Connecting Polynomial Multiplication and Numerical Bases through the Fundamental Counting Principle

I’ve been tutoring a Grade 9 student recently. I love seeing the worksheets and questions her teacher (who I don’t know) assigns. It’s deeply educational to see how another teacher approaches topics and to see how that approach is filtered through the understanding of a student. I feel like I’m learning as much as her each time we meet for a tutoring session.

Two weeks ago, she was learning how to multiply polynomials. Unsurprisingly, she knew an acronym, FOIL, but didn’t know why she was using it, and only had a basic understanding of how to distribute terms. I don’t like to teach FOIL, although I don’t mind the strategy of pairing and multiplying each term. When I teach distribution, I use the following method:

This technique builds on what students already know (distributing a monomial over a binomial)

This technique builds on what students already know (distributing a monomial over a binomial). Once they become fluent with this technique, shortening the approach (and skipping that first step) is great.

Intriguingly, the teacher asked the students to practice multiplying both “horizontally” and “vertically.” The same problem, solved vertically, looks like this (I’ve written out the steps in an exaggerated way, so you can see the progress. When solving, you wouldn’t need to write the question more than once):

I've circled the terms I've multiplied and connected them with their product. Step five is simply summing the different terms.

I’ve circled the terms I’ve multiplied and connected them with their product. Step five is simply summing the different terms. Conveniently, I’ve organized terms in columns according to powers of x, which makes this step particularly simple.

This technique works pretty well, and isn’t fundamentally different from the more common horizontal work. Interesting things happen when you do multiply multiple polynomials though. Using the horizontal technique, we get something like this:

Obviously, once you're proficient with this approach, you can consolidate a few steps.

Obviously, once you’re proficient with this approach, you can consolidate a few steps. That’s what happened with the messy 4 in the third step – my mind was fighting with me, combining terms automatically while I was trying to clearly write one step per line.

In contrast, doing it vertically takes significantly less space:


If you can’t see what I did there, I’ve made you a gif:


Does that pattern look familiar to you at all? Think on it before you read more. What if I turn it sideways?


Ringing any bells yet? Lets change the symbols from Xs and 2s to 1s and 0s:


Anything? One more gif to round out your thinking:


Yessiree Bob. The pattern that you follow when multiplying multiple binomials is the same pattern that you use to count in binary. For higher order polynomials, you’re in a higher base (Trinomials count in Base-3, four-term polynomials in Base-4, etc.). Similarly, the more polynomials you multiply, the more digits your different-base number has ( 3 polynomials yields a three digit number, 4 polynomials yields a four digit number, etc.).

The reason for this connection is the Fundamental Counting Principle (If you have A ways of doing one thing, and B ways of doing another, there are A*B ways to do both things). Distribution requires each term to multiply by all the terms in the other polynomial.  It’s the same combinatoric principle as counting: each position needs to be filled with all possible digits for that base, and unique arrangements of those digits signify a unique number.

I found this realization worth sharing for three reasons:

  1. It’s neat. That’s a pretty good reason to share it.
  2. Vertical multiplication, once I started playing around with it, is way easier for multiplying multiple, multi-termed polynomials. It saves space, effort, and reduces mistakes.
  3. I fight against the siloing of math concepts in education. Seeing deep connections between different topics is pedagogically interesting because drawing students’ attention to those connections can help with both understanding and procedural fluency. It reduces the amount of memorization required by showing how mathematical principles can be applied in a various situations.

Any thoughts here? Further connections that I’ve missed, or mistakes in my work? Lemme know below, yo.

Online Collaboration and Helping Others (as Best I Can)

At the start of ECMP 355 (in the syllabus) Katia asked us to keep track of all of our online interactions. She was kind enough to bold the text. I don’t have any stats, but I suspect that lots of undergrads don’t actually read the syllabus – I’m not one of them! Reading syllabi and filling out my agenda with a semester’s dates is a highlight of my year. However, I ignored her request, so here I was today trying to compile all my online interactions of the last four months.

To all future ECMPers:

Don’t fall into this trap! I recommend a nightly update of your log, with the day’s activities.

Anyway, after two hours, my left hand’s pinky and and index fingers are tired (Can you guess why? Answer at the bottom) but I’ve captured most (not all) of my online interactions. Brief summary:

Google+ CommunitySadly, I had the fewest interactions here, and they were the easiest to locate. Just searched for my name. Thanks Google!

Blogging: This is where I likely lost the greatest percentage of interactions, because it limits how far back you can look at the comments you’ve made. Very poor feature WordPress.

Twitter: It’s easy to find interactions here, but I tweet a lot (imho). So this definitely took the most time. Yer arright Twitter.

If you want to read my log, you can do so here. If you want to read a summary with exemplars, read below.

How Have I Contributed To The Learning of Others?

First and foremost, I feel awkward writing this. It’s like a cover letter – you’re supposed to brag yourself up and that’s just painful. However, it’s even more painful for you to read about how painful the writing is for me. So, I have a compromise! I won’t mention it again, but every time I feel self conscious about what I’m writing, I’ll throw in a little asterisk (*) so you know how I’m feeling. Deal? Kk.

I’ll sort my contributions by medium, but there is overlap.


Blogging, and commenting on blogs, is a useful medium. I personally enjoy it because 1) I can express my thoughts fully and 2) because there’s less pressure time-wise. Unlike Twitter, if it takes me a few days to mull over a post before I respond, that’s perfectly okay. I prefer deep, reflective thinking, so this is great.

