Different Strokes for Different Folks - Differentiated Instruction in Math

In class this week we learned about Differentiated Instruction as it pertains to teaching math.  Differentiated instruction does not refer to my all time favourite Calculus unit, but rather to a method of teaching by which you try to tailor the class to the varying needs and learning styles of the student.

Some people (like me!) learn math best the traditional way: when a teacher stands at the front and explains the concept on the board with examples, then gives problems for students to solve independently.  Other people learn best by discovering concepts themselves through manipulatives (see my last post) and then connecting that visual to a formula and maybe practicing that formula in a group.  Still others would prefer to look at diagrams representing the concepts as they learn.  There are many different styles of learning, and, although it I don't believe it is possible to cater to all styles all the time, there are some excellent ways to incorporate different learning styles into a math classroom.

As we went through the class activities today, I didn't really feel as if they were methods of instruction so much as they were absolutely excellent, elegant ways to check understanding of concepts and help to further knowledge and understanding.

One activity we did involved 16 squares of paper.  Some had rules on them (like # of tiles = 2p+1), some had graphs, some had a table of values, and some had diagrams. We were told that among these 16 squares there were only 4 equations represented.  So, we had to match the squares together and group these squares into 4 groups of 4.  The clever thing about this is that by representing an equation four different ways, it challenges everyone to think about equations in those four different ways.  BUT, it allows people to start wherever they are most comfortable (be it a rule, graph, table, or diagram).  Genius!

Another activity we did was something called Open Questions.  Every student gets a small whiteboard and a marker, and, when the teacher puts a question up on the board, they answer it on their whiteboards  Then they may hold their whiteboards up for the teacher to see, or maybe exchange it with a neighbour to see what each other got.  But what is an open question you might ask?  Well, the best way to describe it is with an example:

Draw a line with a negative slope that passes through the point (2,4).

This question actually has an infinite number of possible answers!  As long as the students draw a graph that has a line with negative slope going through (2,4), then their answer is correct, but the steepness of the slope will vary from student to student.  Can a student get an incorrect answer?  Yes, of course, but then the teacher will be able to see instantly what part of the question the student didn't understand.  If the graph has a positive slope instead of negative, but does pass through (2,4), then the teacher knows that student understands coordinates but is a bit hazy with slope.  Or, if a student has made the line pass instead through (4,2), the teacher will know to review coordinates.

It's a really elegant way of seeing how much students understand without putting too much pressure on them - students generally find this sort of thing really fun and again, it caters to all different learning styles!

One last activity that I want to highlight is a website called Which One Doesn't Belong created by a Canadian math teacher by the name of Mary Bourassa.  The URL is www.wodb.ca. There's also a book full of these puzzles available written by another math teacher called Christopher Danielson. Here is an example of one of these puzzles (it's actually the logo from the WODB website):

This activity is awesome.  The general idea is that each of the quadrants doesn't belong, but for a completely different reason!  For example, the top left one doesn't belong because it has a different font than the others.  This is an amazing activity for differentiation because the students will likely approach the puzzle different ways depending on their learning style, and what they are already comfortable with.  But, by having a class discussion, everyone gets to see why each quadrant doesn't belong and there will likely be many "Aha!" moments as students begin to understand the other answers. There are loads and loads of these on the WODB website and more are added all the time because they encourage people to submit their own ideas.   Many of them are quite difficult - I encourage you to check them out!

I really had a good time in class this week.  I thought that using differentiated instruction in a math classroom was going to be quite challenging, but these activities are wonderful ways of incorporating different learning styles into the classroom and actually getting students (especially those who may not be as keen on math) to (dare I say it) enjoy math class.

Manipulatives - Helpful, or Frustrating?

In class this week, we worked with "manipulatives".  These are essentially toys (like shapes, cubes that join together, peg boards with rubber bands, etc) that can be used to visualize math concepts and formulas, and the use of them is mentioned in the high school math curriculum.

So, what did I take away from the class?  Simple...that I never want to hear the word "manipulatives" again.  OK, I jest, but there is definitely a grain of truth to it.  Allow me to explain.

