I recently wrote an article about podcasts for the Global Math Department newsletter. I had more to write about podcasts, so I'm continuing my thoughts here. There are lots of really well done podcasts focused on mathematics and education and I wanted to share the best of what I've been listening to on my daily commute to and from work. I have just enough of a commute to get through about a podcast each day. I really enjoy how a good podcast make a dull commute alone in a car feel like a road trip with friends.
One new podcast that I started listening to is called Infinite Insights. Duane Habecker and Maggie Peters started this podcast to discuss articles and research regarding mathematics education and how it applies to your classroom. I really enjoyed listening to their first podcast about using math for your Day 1 routines.
Five Fabulous Podcasts Episodes
Instead of just recommending a bunch of podcasts, I wanted to hone in on specific episodes that I think you'll enjoy (assuming you're into mathematics). Check these out:
So what makes a good podcast? I used four criteria for selecting the podcasts above: Length (I find that 30-40 minutes is about right for me), Mathematical Interest (I like to learn new things about math or education), Content Density (I like a podcast to get to its point and not linger on idle chit chat), and Production Value (a good microphone and a sound booth help).
If you're interested in a longer list of math podcast recommendations, David Petro has a nice collection listed on his Ontario Math Links website.
I'd really like to hear about your favourite mathematics or education podcasts. Even better, I'd like to hear about a great episode of your favourte podcast. Please let me know what I've been missing out on.
I recently saw a photo posted on twitter by John Golden. The photo showed the question prompts "What makes a good question?" and "What makes a question good?" on a white board. John said that this was an intro activity to a discussion on questions. These questions really got me thinking and reflecting on my own practice.
There seems to be a lot of discussion about "what makes a good question." There are entire books filled with examples of rich question and engaging problems. Two of my favourites are Good Questions: Great Ways to Differentiate Mathematics Instruction by Marian Small and Good Questions for Math Teaching: Why Ask Them and What to Ask, Grades 5-8 by Lainie Schuster and Nancy L. Anderson. I've seen much less discussion about "what makes a question good." I think it's harder to define the effective teaching practices and routines for asking questions in a manner that makes them effective.
What are the characteristics of a good question? I recently read a post from Geoff Krall in which he wrote about teachers adapting questions they had found to make them even better. He said that to do this, "You start to turn from 'I like this task' to 'What do you like about it?'" I think that this is an important question to ask yourself. Teachers need to think about their students and the context in which they teach to determine what questions are going to best facilitate effective student learning in their classroom. Here are some possible characteristics of good questions that you might consider:
Jennifer Piggott in an article on NRICH wrote, "In essence, rich tasks encourage children to think creatively, work logically, communicate ideas, synthesise their results, analyse different viewpoints, look for commonalities and evaluate findings. However, what we really need are rich classrooms: communities of enquiry and collaboration, promoting communication and imagination." This really resonated with me. It is not enough to have a great problem. As teachers we need to know how to effectively present and lead the exploration of a problem in order to reap its benefits.
There are a number of strategies you might consider when exploring a question:
You Need Both to Succeed
My grandmother was a big fan of the card game pinochle and she taught me to play at an early age. It's a trick taking card game where players work in pairs to score points. I learned early that even the best hand of cards can be beaten by someone who knows how to play well. My grandmother played to win and she would often remind us to "mind your p's and q's" while playing to pay attention and play carefully. I learned a lot of math counting up tricks and keeping score. The important lesson here is that to be successful at pinochle, you have to have both a good hand of cards to meld and play those cards correctly to maximize your score. Just like pinochle, when planning a lesson, you need to consider two important and complementary components. Both finding good questions and using them effectively are equally important to the success of your problem solving lesson.
Have you ever had a student look at you like you're talking in a foreign language during a math lesson? It happens. Sometimes there are gaps in students' prior knowledge. Sometimes students need more time to process a new concept in order to construct and develop their understanding. Sometimes I haven't explained something clearly and I need to think of a new approach to tackle this topic. When I encounter those blank stares, I think, "That's Numberwang."
Numberwang is a skit from That Mitchell and Webb Look, a British sketch comedy show from 2006. If you've never seen it, take a two minutes to watch a video of this skit. The premise of the skit is that while the presenter and contestants seem to understand the rules perfectly, they are completely inscrutable to the viewer. We're left scratching our heads in confusion just like our students sometimes do in class.
So what do you do when you sense that students in your class are not getting it? I suggest that you invest some time to uncover your students' thinking. As Guildenstern implores in Tom Stoppard's play Rosencrantz and Guildenstern Are Dead, "Delve. Probe the background, establish the situation." Take the opportunity check in with students to determine their level of understanding. Here are some strategies you might use:
Reflect and Respond
Once you have a better picture of the misunderstandings and misconceptions that may be present in your class, you can plan your next steps. Was there really a misunderstanding or did you make assumptions about prior knowledge that weren't true? Were just a few students struggling or was it a commonly held misconception? Tracy Zager, in her book Becoming the Math Teacher You Wish You'd Had, writes, "If just a few students were confused, she could work with them individually. If there was a really interesting mistake, or patterns among the misunderstandings she saw, she could use those examples as her next teaching opportunity."
