What Would YOU Do?

In class this week, our instructor presented us with some scenarios that we may face as we go into the classroom as a math teacher.  We were able to discuss them in groups, then have a class discussion on how to solve the problems presented in these scenarios, followed by our instructor going through what she would do (or has done!).  This is a very valuable activity to do with pre-service teachers as it alleviates some of our fears going into the classroom.  I want to share a couple of the more difficult scenarios and what I learned:

1.  "A student comes to you right before university applications are due and tells you they need an 85% in your class for the program they want to get into. They currently have a 72%."

This was a tough one for me because I am a very empathetic person and it would make me feel so bad if a student said this to me, especially if it was accompanied by tears.  My group discussed the idea of looking at the student's marks to see what's going on.  If the student was an 85% student in general, but really did terribly on one unit, then perhaps, as a teacher, you could allow the student to make up the marks for that unit.  However, if the students marks are consistently in the low 70s, then there's not a whole lot that can be done.  The class discussion and instructor comments brought up the concept of being proactive.  Let the students know right from the beginning of the year that these marks are very important, and not a whole lot can be done to adjust marks later on.  In doing so, you are letting the students know to take this all very seriously, and there will be no surprises later on when you tell them that you can't adjust their mark.

2.  "A student comes to you the day before a test and tells you that she overheard Scott telling a friend that he plans to type his notes and formulas into his graphing calculator.  This way, it'll look like he's using his calculator to do math, but he'll really be reading his notes and getting formulas."

My first thought on this was to simply walk around a lot during the test and keep peeking at what the students are doing.  That way, Scott would likely feel quite uncomfortable cheating because he would know at any given moment that I could walk by and see what he's doing.  However, the class did come up with a couple of other possibilities. First of all, the teacher could have all the students reset their graphing calculators to erase everything in memory.  And secondly, our instructor mentioned that one way of solving the problem is to have a classroom set of graphing calculators that are only used for tests.  That way, you know that there is nothing saved on them.

There were a few more scenarios that we went through, and all of them were quite challenging!  I'm so thankful when instructors give us these types of activities.  I don't know about anyone else, but as far as teaching goes, I'm least worried about the actual teaching itself. What I'm most worried about are difficult situations that may arise.  They may have to do with marks, cheating, classroom management, etc, but those are the things that I think will be difficult.  So, the more exposure we can get to possible classroom scenarios and how to deal with them, the better prepared I will be to go into the classroom.

The Log Loop - An Amazing Way to Remember How to Convert a Logarithmic Form to Exponential Form

This week in class, one of my colleagues did a Grade 12 (U) lesson on Logarithms - specifically on how to convert logarithmic form to exponential form and vice versa.  I don't know about you, but it's always been quite difficult for me to remember the formula.  My colleague presented an amazing diagram that helps students (and teachers!) remember how it works:

Gruen, Amy. (2011). The Log Loop [drawing]. Retrieved from: http://squarerootofnegativeoneteachmath.blogspot.ca/2011/02/loop-for-logs.html

I love tricks like this!  If you can simply remember that you start at the bottom of the diagram, then you can remember the formula!  So the answer to the above would be:  2^3 = 8   And technically, the loop also works to turn the exponential back into the logarithm.  In the example I just typed, the loop would start with the 2, go around to the 8, then finally to the 3, which would result in the order seen in the picture with the logarithm.  Awesome!

Secondly, my colleague got us all to play "Log War".  It was basically the card game War, but instead of just numbers written on the cards, logarithms were written on the cards.  So, we all had to solve the logarithms in order to find out who had the highest number in the round.  This is an excellent way to get students practicing!

I will say though (as I always do), that I do feel it's important to assign some textbook homework as well, just so the students get used to writing out full solutions like they'd have to on a test.  Giving them homework will also give them a resource to use while studying for the test.

Having said that, I truly enjoyed this lesson and I hope to use it in my classroom someday!

Find the Mean Like You MEAN It

In class this week, we focused on grade 11 college math.  This is the course that is generally taken by people who know they do not want to go to university, but are rather headed into a college course.  Two of my colleagues presented lessons for this course and I'd like to talk about the lesson on mean, median, and mode.

I really liked this lesson overall!  My colleague began by using the Smart Board to display a worksheet that tackled the definitions of mean, median, and mode, and how to find each.  This gave the students a good review as mean, median, and mode are first taught in elementary school.  At this point she could have just given the students textbook homework to practice these concepts, but instead she used a card game called "Find the Mean Like You MEAN It"

Found at: https://blog.prepscholar.com/hs-fs/hubfs/Body_mean.jpg?t=1517604014667&width=325&name=Body_mean.jpg

She took all the face cards (and jokers) out of the decks so that we (the students) had decks with cards from Ace (one) to 10.  In groups of 3 or 4, we had to shuffle the cards, and deal 7 cards to each player.  Each player had to find the mean of their cards and write it down.  After 3 rounds, each player had to find the mean of the 3 rounds, and the person with the highest mean won the game.

The game was then repeated only instead of using mean, we could use median or mode.  It's an excellent method to get the students practicing!

This was only part of the lesson, so my colleague didn't mention it, but I do hope that students would also be set textbook questions to do.  After all, on a test, you won't have a deck of cards and you might have word problems that will only be solved successfully if the students have had practice with word problems.

I do like the idea of fun and games in math class, but I also stress that rote practice is useful as well.  Completely doing away with textbook questions is not a good idea in my opinion.  I hope to be able to strike that balance in my classroom.  Some topics will lend well to games, others may simply be better learned through pencil and paper practice.  But I've come a long way even to say that, because before this course, I would have said that all topics are best learned through pencil and paper practice.  It's been great learning about the different ways students can learn and practice math in the classroom.

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!