A while back, Pam Wilson shared an old linear matching activity. It had students match up a graph, two points, a slope and two forms of a linear equation to form a set. I really liked it, but it used old calculator screen captures for the graphs. I cleaned it up and ran it with my students. It went well, but I learned that it works better if each type of card (graph, slope, equation, etc.) is printed on one colour of paper so that students have a complete set when they have one card of each colour.
Here is the .docx file and here is the .pdf file. I'd love to hear if you use it and how we could make it better!
I ended my first full unit teaching in a Thinking Classroom (you can read about the beginning of this journey for me here) in my calculus classes just before the March break. I surveyed my students and had mixed feelings about the results. Many suggested 20 minutes of notes at the beginning of class, yet many of these same students also said that they never looked at the notes I post on Google classroom each day. I e-mailed Alex Overwijk to ask for advice, saying that I felt like I was doing something wrong, or at least not entirely right. His response went along the lines of "they don't like being uncomfortable, they don't like having to struggle and you aren't doing anything wrong". I think that what led to me feeling as I did had a lot to do with the very skill-based nature of the unit. They were taking derivatives and taking more derivatives and then they took some more derivatives. This led to groups taking turns doing questions. The questions were not challenging enough to require them to work together as a group to solve them. I'm not sure how to change that for the let's-learn-how-to-take-derivatives unit, but I will ponder that some more before we get there again next year. This past week we have been working on the elements of curve sketching. Using the first and second derivatives to help determine intervals of increase and decrease, local maximum and minimum points, intervals of concavity, points of inflection, etc. I have continued to use visible random groups and they have continued to work on the VNPS (whiteboard/chalkboard) for almost the entire class each day. I have tried to be more intentional about what I do work through with them - mostly at the beginning of the class. The questions have been more interesting and more challenging for them. I am really pleased with their efforts. I am finding that they are putting all the pieces together more easily and that I am also thinking more deeply about the material. And it's fun! At least twice this week the bell rang at the end of my afternoon class without anyone in the room being aware that it was the end of class. They didn't want to stop. It's incredible how much fantastic work they are producing and how well they can explain it all to me. I am not quite sure how much gushing is appropriate, but my students are awesome. I snapped this picture of some of them this morning and it makes me happy and proud to look at it.
Here is the progression I used for the week (apologies - I got lazy and didn't include all the answers). They did not all get through every question yesterday and today, but I believe that they all have a solid grasp of the material. I continue to post filled-in notes at the end of each day should they wish to review the work or try any of the questions on their own.
I was going to blog about the last few days in my calculus class, including "Leibniz Day", but I just can't. I found out at the end of the day today that a student I taught all of last year passed away over the weekend. Sometimes teaching is really hard. Here is the docx version of what I did and here is the PDF version.
I learned a lesson yesterday when my students far exceeded my expectations. One group had completed the entire sequence I came up with to "discover" and apply the quotient rule within 30 minutes of the start of class. This is when I am really glad that I have taught the course many times before and know the material inside out. I let that group start on their homework, something that never really happened during class time even before I switched to a the thinking classroom model, while I found another question worthy of their time. I could have just thrown an uglier question up on the board for them, but I wanted one that would make them think and, hopefully, challenge them. I found a good one and they got back up and worked at it for most of the remainder of the class. The challenge for me is to set up my sequence of questions in a very intentional way, making sure the progression is neither too little nor too much at a time. But I also have to make sure that I have planned enough challenges to keep them going and keep them thinking. Getting that right will take a little more practice. After that class I fixed things up for my afternoon class, adding the new question into the sequence along with another harder question that would make any algebraic misconceptions come to light. It turned out to be a little too tricky for most groups and they all got stuck. I adjusted by giving them the answer so they knew what they were trying to get, but many groups eventually abandoned that question and moved on. After that class I rearranged my sequence again, putting that question last. As I learn to adjust and plan better hopefully I will get better at finding that sweet spot of just-right difficulty progression and quantity.