When I see those "Numberwang" looks I am reminded that even a well planned lesson can sometimes miss the mark. Reflecting on how a lesson went and how I can improve it helps me refine my teaching practice and be more responsive to students' needs. Don't let those "Numberwang" moments go by ignored. Matt Larson, in his August NCTM president's message, wrote "Making mistakes, getting feedback from our colleagues, and making iterative improvement are part of the natural process of continual growth. We should never forget that perseverance isn't just for students—perseverance also applies to us as professionals."
My nephew visited me this summer during a family vacation. He will soon be starting grade 11 (junior) year. He is a hard working student and would like to study mathematics in university. He understands that this is the year when he needs to start thinking about university and scholarship applications. We started chatting about his post-secondary plans one evening and he asked me two questions. The first question was, "What can I do to make sure I have competitive university and scholarship applications?" The second question was, "What sorts of careers are possible with a mathematics degree?" Here is the advice that I offered him.
Break Out from the Pack
As a high school teacher, I have written a lot of reference letters. The hardest reference letters to write are for talented students who come to class every day and work hard but have nothing to set them apart from the other 30 or so students in class. Ask yourself, "What sets me apart or makes me different and special?" Here are some things I suggested to my nephew:
Careers for Mathematians
There are some great careers for mathematicians that don't get a lot of press. Here are some careers that I find really interesting:
So what advice would you give to a grade 11 student who is interested in a career in mathematics? I'd love to know what you think.
A couple of months ago, I posted a list of six exceptional math Ignite Talks. For those unfamiliar with the format, an ignite talk includes presentation slides that automatically advance every 15 seconds. Exactly 20 of these slides result in a 5 minute talk. Since I posted my first list, Suzanne Alejandre (@SuMACzanne) at the Math Forum has been busy posting videos from older Ignite sessions. These playlists are a gold mine of mathematical thought and reflection. Additionally, the Ontario Association for Mathematics Education (OAME) had their 2017 conference ignite talks. There are so many great talks posted recently that I felt the need to recommend an additional six talks. It will take just 30 minutes to watch all six... time well spent.
I've been working on my own ignite talk and have discovered how difficult it is to craft one. There is a lot of reflection and thought involved in refining what you are really passionate about as an educator, determining how to explain it clearly and figuring out how to make it entertaining. At this point, my talk is a still just an organized collection of notes, ideas and images. I don't have any plans to actually present this ignite talk, but I feel that the process of creation and reflection is very worthwhile.
June is here and the end of the school year in Nova Scotia is rapidly drawing near. In grade 7 math classes, it is time for students to break out their geometry sets and practice geometric constructions. The outcome for this unit (7G01) states, "Students will be expected to perform geometric constructions, including: perpendicular line segments, parallel line segments, perpendicular bisectors and angle bisectors."
The intention of this outcome is for students to be able to describe and demonstrate these constructions using a straight edge and compass. In developing the understanding of these constructions, students are exposed to a variety of methods and tools such as paper folding, Mira, and rulers. I've written about geometric constructions in the past, so I won't get into a discussion of the merits of Euclidean constructions. My concern is that this unit could turn into a series of steps to memorize for a short list of basic geometric constructions that lacks coherence and context. If you really need to construct a heptadecagon using only a straight edge and compass, I'd expect you to look up the steps, not have them memorized. Additionally, in my opinion, the exercises in the student textbook are routine and dull. I think that this is a unit that has a lot of potential for student engagement but could easily become tedious.
Dan Meyer gave an inspiring ignite talk titled "Teaching the Boring Bits" at the 2013 CMC-North conference. In this talk, Dan encourages teachers to engage students by creating an intellectual need for new knowledge. Teachers should strive to provide students with a reason to want to know new mathematical skills and methods.
A grade 7 teacher that I know thought that incorporating Islamic geometric designs into this unit would give a purpose and context for doing geometric constructions. Another factor in selecting this context was that she has a number students who are recent immigrants from the Middle East in her class. We brainstormed some ideas and developed several activities to infuse into this unit that might help give this outcome some coherence and allow students to be creative and artistic. The teacher started by using a template and pattern from Eric Broug's School of Islamic Geometric Design. Students used the template and followed the instructions to create and colour their designs which were then tessellated in a grid to make a group composition on the bulletin board.
