Last December, I discovered the GCHQ 2023 Christmas card. I printed it off and took it with me while visiting relatives over the Christmas break. I started working on it during the plane ride and continued working on it throughout my week-long visit. Some puzzles I managed to solve quickly. Others kept me engaged and thinking about them over several days. As a Canadian, a few of the word puzzles were an extra challenge because they referred to UK locations or people. It was quite a thrill to put all the solutions together and solve the final puzzle. This year's GCHQ card has just been released on December 11th and I'm planning to once again bring it with me on my winter holiday travels.

I decided a holiday card would be a great activity for my high school math club. The card is designed with grade 10 students in mind, so the puzzles are tailored to their skill level. My hope is that students will solve some puzzles at school and take the rest home to share with their families over the holidays. Recognizing our school’s diverse student population and varied celebrations, I focused on creating puzzles with general winter themes rather than centering on specific holiday traditions. The card is designed to fit on a single 8.5" x 11" sheet of paper, folded in half. Creating the card and its solution set took more time than I anticipated, but I hope the students enjoy it. Here is a link to the pdf version of the card.

There are some other great holiday cards online that I was inspired by. Check them out as well:

• Matthew Scroggs Christmas Card 2024 - Matthew has designed and shared these cards for a number of years.
• The Maths Ed Ideas xMaths Cards - There are two xMaths cards shared here. Both contain a tone of puzzles.
• GCHQ Christmas Card 2024 - Some serious puzzling.

This year, I’ve been leading a math club at my high school, where we meet once a week during lunch. Each session, we explore a variety of mathematical puzzles, problems, and art.

​Some weeks, we collaborate on solving challenging problems from past math contests, sharpening our problem-solving skills. Other weeks, we dive into math games like Prophecies by Ben Orlin or Box the Numbers by Dan Finkel, which combine strategy and mathematical thinking in fun ways.

​Last week, we created mathematical art using knot tiles, inspired by Day 22 of Annie Perkins' #MathArtChallenge. This particular activity was developed by Dave Richardson as an extension of a previous challenge. To make it accessible, I adapted Dave’s challenge into a handout for the students to complete and color.

The reason that I like this activity so much is that it sparks curiosity and questions about lots of different areas and topics in mathematics:

• Knot Theory - How does knot theory apply to the tilings we've created? What types of knots are these? 
• Statistics and Probability - How many loops do we create? What is the average length of the loops created?
• Binary Number System - How do we represent base-10 numbers as base-2 numbers? 
• Tilings and Tesselations - What other types of tiles could we use in this activity? We could explore Truchet tiles, Wang tiles, Girih tiles, etc.

These knots might spark you to explore other ideas as well. I thought about how these knot tilings are similar to QR codes in that it encodes information in two dimensions.

As the holidays are rapidly approaching, I create a tree shaped version that teachers could use as a classroom activity in the final week of school before the holiday break.

Mediocrity - This is a game from Ben Orlin's book Math Games with Bad Drawings (which I highly recommend). Split the class into an odd number of teams and have each team choose a number between 1 and 30. The team with the median number gets that many points... after playing a number of rounds (corresponding with how much time you have left in class), the team with the median score wins! 
https://mathwithbaddrawings.com/wp-content/uploads/2020/10/Game-17-Mediocrity-1.pdf ​

Pico, Fermi, Bagel - The classic game of "guess my number". Mastermind is a great game... just switch out the colors for numbers and now you have a game for math class. I originally saw this game in the book Math for Smarty Pants by Marilyn Burns (1982) however the game has been around since at least the early 1970s. I built a Desmos Classroom version of this activity but it is just as much fun to play on the board with the whole class.
https://sites.math.washington.edu/~mathcircle/mmc/mmc2010/PicoFermiBagel.pdf ​

All Ten (Beast Academy) - Last year in my class, I had students do this activity individually on their cell phones. Now we all do it as a class. You could project the computer screen on your white board (https://beastacademy.com/all-ten) or go low tech and just choose 4 numbers to write on the board. Ask students to come up with an expression to equal each of the integers from 1 to 10. If the class comes up with multiple expressions, debate which one is the most elegant and mathematically beautiful. I saw a great post from Jenna Laib of an elementary classroom using this game on their whiteboard.