Rheanne blogged about ability grouping of students, a contentious issue, and I’ve got opinions! But they’re not easy to express. Here’s the opening to my response:

Read the rest here

Read the rest here

I’m setting up a fairly complex argument, largely drawn (in structure, though not message) from this Keith Devlin blog I’d read a fortnight earlier. That depth of thought*, that sort of connection between materials, is fairly natural in blogs as spaces. In this case, nobody has yet responded directly to my thoughts, but I don’t consider that necessary. When people respond, it’s validating, but I hope that people have read my thoughts and reflected on them – I don’t need their validation, although their input is much desired. Tori captures this in a post lower in the comments when she says, “I’m not sure that I have an answer to the questions that have been posed in this discussion. However, I love that they are being asked.”

A lot of discussion in this class has focused on Digital Identity, specifically looking at exposure to sexual material for youth. Ryan posted a lovely blog about the documentary Sext Up Kids, and the discussion in his comment section was exhilarating. I attempted to take that discussion away from the “shocking” nature of the documentary, and away from the way things were “in the good old days” and transition it towards a solution-based discussion. I did this by bringing in Sex-Ed, and what sort of progressive Sex-Ed is both necessary, and acceptable, for students living in a digital environment, at various ages. Ryan thanked me for my thoughts* and built on them, asking about the ways to incorporate this sort of discussion into pre-service teacher training.

Cole made a blog post addressing strategies and supports for students on the autism spectrum, and he provided a link to some lovely resources. This semester, in my Ed Psych class on Exceptionalities in students in Canada, I learned about person first language, referring to, for example, a student with autism, rather than an autistic student. I was happy to share that idea with Cole, and spread what learning I could.

The last blog I’m going to mention belongs to Bryan Penfound, an instructor at the University of Winnipeg (in Math and Math Ed). He’s going to come up a few more times in this post because we’ve had several in-depth exchanges about math pedagogy this semester. In this post, he was writing about some math work his students had done, and how he analyzed and detected errors. We had some back and forth in the comments, and I suggested he partner with a local math teacher so his pre-service teachers could try analyzing real student work. He seemed to like the idea:


No favourites. I couldn’t do it. Too self serving*****

Google+ Community

I didn’t find Google+ particularly useful, so I didn’t use it often. However, two attempts I made to help peers in that forum were:

1) Tori has become interested in Social and Emotional Learning this semester, so when she asked the community about emotional literacy, I responded as best I could.

2) Cameron, early in the semester, posted about the rejection of Learning Styles by recent research. Since my twitter feed had been exploding with overly-vitriolic comments about just that, I shared my thoughts, from what I’d seen. Katia brought in a neat article distinguishing learning styles from multiple intelligences (which also make people angry). I definitely felt some mutual growth through these exchanges.

3) (Bonus!) Curtis had been torturing Katia over her fear of puppets. I waded into the discussion, clarifying that, on a scale of terror, Marionettes > Puppets.

Haunting Eyes. From Czech Marionettes

Haunting Eyes. From Czech Marionettes


I went through my log and selectively chose what I felt were the best tweets where I helped people. I now have 24 tweets queued in tabs. It’s too much to put here, but there’s a lot of stuff I want to share. I’m going be incredibly brief on each tweet that I share, and maybe an extra sentence or two on the absolutely essential ones.




Justice Sinclair (now Senator Sinclair) retweeted and commented on my blog about his lecture. The next day, he tweeted it again. Spectacular and strange feeling.







During an EMTH class we were discussing useful software for drafting math assignments, and didn’t know any intuitive options. I sent a tweet to Desmos, and the CEO got back to us within 10 minutes. Our whole class benefited from that.









I continued a discussion with Michael Pershan, around an article about math anxiety, with Bryan Penfound.










I offered a counterpoint article to discuss growth mindset with Tori, Rheanne, Brea, and Lydia.









I set up a date with Leanne to help her through a more positive coding experience.










I prepared a screencastify on how to make sub-menus in WordPress for Brea and Tori.







I encouraged Regina to participate in Earth Hour. Brea told me she saw my tweet and shouted, “We need to shut off the lights, now!!!” and rushed around to get it done in time.







I was pulled into an (admittedly very productive) #SaskEdChat by Tori, discussing typing skills for students.






Responses and Interactions

I want to showcase three specific interactions that I felt had an impact, or in which I felt significant growth and learning:

First, I tweeted a photo of a mail-out from the Sask Party which led to a back and forth between Alec and I on bias in graphing, and the way teachers can tie local events (like a provincial election) into the curriculum. I had too many thoughts to share, so I had to turn the conversation into this blog post, which then got comments on it, further refining my thinking. The discussion and learning evolved naturally, and mutated through different spaces as required, but it all started from me taking a photo of an ad that came in my mail, and that felt very authentic.

Secondly, I had a lot of issues with this course, especially in the early months, surrounding the normalizing forces that happen in online spaces. I blogged and tweeted about it, and got great feedback and thoughts from Alec. It was nice to have a reasonable approach to what could be considered public criticism. I was nervous going into the blog post, but the results were excellent. Alec didn’t necessarily agree with me, but he was human and polite in responding to real concerns.

Finally, I sent this tweet to Alec and Katia requesting class be cancelled so we could attend Justice Sinclair’s lecture:


I got a lot of support from peers, and eventually, Alec and Katia agreed and we went to the lecture. I certainly wasn’t solely responsible for this happening, and a lot of credit should go to the instructors for agreeing to the change of plans, but I feel like I played a role in making this happen* and I would consider that lecture a pivotal point in the course, and certainly memorable for a lot of students. There were a lot of magnificent posts and reflections that came out of the experience.


