Sine Law

Jennifer Wilson wrote a post about productive struggle that you can find here. I did my usual trick of sending myself the link. I have a filter on my school email that automatically labels emails from me and files them away under "Twitter" so that I never actually see these emails. Thankfully I keep vague recollections of what I have tagged and earlier this week I knew to search my "Twitter" label for something interesting relating to sine law.

I stole this from Jennifer's post, having first shown my students the picture of the riverbank and surveyor and the diagram without any numbers. Straight from Jennifer's post.

Students worked in groups on their big whiteboards to find the width of the river. There was definitely some productive struggle going on and a lot of good mistakes were made along the way. I love that students try to use other mathematics they know, in this case Pythagorean theorem and right angle trigonometry. There were some good conversations around when those apply and why they don't in this case (with the triangle as drawn here). One group split up the 107° angle into a 90° and and a 17° angle (below). They struggled a lot and did some good math along the way. When they realized that they needed to create an altitude instead, they were quick to solve the whole thing.

Groups, one by one, figured out that they needed to add the altitude from S2. Some made mistakes along the way...

But they eventually got there!

Once they had found the width of the river, they had to work with an oblique triangle that did not have any numbers on it (again stolen from Jennifer's post):

Find an expression that represents the length of side c, I said. I got a lot of, uh, interesting looks. Most did not want to work without numbers. Didn't know where to start. They struggled in their groups for a bit then I told them to finish it for homework. That was their only homework.

The next day, we worked through the development of the sine law together and it went fairly well - better than in previous years, I think. 

Working in groups on the big whiteboards has really made a difference to the culture of my classroom. Students are willing to try things and make mistakes and they persevere. That makes me happy.

As a side note, none of my students thought of solving it the way Jennifer showed on her blog here. I will have to make sure I work that in next time around. 

Rethinking Tests

There have been a few tweets lately that have made me pause and think about tests. Even in the course where I spiral through the curriculum and no longer have units, I still give tests. I will admit that I'm not sure how to evaluate without using tests given the time constraints in place and the rigidity of the department in which I teach. I would love to sit down with each student and ask questions to elicit what they understand and where they are struggling, but I don't know how to manage that during class time. I look at Alex Overwijk in awe as I know he does this successfully. 

The tests I give, even in my spiralled class, look fairly traditional, but the way I "administer" (that sounds so formal when, in fact, I spend the entire class running around from one student to another) is likely not. The students in this class, grade 10 applied, all get two classes to finish the test. I take the time pressure off for everyone. I also don't let anyone hand in a test without answering all the questions, which means that I prompt as needed. I write whatever I ask or say in a pink or orange pen on their test paper and take it into consideration when I am marking the test. This means that the students who say that they don't care or that they have had enough still have to do work. I don't accept them opting out, especially since I know that they all can be successful. (So why am I testing them?) I don't have the same flexibility in my other classes though...

(I own this shirt)

In my academic classes (MPM2D and MHF4U this semester) I like to do stations for review the day before a test. The questions come from last year's test. I know the old tests are out there, some students have them, others do not, so I feel that this helps level the playing field. They work on the questions in random pairs on big vertical whiteboards and show me only the answer (unless the question asks for a proof in which case they bring the whiteboard to me!). They get a sticker when their answer is correct. Stickers provide motivation in ways I still do not fully understand. Here's the thing - when they do not get the correct answer I try to give them some feedback that will help them find their error and fix it. For example, if they had to determine the sine equation given certain information, I would tell them that the amplitude and vertical translation were correct in their equation, but they needed to look at the phase shift and period again. If they came back and the period was still incorrect, I would ask more specific questions about the problem to help draw out what they know about period then have them work on the phase shift again. This process of letting students correct their work in real time seems incredibly valuable to me. So I wonder why we cannot do this during a test. I realize that the logistics of attempting to do this with 30 students in 75 minutes do sound challenging. But I take issue with the snapshot of learning that we get with tests. Have you had a student do poorly on a test despite knowing that they understood far more than they were able to show you on the test? I have. Over and over.

I have started caring less about what I am supposed to do and more about doing what I think is right. When students don't succeed the first time, I give them another opportunity to show me what they have learned. After the 2nd grade 10 academic math test this semester, one student in particular had trouble with several questions bringing this student's  overall level on the test below where I thought it should be on this material. I had the student come in at lunch and we talked about how to tackle a couple of the questions. We had a good discussion - I found out what really was not understood and what this student could do given how to start the question. I think this was such a positive step forward for both of us. The student cleared up some misunderstandings and built up confidence by being able to show me all the things that they could do, but hadn't. I learned more about that student's understanding and what they really knew and showed that the learning, not the test, was the important part.