Today was day 3 of running my classroom as a thinking classroom - visible random groups (of 3) working on vertical non-permanent surfaces. We started with a quick demonstration to show that the derivative of a product is not the product of the derivatives. Students then worked through a product rule "discovery" activity that I have been using for years. You can find it here. I wish I could give proper credit for it, but I do not remember who shared it with me.
Here is the setup:
Students worked out expressions for length, width and area and for their rates of change before completing the following table.
Their job was to work with the numbers they came up with in the table to figure out a pattern that worked for each row - that would be the product rule. It is always interesting to see who comes up with it quickly and who takes a little longer (often because they are trying really complicated things!).
Once I felt like the majority of students had found the pattern, I sent them off to their VNPS to work through today's sequence of questions.
The last of these asked them to come up with the product rule for three terms. I loved what some of the groups did. They extended the introductory activity to 3-D, added height and worked through the numbers again! (sorry that the picture quality is terrible)
Here's a group that came up with a conjecture for the product rule with three terms and tested it out. I would like to say they did this instead of asking me if they were right, but they did jump right to it when I said they should check it for themselves.
It was really exciting to see such fantastic work, at such a high level from all my students. I love how they trust that they will be able to tackle all the questions I give them and believe in themselves enough to try. P.S. All of my planning is on one getting-bigger-by-the-day Word document. I'll post the whole thing at the end of the unit.
After last week's workshop with Peter Liljedahl I decided to go full-on thinking classroom in both my calculus classes. I told them that they wouldn't be taking notes today and that they would be working in groups at the boards around the room. We talked a little about what the derivative function is and how we find it, along with the issues that would arise if they tried to find the derivative of y = x^729 from first principles. Next they each chose a card to determine their group. Off they went to their whiteboards/chalkboards and they started on the first question. Even though many had already been told the power rule, I made them "convince me" (and themselves) by finding each derivative from first principles. Here is the order of the questions they did:
They noticed patterns in parts (a) and (b) and were able to explain why the derivatives of (c) and (d) were the same as (a). Part (e) went better than expected and generally confirmed their conjectures. The results from parts (f) and (g) were confusing for many and I found that it was helpful to rewrite the question and get them to write the answer in the same form in order for them to see the pattern still held. They got stuck trying to do part (h) from first principles so needed to find the derivative another way. At this point we stated the power rule as a group and turned to proving it. In the past, I have gone through the proof with my classes and many students' eyes have glazed over as they completely tuned me out. This time I gave them the expansion of x^n - a^n, we talked about how many terms there would be in part of it and let them try the proof. At least one group in each class finished the proof on their own! And all groups made good headway with it which helped them stay engaged when I showed them the full thing. I think they thought it was kind of cool! I gave them two more questions after the proof:
The first was no problem and the second was done incorrectly by almost 100% of groups. We stopped there for today and I asked them to write down a summary of what they had learned. I didn't do anything else to close the lesson as I felt like it wasn't needed. Here is the sequence for tomorrow:
Overall I thought today went well. I have done enough of this type of work with students that I was very comfortable and my students were great. There were a few times when I took a marker (there was only one marker/piece of chalk for each group) and handed it to a particular student, but in general they took turns doing the questions. Those not writing the solutions were watching what was going on, looking for errors. There were some good discussions going on today, but I anticipate more tomorrow due to the nature of the questions. There were some groups that would call me over to check their work, but they got a lot of "What do you think?" and "Convince me" and "Are you sure?" so I suspect that will diminish as we continue. I had to ask a few students to put their phones away, but it was not really an issue. They all did math and were all thinking and even those who came in knowing the power rule learned something new.
As I wrote in my last post, I am jumping into the full VNPS-VRG-thinking-classroom in my calculus classes. I have two classes - one in the morning, the other in the afternoon - so I will be able to tweak my plan in between and hopefully really make progress with both my planning and implementation. We have finished our first unit on limits and introducing the derivative function. On Wednesday we will start with the derivative rules (power, product, quotient, chain) so that's what I'm trying to plan out. Sheri Walker and I thought through this progression together yesterday so I've tried to tie things together and include some of what I expect to see. The intro will be to the entire class, the sequence are the questions I will give each group of 3 at their whiteboards/blackboard. Groups should progress at different rates so my job is to circulate, observe and help keep them in flow. The last question (part i) may be for the speed demons or for everyone - we shall see! I am also trying to keep in mind what comes next.
