Our grade nine students wrote their Grade 9 Assessment of Mathematics (EQAO) last week. Often during this time I reflect on the process, because really what else are you going to do for two hours while supervising. This year my thinking wasn't about the pros or cons about the test but rather the way we evaluate it. The test is sent off to be marked provincially but before that happens schools have the option to evaluate it in order to include some or all of the mark as part of the student's final grade. The thinking here is that if it counts for something then perhaps students will take it seriously. At my school we count the test for 10% of a student's final grade. Then about a week later they will write the final exam that counts for 20% of their grade.

The test consists of two booklets that each must be completed in an hour. Each booklet is made up of 7 multiple choice questions, followed by four longer 'open response' questions then finishes with 7 more multiple choice questions. Once the second booklet is completed students are asked to complete a questionnaire.

My observation has been that more often than not students come into the test under prepared and it serves as a bit of a wake up call to them. They then (hopefully) use the remaining classes to prepare for the final exam.

This year I have decided that I am not happy with counting the test for any portion of the students' final marks. In fact, my students did so poorly that after the fact I told them that I was not going to count it at all towards their mark and here's why:

1. Time

Many students did not have time to complete the test. They had an hour to complete each of the two booklets. For the second year in a row my strongest students did not complete the booklets on the first day. These students were very concerned about the impact it was going to have on their overall grade. Rather than providing incentive to do well it caused a great deal of anxiety. As a math teacher my goal is to help students reduce their anxiety towards math not contribute to it. I also try to evaluate what a student knows and does not know. If a question is left blank I have no idea if it was because the student ran out of time or because they did not know how to do it. By removing time from the equation I can make a better judgement of what the student know.

2. Multiple Choice

I have decided that I disagree with the multiple choice questions. They obstruct my view of what the student does or does not know. Some students will get the correct answer by guessing. Others will get the incorrect answer by guessing. In either case, I am unable to see the process that allowed them to arrive at their answer and as a result I am unable make a true judgement of their understanding of the material.

3. Feedback

I don't know much about the official feedback students get so if I'm wrong here let me know. I believe that tests get marked in the summer (the rest of the cohort will write in June) and a mark is returned to the students in the fall. This is far from immediate feedback and is anything but descriptive. Not very useful in my mind. As a teacher I can mark the work, but I'm not allowed to copy anything. This means that I can't show students where they went wrong. I can tell them that they messed up on the bicycle question but unless they can see where, I'm not sure that's useful.

4. Justification

I'd be hard pressed to justify any mark to a student or a parent given that the tests get sent off, never to be seen again. Students should be able to look at their marked work and question my judgement, which is sometimes right and sometimes wrong. In fact, I enjoy when students start questioning my evaluation as it often brings out what they truly meant to write or allows me to better understand their misconceptions.

5. Rationale

When students ask why the test has to count for a portion of their grade I struggle to give a valid reason. I typically say something along the lines of "If you're going to spend two days writing it, we may as well give you some credit for it". It's not an answer I'm comfortable with but it's all I have. One of the reasons I'm not comfortable with it is that the vast majority of my students perform much worse on the test than they do on the final exam. We could probably discuss what that says about my teaching, but let's save that for another post. The real reason that we count the test as a portion of a student's grade is that we believe that this will make them take it more seriously, which means they will perform better, which will make the school look better. Given that twelve out of eighteen students in my colleague's class said on the survey at the end that they didn't know if the test was going to count (and yes he did let them know on numerous occasions), I'm not sure that counting it is a good motivator. Besides, is this in the best interest of the student or the school?

I'm curious to know whether counting the test as a portion of a student's grade makes them perform better. What does the data say? Do schools that count the test outperform those that don't? Is this information publicly available? Are there any schools that don't count the test? Or does everyone count the test so that they don't look bad? Is this in the best interest of the students?

What does your school do about EQAO testing in grade nine math?

## Monday, January 19, 2015

## Tuesday, November 18, 2014

### Triangles in Scratch

The video below is a follow up to the one I made here. This one is not that much different. Instead of drawing a square I show how to draw an equilateral triangle. To me the interesting part of the tutorial is having students play around in scratch to see if they can create an isosceles or scalene triangle. I haven't had a chance to do this in a class yet, but I'm hoping that there will be a lot of trial and error and eventually some students will get close enough. Close enough that they will be able to answer the questions "What is the sum of the angles in any triangle?".

The second challenge is to have students draw a regular pentagon, hexagon, etc. In order to do this they will have to determine the sum of the angles in each of these figures. We've done enough visual patterns that I hope this will be easy for them. I'm hoping this will be a fun way to cover some of the geometry in the grade nine math courses.

Give it a try.