With what have other HS math teachers replaced tests? I would love to hear options. I teach at a school where the pacing guide is king, so I have an uphill battle, but I think this is the next goal in improving how I teach. I need help and would love to hear your ideas. Thanks in advance.

Would You Rather?

Along with warm ups from Estimation 180 and number talks, I have started using questions from Would You Rather? to start my grade 10 applied math class. This site was started by John Stevens and is a treasure trove of interesting conundrums.

The premise is simple: students have to make a choice between two options and support their choice with mathematics. I randomly place my students into groups of 2 or 3 and each group works on a large whiteboard. Here is the first one we did:

One of the reasons I love these is that it makes students think about what they need to know in order to choose the better option. I don't tell them anything except that they are allowed to use their smart phones if they need to look up information. So they have to figure out what they don't know and find it before they can start calculating. In this case they looked up (or asked Siri) the mass of a new nickel and of a new dime. Some worked with the mass of American coins, others with Canadian coins. It didn't matter. (Interestingly, the ratio turned out the same.) They worked at it and, because they were in groups, they could help each other through any misconceptions. These are also short enough that no one complains about who they are working with and they can stay focused through to a decision.

Today we did this one:

Most groups did well with this one and all chose the same option, with some mathematics to back it up. I then asked them how long a work day is, to which the unanimous reply was 8 hours. I asked what if they weren't paid for their 30 minute lunch - would they still chose the same option? They worked it out and this led to a quick discussion about asking good questions versus making assumptions.

There are many Would You Rather? questions that involve converting units which is part of the measurement curriculum expectation. I love that there is a context for converting and that I can incorporate it 5 minutes at a time. I hope that, at least once, my students will talk about the interesting question they did in math today to a friend or family member. I hope that this helps them connect to math and helps build their questioning and reasoning skills.

Equations of Circles

In grade 10 academic math we look at midpoint of a line segment, distance between two points and then equations of circles. But only equations of circles centered at the origin (sigh). This is a one day thing, that has been quite confusing for some students. I should note that these students have worked with linear equations a lot, but not much else, so these are very different looking equations. In past years, there has confusion despite my best efforts to connect the equation to the Pythagorean theorem/distance formula. I decided to add a little intro activity this time around:

They worked in groups of 4 and got a quick refresher on finding the length of a line segment. They also quickly figured out that the points collectively were leading to a circle. We then defined a circle. They came up with all kinds of properties of circles. When I could, I would provide a counter-example, like a shape that is round, but not a circle. We honed in on "all the points are the same distance from the middle" which we turned into a mathematical definition. Then we "developed" the equation of any circle centered at the origin.

Next, I hopped on to Desmos and asked them them what to do with the equation to make my circle have a radius of 6, or 3, or 8, or 3.5. They got it. Have I mentioned lately how much I love Desmos? I also showed them how to make the circle "move". We had looked at linear equations in the form y = a(x - h) + k, so it wasn't a huge stretch (no stretches involved, actually!) to perform horizontal and vertical translations on our circles.

When we talked about how to tell if a given point is inside, on or outside a circle and they really got it. I love how making a little change can make the rest of the class become seemless.

Oreos, Candies, Chocolate and Chocolate Milk

Sometimes life gets in the way of blogging so this post has morphed over the past week. One of my goals this semester is to add activities to my grade 10 academic class. Here is how things have played out for the first unit on solving systems of linear equations.

I wrote about how I changed my introduction to solving systems by elimination here. The next day they worked on the Oreo problem. I stole this from Nathan Kraft (here) which is why I haven't blogged about it myself. I introduced this activity last year and since it is awesome, I continue to use it. Students have to figure out whether the wafers or cream centre of an Oreo has more calories. (But really, you must read Nathan's intro to this.) Here is the information they are presented with in order to solve:

They worked on the big whiteboards and some even presented their solutions to the class. Note that Canadian Oreo packages have 2 cookies per serving for both regular cookies and Double Stuf cookies, which is not nearly as interesting as the ones shown. I tell the kids this once they have answered the question. And, yes, I do bring in Oreos for them.