I feel that throughout this course, I’ve been an active participant in online spaces, and I’ve done my best to further the learning of my peers. One advantage to my delayed approach to my log is that I got the opportunity (was forced) to reread all of my interactions for this course. And so, tonight, I replied to a tweet I sent in the first week. Brea had asked for some educators to follow and I recommended several. Tonight, after my experiences of the semester, I sent along several other, new recommendations. As I’ve written this (epic of a) blog post, every one of them has liked that tweet, and I got the following very kind words back from Bryan Penfound:


And that’s about as pleasant a conclusion as I can manage.

Answer: Control-C and Control-V, over and over again, with links to my interactions

Sext Up Kids, Shock Therapy, and Solving an Epidemic of the Soul

In my creative writing class, with the lovely Melanie Schnell (review of her novel here!), we’ve discussed “character as desire.” The big idea: a well written character has a deep, driving desire. This influences every action, every thought. However, the desire may be abstract. For instance, they may desire “self-respect,” but you won’t see them walking around saying that. Instead you’ll see them trying to reach short term goals, like winning a football game, or learning a new skill, or repairing a damaged relationship. How they express their desire will change with the situation, but they’re always driven by that deep goal.

Speaking of desire, a timely poem excerpt.

Speaking of desire, a timely poetic excerpt.

This concept of “character as desire” has helped me to resolve a conflict I’ve been struggling with for almost a decade.

My Conflict

My opinion since I started teaching youth (circa 2006):

  • “Wherever I go, kids are all unique individuals, but for the most part, they’re more similar than they are different. I really doubt they’re that different than kids have always been.”

Every demagogue/ letter-to-the-editor writer ever:

  • “Kids these days! What punks! What rapscallions!”
  • “Back in my day, kids respected their elders!”
  • “Generation X (or Y, or Z, or Millennials, or Hipsters,  or whatever) are the worst generation yet! They’re completely morally corrupt! They can’t even sign their name in cursive writing!”

To the authors of these ideas:

  1. Who raised the generation you despise? Was it you? Should you take some blame here?
  2. If you shouted less (the exclamation marks above are not exaggerations) I’d be more likely to respect, rather than dismiss, your opinion.

Some examples of the corruption of youth throughout the ages:

The Time cover is from 2013. Quotes from Mental Floss.

The Time cover is from 2013. Quotes from Mental Floss.


So who’s right? Have we as a species been devolving, generation after generation, for millennia? Are the most recent generations truly the worst yet? Or are kids still kids as they’ve always been?

My latest theory is that these view points are all true, but at different levels:

  • The deep desires, the underlying aspects of being a child, haven’t changed. Youth still want:
    • Support
    • Love
    • Independence
    • To test boundaries
    • To be with their friends
    • To have social acceptance
    • To make a difference
  • Youth meet these goals in new ways as our society changes and as their situations change.

It comes back to “character as desire.” The character of youth hasn’t changed – the expression of their desires does change with time. To preceding generations, these new expressions may appear morally decrepit.

Sext Up Kids

I chose to watch this documentary for class (one of two options). While watching it, I came to the above realization. You wanted proof of cross-curricular thinking being beneficial? This is it.

I assume everyone prefers logical proofs to be expressed symbolically. That's not just me is it?

I assume everyone prefers logical proofs to be expressed symbolically. That’s not just me is it?

Youth still have the same underlying, deep desires. The way they try to meet those desires has changed. As the documentary points out, access to pornography is up, sexualization of youth is a reality, and there are disturbing trends towards sexual violence in youth. So what’s our solution?

Shock Therapy

Adults, including in Sext Up Kids, focus in on the elements of youth culture that they find most disturbing, most shocking. For these elements, they attack youth (with the best intentions!) decrying the generation. They ask things like:

  • “How could you even think to send a photo like that!?”
  • “Why would say that to a girl?”
  • “Do you have no respect? Did I teach you nothing?!”

I call this shock therapy. It has nothing to do with electricity. Adults think that by focusing in on the elements that they find most shocking, they will be able to change the behaviour of youth. As one teacher in the documentary lamented, kids are desensitized to sexual imagery – they aren’t shocked by it at all (unlike, presumably, he is).

Why Shock Therapy Fails

When adults focus on the “shocking” elements of culture, they are thinking about the way their own generations expressed desire, at a surface level. Recall those deep desires of youth, the ones that haven’t changed in generations. Those still need to be addressed in today’s youth. The current generation doesn’t respond in the way their parents did because they live in a different time, and so their desires manifest in unique ways. We can’t tell stories from our own youth and expect children to relate to them perfectly – their world is different. Our stories are disconnected from their reality.

An Epidemic of the Soul

The discussion around the sexualization of youth culture is deeply tied to generational moral perspectives. Many adults can’t accept the level of youth porn use, or sexting, or sexual desire because they view it as morally corrupt. This is an ineffective way to communicate and influence youth behaviour. We can’t present moral arguments, because morals change.

This is not to say that the sexualization of youth isn’t a problem – it is. Issues are developing, including sexual violence, child exploitation, and poor social development. However, to effectively address these challenges, we need to change our approach. Instead of attacking sexualization as morally unacceptable, we need to remove stigma, and open ourselves to meaningful conversations about sex and sexuality with youth. We need to address their deep desires, the ones with which youth have always struggled. We need to change our language, away from shock and revulsion, and towards acceptance and support. Overall, we need to remember: our goal is not to make youth into clones of ourselves. Our goal is to help them grow and thrive in the world we’ve left them.

