) the second one, and change it from division to multiplication.  Like this:

(https://www.coolmath4kids.com/math-help/fractions/dividing-fractions)

But then she posed the question..."WHY?"  Why do we invert and multiply?  And none of us knew the answer right off the bat!  We were likely all thinking, "Because that's how it's done."  I mean, as the title of the blog says, "Who are we to question why, just invert and multiply!" Seriously though,    speaking for myself, I have never even really thought about why we invert and multiply to divide fractions.  So I decided to do some research and find out.  There is a wonderful page I found at Mike's Math Club (http://www.mikesmathclub.org/div_fractions.pdf) which explains it perfectly.  The images below are from that PDF.

Firstly, it's worth the question... If we can multiply fractions by going straight across, why can't we just divide fractions using the same method.  And the truth is...you can!  In fact, if the fractions have a common denominator, then it's a very efficient method!  For example:

And even if there isn't a common denominator, if the two numerators and two denominators divide nicely, then it's still faster to just divide straight across:

So, now then, what happens if the two numerators and two denominators don't divide nicely?  Well, you have two options:

1.  You could manipulate them to have common denominators, then use the method above.

2.  You could still divide them straight across using the following method.  Oh, and by the way, an "identity element" is something that, for a particular operation (like multiplication, division, etc), returns exactly the same answer that was there to begin with.  So the identity element for addition is 0; for multiplication, it's 1.

So there you have it! 

(http://www.funnyism.com/i/funnypics/when-you-finally-understand-a-mathematical-concept)

I truly love finding out the "why" of the simple math tricks we were taught in high school.  I actually felt quite excited when I realized that you totally can divide fractions straight across just like with multiplication!  Honestly, I'd just never really tried - when you're taught to "invert and multiply", or, as my math teacher said, "Keep Times Flip", it's just what you do.  But now I know why it works...and so do you!

Is the "New" Way of Teaching Math Really Better?

This weekend I attended a math conference as part of my "teaching math" course.  There were some really interesting things I learned, particularly regarding technology in the classroom, but there were also some things I really didn't like regarding how to teach math.

This post may end up being a big rant, but sometimes a good rant gets everyone thinking.  I want people to think about the way some teachers are suggesting that math should be taught, because I think it's just not working.  Allow me to explain...

At this conference were two very competent math teachers who had some very innovative ideas on how to teach math.  Unfortunately, I really just didn't agree with them.

What I want to focus on here is the concept of giving problems to students without giving them the tools to solve them just to see if students can innovate ways to solve the problems.  Here is an example that was given to us at the conference:

Mrs. Lin walks into the Sweets Emporium and buys 3 candies and 4 chocolates.  It costs her 26 cents.  You walk into the same store and buy 7 candies and 2 chocolates. It costs you 24 cents.  What is the cost of the candy and the chocolates?

This problem is intended to be given to students who are not yet well versed in the method of using algebra to solve this type of problem.  They are supposed to be given the candies and a bunch of pennies so they can figure out the answer visually.  This is all well and good, but as far as I'm concerned it's a complete waste of time.  Sure, we could give the students half an hour to mess around with the pennies until they come up with the correct answer.  OR, we could give them a 10 minute lesson on algebra, give them the tools they need to solve the problem, and then let them have it at - they'd be done in half the time!  The students who prefer algebra will solve it that way, and, to be honest, the students who aren't strong in algebra will solve it visually anyway.

They are giving students problems without first giving them the mathematical tools to solve them!  This is completely backwards as far as I'm concerned.

The thing is, if math had been taught to me like they're suggesting to teach it to students now (ie - backwards), I think I would have hated math...  I, the person who loves math, would have hated it.  And that's sad to think about.

It has been proposed to me that maybe I don't like this way of learning math because I wasn't taught that way, but I think that's just not true.  I can easily think of two examples of how I was taught in high school that could have been improved.

1.  In English we had to read Shakespeare.  I wasn't a big fan of the old-style writing and so I found it difficult to follow along.  Someone in one of my university classes the other day suggested an amazing method of using a diagram on the board to join different characters together visually with pictures and words to help people remember who's who.  I think this would have helped me immensely and would have been a much better way than just having us read the book out loud.

2.  In French classes in high school, we spent a huge amount of time on grammar and vocabulary, but very little time on conversation.  This, I believe, was a mistake.  We should have spent way more time on conversational French as this is what is needed to be able to speak the language well.  I think 50/50 would have been a good proportion.

So, there you go.  It's not just how you were taught that influences what you think works well and what doesn't when it comes to teaching.

The other big thing is, this way of teaching math is not how real life works.  When you get hired for a job, they don't sit you down, give you a problem and tell you, "Now we're not going to give you the tools yet, we're going to see how well you do on your own," then come back a half-hour later to see how you're doing with your problem solving and then give you the tools.  NO, they will give you the tools, equip you as best as they can, then let you do your job (time is money!).

Some may say that this method of having students problem solve without the tools may help in everyday life. But I beg to differ.  If you have a loose doorknob, you're not going to MacGyver a solution.  You're going to either Google it, or you're going to get yourself to Home Depot to ask an expert what to do.  And what will they say?  Not, "Why don't you go home and ponder it for a while.." No, they will lead you to the tools necessary to fix it!

Anyway, I'll end this rant now.  To be honest, I know these teachers are very good at what they do, and they wouldn't be teaching us this method if it didn't work for their own students.  However, for the reasons mentioned above, I must, respectfully, disagree.

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.