I also added in a candy lab when we got to solving word problems. I stole this from someone who posted on Twitter. I sent myself the link but it didn't have a name associated with it, nor does the Google doc. If someone knows to whom I should attribute credit, please let me know! I started the class like this:

They were fairly (!) excited when they saw the word candy last period on a Friday afternoon. I randomly picked names from my tin of popsicle sticks to make groups of 4. Each group received a brown paper bag upon which I had written a letter and a number. The letter identified the bag and the number indicated the total number of candies and chocolates in the bag. I borrowed kitchen scales from our science department (thank you!) so that they could weigh their bag of candy. I wrote the mass of 1 candy (7 g), 1 chocolate (13 g) and an empty bag (8 g) on the board. And they were off! They did good work and once they had an answer we did the reveal - I opened the bag and counted out the candies and chocolates. They weren't perfect, but they were close and their work was excellent. I also had a few more challenging bags that had candies, chocolates and granola bars in them with an additional hint written on the bag. I will add pictures...sometime! As a side note, a greater total mass in the bag led to a better result. For next time...

Next in the word problem collection were mixture problems. My students have always found mixture problems to be particularly confounding. I thought I would use a demo to help (the idea, again, stolen from the Internet). A litre of milk appeared along with chocolate syrup, and the excitement was palpable! 

I told them that Noah only put 1 tablespoon (15 ml) of chocolate syrup with 250 ml of milk, while Isabelle puts 4 Tbs of chocolate syrup for the same volume of milk. (This is a lie. They would both put as much chocolate syrup as they could.) I made their respective chocolate milks for the class and showed them to all the students. They could see the difference between the two as one was much darker than the other. We calculated the percent of chocolate syrup for each. Then the question:

Fake context for sure. Jacob would want more chocolate syrup than Isabelle and Noah put together! But my students were engaged. They were paying attention. They, for the most part, wanted to know how to do this. And we did. Then, they worked through one more on their own:

I find that I am not worried about time, despite having to stick to a pretty regimented schedule. I believe that it is better to do one example where are all students engaged rather than 3 traditional ones where several (many?) of the class is not really paying attention.

What else have I changed? Homework. I do give homework in this class (but not in grade 10 applied math), which I check daily for completion. I believe that some practice is important in consolidating the material we have (un)covered and hopefully a deeper understanding can be developed, at least some of the time. But while looking for midpoint activities I found an old post from Dan Meyer about homework (or not giving homework) which included a suggestion from someone about modifying how homework is assigned. I liked the idea so implemented it the following morning. I am breaking up homework into basic, regular and challenge questions and have asked students to do 2 of the sets. The students who are confident with what was done in class can start with the regular set and move on to the challenge questions. Those who are a little less solid on the material start with the basic questions and then do the regular. It looks something like this:

I still think there is too much homework there, but I am working on that. The feedback I have received from students has been positive. I did feel the need to point out that after doing the regular set, students should not do the basic set in order to avoid the challenge questions!

Overall, I think I have made some positive changes. Not a huge overhaul like I did with grade 10 applied last year, but a move in the right direction nonetheless.

Solving a System of Linear Equations

This post is a bit of a repeat as I took something I did with my grade 10 applied students last year and am using it for my grade 10 academic students this year. We have solved systems of linear equations by graphing and by substitution. Before doing elimination, they normally do this investigation on equivalent systems that takes a very abstract look at why you can multiply equations by a constant and add or subtract equations. In the past students have done the investigation but really did not get anything out of it (other than confusion). I have never liked it, but it was one of those things that all the other teachers were doing, so I assigned it too. No more! This is what we did instead:

The pictures were key to some students' understanding. There were clearly 2 more coffees on the top line and that was the ONLY difference, so those 2 coffees had to account for the difference in price. From there they could work out the price of 1 coffee, then of one doughnut. Then we "translated" it into something more algebraic:

It took a little prompting for them to come up with the idea of subtracting. I asked what they had done with the costs to get them going down the right path.

Next:

I loved hearing a student come up with the idea of doubling the first order. This now gives us an equivalent system (that has meaning) where the number of cookies is the same in both orders so the difference in price is due to the extra latte.

Their homework was to finish this question and do one more that involved multiplying both equations. I really think this method brings meaning to the whole process and will make tomorrow go really smoothly.