Curricula as Permissive Documents?


“What purpose should schools serve?”

This question is the dark matter of debates in education: it’s rarely detected, yet it makes up five sixths of the conversation. The debates themselves take many forms: traditional vs progressive in math ed; inclusion vs separated special ed; strict academics vs teaching for social justice. However, underneath, they are all asking some version of this question: what purpose do we want our schools to serve? What are the priorities of education?

I was discussing graphing bias and provincial elections with Alec on Twitter and, as inevitable as gravity, the purpose of education drew close to the surface:

See the full conversation here

See the full conversation here

I always ask, when people say things like this, “well then, why don’t we change the curriculum to include these important things?” It’s a naïve question – there are so many reasons. Political will, disagreement on priorities, cost, etc. However, this does raise fundamental questions about the nature of curricula.

In the Canadian Forces, we have the concept of a permissive document – a document that gives permission, and without that permission, actions are verboten. A great example are the military dress regulations – if an article is not listed as permissible for wear, you cannot wear it, no questions asked.

Are curricula permissive? In my experience, most teachers don’t think so. We have no compunction about doing things outside of the curriculum, if we believe them be beneficial to our students. In that sense, we treat curricula as baseline, and we teach beyond them at our own discretion.

Note: some teachers complain that curricula are so full, they have no time to do anything extra. In most cases, if these are math teachers, they’re conflating curriculum with a textbook.

This is not curriculum - it's a resource. Adjust your complaints appropriately. (Image from Pearson)

This is not curriculum – it’s a resource. Adjust your complaints appropriately. (Image from Pearson, w/o their permission)

I believe in centralized organization as a genuinely efficient system, if implemented well. I have an issue with teachers presenting whatever additional education they choose, without guidance, despite my faith in their professionalism. So here’s my proposal for redesigning how we think about curricula:

  1. We release baseline, academic curricula. These are very similar to current curriculum documents, in which teachers are mandated to cover certain academic topics, but are still given flexibility in how they approach it.
  2. We also release guidance curricula for things like behaviours, social justice, citizenship, leadership – whatever social norms we as a society decide to promote. These documents require teachers to promote positive social norms, but are much more flexible about approach and how much material needs to be covered.

The reality is, teachers already do the second part. Denying that is futile. Many, many teachers do a great job, and it’s important that centralized bodies don’t decrease their efficiency. However, providing guidance and support for teachers who struggle with this part of their job (you could call it the dark matter of the teaching profession) is the role and purposed of centralized agencies. So why don’t we own up to our own realities and support our people in meeting these goals?

Did I make you angry? I would love to read your critiques below.

Listening, Not Speaking

On Wednesday last, my ECMP class was cancelled for the express purpose of attending Justice Murray Sinclair‘s lecture on the legacy of Residential Schools (recording here). That idea is outside the scope of your average university professor – what could be more important than their lecture? –  and it’s one of the reasons I’m taking my Education degree at UR: as a faculty, it cares about reconciliation.

When someone says it better than you, don't fret. Just quote them.

When someone says it better than you, don’t fret. Just quote them screenshot their tweet.

The Justice was an excellent speaker. He was warm, funny, clear, concise, and meaningful – all the things I hoped for. However, while I went to the lecture to hear him speak, I left thinking about how he listened. As a justice and as the head of the TRC he has spent a great deal of his professional life listening to people, and does a spectacular job.

I couldn't help thinking about this comic at the end of the night, from The New Yorker.

I couldn’t help thinking about this comic at the end of the night, from The New Yorker.

Justice Sinclair spoke for less than an hour. He was preceded by three layers of introductions, and followed by question-speeches from the audience. Throughout the evening, he was patient, conscientious, attentive, and never once interrupted – no matter how meandering a question. He had an air of empathy, he made each speaker feel welcome, and he ensured each knew how much their individual stories mattered.

So often, when we try to advocate, we speak for a person; what we need to do is get out of the way and let them speak for themselves. Justice Sinclair is an exemplar of allowing others to speak, and letting their voices be heard. One day, I hope to be able to listen as carefully, and with as much empathy, to my students.

A Reading Week Smorgasbord

A smorgasbord from Photo Credit: Average Jane via Compfight cc Do you want to eat everything here? Of course you do. Should you? Probably not.

A delicious smorgasbord from Average Jane via Compfight cc.
Do you want to eat everything here? Of course you do. Should you eat everything here? Probably not. I guess.

It was reading week, so I read a lot. Not textbooks, or things for school. Just musings from around the internet. I’ve compiled them all here. However. It’s too much to eat all at once. Don’t try. Treat it like the smorgasbord above – pick and choose what looks tasty and move on. No regrets.

To help you out, I’ve organized a bit:

Appetizer: Education Articles

Main Course: Math Education Articles

Dessert: Miscellaneous Articles (spoiler alert: we’ve got a mathematical proof of how outstanding you are, an exceptionally voiced post on the realities of living in our patriarchy as a woman, and a l’il Ted Talk explaining how too many choices (like this smorgasbord) is detrimental to your life. I thought we’d end on a depressing, meta-cognitive note 😀  )

Education Articles


Cognitive Load Theory is Wrong??