I've started my 20th year as a teacher and I continue to grow and improve as an educator. Reflecting on my teaching practice on this blog is an important tool to support that growth. Like writing in a Field Notes notebook, "I'm not writing it down to rememver it later, I'm writing it down to remember it now." The physical act of writing down a reflection requires you to pause, organize and summarize your thoughts.

I attended John Golden's (@mathhombre) session on Math Games for Secondary Learners. The room was packed with more than 150 educators. He lead the entire room in a game of Four Strikes (https://marilynburnsmath.com/games/four-strikes-and-youre-out/). This is a game from Marilyn Burns that can be adapted to be used in nearly any classroom. The entire room was completely engaged in this simple number sense game. A great game that requires no prior preparation and can be used at any time. I'm definitely going to be using this one with my class. 

Paige Sheehan (@MrsSheehanMath) presented a session on Diversifying Your Assessments. She shared a number of ideas on alternative types of assessments and assessment questions. I really liked the idea of asking students to provide an explanation along with their answers to multiple choice questions. While I often do this in Desmos Classroom, I haven't yet used it in paper assessments. This strategy allows students to explain their thinking and show their understanding when answering multiple choice questions.

In the session Lab Calculus, Kevin Bartkovich and Jess Emory described their Calculus course at Phillips Exeter Academy which is based on numerical methods, problem solving, investigation and modeling. I really liked the project they described on the Gini index. This is a measure income inequity in a population. Using census stats, we can use integrals to calculate this index using a Lorenz curve and trapezoid approximation. This activity demonstrates a practical application of integrals and might prompt a discussion of the implications of income inequity in a community.

Some sessions don't focus on specific classroom activities but instead inspire us to think about how we can improve our current practice. They might help us build stronger classroom communities or foster joy in our mathematics classrooms. One of my key takeaways from NCTM was not just new activities, but the connections I made with other teachers and conference attendees. I was able to arrive early and attend the Desmos Classroom Day. This was a chance to see what is new at Amplify/Desmos as well as connect with some really amazing educators. I had the chance to talk and collaborate with educators from all over North America. I hope to apply what I’ve learned to become a better teacher and share these insights with my colleagues to support them as well

This year I taught a class of Mathematics 10. Since the very beginning of the year, I have  presented a daily warmup question to engage students from the moment they enter the classroom. By projecting a warmup on the board as students arrive, I set the expectation for immediate engagement. While not every student starts working on the problem right away, enough do to make this routine valuable.

The idea for daily warmups came from Geoff Krall, who shared a year's worth of warmup questions on his website. Throughout the year, I've experimented with various types of warmups and gained insights into facilitating them effectively and identifying which ones most engage students.

At the start of the year, I used non-curricular warmups like general math puzzles (e.g., All Ten, Nerdle). These were instrumental in building connections with students and encouraging a playful approach to mathematics. As the year progressed, my warmups have generally fallen into four categories:

• Formative Assessment: Questions based on recent lessons to gauge understanding.
• Discussion Prompt: Questions designed to provoke thought and class discussions, such as "Which One Doesn't Belong?" and "Would You Rather?"
• Retrieval Practice: Opportunities to recall skills from earlier in the year or previous years.
• Choice Task: Questions with varying levels of difficulty, allowing students to challenge themselves in a low-stakes environment.

Among these, students have indicated that retrieval practice best supports their learning and understanding of the course material. This might be because I emphasize its role in promoting long-term retention of skills. They also said that they find discussion prompts the most engaging and enjoyable.

If you’d like to check out my daily Math 10 warmup questions, they are organized by unit here:

• Measurement
• Trigonometry
• Finance
• Relations and Functions
• Polynomials
• Roots and Powers
• Linear Functions
• Systems of Equations

As a side note, I've been trying out AI as a writing partner/editor. I wrote this blog post and then asked AI to edit it to make it more clear and consise. I think it did an okay job at this. It did not do such a great job when I asked it to create an image for the post of students working on a system of linear equations at the board in a dynamic classroom setting. Despite several attempts, the math on the board was always psuedo-math gibberish.