Good Things from Day 3

In Advanced Functions today, I used the Popsicle sticks for the first time. I have been reading Embedded Formative Assessment for a while now, and this is a suggested strategy when questioning students. Whenever I had a question, I drew a name written on a Popsicle stick from my Starbucks tea tin (I finished the tea long ago) and that student had to provide an answer. I did try to give leeway for some questions to help them feel less anxious about the process. I think it went well - I even said that they could only raise their hand when they had a question. I also found myself saying "Convince me" many times throughout the day. I may not be reinventing the wheel in this class, but I am still trying to make important changes.

My grade 10 academic crew wrote a quiz at the beginning of class. I wanted to see if they were solid on graphing lines (most are) as we head into solving systems. I added a question, though. I wrote something like "Convince me that your graphs are correct.". I wanted them to use another method to check their work. If they used slope and y-intercept to graph, then they could check a couple of points or find the x-intercept, if they found the x- and y-intercepts to graph, then they could rewrite the equation and check the slope, etc. I am trying to get them to reflect on their work and try to think of multiple ways of solving problems. Anyway, I liked the results and will continue to ask similar questions.

The good thing about my grade 10 applied class is that I have some idea what I'm doing this time around (!) and know where I'm heading. I have a better sense of where I can push them a little more, and where I need to give them more time to absorb concepts.

And there was a double dose of Desmos along the way, which is always a good thing.

Here we Go!

Today was the first day of the new school year. Last semester I blogged daily about my grade 10 applied experience as I was spiralling through the curriculum with activities for the first time. I don't plan on blogging daily again, but really do feel that it helped me reflect about my practice, which I believe is really important and often neglected aspect of our profession. So my plan is to blog when I do something that I think is interesting, however often that is.

In Ontario, most schools are semestered so we teach the same students every day for half the year, then get a new crop of kids in February. We see each of 3 classes for 75 minutes and have a 75 minute prep period each day (when we are not supervising or covering someone else's class). This semester I have MHF4U (grade 12 advanced functions), MPM2D (grade 10 academic math) and MFM2P (grade 10 applied math) first semester.

At the end of last year, one of the cards I got from a graduating student said something to the effect of "I will never forget the first day of math class in September". At the time I read it, I had no idea what we had done back on September 3rd! I did figure it out though - we did the Marshmallow Challenge. And we did it again today in MHF4U! I stole this from someone (thanks and I'm sorry I don't remember who you are). I put students in random groups of (mostly) 4 and gave each group a large whiteboard.

I actually gave them 20 minutes to build as I remember time being really tight last year. Here are some of the final products. These are the sturdiest ones:

and here is the winner:

It was really interesting to watch them interact and I loved hearing the numerous "What if we ...". I was impressed with their efforts and collaboration. We also talked about mindset and learning from mistakes. It was a good first class.

Next up, grade 10 academic. I have done the "Don't Lose Your Marble" activity on the first day of MPM2D for ever (well, almost). Students have to find the relationship between the height of a ramp and the distance a marble will roll. I like this activity because I think it gives students a chance to remember something about linear relationships in a low pressure situation. (Note: I know that it is actually a quadratic relationship and I will tell them that tomorrow. For the small data set that they collect, a linear model works quite well.) They work in groups and help each other remember things like dependent & independent variables, slope, etc. And I get to observe them which is a great way to start to get to know them. I plan on doing many more activities with this class this year so we are off to a good start.

Last period of the (very hot) day was grade 10 applied. The girls all sat together and the boys did the same. I moved a few people around to even out the groups. We talked a bit about spiralling and the fact that we will be working through a lot of activities, which was met with a positive response. Then I gave them Fawn's Noah's Ark problem to work on in groups. I think my next comment is a reflection of why many of these kids are in the applied (vs. academic) math. It is like they have have the curiosity zapped out of them. It is sad that they gave up so easily, or tried to. That they did not want to solve the problem. That they thought they were not smart enough to solve the problem. I continued to encourage them and kept saying "Convince me!" when they came up with an answer. I tried to point out some of the good strategies they were using and nudged them in the right direction if they were really stuck. No one finished the problem in class so I said there would be a prize if anyone comes in with a correct, well written up solution tomorrow. We'll see. There is always a lot of work to build up the confidence in many of the students in MFM2P. I believe they can all succeed and will work toward having them believe the same.

I did accomplish my day 1 goal: learn the names of ALL my students. Now to finish planning for day 2!

A Summary of Spiraling through the Curriculum with Activities

At TMC14, Alex Overwijk and I shared our experience spiraling through the curriculum using activity-based teaching. I'm going to attempt to summarize what that entails. The PowerPoint is on the wiki.