I’ll pick up where my last article summation blog left off, with CLT. Another blog from Bryan Penfound, who frequents the topic. He’s offering a summary of a new article suggesting a re-conceptualization of CLT. What’s beautiful about CLT is how unattached it is from pedagogy. Most pedagogical arguments are between people so fixed in their viewpoints, they won’t even consider alternatives. Some of these people use CLT to defend their perspectives. However, the actual researchers in CLT are not so fixed, and as empirical evidence changes, they change their theory to adapt. It’s wonderful seeing science instead of ignorance at work. This blog shows how CLT can still be accepted, but a strict, traditional instruction style is not essential at all times.

Learning is Liminal

This article does a lovely job exploring why education (for learners) is a scary, unknown world, and why students seem to forget things between classes. It also gives a strong image of an inexorable tide of knowledge rising, that can be very comforting for a teacher. I do want to problematize that image a little bit. If it’s accurate, the students have very little control – they’re going to be overtaken by the waves, by the tide. So we’d best be certain we’re teaching the right things – because the ocean is dangerous. We can drown our students by teaching the wrong things.

Howard Gardner: ‘Multiple intelligences’ are not ‘learning styles’

There has been a lot of anger flying around Twitter this week, decrying the villainy of Learning Styles. Allow me to summarize (instead of linking, because I believe our lives are better off without reading vitriolic attacks): Learning styles don’t exist, they’ve been disproven by science, please stop teaching them in school/teacher education programs. It’s true, Learning Styles don’t exist. However, a lot of people are taking that to mean: teach everyone without differentiation, and the best way to teach a topic is inherent to the nature of the knowledge. I was in a discussion with Cameron Mohan about this when Katia Hildebrandt directed us to this article. It’s written by the developer of Multiple Intelligences Theory, and distinguished MI from Learning Styles. He offers a nuanced, calm discussion based on empirical evidence, rather than just attacking Learning Styles as evil.

Term papers? These science students write Wikipedia pages instead.

The title says it all. Undergrad students are writing Wikipedia articles under the supervision of their prof, rather than term papers. ECMP 355/455 talk a lot about building an audience for student work. What could be a more authentic use of research than posting on Wikipedia (also, – building Digital Citizenship anyone?) I want to do this with high school students. I think it would be spectacular. There are several ways to take it. I think a meaningful route would be conducting local research. Pairing with local historical societies and the like to create and edit pages for smaller, localized content.

Off-Task Communication is Spot-on for Collaborative Learning

Education has often put a lot of focus on “on-task activity.” It’s the measure of classroom management and productivity. It’s how you tell a good teacher from a poor teacher. In the States, it can help determine a teacher’s salary. And, just as everyone who has ever worked in a group knows, the conversations you have off-task are essential to your collaboration. It builds trust, respect, and a desire to collaborate. It helps children develop socially. This article summarizes these thoughts because a recent study has given evidence to support what we all already knew – you work better, especially long term, if your team has strong dynamics, and those are built through off-task communication.

Math Education Articles


London maths teacher on shortlist for million dollar teaching prize

Like flipped classrooms? Love Khan Academy? Want to earn 630k quid? Colin Hegarty has got you beat. London math teacher has made 1000+ math instructional videos and put them online. I haven’t investigated too deeply, but I believe they’re open to the public, so you can use them in your classroom if you choose. For North Americans, pro-tip: math=maths.

Ed Begle’s First And Second Laws Of Mathematics Education

These are so good, I’ve gotta repost them:

  1. The validity of an idea about mathematics education and the plausibility of that idea are uncorrelated.
  2. Mathematics education is much more complicated than you expected even though you expected it to be more complicated than you expected.

I’ll just leave them there. Genius.

Visual Patterns – Who Needs Them?

This post by Michael Pershan is a bit too dismissive of visual patterns imho. I’m all for providing multiple points of access into a math problem – and a visual, relational pattern is one point of access from which a lot of students can enter the math. So, are recognizing visual patterns a goal? Yes! Are they the end goal? No. But that doesn’t devalue them. Every actor on a stage matters, even small roles with few lines; visual patterns are part of pattern recognition in general, and they have intrinsic value.

Miscellaneous Articles


Theorem: You Are Exceptional

Keith Devlin has opinions – he calls them his Angles, because, you know, he’s a mathematician. In this post, he makes an argument about human exceptionality based on the ratio of perimeter hyper-volume to total hyper-volume in a 200 dimensional hyper-cube. Sounds thrilling right? It is to me, and if it doesn’t sound so to you, try it out anyway. He explains his logic very clearly, even to a mathematically lay person. As for its accuracy? I think it’s neat. I’m not sure it’s mathematically sound (I question the assumption of independence, and thus orthogonality, of the 200 human performance characteristics.)

To Men I Love, About Men Who Scare Me

Read it. If you’re a woman, it won’t be a surprise, but you’ll appreciate the clarity of expression. If you’re a man, don’t put up your defences. Don’t try to make excuses. Read it, and try to accept it. Try to get inside it, and let it get inside you. Try to understand the message that’s here. It’s very important.

The Paradox of Choice

A TedTalk to finish up this smorgasbord. He suggests that Choice and happiness follow a parabolic structure, like this:

Doesn't this graphical representational help it make more sense? I think so. Doesn't mean it's my Learning Style though.

Doesn’t this graphical representational help it make more sense? I think so. Doesn’t mean it’s my Learning Style though.

I can connect with this. I’m happier when I leave my phone at home, and go do things without it. I have fewer choices, and I enjoy the ones I make more. I’m going to see if I can adjust my lifestyle in some small but meaningfully ways according to this principle.