For our class activity, we split up into groups and a different set of manipulatives was put on each of 5 tables.  Along with the manipulatives there were a couple of sheets of paper. One was the instructions for what we were to learn and discover, the other was a question we could look at from a teacher's point of view.

I learned very quickly as I moved from table to table that using manipulatives to learn math makes my brain hurt.  I just don't learn that way!  True, there were a couple of tables where I was able to connect the manipulatives to the concept, but for the most part I just really didn't enjoy myself at all.

At one table, we were supposed to be working with fractions by creating shapes (like a parallelogram or triangle) that were, say 1/6 red, 1/3 green, and 2/3 blue using the different shaped blocks on the table.  Sounds fun, right?  Well I thought it was difficult and frustrating!  In fact, at one point I said to my colleagues, "I'd rather be doing Calculus."  And this was likely an activity that would have been done with students in elementary school!  I would much rather have just worked with fractions the traditional way - doing drills and written problems with them.

At another activity, we were playing with algebra tiles.  I'd only really seen pictures of those.  They are essentially different coloured tiles that can be used to represent algebraic equations like 3x + 5x or (2x+1)(3x-2).  I didn't know how they worked and my colleagues weren't sure either; our job was to work it out.  Even when I began to grasp how they worked, I ended up looking at these tiles thinking that this is NOT the way I would want to learn.  If I want to solve the two equations I put above, I'll just add the first one and FOIL (First, Outside, Inside, Last) the second!  I don't need complicated algebra tiles to do that.  In fact, I found myself just wanting to mess around with the tiles instead of trying to learn.  I actually took the tiles and started literally spelling out the phrase 3x + 5x using the long skinny tiles.  I was trying to amuse myself and get a laugh.  You know, "I am representing the equation with tiles, like the instructions say!"  Just not the way it was intended - I was being creative!

At the last table, we were given 11x11 pegboards and a bunch of rubber bands.  We were asked a question about a farmer who had a certain length of fencing, and who wants to build a rectangular field.  The question asked what the possible dimensions of the field could be and which dimensions would give the maximum area.  So what did I do?  Well, I took a piece of paper and began to draw rectangles with different side lengths to show the different possibilities.  From there, I could have simply done some area calculations and, by trial and error, found the field with the maximum area. (What I actually did was use Calculus to solve it).  I know I was supposed to be using the pegboards and rubber bands, but again, all I wanted to do was play with the things on the table. I actually picked up two pegboards and pretended to play Battleship!  People....I am 36 years old - I am not a little kid who just wants to mess around - I'm usually very good at "just getting things done" while in class, but this was a challenge!  It was so hard to focus!

The first thing this told me is that manipulatives do not help me to learn math.  I would far rather be taught the formulas and procedures right off the bat, and then maybe be shown a visual to have a second way to look at it.

The second thing this told me is that not everyone learns like me.  There will be students for whom manipulatives are the best thing ever to help them focus and understand math. Maybe those students feel the same lack of focus during traditional learning that I did while trying to use manipulatives.   So, I know at some point when I'm teaching math, I'll have to "teacher up" (as opposed to woman up or man up) and bring out the manipulatives.

However, I'm sure that I represent a good chunk of students who just can't deal well with manipulatives. So, if I do have a particular lesson which lends well to using manipulatives, this is what I might do:

I would figure out which students in my class learn best with manipulatives and which students learn best the traditional way.  I might have done this beforehand with observation over time, talking to the students, or getting them to experiment with the idea of manipulatives to see how they feel about it.   Then, I would split the "manipulatives students" into groups and they would go to the few stations set up around the room with instructions on what to do.

Next, I would instruct the students who would like a traditional lesson to come close to the board (the "formula group"), where I would teach the concept the traditional way.  After the lesson, I would assign those students practice questions to do while I walk around the room to see how the manipulatives groups are doing, and assist if necessary in understanding the connection to the concept.