Let me start by giving credit where it is due. Alex and Bruce McLaurin (@BDMcLaurin) started spiraling at their school 5 years ago. They are the experts on it now and are sharing the experience with teachers far and wide. I got involved this past school year when I spiraled my grade 10 applied math class, second semester. Alex wanted to collaborate on this with me and Sheri Walker so we met 3 times during the semester to figure out what we wanted to do, try some things out and plan each cycle. We took the lead from Alex on which activities to look at and added a few things of our own. This was truly jumping in the deep end of the pool, but I firmly believe it is the only way to do this successfully. At the end of our session Alex said that if you try to do it slowly and incrementally, the kids will drag you back down and you will go there because that is your comfort zone. I agree, so jump on in!

A bit about grade 10 applied math. The students in this course have generally been unsuccessful at some point in math and most do not like math. There are a lot of behavioural issues and I would guess that in the average class, over half of the students have an IEP. Asking these kids to sit and take notes then do homework is a recipe for frustration all around. Instead, we tried to get them up and moving, doing math in context, even if it was a contrived context, and assigned no homework. There was no point assigning homework - those who would do it are the ones who didn't need to and the ones who needed to wouldn't do it. So no homework, no textbook; we created worksheets for practice which some students finished and others didn't. They did what they could in 75 minutes each day, which was different for each student.

Here is how it was structured. Over the course of the semester we did 4 full cycles and a bit extra at the end. This means no more units. Each cycle covered each of the overall course expectations (they start on page 53) and we tested all of these at the end of each cycle. 

We uncovered the material through activities. Hands-on as much as possible, with manipulatives, graphing technology and whatever "stuff" we needed - spaghetti, pennies, linking cubes, algebra tiles, cups, etc. The activities started out very scaffolded and become more open-ended and richer as the course progresses. I blogged about it all starting here and I will link to Alex's blog posts below.

The first cycle started with the 26 Squares activity which took about 3 weeks. We hit all 3 strands of our curriculum. Next we did some toothpick activities to work on linear modeling and equations. We did Andrew Stadel's File Cabinet 3-act next to hit on some of the measurement strand. We used Smarties, Jujubes and pennies to solve systems of equations. That is where we ended cycle 1.

Please note that these are Smarties:

These are actually called rockets:

During the 2nd cycle we did Spaghetti Bridges for linear modeling/solving, a series of activities including High Fives and Frogs for linear and quadratics, found inaccessible heights for similar triangles and trig along with a roof truss task for the same expectations, then I did some work with surface area/volume of prisms and pyramids (I think Alex and Sheri did something different). Then came test #2.

The following chart, although hard to read, gives you an idea of the number of activities and which expectations they are hitting.

Alex also did a card tossing activity which we got participants to try out at TMC14. It is a lot of fun to do and hits a lot of curriculum expectations. I find that the more of this I do, the more I can see how to extend an activity to get more out of it.

(thanks to Nathan Kraft for the photo)

Throughout the course students saw the same concepts several times. For example, in cycle 1 they solved systems of equations only with manipulatives. They did not even write equations to represent the situation. In cycle 2, they did the same but also started to write something to represent the situation - some wrote equations, some used symbols or pictures. In cycle 3, they learned how to use elimination through a very logical progression of questions that made sense in context. And we reasoned through the answer. They may not have been able to present a beautiful algebraic solution, but they could solve the system and could explain to you how they did it. They were not following an algorithm - they understood what they were doing. In cycle 4 they solved systems to work through the cup stacking activities. We created a solid foundation and built upon it each cycle.

The benefits from the PowerPoint are shown below, but one of the big ones for me is time - you have time to get through the course - in fact they see most of the the curriculum in the first 6 weeks; you don't need to feel rushed to get an activity done - if students need an extra day, take it; if someone is away for an extended period of time, they miss an activity or maybe 2, but we will cycle back and see it again.

As a side note, I got comp books for my students because I enjoy structure a little more than Alex. (By "enjoy" I really mean "need".) For each topic I would print out an example or fill-in-the-blank type of sheet for students to glue in their comp book and fill in. This way they all had a resource to refer to. Some students took their comp books home to help them study for tests and they were allowed to use them for the end-of-year summative task.