Two Weeks of Thinking – 1 Feb 16

Over the last two weeks, I’ve been reading a lot of articles on education (especially math ed) and I keep coming back to Cognitive Load Theory (CLT). I had heard of it many times through my PLN; Dan Meyer seems to mention it especially frequently. But I hadn’t understood it until recently.*

Here’s an entirely unsatisfactory description of CLT that I wrote for a non-education friend (who happens to sail):

We have working memory and long term memory. Our working memory is only so big – it can’t hold very much at once. So  our brains evolved to take several basic facts and combine them into one idea. That idea is called a schema. Here’s an example: if, when sailing,  you want to pass bow through the wind, as the skipper, you have to stand, duck, switch the tiller, switch hands, straighten, etc. So many things! So we learn them, and combine them all into one schema (we call it a tack). Then, your working memory can draw up the idea of a tack, and still have room to think about other things (like tactics, or roll tacks, or double tacks, or not dropping your crew overboard). So that’s how brains work. Cognitive Load Theory looks specifically at how knowing this should change instructional methods.

As I said, not satisfactory. If you want a quick summary, go read this article:

Fred Paas, Alexander Renkl & John Sweller (2003) Cognitive Load Theory and Instructional Design: Recent Developments, Educational Psychologist, 38:1, 1-4, DOI: 10.1207/S15326985EP3801_1

Available from the UR library, for those with access. I find the writing extremely concise and efficient. A pleasure of an article actually.


Other Readings (on CLT):

Germane Load: The Right Kind of Mental Effort?

I read this article before the one I mention above. It also summarizes CLT, and does a fair job. Reading it placed me in a position where I felt able to do some real research and find peer reviewed articles. A confidence builder of sorts (Also, pretty good for those who want a summary but don’t have access to online journals. By the way: have you tried the public library near you?)

Reflections from Teaching: When are Learners Novices No More?

This article extended my thinking on CLT. Novice learners and expert learners need to be taught it different ways. Many traditional math instructors (who favour direct instruction, worked examples, etc.) believe their students to be novices. According to CLT, their pedagogical approaches are appropriate and more open ended approaches (inquiry, opened ended tasks, self-discovery) are inappropriate for novice learners, because of the extraneous load it puts on working memory. This blog posts very carefully critiques our concepts of our students – what makes them novice, and when are they novice no more?

Novices and Experts Cannot Think in the Same Ways

More thoughts on the novice vs expert debate in CLT, with extra focus on differentiated instruction. Makes compelling points about scaffolding learning so that students progress towards expert status.


Further Readings (Not on CLT):

On Praise: Carol Dweck and Beyond

This carries on my uncomfortable learnings from two weeks ago, when I posted about the flaws in Growth Mindset pedagogy. It questions whether praise is always a positive force on learning.

It reminds me of cycling laws (bicycle, not motorcycle). First, I think every person on a bicycle should wear a helmet. Second, I am completely against laws that make wearing a helmet mandatory. This doesn’t come out of any kind of libertarian drive; rather, studies have shown that mandatory helmet laws make it more dangerous to cycle in places that have adopted them. There are many potential causes, but the most likely is the sharp decrease in biking population as a result of mandatory helmet laws. With fewer cyclists on the roads, drivers lose awareness, get less exposure; cities put in less infrastructure, do a worse job of maintaining it. The unintentional consequences of bicycle legislation are mirrored in this article – the unintentional outcomes of giving praise.

How ‘Progressive’ Education Patronizes the Poor

Speaking of unintentional outcomes…

This article is controversial, no doubt about it. It makes the argument that treating children holistically (with too much giving of responsibility and control in the classroom, and not enough focus on drilling educational knowledge) is ruining them, and the world. Not to put it too lightly. Another step in my attempt to read and understand other perspectives in education that seem anathema to my beliefs.

Stop Talking About Teachers As If They’re Missionaries

This article is also about the importance of teachers as educators, rather than as a balm to all societal needs. However, it’s not quite so drenched in vehemence as the last one. I pull this quote, because it stood out to me:

But the biggest transformation will be cultural. As more people see teaching as prestigious, other magical changes follow: Parents begin to trust teachers a little more. Taxpayers start to believe their money is well spent. Politicians step aside so teachers can shape what’s taught and how. Most important, kids notice, too. When they hear stories about how hard it is to become a teacher and see the respect with which teachers are treated, students start to infer that school isn’t a joke after all—that when adults say education is important, they might actually mean it.

My biggest fear reading this article is confirmation bias. I’m worried that I like the article, and it seems great, only because I already believe what it’s saying. When I read things like this, and then feel this way, it only motivates me more to try to gain additional perspectives, especially those I don’t like.


*I don’t understand it yet. It’s complex! But I now know the terms, some of the perspectives, and I have the ability to understand and engage in discussions on it. Meta-question! Am I still a novice with respect to CLT?

Pleased to make your acquaintance, Blog-Friend


How are you? I’m well, thank you. Formalities aside, here’s a brief intro to me!


I was born and raised in Pictou Nova Scotia, a town of ~3000 people. It’s on the north shore of the province – if you want to get to PEI by ferry you go through Pictou!

The stunning Maritime provinces. NB and NS are cropped, PEI is not. It's hard not to fit PEI in a single screenshot.

The stunning Maritime provinces. NB and NS are cropped, PEI is not. It’s hard not to fit PEI in a single screenshot.