Finally, once all students have some grasp of the concept, I would ask a representative from the formula group to come up and explain what they've learned about the formula and procedure to the manipulatives group.  Likewise, I would have a manipulatives group representative come up to explain to the formula group the connection between the visual representation and the formula.

That way, all students leave with similar knowledge - the manipulative group will have learned the formula, and are now comfortable using it because they were able to discover it visually.  And the formula group not only knows the formula, but now has a visual to connect it to, which furthers their thinking and understanding.  It's the best of both worlds.

Now, that doesn't mean I'll be using manipulatives too often in my high school classes, but it is something I can think of adding to my repertoire when it fits well with a concept.

The class activity we did really, if nothing else, highlighted two very different ways of learning of which I'm very much in one camp.  It emphasized to me that although there will be students who learn like me, not all students learn that way, and I need to try to keep that in mind when I'm teaching, whether that means using manipulatives, or maybe just enhancing my more traditional lessons with visuals.  The possibilities are endless.

Math Curriculum

As part of my "teaching math" class, we need to write a weekly post reflecting on the topics covered in class for that week.  This post will cover my thoughts on the Ontario math curriculum and will be of most value to those in the teaching profession.

In class this week we talked about the Ontario math curriculum.  At the beginning of the class, our instructor put up links to some recent news articles regarding how the powers that be in Ontario are planning on revising the math curriculum right from K-12, and how they are planning to change the report cards.  After reading the article on the curriculum revision, I must say I felt both happy and frustrated.  More on that in a moment.

During the class, we did a group activity involving the high school math curriculum. We focused on a few courses including Grade 10 Academic, Grade 11 Workplace, and Grade 12 Data Management to name a few. The activity involved writing down all the verbs (like determine, solve, etc.) and tools (like software, graph paper, etc.) mentioned in the curriculum.  This activity required us to read through the curriculum and gave us all an idea of the overview of each course.

After considering the high school curriculum and remembering some past tutoring experience I've had with high school kids, I got to thinking about the curriculum in general; more specifically the elementary curriculum, since the elementary curriculum is what prepares students to take on high school.  I'm not as familiar with the specifics of the elementary math curriculum, but I have a general idea of what it's all about.

The article I read stated that EQAO (the standardized test in Ontario) scores have been quite bad in math, particularly at the grade 6 level.  The question of course is whether there is a flaw in the curriculum, or whether there is a flaw in the test itself.  In all honesty, it's probably a little of both, but I lean more towards the curriculum being the problem.

I went through elementary school from the mid 1980s to the mid 1990s, and, as such, I was taught math very traditionally through drills.  We would learn the adding algorithm and then apply it through plenty of worksheet practice.  Likewise, we were instructed to memorize the times tables from 1x1 right up to 12x12.  Unfortunately, when they revamped the curriculum in the late 90s, from what I understand, a lot of these basic fundamental drills were removed.

The curriculum focused more on problem solving, helping students to construct their own knowledge, and how math could be applied in the real world.  Helpful, yes, but without the background of consistent practice (and memorization) of the basics, how will students be able to focus on more complicated concepts?  In fact, Robbie Case, a famous researcher in the field of cognitive child development found that memorization actually allows the brain to free up capacity for other tasks (Case, 1992).

My 8 year old son is in grade 3, and yesterday he came home telling me he didn't understand what they did in math class.  They were doing adding of 3 digit numbers (which he can do the traditional way with no problems), but they were using an open number line to show the addition.  

Here's an example of what that looks like:

I took a look at this and thought, "Wow, that's unnecessarily complicated."  I went through the concept with my son and, while he did understand it, I think he felt like me.  I believe that, like me, my son learns better through algorithms and formulas than alternate strategies.

I became concerned that not enough emphasis was being put on the concrete algorithm of adding, so I actually had a good chat with his teacher today about the curriculum.  I know she has to follow it, but I just wanted her thoughts.  She reassured me that students are taught both the algorithm and the other strategies, and she also brought up a good point.  Apart from sometimes being more practical when you're standing in the grocery store and needing to add things together without a paper, pencil, or calculator, alternate strategies are also a way that students can check their answers.