Later in the unit, students were challenged to construct eight-pointed stars using geometric constructions without a template (although a template could be used for students that needed additional supports). Creating this design using a straight edge and compass required students to perform the majority of the constructions required by this outcome. Students also had the opportunity to use their creativity to personalize their design and make it unique.
A number of students were very interested in creating designs of increasing complexity. They were able to pursue this to apply their geometric skills to create some very impressive designs. Have you used any creative or artistic activities to teach students geometric constructions? If so, I'd appreciate hearing about it!
A few colleagues and I recently initiated a bit of self directed professional development. We decided to watch a selection of math ignite talks as a focus for discussion regarding mathematics instruction and educational practices. In preparation for this, I reached out the the #MTBoS on Twitter to ask for suggestions. I was pleased to get some valuable feedback.
We took the feedback we got from our Personalized Learning Network (PLN) as well as sharing our favourites with each other to create a playlist of math ignite talks. A big thanks goes out to The Math Forum at NCTM for having such a well organized and easy to access YouTube channel containing math ignite talks from numerous conferences. What are your favourites?
Has watching these videos inspired you to create your own mathematics ignite talk? If it has, and your looking for advice, look no father than Robert Kaplinsky's web page. He has a post titled, "How I Prepare for an Ignite Talk" that will provide you will lots of tips and ideas.
Update: I've added a part 2 playlist of six additional math ignite talks here.
My son and I were looking through an issue of Fun to Learn Friends magazine recently and we ran across a game called "Spring Bingo". He was quite interested and wanted to give it a try. He is in grade primary and can subitize the pips on a six sided die as well as confidently sum the values on two dice. Just the skills we need to play this game.
We played several times. At the end of most of the games, my son got frustrated trying to roll either a 2 or a 12. He doesn't quite understand why it takes so long to roll one of these numbers. George's bingo card has both of these numbers out of the eight numbers on the card. This seems unnecessarily cruel. Ted's card has the 2 but not the 12. This made me wonder if this is a fair game or if one of the cards has a better probability of winning. Time for some math...
It appears that Card B is slightly better than Card A. I'm not sure if this would make a significant difference in the outcome of the game (i.e. rolling all the numbers on your card before your opponent does). I wonder what the average number of rolls it takes to complete each bingo card is? How much of an advantage does going first in this game give? If the player with Bingo Card A goes first does this equalize the advantage of the better Bingo Card B? These would be great questions for the Mathematics 12 research project outcome (MRPO1).
To try to answer some of these questions, I thought that writing some code would be helpful. I know a grade 8 student that completed the Introduction and Intermediate Programming with Python courses from Art of Problem Solving. I contacted him and he graciously created a very nice Python program to simulate this game for me. I modified the code a bit so that it just plays one bingo card and counts how many rolls it takes to complete the card. The average number of rolls in 100,000 games for "George's Card" was 58. The average number of rolls in 100,000 games for "Ted's Card" was 48. I was a bit surprised that there was this much difference. Playing with Ted's bingo card appears to be an advantage.
Next I modified the code again so that it plays the game with George going first to count how often George wins. In 100,000 games, when George went first, he won 50703 of the games. I again modified the code so that Ted plays first. In 100,000 games, when Ted went first, he won 50806 of the games. It seems that going first is an even greater advantage than having the better bingo card.
What I really like about this game is that there are mathematical outcomes that can be addressed with this activity across a wide range of grade levels. At younger ages, students are practicing subitizing and adding numbers. One variation of this game is to play it solo. This might be a nice option for a math station. I found several examples of "Roll and Cover" games where students have a sheet of paper filled with numbers (from 1 to 6 or from 2 to 12) and students roll the die or dice and cover the number (with a token or a bingo dauber) when they roll it. Just do a Google search for "roll and cover math game" and you'll find lots of examples posted online.
For students looking for an opportunity for enrichment, they can make variations of this game. They could also write computer code to simulate this game (using Scratch or Python or some other language). They could also do some statistical analysis of the game to see how fair it is. There are so many options with this simple game.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 1 N02 - Students will be expected to recognize, at a glance, and name the quantity represented by familiar arrangements of 1 to 10 objects or dots.
Mathematics 1 N09 - Students will be expected to demonstrate an understanding of the addition of two single-digit numbers and the corresponding subtraction, concretely, pictorially, and symbolically in join, separate, equalize/compare, and part-part-whole situations.
Mathematics 2 N10 - Students will be expected to apply mental mathematics strategies to quickly recall basic addition facts to 18 and determine related subtraction facts.
Mathematics 5 SP04 - Students will be expected to compare the likelihood of two possible outcomes occurring, using words such as less likely, equally likely, or more likely.
Mathematics 6 SP04 - Students will be expected to demonstrate an understanding of probability by: identifying all possible outcomes of a probability experiment; differentiating between experimental and theoretical probability; determining the theoretical probability of outcomes in a probability experiment; determining the experimental probability of outcomes in a probability experiment; comparing experimental results with the theoretical probability for an experiment.