You may also know it as the filming location for Mr. D. They gotta make money in the summer when they're not bringing in tuition.(Photo Source)

You may also know it as the filming location for Mr. D. They gotta make money in the summer when they’re not bringing in tuition.(Photo Source: Chronicle Herald)



I did my first degree at the University of King’s College in Halifax NS. I completed a Bachelor of Science, major in Math, in 2013. As a liberal arts college, my experience there really helped guide me into pairing Math and English as my education major and minor.









Although I’ve taken a hiatus during my school years at University of Regina (I wanted to be able to focus on my degree) I am a reservist with the Royal Canadian Navy. Specifically, I’m a member of COATS (Cadet Organizations Administration and Training Service). What that means is, as a Canadian Forces officer, I work to train sea cadets, aged 12-18; four general focal areas are citizenship, leadership, instructional technique, and physical fitness.

As a speciality, I’ve trained as a sailing instructor. Every summer since 2009 (including my years in Regina) I’ve work aboard HMCS ACADIA in Digby Nova Scotia to deliver their sail training program. Since 2013 I’ve been a Learning Facilitator with Sail Canada, allowing me to train and develop sailors who want to learn how to coach the sport.

Coaching out of Shearwater Nautical Centre, near Halifax

Coaching out of Shearwater Nautical Centre, near Halifax

My pal, Capt Nicholas Foran (Lt at the time, for those paying attention) and I in downtown Halifax

My pal, Captain Nicholas Foran (Lieutenant at the time, for those paying attention) and I in downtown Halifax













More specific to ECMP 355, here are my top thoughts on Edtech:

  1. Get over yourself. Technology is here, it’s part of life for both youth and adults in Canada. Sticking your head in the sand won’t change that.
  2. Technology has inherent risks, and potentially high rewards. Since it’s part of modern life, education should teach best practices around it, the same as we do for relationships, sports, and other activities on the risk-return spectrum. See point 1, and stop ostriching.
  3. Technology does not automatically make all previous innovations and techniques obsolete. Incorporating tech for the purpose of incorporating tech is a disservice to your students, and just because you have tablets doesn’t mean you can skip forming relationships.
  4. You will not be the technology subject matter expert in your classroom for long, if ever. Accept that now and make it part of your continuous learning. Walk-the-talk on lifelong learning, lead from the front, and show your students how to be an engaged amateur.

Connect with me here at WordPress or on Twitter. Always happy to chat.

My Week in Review – 17 Jan 2016

Here are some of the articles that caught my eye over the last week:

Zombie Ideas Again: The Learning Pyramid

This one came from Katia but had been on my mind since an Ed Psych Class on Tuesday. This classic image was included in our first week of handouts

learning pyramid

I looked at it, and said, “oh yeah, this thing. I’ve seen it before.” and quickly closed the link. About 20 seconds later, I asked myself, “I wonder if that pyramid has any scientific backing?” so I reopened the link and saw the source at the bottom, National Training Laboratories. Certainly sounded official, so I let the idea slip from my mind.

Enter this article. It solidified my opinion that there’s probably not much peer-reviewed backing for the learning pyramid. On the other hand, the article only cited two blogs and a short primer….. which referenced one of the two blogs. When I see a neat little cycle forming among references, my distrust metre cranks up a few notches.

My decision: Stop referencing the learning pyramid. However, I’m also not going to go around lambasting it. I’ll excise it from my pedagogical considerations.

The collapse of parenting: Why it’s time for parents to grow up

My sister tweeted out this link. It suggests, in rather strong terms, that parents today are spineless neophytes who’ve abandoned all control over their children and the result is kids being “overweight, overmedicated, anxious and disrespectful of themselves and those around them.” I have two, differing opinions on this:

From military experience, I’ve seen new leaders who don’t know how to give an order. It causes confusion and ineffective task completion, as well as frustration for both the leader and their subordinates.

On the other hand, this articles refers to parents “at risk of losing primacy over their children,” and “uncomfortable. . .in their position as the ‘alpha’ or ‘pack leader’ or ‘decider’ of the family.” There’s a lot of pack animal language. I’m no expert on child development – it would be an overstatement to call me an amateur. But until I read some very firm, scientific literature in support of the idea, I’m not convinced that A) humans are strictly pack animals or B) that the only way to effectively raise a kid is with an iron first, Alpha, pack-leader mentality.

Study: Feedback can hinder kids’ math outcomes

This one came to me over twitter from Michael Pershan (follow him if you don’t and you care about math ed). The title is misleading – what it really means is “feedback in the middle of the task.” It didn’t really assess in any sort of longitudinal study about the long term effects on not giving feedback – it was looking at giving feedback for each question as students worked through repetitious math problems.

This definitely raised questions for me about tech incorporation in a math classroom though. Almost always, new apps are heralded as “able to give instantaneous feedback to students.” This study  suggests that may not be a good thing for students. Deep thinking, learning, conceptual understanding – these things seem more important to me than feedback the moment a student makes a mistake. I believe error recognition and self-correction to be an essential part of doing good mathematics, and you can’t develop those skills if you’re not given time to think about if you made an error.

The perils of “Growth Mindset” education: Why we’re trying to fix our kids when we should be fixing the system

This article (from Katia again) hurt. It cut me in the way only coming face to face with your own assumptions can. I’ve liked growth mindset – I’ve practically preached it to students before. But I had never equated it with “pulling yourself up by your bootstraps” or generally conservative dogma. This article caused a lot of introspection.

However, it does not conclude that growth mindset is a bad thing. It simply isn’t enough, nor is it an excuse to teach poorly. I’m going to be doing a lot of thinking for the next few months on this.