Having said all that, after reading the article I was happy that the curriculum is going to be revised.  I'm hoping that the new curriculum will add back in some of the pure math drills that were taken out (and especially memorization of times tables), while still keeping alternate, practical strategies.  However, I'm also frustrated that my son will go through much of elementary school without these drills (curriculum takes a long time to revise). I will continue to supplement his math learning with drills at home (much to his chagrin I'm sure 🙂).

Anyway, those are my thoughts.  Please feel free to chime in with your own thoughts.

References:

http://nationalpost.com/news/politics/ontarios-math-scores-started-declining-as-kids-took-the-new-curriculum-according-to-eqao-data

https://beta.theglobeandmail.com/news/national/wynne-ontario-math-curriculum-changes/article36192881/?ref=http://www.theglobeandmail.com&

Case, R. (1992). The Mind's Staircase: Exploring the Conceptual Underpinnings of Children's Thought and Knowledge. (pp. 32-33) Mahwah, NJ: Erlbaum

High School Math: The Journey

The journey through high school math is long, but fascinating.  I know, not everyone will agree with the fascinating bit, but there are ways to make it intriguing.  I thought I'd write a poem to introduce the basics of the journey and what it entails.  Hopefully it will inspire students and teachers alike!

Welcome to the big wide world
Of Ontario high school math
I hope this prose will help you
To navigate this path

First there comes Grade 9
Where you'll get straight into angles
First differences and slope
And the lengths of right triangles

Grade 10 brings trigonometry
Some guys long after Noah
Worked out trig ratios while staying 
At Camp Soh-Cah-Toa

In grade 11, functions
Finance, sequences, there'll be
Your knowledge of math will increase
Exponentially

In grade 12 there's Advanced Functions
Your peers will say just one thing:
"I see you have some graph paper
You must be plotting something..."

There's also Data Management
And Calculus & Vectors
Those last two come together
To make one course with two sectors

Calculus, my favourite
There's so much joy that's in it
Not everybody will agree
And I know, there's a limit

The intro's done to high school math
These puns you may well hate, but
'Tween good and bad math jokes it's hard
To differentiate

My Math Story

In this post, I want to tell you my math story.  Everybody has a math story.  That is, your experience with math from when you were knee high to a grasshopper right up until this very moment. 

As part of my first "teaching math" class, our instructor gave us an empty graph and asked us to label the axes however we liked and create a graph based on our math journey.  This is an Excel version of what my graph looked like:

As you can see, I put my age along the x-axis and my love for math along the y-axis (on a scale from 1 to 10). 

In elementary school, I don't remember having any strong feelings about math.  I didn't hate it, but I don't remember saying that it was my favourite subject either. I think I would describe math in elementary school as "unremarkable".  On the graph, I put that time in my math journey at a solid 6.

When I got to high school, I started to really enjoy math.  I can remember going through grade 10, 11, and 12 math and almost looking forward to doing my homework.  It was in OAC (yes, I'm old) that my love for math peaked.  I wasn't as keen on OAC Finite Math (now grade 12 data management), but I absolutely loved Calculus.  At the time, Calculus was a separate course to Vectors, and, as such, the course covered a lot more than is covered now in grade 12 Calculus.  (I truly lament the loss of topics such as implicit differentiation, related rates, and basic integration in high school Calculus, but I'll have to save that rant for another time.)  I can remember making up derivative questions to solve for fun.  I know, I know, you can laugh.  But it's true!

It was during my final year of high school that I made the decision to take math in university.  I thought, "Wow, if math is this amazing in high school, how much more amazing will it be in university?"  Unfortunately, it didn't turn out that way.  First year university, and even second, was pretty good.  In fact, there was a first year algebra course which focused on matrices that I really enjoyed!  Calculus was good, but it went super fast and it wasn't always easy to keep up.  Having said that, I was ready to keep going and see what else I could learn.  I hit the wall in third year.  It was all theoretical math!  That is, no more concrete steps to solve problems, just proofs, proofs, and more proofs. I learned very quickly that the concrete math was what I loved.  Stats provided some respite, and I did quite well in those courses, but stats was never something I was too interested in. 