You may notice that I have a number of unanswered questions. If you can help me think through those, I would be most appreciative. My plan is to edit my document after I have done both classes and then post the new file in case it might be useful to others.
I was fortunate enough to be able to attend a half-day workshop on Thursday with Peter Liljedahl. I first heard about Peter's work a few years ago after he had spoken at a conference in Ottawa. There was much buzz from those who attended about VNPS and VRG, most notably from Alex Overwijk. Al was happy (!) to share all he had learned about vertical non-permanent surfaces and visible random groups (he may or may not have stopped strangers on the street to tell them about it). He has become Peter's #1 fan, even entitling the Ignite session he did last year "Things Peter Says". Although I had heard the vast majority of what Peter shared with us, there were a couple of important puzzle pieces that got filled in which make me believe that I can create what he refers to as a thinking classroom. This post is not intended to explain it all to you - for that you should visit Peter's website here or read this post from Alex. I have used VNPS in my classroom for a few years now, but not every day and not instead of teaching/facilitating lessons. I use them for review stations and for tasks. I do visible random groups whenever I do group work, so that's not new for me (best group size = 3). I also spiral some of my courses with lots of activities, so I think that in many respects I already have a thinking classroom. My students tell me that even when I teach the same lesson as other teachers, I do it differently - I make them do the math, I don't just give it to them. But one can always do better... What convinced me was hearing about an actual curricular example of how to structure VNPS. Peter only spoke about students learning how to factor for a couple of minutes, but with enough detail to make it all click for me. I will attempt to share how I see it all working, but apologize in advance if my thoughts have still not gelled. VNPS is a means to a goal, not the goal itself. It is, according to Peter's research, the most effective vehicle to creating a thinking classroom. One where all students are engaged in meaningful mathematics - doing the math, not watching someone else do it. They are learning to because autonomous, to look for the next question and persevere when they get stuck. The teacher's role in this is hard to nail down - you need to adjust to what you see constantly. It's structured chaos at its best. And I don't think it can be successful without really good planning (and I would highly recommend the book "5 Practices for Orchestrating Productive Mathematics Discussions" to help). Thursday's experience helped me see how all the pieces fit. The structure looks like this:
Start with a quick (~2 minute) lesson or prompt to activate prior learning or give students enough to build upon. Instructions should be oral as much as possible. If you give instructions in writing students have to decode them individually, whereas if they are oral instructions, students immediately start talking to each other.
The questions they are working on must be sequenced in a logical way to develop the skills while keeping all students in "flow". These don't have to be incredible task questions - they can be everyday textbook questions. The importance of proper selection and sequencing is huge!
There must be a lesson close that will level the class to the bottom. This could be a full-class debrief (going through a different example) or a gallery walk that is also thoughtfully sequenced.
I spend part of Friday with Sheri Walker working on sequencing questions for calculus (she was gracious enough to work on topics that she has already covered). Although not finalized, I think it helped us both think through how to make this work.
I still have lingering questions/concerns.
I worry - maybe that's too strong of a word - about the introverts in my classes. Especially the shy introverts. Because I am one, and I know how exhausting working in groups where you may not be entirely comfortable with the material or people can be. I am already purposeful about making my classroom a safe space for learning, which includes making mistakes, but I still worry...
I wonder about the number of markers and who is using them. We worked through two problems on Thursday and each group only received one marker. There were no rules around who should do the writing, but Peter was going around the room taking the marker from some and handing it to others. I was very aware of how long I had the marker when I was in my first group and did my best to always put the marker down when I had finished with it. It is much easier to pick a marker up than to take it out of someone's hand. There were questions about all of this and suggestions of either using a timer so that each person had the marker for an equal-ish amount of time or that the person with the marker could only write others' ideas. This takes me back to the introverts issue - I would hate to be the one who had to write someone else's ideas if I didn't understand them. But I also know that I did not touch the marker in my second group, so it's not that hard to step back a little which is not what we want.