My son's elementary school had it's "Spring Fling" this weekend. This is an annual fundraiser for the school that includes lots of carnival style games for kids to play. I volunteered to run the Plinko game. At this station, players could pay one ticket to drop 3 chips on the Plinko game. They added up their total score to determine their prize. If they scored a total of 3-6, they won a small prize, 7-10 won a medium prize and 11-15 won a large prize.

I volunteered for this station because, as a math teacher, a bit of mental math makes adds to the excitement of a carnival game. Also as math teacher, I sensed that this prize allocation was going to be problematic. I remember attending a fantastic NCTM session presented by Bowen Kerins titled "The Mathematics of Game Shows" which featured Plinko probabilities. The distribution of the numbers in the plinko game would result in 1 and 5 being most often scored. Averaging 3 chips and we were going to have a lot of "medium" prize giveouts. I sensed that this would be a oppotunity ripe for data collection. During my 1.5 hour shift I had 49 plays of Plinko and I recorded the results of every game.

As predicted, this resulted in more 1 and 5 than the other three numbers. Interestingly, the inner slots turn out to be similar in this experimental data. As the height of the board is increases, the probability that the three inner slots will be about the same increases. According to a paper I read online, "this change indistribution is mostly due to the chips bouncing off the side of the board and appears to occur when the height of the board is three times the width." I started calculating the probabilities by hand but quickly realized that technology would be a better way to handle this so I entered some equations in a Google Sheet to find the theoretical probabilities of this specific Plinko board assuming each chip has a 50% change of dropping either left or right of each peg.

Anyway, the kids had fun and won lots of prizes and I had fun collecting and analyzing data. Perhaps I'll have to put this all together for a NS Math Circles session on carnival games!

About a week ago, I gave my students a handout to practice multiplying polynomials. It was just a quick check in to practice some of the skills we were learning in class. We were just starting this unit and some students already had some prior knowledge so I decided to give three different levels of challenge to choose from. It was a Friday afternoon last class so I didn't have high expectations for the level of engagement. 

To my surprise, nearly every student was working diligently on their chosen level of problem. Many students chose the most difficult level. About half chose the "Spicy" questions while the other half chose the "Mild" questions. A couple of students chose the "Medium" level or selected a few questions from several different levels. Letting the students choose their level gave them some agency in their learning. A few students who picked the "Spicy" questions worked hard to master the challenge. They asked lots of questions but felt really good about their accomplishment when they succeeded.  

I collected these sheets looked through them. I didn't mark them but instead used them to identify and share common misconceptions about factoring. I used examples from the students work (rewriting them in the style of "My Favorite No" from Leah Alcala) as the starting point for the next day's lesson. We discussed what we saw that was wrong but also identified some parts that were on the right track as well.

I've been trying out different ways to offer more choice to students. It is important to offer a challenge for students that are ready for it but not to overwhelm students that are still learning. Students in my class come from a number of different feeder schools and have a wide variety of past experience. I created a factoring "scavenger hunt" activity last week for students with two levels of challenge side by side. Students could select the "mild" side if they wanted basic practive or the "spicy" side if they were ready for additional challenge. About 20% of students chose to do the more challenging side. Everyone in the class was very engaged and was able to complete the hunt with instant feedback (if they couldn't find their answer on another sheet, they knew they had made a mistake and worked to correct it).  

You might also notice letters on each question. I also added a ciphertext message for students to decode when​ they completed the loop. I've also been including choice in warmup questions and chapter review questions.
​
Let me know if you have a favourite way to offer students choice in their work and practice.

When I first started teaching, reviewing for a test meant one thing. I would stand at the front of the room and reteach each unit, section by section. I would carefully annotate important points and questions that would likely be on the assessment. After reteaching the material, I would hand out a large packet of questions for students to work on. ​

I found that while this helped some students, many students were either bored or confused. If they didn't understand it the first time, hearing me teach it again the same way probably wasn't helping. If they already understood the material then listening to me teach it again was not helping them learn it any better. I recently ran across a couple of blog posts from MathMedic reminding me of why I do things differently now.

• Reviewing without Reteaching Part 1 describes the problems of reteaching
• Reviewing without Reteaching Part 2 describes some alternatives

My Mathematics 10 class recently reviewed for a unit test and now we're reviewing for a cumulative assessment on the first half of the course. I typically start with some individual self-assessment where students can review outcomes and identify areas where they need additional practice or review. Next I give students a variety of ways to practice the areas they identified for themselves. Then I like to finish with a (hopefully) fun review game to wrap things ups.