In fact, I still have a poem that I wrote during my time at university.  Here is an excerpt from that poem which shows how I felt at the time:

Math was great in high school
That's where it reached its peak
'Cuz here in university
It's not math, it's Greek

Delta, pi, and epsilon
Sigma, phi and theta
Where alpha is preferred for a
And b's not b it's beta

In short, math became a chore and a source of frustration, rather than a joy.  No wonder I didn't feel like going on to become a math teacher once I graduated in 2003 - I was all math'd out!  I ended up going back and doing a second degree in Linguistics.  Linguistics was amazing, but I didn't end up going into a Linguistics related job.

Fast forward to 2015.  I had 12 years to reminisce on my math journey.  I came to realize that, despite my experiences in university, my love for high school math remained.  I was talking to my husband and I mentioned that I should have become a math teacher.  We ended up agreeing that it's never too late.  I threw myself into volunteer tutoring at the library, and I volunteered at two different high schools in the classroom.  These experiences served to solidify my desire and resolve to become a high school math teacher.

And here we are now, in 2017, and I have officially begun my journey to become a teacher.  My love for math is now back at a high point and I'm so excited to see what the future holds!

That's my math story and I'm sticking to it.

Welcome to "Teaching on a Tangent"

Welcome to my blog!  This is my first post and I want to take the opportunity to introduce myself.  My name is Rachel and I'm currently a teacher candidate in Ontario, Canada, and I'm navigating my first year of teacher education.  My goal is to become a high school math teacher (yes...math...I love math!).  One of my classes focuses specifically on teaching math.  In this blog, I will talk about my journey through this class; things I'm learning, things that may surprise me along the way, and just some general thoughts on math teaching that I want to share.  I hope to not cause any division (except long division, and, occasionally, synthetic division), but rather I hope to add to the general math community with my musings.

In this post, I want to focus on the name of my blog and why I chose it.  So, why "Teaching on a Tangent"?  Well, as you'll discover about me (and may have already guessed), I love math puns and math jokes, so, naturally, the title of my blog had to function well in this area. 

But there's more to it...

For one thing, I will literally be teaching on tangents at some point during my career.  The simple definition of a tangent in math is a line or plane that touches a curve but does not intersect it.  It's essentially a line that has only one point in common with a curve but does not cross over that curve; if you extend the tangent, it will never touch that curve again.  We generally speak of these curves as functions that are defined by equations (such as y = x²).  If we plug numbers into the "x", we can find the corresponding "y" points and plot the equation on a graph.  This visual representation will be the same every time you plot it - it doesn't change. There are, however, an infinite number of tangents (since there are an infinite number of points on the curve which a tangent can touch).

Here's a photo of a tangent (courtesy of http://study.com/academy/lesson/tangents-definition-properties-quiz.html):

This brings me to my next point.  I want to teach on a tangent.  Notice I didn't say "off on a tangent", I said ON a tangent.  I want to start with a concept and general method for teaching the concept, but I don't necessarily want to use the same method that's always been used simply because "that's the way we've always done it".  I want to see if it's necessary to deviate from that "equation" to make the lesson better.  That is, I want to find a starting point on the equation, and I want to ride the tangent; I want to find ways to make the lesson more interesting and more effective.  

However, keep in mind that any tangent does indeed have one point in common with the equation of the curve.  To me, that represents the fact that sometimes the classic way of teaching a concept is still the best way.  I'm not obligated to change lesson plans just for the sake of change; the change must make sense.

I hope I've been able to accurately communicate my reasoning behind the naming of my blog.  I do, however, realize I've talked a fair bit about graphs in this post.  And I know, graphs aren't necessarily everyone's favourite topic.  As far as I'm concerned, I love trigonometry, I love algebra, and I could do calculus all day.  But graphing...well - that's where I draw the line.