Formative Thoughts on Assessment and Evaluation

My philosophy of assessment and education has been primarily forged by my coaching experience and deepened by my study in Education. Yogi Berra offered a pithy quote that summarizes my thoughts nicely: “it ain’t over ’til it’s over.” I believe learning, of any skill, process, or concept, to be a continuum. We all start somewhere, and end somewhere – the role of an educator is to help students progress as far as possible in the time that we’re given. To accomplish that goal, we use three types of assessment. Diagnostic assessment is used to determine the starting point of students; formative assessment is completed daily to set goals, work towards them, and deliver feedback for the next day; summative assessment only occurs at the end of a course. Summative assessment, in this model, is not part of the learning process – it’s a way of summarizing learning when reporting to outside agencies. In a strictly regulated learning environment, like our public education system, summative assessment needs to be completed more often to satisfy the supervising outside agencies.

During pre-internship, I used all three types of assessment, but I didn’t design them all. Most of the summative assessment I implemented was created by one of my co-ops. For diagnostic tools, I used quick-writes, brainstorms, and think-pair-shares. The summative tools I used included tests, quizzes, presentations, and written narratives. The majority of my tools used were formative; I used many different worksheets, individual and small group conversations, monitoring sheets and anecdotal records, peer-feedback with rubrics, defence for credit forms, written reflections, Kahoot, and class discussions.

My implementation of assessment tools was most effective in my English A30 class, for several reasons. The most important factor was my pre-planning. I was told to do a unit, so I made a unit plan. I built in assessment pieces, and used them to rework my plan as the unit progressed. In comparison, I planned my math classes day by day; I didn’t plan assessment far in advance, and this limited my effectiveness. I was also given significantly more freedom to do my English unit plan however I wanted – I was more strictly bound to teach math traditionally. In a traditional math classroom, descriptive feedback doesn’t really exist, and almost all assessment is summative.

I directly involved my students in the assessment process in my English class. We built a rubric together, they marked each other using the rubric, they submitted defence for credit forms with an argument for why they deserved the mark they chose, and they had a chance to omit one of two marks for the unit at their discretion. However, when I tried to involve my math students, in a minor way, I was shot down. They worked on a math riddle, and I asked them if they’d prefer it to be formative or summative (in student language). Because they found it easy, they requested it to be a summative grade. However, since that math class had more than one section, and the other sections didn’t do the riddle, it could not be included in the grade-book of just one section. School policy actually seemed to purposefully limit DI.

My main English project was fairly open. Students had to create their own myth, but the presentation option was open. They could submit a written copy, or present to the class. One group decided to film a video. This allowed students to express their learning in a medium they found comfortable, but still tied to the ELA curriculum. In math classes, almost every piece of work we did could be done as an individual or in a small group. Timelines for assignments and tests were relaxed; students would frequently come back on another day and finish work during lunch, or during their tutorial periods. I helped scribe for a student with broken fingers, and spent time reading questions to students who had difficulty making sense of word problems.

I found pre-internship felt like a snippet of internship – I went in, started teaching, meeting students, forming relationships. Then, after 3-weeks, I was whisked out, so I didn’t have to deal with the consequences of all the mistakes I made. It felt like a chance to make all kinds of mistakes, to prevent me from being stuck with them for four months in the fall. I hopefully got a lot of the bugs out of my system (though, of course, other bugs will surface). The biggest barriers I predict for internship will be the restrictions placed on me as a math teacher. If my internship school is like where I went for pre-internship, I can’t design my own summative assessment, all evaluation is standardized across the district, and I’ll be expected to teach topics in a particular order, following the textbook.

I don’t want to spend the last four months of 2015 frustrated and champing at my bit. My plan is to approach variety from a slow and steady perspective. Get to know my co-op,  build trust, and make small changes as I go. If I can develop mathematical curiosity in my  students, and facilitate meaningful discussions around mathematical topics, I’ll be happy. Overcoming fear of math is one of the essential steps of building conceptual understanding. If I have to do some workbook problems along the way, then I will prepare myself for that, mentally.

3 Key Learnings

Take it slow: In the span of 4 days, I had my English class design a rubric, analyse a myth, complete a group presentation, provide their peers with feedback, argue for a mark, and offer suggestions to improve the rubric. It was too many new ideas, too fast. I need to scaffold their learning a bit more. Use a simpler project for their first peer feedback; allow them to design a rubric that I use to mark them; argue for grades later in the semester. Overall, students enjoyed the experience, but I want more than simple enjoyment – I want deep learning.

Diagnostics catch my assumptions: I did several quick-writes from students, and I learned so much. Several students didn’t know their peers; group work was the first time some students had socially interacted with their peers; some students were frustrated by the idea of Aboriginal myths because they had taken Native Studies 30; some students requested adaptations for their learning of which the co-op knew nothing. I did rote diagnosis – now that I’ve experienced the value, I plan to do much more.

Specific direction is required for open assessment: I want students to have variety in their assessment, and have several ways to approach assignments. Being vague and unspecific does not achieve this goal. Especially with students trained to anticipate what a teacher is thinking, who aim to meet unspoken expectations, this confuses and frustrates them. For an open assignment to be effective, I need to give even more, clear direction.

Although my philosophy on assessment and evaluation hasn’t changed much this year, my language when discussing it and my approach to achieving it have. It’s beautiful to be able to consistently learn so much. It makes the future look so much brighter, because my learning will have no end.