I don't know if I can make this work with my grade 10 applied class. MFM2P is generally made up of students where one half to two thirds have IEPs. Many require written instructions. Many cannot work in groups, only pairs. Many cannot work with certain other students in the class. Many (most?) hate math and are often very unwilling to do any work. There seem to be so many obstacles with that group, that I'm not certain this is the way to go. Spiralling with activities has really helped with engagement and success, so I think the VNPS may continue to be an every-so-often thing. If you can convince me otherwise, I would love to hear your thoughts!
Peter says we shouldn't give students notes. I agree, however, I will still continue to post notes on Google Classroom in calculus because there would be a mutiny if I didn't. There are only so many battles that I will take on! I like the idea of finishing the "lesson" at around the 50-60 minute mark, doing the lesson close then leaving ~15 minutes for students to write down, in their own words, what they have learned. They will then be able to reference my posted lesson with examples as they need.
There is so much more to all of this, but this is where I am for now. I am not promising to blog every day, but I will write about how it's all going. I would love to hear your thoughts in the comments. Thanks.
I spent a lot of time thinking about what activity I should do with my grade 10 applied students on the first day of semester 2. I wanted them to be engaged in mathematical thinking, preferably with something hands-on (but nothing that would cause complete chaos!) and I wanted them to work with someone else in the class. What I ended up choosing was Skyscrapers from BrainBashers - here is the link to their site. These are logic puzzles with only a few rules:
A completed board would look like this, where the numbers in the grid represent the height of each skyscraper:
I first learned about these puzzles from Alex Overwijk last year and I'm fairly certain that he heard about them from Peter Liljedahl(I spelled that correctly this first try!). Alex let us try them at our math PD day last year using linking cubes as the skyscrapers. I set up the skyscrapers ahead of time for my class, using a different colour for each height so that it would be easy to see if they had more than one skyscraper of a particular height in the same row or column. It turned out that I found this feature more useful than them as I circulated and checked their work. Each pair got their first puzzle and these:
Here is an example of one column. We reasoned through the fact that there is only one way to place the skyscrapers if you can see all 4.
Here are a couple of views of the completed puzzle:
I had printed out 6 different puzzles for them to work through, each on a different colour of paper so that I knew which one they were working. I also had the solutions printed on the same colour of paper to make checking their work faster. I checked the first two and then let them go. I should have printed some harder ones as some groups flew through these. I did have blank ones for them to make their own, but I really think this could have been a much richer experience if they had tried some of the harder ones, like these:
This is what I tweeted out:
What you may not realize is that very few grade 10 applied students ask for more "work", so it was awesome that they wanted more! Day 1 was definitely a success. I got to interact with all my students and see a little of how they think, whether they are able to follow directions easily, and how they work with others. It was a good day and a great start to the semester.
This post is long overdue. I knew I needed to write it last May and am finally making the time. Way, way, waaaayyyyy back when I was doing my education degree, my senior (grade 10-13, yes, grade 13 existed back then) math instructor was amazing. He drove about an hour each way to teach us twice a week. He brought in graphing calculators and taught us how to use them and how to teach with them. This was pretty incredible as it was 1994 (I told you it was as long time ago!). This laid the foundation for me to become a national instructor for TI a few years later (I have since resigned - my heart belongs to Desmos). This instructor also had us create "backward problems". Instead of just asking a question, we started from the answer and turned the question around. He helped me think in a different way and really turned around (pun intended) my idea of what assessment questions can look like. So a public merci! goes out to Rodrigue St-Jean for helping me start my career in a positive way. I am grateful and a better teacher today thanks to you.
Back in September, I decided to try a new way of sharing some of the cool stuff I hear about online with the math department at my school. I call it "Math Minute" and this is part of the original email I sent them:
"Hello fellow mathies,
I thought I would share some of the great things I come across on Twitter and on the blogs I read. It might be a cool activity or link to an article or blog post, but should only take a minute (or so) to read - hence the Math Minute title. I'll do my best to send a Math Minute out once a week, however please feel free to let me know if you would prefer not to receive them."