The second challenge is to have students draw a regular pentagon, hexagon, etc. In order to do this they will have to determine the sum of the angles in each of these figures. We've done enough visual patterns that I hope this will be easy for them. I'm hoping this will be a fun way to cover some of the geometry in the grade nine math courses.

Give it a try.

## Monday, November 17, 2014

### Playing With Rectangles

I'm currently teaching my grade 9 students about linear relationships. We create scatter plots, draw lines of best fit, use the information to make predictions and so on. As we came to the end of the unit I felt as though we hadn't done enough. I felt that somehow it would be far more interesting if we could connect this section of the course to another section. The measurement unit seemed like a simple connection.

I gave students 12 straws and asked them to find the rectangle that would give the largest area. They messed around with the straws, made tables and graphs and determined from their graphs what the largest possible area was. Next I gave the 12 linking cubes and asked them to create the rectangle with the smallest perimeter. Again they played, created tables and a graphs but the solution wasn't as obvious.

None of this work is ground breaking or much different from what I have done in the past. The only differences were that I cross pollinated (some might say spiralled) the units. I think this helps show students that mathematics is interconnected, that it's possible for units to have a common thread. The other difference was that I physically gave them objects to manipulate, which is different from how I taught this before. In the past I would have them draw out rectangles. I think something gets lost here. It was very obvious to students what was going on when they were manipulating the physical objects.

Although neither of the graphs were linear it was useful to create them and to discuss what type of correlations there were and read information off the graph. We will revisit this concept again in the measurement unit. I look forward to seeing how well they retain the information.

As a side note, the graph of the maximum area was a parabola as expected. Without thinking too much about it, I expected the perimeter to do the same. As I saw students' graphs I wondered why they were the shape they were. I did the algebra and recalled that the resulting function was a rational function. Looks like a good problem for the grade 12 students. Tomorrow: Determine the function that minimizes the perimeter of a rectangle.

## Saturday, November 15, 2014

### Let's Start Coding

The second week in December is Computer Science Education Week (CSEd Week). I get really excited about this because more teachers start talking about coding. The trouble with this event is that it often seems to be a one off event. Teachers give up an hour of their curriculum time to participate and once the hour is done they tend to move on. It's great that they participate but it could easily become part of any teacher's curriculum. I realize that it's "one more thing" to do but it can be very engaging for students and I believe that it really helps develop skills that are useful in many disciplines both in and out of the class.

Why are teachers not extending coding into their regular routines? I think for many of them it's about comfort. The Hour of Code tutorials are great. They are well laid out and could be coordinated by anyone. If you're going to start building coding into your classes, however, you need to understand the tools a bit and you also have to find a way to weave in some curriculum expectations. No small task.

Brian Aspinall is a teacher who has not only embraced coding in his class but has taken it upon himself to help other teachers see how their students can code and meet curriculum expectations at the same time. He uses Scratch, which is really easy to use. You can find his videos here.

Brian's work had inspired me to play around more with Scratch and to find more ways to work it into the curriculum. I've decided to create videos that introduce some coding ideas but also create a challenge for teachers (or students) to work through. Hopefully, some teachers out there will decide to follow up on the challenges. If not, at the very least I will have thought more about how coding can be woven into my courses.

If you're interested in integrating computer science into your curriculum check out the #CSk8 hashtag.

Here's my first video.

## Friday, November 14, 2014

### Trigonometric Regressions with Desmos

I decided that this year I was going to make trigonometric modelling a little easier for my students. I've used Kate Noak's Moon Safari in the past but I found that something was lost using a graphing calculator. The process of entering the data is time consuming and causes some students to get turned off the activity before they managed to get to the good stuff. This year I decided to give Desmos a try.

I entered the data into a Google spreadsheet that I made public. I gave students Chromebooks and had them open the spreadsheet. They copied the data from the spreadsheet, then pasted it into Desmos (yup just two steps) and then got to work trying to determine the equation that best modelled the situation. I really liked that they could see the graph as they modified the equation and see the coefficient of determination to help them determine if one equation was better than another. When they were done they could get Desmos to do the regression to see how close they were.

Labels:
desmos,
Lesson,
modelling,
trigonometry

## Saturday, October 4, 2014

### Dismissing Long Division

Every year I teach grade 12 students how to divide polynomials. I always start by reviewing long division with natural numbers. As I put an example on the board, I'm always met with groans and comments such as "I never learned this. I was taught it but I never learned it.". I always reassure students that it will be much easier now than when they were in grade 4. I'm blown away by how many students hate long division and these students are our best math students. If the majority of the best math students hate long division, what do the rest of the students think?

Every time I teach this lesson I can't help but think that students in grade 4 aren't really ready for long division. It's a long algorithm that likely makes little sense to them. Perhaps they aren't ready for it cognitively. I'd even go so far as to say that although many adults can perform the algorithm, how many of them can actually explain why they perform those steps?