I hope this helps shed a little light on what is involved in spiraling through the curriculum with activities. Please throw any questions my way in the comments. Perhaps the strongest endorsement I can give is that I will never teach this course in a "traditional" way again, and that I have volunteered to teach it (the course that no one wants to teach) both semesters next year.

Why I Go to Twitter Math Camp

It’s funny – a year ago, after TMC13, I wrote a blog post entitled “TMC 13 - Minus the Math”. I find myself again not wanting to talk about the math that surrounded me at TMC14. It’s not that there were not great ideas presented or innovative ways of doing things, but more that the reason TMC is so special for me is due to the interactions that happen around and outside the math. Being constantly surrounded by up to 150 people for 4 days should sound a little like hell to an introvert like me, but it is truly the most deeply fulfilling professional, well, anything I have taken part in. I have presented workshops in a lot of places, in a lot of formats, and other than Exeter, nothing even comes close to TMC. The connections that exist before those magical 4 days get strengthened and new ones are formed, and (this is the thing), they continue to grow after we all head in different directions geographically. Each of these connections helps make me a better teacher as they encourage me to share, try others’ ideas and stay connected to the greater math world around me.

I cannot possibly do justice to how I feel about my tweeps. The fact that I feel comfortable even calling them my tweeps is remarkable. I do not ever make assumptions that people are going to like me or think highly of anything I do, but I feel that some teachers on Twitter really do value me and my work. This is remarkable.

I want to mention a few people that had an impact on me at TMC14. It was such a pleasure to have spent time with many others, but this post might never end if I list you all…

Pam Wilson (@pamjwilson) may be the nicest person on the planet. She makes me want to be a better person and teacher and continually motivates me to do more math ed reading to continue learning. She is amazing. She also reads my blog and has always been kind and supportive which means so much to me.

I was actually too shy to talk to Nathan Kraft (@nathankraft1) last year, but am really glad that I got to spend some time with him this year. He is genuinely a good person and I feel fortunate to call him my friend.

I enjoyed a conversation with Viktoria Hart (@Viktoriahart) and the always awesome Justin Lanier (@j_lanier). Viktoria has 1 more year of school left before she becomes a teacher. Wow! She is going to kick ass when she gets in the classroom.

MaryAnn Moore (@missnarymm) was kind enough to chauffeur us around. She is simply lovely and I look forward to continuing to get to know her on Twitter. She said it well when she tweeted this:

I had the pleasure of two dinners in the company of the Memphis trio: Kevin Mattice (@kjmonopoly), Matt Bigger(@mwbigger) and Seth (@melroseharkins) along with Levi Patrick (@_levi_) from  Oklahoma City. I was lucky enough to get to work through Park Central (or was it Central Park?) with Levi and have to tell you that he is sharp, funny, kind and passionate about what he does.

Kate Nowak (@k8nowak) was kind enough to help us find a quick lunch on Friday and it was really refreshing to talk to her. I have admired her work from afar and feel fortunate to now have gotten to know her a little bit. She later tweeted that she would like to know more about the spiraling work that Alex, Sheri and I have done so I sent her the link to my blog. I think this may be the best tweet I have ever gotten:

I can’t not mention getting to see, talk to and hear Eli Luberoff (@eluberoff). He continues to impress me with his passion and commitment to making a product that is as good as it can be for students and teachers.

Most notably though, I brought Alex Overwijk (@AlexOverwijk) and Sheri Walker (@SheriWalker72) with me to TMC14. There are now two people that I collaborate with at home who get “it” – the “it” that makes TMC so special. The passion of all the educators, the friendly nature of all who attend, the collective need to improve as teachers that is second to none. I got to share that with them and that is huge. I am so glad they were able to make the trip with me. I love them both and they defined TMC14 for me.

There was a chunk of time that was not my finest hour, so to speak, and I appreciate all the concern that was expressed. I should know that I cannot be “on” for 3 days straight, especially on little sleep, especially, especially with no lunch on the day I am presenting. The intersection of those (and a few other) things left me unable to give any more when asked to do something that would have been very difficult for me even on a good day. I hope I didn’t offend anyone and am truly sorry if I did.

I leave TMC renewed in my passion for teaching, knowing that I belong to an amazing group. Being part of this community makes me feel complete. Thank you to Lisa and Shelli for making this possible. You will never truly know how much this means to me.

See you at TMC15.

Not a TMC14 Post!

I am really just testing out a theory here. While I do that, here is a picture of one of my dogs.