For our Relations and Functions assessment review, I started with an overview of the outcomes for the unit so students could identify areas where they needed additional support.

Another way we reviewed was to have students in small groups at vertical whiteboards make a list of the most important topics, formula and skills from each unit. This ends up being a study sheet. After groups complete it, they share with the class to see if any groups has something that would benefit everyone. I then collosidated all of these into one class sheet that could be shared on our Google classroom.

After this, I gave them some options to reivew and practice the areas they identified as a challenge. Sometimes this looks like the day in class we practiced domain and range.

Students that want to hear the topic being taught again have lots of options online. There are tons of YouTube channels full of teachers providing lessons on specific topics. I recently found out that a teacher from my school has his own YouTube channel with videos from our curriculum (way to go Mr. Boudreau!). There are also videos on the Nova Scotia Homework Hub, the CEMC Waterloo Courseware site, and many others.

To finish off, I like to play a class game where students work together to study and have some fun while doing it. A game like math basketball is a favourite in my class. We've also done self checking activities like Add 'Em Up or a Scavenger Hunt. Today for the last class of the semester before our assessment, we had a "math market" activity. Students can "buy" questions of different difficulty levels and topics from the market and "sell" solutions back for a profit.

Hopefully students learn lots and find class challenging. I still have lots to learn about teaching and am really trying new things out with my class this year. Let me know if you have a favourite review activity I should know about.

I must admit that I look forward to the beginning of advent and the start of a good advent calendar. All those little doors waiting to be opened. My son and I have a tradition of starting the mornings in December with a Lego advent calendar. This year we have also gotten an electronic games advent calendar from Eight Innovation that looks like it is going to be a lot of fun.

I also really enjoy exploring online mathematics advent calendar of which I've found several. Here are some of my favourites:

• Nrich Maths (both primary and secondary) - twenty-four challenging math problems to explore.
• Mathigon Puzzle Calendar - You can go back and explore previous years questions here as well.
• Nerdle Game Add-vent - A special series of Nerdles games for December.
• Transum Advent Calendar - lots of festivity here.
• NCTM December Calendar problems - Karen Campe posts older NCTM calendars of problems from the Mathematics Teachers magazine. Not strictly advent themed but lots of fun to solve.
• 2023 GCHQ Christmas Challenge - Not technically an advent calendar but some Christmas puzzle fun from GCHQ. 

Do you have a favourite advent tradition (mathematical or otherwise)? Let me know about it.

I have an object hanging from my wall at school. Many people don't even notice it or consider why it might be a strange object to see in a building with a firm foundation on dry land. It is a ship's clinometer. A relic of my past days as an officer in the Navy. For me, it serves as a metaphor, a reminder each morning as I walk into the room.

Before I became a teacher, I served for seven years as an officer in the Navy. During this time I spent many nights sleeping on board the ship while docked in port as the Command Duty Officer (CDO) while the Commanding Office was at home. A sort of "in loco parentis" responsibility to literally keep things afloat. Part of this responsibility was to report to the Commanding Officer as he arrived at the ship in the morning. One of the most visible indicators of a ship's health is it's list and trim. If it is listing to one side or the other you know that something is amiss. I knew that the CO would be carefully scrutinizing the state of the ship as he walked down the pier. If the ship was visibly listing, the CO would want to know what was going on... and it probably wouldn't be a pleasant morning conversation. 

I have this clinometer on my wall for two reasons. The first is as a reminder to consider, what is really important today? What really needs to get done to keep things on an even keel and what are the things that could wait? What are my true priorities for the day both at work and at home? The second is as a reassurance during stressful times... everything is level and its going to be okay. A bit of self-care to maintain a positive attitude. A reminder to stay positive and find you marigold (if you haven't read this article, I highly recommend you pause now and do so). I guess there is a third reason to have this on my wall... it is a bit of a conversation piece and can spark some interesting discussions.

How do you establish your priorities for the day, week or school year? How do you remember to focus your efforts on what you have influence over and let slide the things you have no control of?