I'd like to see long division scrapped from the elementary curriculum. Instead I'd like to see students focusing on understanding what division really means in a wide variety of contexts. Clearly it's a skill that my division-phobic students have not used since grade 4 so what's the point in teaching it then? Often when I mention this in conversations I get the reply "How will you teach them to divide polynomials if they can't do long division?". This argument seems incredibly weak to me. Does it make sense to teach students something in grade 4 and then ask them to recall it eight years later, without having used since grade 4? Why not just teach the algorithm in grade 12 when students have a better mathematical background and can understand what is being done rather than memorizing an algorithm that is likely to be forgotten?

As a side note, I really like James Tanton's representation of long division here. I think that conceptually it gives a better understanding of what is going on than the traditional algorithm.

What are your thoughts on long division?

Every time I teach this lesson I can't help but think that students in grade 4 aren't really ready for long division. It's a long algorithm that likely makes little sense to them. Perhaps they aren't ready for it cognitively. I'd even go so far as to say that although many adults can perform the algorithm, how many of them can actually explain why they perform those steps?

I'd like to see long division scrapped from the elementary curriculum. Instead I'd like to see students focusing on understanding what division really means in a wide variety of contexts. Clearly it's a skill that my division-phobic students have not used since grade 4 so what's the point in teaching it then? Often when I mention this in conversations I get the reply "How will you teach them to divide polynomials if they can't do long division?". This argument seems incredibly weak to me. Does it make sense to teach students something in grade 4 and then ask them to recall it eight years later, without having used since grade 4? Why not just teach the algorithm in grade 12 when students have a better mathematical background and can understand what is being done rather than memorizing an algorithm that is likely to be forgotten?

As a side note, I really like James Tanton's representation of long division here. I think that conceptually it gives a better understanding of what is going on than the traditional algorithm.

What are your thoughts on long division?

## Sunday, September 14, 2014

### Troubled Start

This year, for the first time, I was not at all excited to get back to school. I'm not talking about the traditional 'The end of summer is near. I don't want school to start' kind of apprehension. For whatever reason I was not interested in teaching this year. It seemed that I had nothing to look forward to. To put this in perspective I'm usually excited to get back to school and start teaching again. I've always found something to look forward to. This is what has allowed me to enjoy teaching for all the years.

What was different about this year? I spent a lot of time thinking about it and didn't come up with much. Was this the mid-career lull for me? Was I about to become an old crotchety teacher that didn't care anymore? I figured that once the fist few days were finished I'd be back in the groove. That didn't happen. Was the problem that I'm teaching the same courses and I'm happy with the start of those courses? Same old, same old?

It wasn't until midway through the second week that I may have stumbled on a possible explanation. I'm teaching a grade 9 course. I've never met most of these students before. My grade 11 class has a small number of students that I've taught before but I only taught them for a couple of month, two years ago, before I took a leave. Finally, I have not taught any of the grade twelves, in my class, in grades 9, 10 or 11. I really don't know any of my students very well.

It occurred to me that teaching is as much about building relationships as it is teaching content. It's about the student who is really good at math and finding a way to challenge her. Or discovering the student who is very capable but lacking self confidence and helping him develop that self confidence. It's about working with those students who need extra help on a regular basis and getting to come in for that help. And the list goes on and on.

Now that I have gotten to know my students, a little, I'm excited to help them reach their goals and help them be successful. I'm excited to further those relationships and help them (as well as myself) grow. Here's hoping for a great semester.

What was different about this year? I spent a lot of time thinking about it and didn't come up with much. Was this the mid-career lull for me? Was I about to become an old crotchety teacher that didn't care anymore? I figured that once the fist few days were finished I'd be back in the groove. That didn't happen. Was the problem that I'm teaching the same courses and I'm happy with the start of those courses? Same old, same old?

It wasn't until midway through the second week that I may have stumbled on a possible explanation. I'm teaching a grade 9 course. I've never met most of these students before. My grade 11 class has a small number of students that I've taught before but I only taught them for a couple of month, two years ago, before I took a leave. Finally, I have not taught any of the grade twelves, in my class, in grades 9, 10 or 11. I really don't know any of my students very well.

It occurred to me that teaching is as much about building relationships as it is teaching content. It's about the student who is really good at math and finding a way to challenge her. Or discovering the student who is very capable but lacking self confidence and helping him develop that self confidence. It's about working with those students who need extra help on a regular basis and getting to come in for that help. And the list goes on and on.

Now that I have gotten to know my students, a little, I'm excited to help them reach their goals and help them be successful. I'm excited to further those relationships and help them (as well as myself) grow. Here's hoping for a great semester.

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