This was followed by a short description and the first link that was all about Desmos card sorts. Here is the link to the Google doc I am using to keep track of what I have shared. This is a sample (since it had no link): "Week #8:
I have been using the box (area model) method for multiplying and dividing polynomials for a while now. I like it because there are no tricks involved and students can see (and hopefully understand!) why they are doing what they are doing.
Below is a sample of factoring a non-monic trinomials. There would normally only be one box, but I tried to make my steps understandable for you. I have to say that I love algebra and decomposition and I have been friends for a long time, but I love the box for these. Try it out!
Mary" Although I have had little feedback from these Math Minutes <insert sad face>, I thought I would share what I've done in case someone else is looking for a way of sharing ideas.
Sometimes it's the little things that make a difference... Thing 1: As I was writing names next to questions on the board last week I realized that I was getting uncomfortable. Let me back up. I was randomly choosing the student who would write the solution to each question by selecting a Popsicle stick from my (Starbucks tea) tin which contains one stick with each student's name on it. When I chose a student who often struggles to solve a tougher question, I got uncomfortable. And I realized how in the past I would purposely assign those questions to students that I knew could complete them. I'm not even sure I did this consciously, but I am sure I did it. Using a random method of choosing names forces me to give all my students the opportunity to rise to the challenge. Given that I always tell my students that I appreciate it when they make mistakes because everyone can learn from them (and I always give them the option of purposely including mistakes when they write solutions on the board), I have no reason to worry about what solution they write. Noticing that it bothered me was a good reminder of why I use Popsicle sticks in my classes. Thing 2: Review day (before a test) is usually a day when I do stations with my students. I stick questions from a previous year's test up around the room, randomly pair up students and have them go around the room working on big whiteboards. Each pair gets an answer sheet, the right-most column of which is labelled "For Administrative Use". I could simply post the test answers/solutions somewhere in the room for students to self-check, but instead I have them come show me their answers. If an answer is correct, they receive a sticker in the right-most column and if it is incorrect I send them back to try again. If they return with another incorrect answer to the same question I may get them to try a third time or have a conversation with them about what they tried so that I can (hopefully) ask a question that will help redirect them. I ensure that they eventually get to the correct solution and receive that sticker. It may seem silly, but the motivation provided by those stickers is huge. My students are engaged in meaningful mathematical discussions which sometimes turn into arguments as they work through the station. They are talking about the math and helping each other understand the material in greater depth. They make mistakes and figure out where they went wrong before they take the test. Thing 3: When my students are working on a practice question as a class, I often walk around the room to check on their progress. I bring along my happy face stamp (or stickers) which I use if their solution is correct. They love this. I get a good sense of how they are doing by the number of stamps/stickers I have given out, but also get the opportunity to help those who are stuck by asking a question to get them unstuck. I'm still working on ensuring that my questions are not leading questions... It's a tiny bit of one-on-one time with each student that gives me a window into their thinking. I am certain that all teachers have a multitude of little things that they do which make their classroom unique and better. I would love to hear some of yours in the comments.
This semester I taught two sections of grade 10 academic math, both of which I spiralled. Although I have taught this course for many years, this was my second time spiralling it. I blogged my way though last year's journey, starting here. I thought it might be worth sharing what I did this time around as the changes resulted from my reflections the first time through the course. The one big change was doing more quadratics earlier in the course. About half the course is quadratics so I tried to devote about half of each cycle to quadratics. We factored starting in cycle 1; the rationale being that factoring doesn't always stick so multiple exposures to it (and lots of practice) should help students better retain how to factor. Cycle 1:
Not-really-a-cycle Cycle 4:
We managed to finish the content on December 22 so when we returned from the break there was a sufficient amount of time to review each strand and have an optional test for each strand. These tests covered 9 of the 10 curriculum expectations and each expectation was optional. Each student could choose which expectation they wanted to show - some did none while others did all 9. I modified some of the homework sets from last year, but continued to spiral the homework as well. I tailored it to include questions that I know many students needed to practice, but always kept it to one page. I'm sure there were other differences that I cannot recall at the moment. If you have questions I would be happy to answer them in the comments.