Mathematics 7 SP06 - Students will be expected to conduct a probability experiment to compare the theoretical probability (determined using a tree diagram, table, or other graphic organizer) and experimental probability of two independent events.
Mathematics 8 SP02 - Students will be expected to solve problems involving the probability of independent events.
Mathematics 10 Essentials G1 - Express probabilities of simple events as the number of favourable outcomes divided by the total number of outcomes
Mathematics 12 P03 - Solve problems that involve the probability of two events.
Mathematics 12 MRP01 - Research and give a presentation on topic that involves the application of mathematics.
I know many teachers who say they wish they could go to a math conference but they are not sure of the logistics of applying for funding, applying for leave, getting sub coverage, etc… I have gone to many conferences over the past 10 years and always return feeling rejuvenated and excited to do my job!
I have attended the NCTM Annual Conference although I may try a regional conference in the future. The following is based on my own experience; please review your school board and Article 60 guidelines before making plans.
Things to consider before booking your conference:
So now you have decided the conference, the city and travelling companions, here are some of the logistics of applying for funding:
Once you have gotten approval for leave and have confirmation on how much you will be reimbursed in expenses, it’s time to start planning your trip! I often scout out hotels and flights even before I have completed the steps above. My preference is to stay at one of the conference affiliated hotels. There are complimentary shuttle busses that run from the hotels to the conference centres all day. This is a big bonus in my eyes. Sometimes, I have used the shuttle busses just to get to another part of the city! You can find all the information about accommodations on the NCTM website. The best hotels (location or price) usually book up early. If I don’t have all the proper approval from my principal yet, I still reserve my hotel room. I can always cancel the reservation if plans fall through (unlike conference registration and plane tickets. Wait to book those!).
Take note of the early bird registration dates for the conference. It will save you about $50 if you register before a certain date. If you know of a bunch of people who are attending the same conference, investigate group rates. Also, I usually scour the internet for NCTM conference registration coupon codes. I once saved myself fifty dollars by using a random code I found online.
So now that you have gotten approval for leave and funding, registered for conference, reserved your room and booked your plane ticket, you need to plan your time at the conference. With thousands of sessions to choose from, selection can feel like a nightmare (similar to those many of us get in the days before a new school year begins). For this reason, you want to choose wisely. Like the first days of school, this planning you do in advance will certainly impact the days ahead.
More on strategies to navigate the conference booklet and choosing your sessions coming soon!
For anyone interested in data visualization, the r/dataisbeautiful subreddit is an amazing place to explore (you can also follow @DataIsBeautiful on Twitter). This subreddit has an active community of posters who create and discuss a wide variety of data visualizations. Shortly after the first successful landing by SpaceX of a reused Falcon 9 rocket, an incredible feat of engineering, the visualization below was created and shared by reddit user Brenden2016.
Even more interesting than the visualization was the discussion that followed. Over 200 comments (as I write this) have been posted regarding this visualization and suggestions of how to improve its powerful story of the increased efficiency and success of SpaceX Falcon 9 landings. Several issues emerged:
Additionally, two users created and posted alternate versions of the graph. One user, nicocote, created a double bar graph using additional launch data that didn't appear in the original. Another user, azura26, created a step graph instead of a line graph.
The full list of Falcon 9 landings can be found on the list of launches on Wikipedia. I took this data, summarized it and put it into a Google Sheets document to share (https://goo.gl/Ihr2Io). Please feel free to use it to generate your own data visualization or use it with students.
In the Classroom
This set of data appears to be a nice one to use in a classroom. It is a fairly small data set, is current and from the real world and (for me at least) is an engaging and interesting story. In a grade 8 classroom working on data presentation, I might start by showing the students some dramatic video of Falcon 9 boosters both crashing and landing successfully. Fiery explosions always make an impression! Then we could look at the data showing how SpaceX has done an impressive job of learning from their mistakes (and perhaps how this relates to a growth mindset in the mathematics classroom). We could then look at the data visualization from above and brainstorm, as a class, what features of the graph are positive and which could be improved. We could also answer the question, "what is the story this graph is trying to tell?" We could then break into groups, review that data and see if the groups could come up with a different way of telling that story using the data. Perhaps a different type of visualization or even an infographic could be created.
What would you do with this data in your classroom? Please let me know.
Nova Scotia Mathematics Curriculum Outcomes
Grade 8 SP01 - Students will be expected to critique ways in which data is presented.
Mathematics Essentials 11 F2 - Select an effective data display for a given set of data and explain the reasons for the choice.
Mathematics at Work 11 S01 - Students will be expected to solve problems that involve creating and interpreting graphs, including bar graphs, histograms, line graphs, and circle graphs.