TMC14 - Thinking Through Day 1 of Algebra 2

I feel very fortunate to be back at Twitter Math Camp. It is so great to see the Twitter friends I made last year and to meet so many fantastic new (to me) people! For the morning session (2 hours/day for 3 days), I chose Algebra 2. Being Canadian, we do not have a course called Algebra 2. In fact, the content from Algebra 2 fits in grade 10, 11 and 12 courses within our integrated Ontario curriculum! Therefore talking about the flow of the course or the connections within the course is a little less relevant for my situation. However, I am taking pieces and figuring out where they do fit and how I can adapt them to help my students make more sense of math.

Glenn Waddell (@gwaddellnvhs) spent a large portion of the first day showing how he ties all of the algebra 2 functions together through a common algebraic form. 

The one of these forms that most teachers likely don't use is the linear one: 
y = a(x - h) + k. 

Glenn has blogged about it here. I like the way this connects to the other equations which we do use. I like it, yet it bugs me and I'm not sure why. It makes a lot of sense and is a very useful form. 'a' represents the slope (or rate of change) and (h,k) is a point on the line. Slope-intercept form is great for graphing, but Glenn's form (I'm not sure what to call it) is so much more useful when trying to find the equation of a line given the slope and a point on the line or two points, etc. Once you determine the slope, substitute the point for h and k and you have finished! Here is an example using Desmos.

The more I think about it, the more I think that my discomfort with this form is simply that it is not what I am used to seeing, and perhaps, that it can be simplified (and I kinda like my functions to look tidy). But it really is a smart way of tying together so much of what we do with functions and their graphs. I think I will work with this at the beginning of my grade 10 academic class in September. They will all know y = mx + b and will be working through transformations of quadratics before long, so it seems like a natural fit. Thanks, Glenn!

Change is Hard

In my last post I eluded to the fact that getting away from a lot of "direct instruction" (there are so many connotations of that terms) is hard. I teach in a department where everyone is expected to teach the same thing, in the same way, on the same day. This stemmed out of necessity. The school population used to be much smaller so when there were 2 sections of a course being offered, the teachers would work together, taking turns making up lessons. They would both teach the same thing on the same day as they had both contributed to creating those lessons. We still do this and also take turns creating tests and test on the same day. But times have changed. Our school population is over 1100 and there are generally 2-4 sections of each math course each semester. However, getting away from the SMARTboard lessons we all share is not an easy task. The thing is that you can't formally teach a lesson at the board (show concept, work through examples, repeat) AND do activity-based learning. Both require time and I can't do justice to an activity if I also have to "teach" the lesson. In MFM2P it was rare for me to do any direct instruction, and when I did, I disliked it. You may be able to cover more, but many of the students are not taking that journey with you. How many times have you found yourself saying "but I taught this!" when students don't know how to do something? Just because we teach it does not mean they learn it. Why it has taken me so long to figure this out is beyond me. Anyway, I want to change the way I teach in my academic classes, but I don't want to rock the boat too much. I value the collaboration I have had with some of the teachers at my school and don't want to upset anyone. What works for them is great and I don't wan to change anyone else (except if they are teaching kids in the MFM2P stream!), but I know that I could connect with more students and do a better job of engaging them all. And I'm okay with doing more work...

So my thought is to stay on track with testing at the same time as the other classes (which rules out spiraling), but do activities for the majority of the days leading up to the test. There will be some skills that I will have to teach, but I'm going to work on making that the exception, not the rule. I will also blog about what I'm doing so if anyone wants to try what I'm doing, it'll all be there. Change is hard, but change can be very, very good!

Reflecting on 2013-2014

In the fall of 2013 I started trying a few new things in my classroom. I did counting circles with my grade 9s (MPM1D) and I think that really helped set the culture of my classroom. Some students loved it (perhaps because they didn't consider it "work"), while others hated it. It took me outside my comfort zone, but I know it was a good thing. Making everyone say something every day is big. I try to involve all students every day, but this forced the issue and may have made some more comfortable to share ideas along the way. I hope it helped improve some students' number sense too. I tried to do math talks with that class, but got swallowed up in the "I have to cover this curriculum by this date so I don't have time to do anything extra" mentality. I need to lose that! I did throw in more activities and tried new things, but not enough. In the fall I also had a grade 10 academic class (MPM2D) and a grade 12 Advanced Functions (pre-calc-ish) class (MHF4U). I also tweaked them (I have taught all these courses before) and added cool stuff where I could make it fit, but still, there is much more I could do. In all of these classes there was still way too much of me teaching and them taking notes, then working through examples together. This, I feel, needs a whole separate blog post...

Second semester I had a grade 10 applied math class (MFM2P) along with two sections of grade 12 Calculus & Vectors (MCV4U). The Ontario math curriculum documents are available here: grade 9 & 10 and grade 11 & 12. I have to admit that I didn't do too much to make MVC4U better - a few added investigations, a few more station activities, more use of whiteboards, but not a big overhaul. It works and it is the course that I'm least likely to change in a significant way. The grade 10 applied class was a whole other story. I ditched pretty much everything I had done before and started fresh. I was very luck to have the opportunity to work with Alex Overwijk and Sheri Walker throughout the semester and successfully spiraled through the curriculum with activities. I blogged about it all, starting here, so I'm not going to go into the details of what I did. I will say that it was exhausting and exhilarating and at times frustrating, but it was good. My students learned math, despite not wanting to, in many cases. One student in particular complained about having to do work, said things like "can't we just have a lesson?" and often said that he didn't know how to do whatever we were doing, but with a little nudging always got there. He told his guidance counsellor that I "wring the math out of him". I like that. And it is an apt description of what went on in that class. They often tried to quit but I never let them. They may not have worked as hard as they could have, but they all worked every day. They were not spectators. They learned to persevere. It was not perfect, nor will it ever be, but I think it is on the right path. This is what they left on the board for me:

(yes, some of them called me "b-dawg" - so not me, but I grew to like it)
They were a great class and I feel fortunate to have spent the semester with them.

Next year, I volunteered to have a grade 10 applied math class each semester as I am very invested in teaching this course this way. I am looking forward to actually having some idea what I'm doing from day to day (!) and learning from what I've done. Blogging about it has already proven itself useful to my two colleagues who will be teaching it the same way next year along with me. I'm sure I will also be reading my posts to remember what did and did not work. There are definitely some things that I will change, but I am so thankful to Alex for sharing his time and work with me.

I also want to make changes in the other courses I will be teaching. That goes back to needing a separate blog post so I will leave it at that for now.

Graduation 2014

Today, the class of 2014 graduated. It was a nice ceremony that was clearly well planned as it ran seamlessly. I taught many of the students in this graduating class of 227 - I think I taught 139 of them. Many I taught twice or even three times. It is remarkable to have been there with them on their first day of grade 9, then be there with them on their very last day of high school. You would think that with 20 years in I would be used to commencement ceremonies, but I am usually at the Exeter math conference for the last week of June so I have actually attended very few graduations. Although not thrilled to have missed the conference this year, I was happy to have the opportunity to see this class graduate. These are special kids and I feel fortunate to have seen them grow up and to have helped them learn along the way. However I find goodbyes difficult (just ask @fawnpnguyen!) as I am such an emotional person. So today was tough for me. I'm an introvert so I'm not good with crowds of people, but I did find a few students that I needed to say congrats and goodbye to. I got asked by a few to have my picture taken with them (which, if you know me, is a big deal). Sadly though, I avoided going up to a few students because I knew I would start to cry. The last student I did talk to got the tears going so I had to disappear. So to all of those that I missed I say congratulations and I am so very proud of you. I will miss you and West will not be the same without you. I wish you all the best down the road...

Day 85

Day 85

The school is quiet. The students have all left. It is the end of the last day of classes for the 2013-2014 school year. Traditionally, not many student show up for class on this day - it is "beach day" for the seniors. So I was a little surprised that among my 6 or 7 MFM2P students that were here today, was one who has missed a lot of school. And this student spent a good chunk of today at my desk working through math with me. Seriously working. Didn't go hang out with friends at lunch in order to finish the sheet we were working on. Beyond anything else today, I am immensely proud of this student. I got a lot of really lovely cards from grade 12 students today, which truly filled my heart, but my 2P kid who gave up half of the day to do math, that was the icing on the cake. This is why we are teachers.

Day 84

Day 84

Today was pretty quiet as they reassessed on those curriculum expectations where they had yet to meet expectations. I loved how some used manipulatives and graphing calculators and really tried their best. I am dismayed by those who did not...

On another note - here are a couple of good (and positive!) blog posts about MFM2P:
* from Alex Overwijk, a post about introducing factoring by talking about area and length & width.
* from Heather Theijsmeijer, a post about a student-created scavenger hunt across classes.