Tim Hortons and proportions: When half-sugar isn’t really half at all

TimsLike many sleep-deprived Canadians, I begin my morning routine with a rejuvenating cup of coffee. Whenever I’m too disorganized to make one at home, I’ll order one from Tim Hortons (for my non-Canadian readers, Tim Hortons coffee is basically as popular in Canada as water is to fish). Lately, I’ve been trying to cut down on the amount of sugar that I take with my daily cup of Joe. I’ve historically taken my medium-sized java with one serving of sugar. Then one day, I cut it out entirely, and life became really awful. So, I now order my coffee with a half-serving of sugar as a compromise between health and happiness. I know I’m not the only one on the planet who orders half-sugar, because there’s a dedicated “half-sugar” button built in to the Tim Hortons computerized register. However, the manner in which this half-sugar is served got me thinking: since the Tim Hortons workers push a button on the automated sugar dispenser corresponding to the size of the cup, how would they account for a half-serving? Do they simply dispense sugar that’s meant for the size below instead? Two sizes below? Just what’s going on back there?

After some investigative work, including a source from the inside (i.e. a relative of mine currently works at Tim Hortons), there is no hard-and-fast rule for half-sugar orders. Some workers may press the size below, or two sizes below – it just depends on the person. So, if that’s the case, is half-sugar really ever half?

Here’s what I’ve been able to uncover using Tim Hortons’ own Nutrition Calculator. Below is a chart of coffee cup sizes, the number of grams in a full serving of sugar for each cup (this would be relatively consistent as it is dispensed by a machine), the amount of coffee per cup size (of course, this will vary depending on the actual pour), and a ratio of sugar-to-coffee (the greater the number, the sweeter the drink):

Tims Sugar
Source: Tim Hortons Nutrition Calculator

Before we even talk about the half-sugar serving issue, notice that a small cup of coffee is, theoretically, about 15% sweeter than the other coffee sizes. A 6g serving of sugar for a small cup would actually make it more in line with the other drinks (the sugar-to-coffee ratio would be 0.210 g/mL). Just thought I’d point that out.

Okay, back to the half-sugar. Let’s assume that I get my typical medium-sized coffee. If the Tims server accommodates my half-sugar request by dispensing a small coffee’s worth of sugar, I’m not actually getting half at all (7g vs. 9g of sugar). It’s really more like three-quarters sugar (78% actually). As a guy trying to limit my sugar intake, this is simply outrageous! When I, a math guy, say half-sugar, I’m expecting half. Period. Unfortunately, I’m unable to comment on the results if an extra-small coffee’s worth of sugar is dispensed, since Tim Hortons no longer serves nor provides nutritional information on extra-small coffee (nonetheless, the button on the sugar dispenser for extra-small is still present). If we were to look at a large or extra-large coffee cup, however, not only does the size below not meet the half-sugar requirement, the sugar for a cup two sizes below is still not half either! Therefore, it’s reasonable to assume that an extra-small sugar for a medium coffee also wouldn’t work. Regardless of the half-sugar dispensing technique (one vs. two sizes below), half is never half! If this is not a scandal of epic proportions (pun intended), I don’t know what is.

Do you take half-sugar with your coffee? Are you as outraged as I am? Now that I’ve brought your attention to Sugar-gate, help me raise awareness of this coffee calamity by emailing Tim Hortons and voicing your utter disgust with this mathematical misinformation! Send them the link to this blog post and let them see for themselves the sugary scandal that they’ve created. Tweet them with the hashtag #HalfMeansHalf and let them know that you won’t stand for this any longer! I’m so angry right now! THIS IS SPARTA!!!

Are giving grades the problem with education, or is it how we generate grades?


Last week, there was a provocative piece in the Globe and Mail regarding the practice of grading and its effect on student learning. The author, Elyse Watkins, feels that grading negatively affects those students who need more time to learn, and grades have essentially become a currency that teachers and professors “pay” students for good work. She believes that alternative methods of assessment other than tests and assignments, as well as alternative teaching structures will promote lifelong learning for students. If you read the comments at the bottom of the article, you’ll see many people, particularly those in post-secondary teaching, vehemently disagreeing with her. From my perspective, the practice of grading itself is not the problem. Rather, it’s how educators promote learning and generate grades that should be carefully analyzed.

I agree that it’s important to have alternative ways of assessing students. In Ontario, grades are supposed to be assigned to students as a judgement on their learning and understanding of course concepts. That judgement should be informed by all sorts of means. Tests, quizzes, and assignments are the traditional items that teachers use to see how much students have learned. However, teachers are encouraged also to use observations of and conversations with students to further identify the amount of learning that they have attained. So, for the student that has difficulty with test-taking, but can demonstrate their knowledge brilliantly with a one-to-one conference, such alternative methods of assessment are to their benefit.

Teachers should also be assessing students throughout the learning period and providing corrective feedback so that when judgement must be passed, students will be ready to prove their learning. For me, if I’m giving a test after, say, 15 classes, I will have checked throughout the 15 days that students are learning or not, and provide feedback and opportunities to get better and better. I don’t give grades during the learning period – that`s the students’ time to figure things out, and they shouldn’t worry about making mistakes during that time. I don`t wait for test day to see whether students “get it” or not. By then, I will have checked and helped students many times over.

Grades should never be considered currency, as Ms. Watkins puts it. Unfortunately, it happens sometimes. I overhear teachers giving students grades (or taking marks off) according to behaviour. Since grades should be a judgement of learning and not how well a student is behaving, this is clearly a misuse of the system. I have also seen teachers dangle marks like bait, making students complete assigned work to get them. Again, since grades are a judgement on learning, there are other ways to see student understanding. It doesn’t have to be a transaction of work for marks. Finally, grades are used to differentiate higher-achieving students from lower-achieving students for many reasons – admission to post-secondary schools, scholarships, or employment. Do I believe that grades shouldn’t be used as a tool to choose the best candidates? No, but it should be only a part of a more holistic way of evaluating a person. Grades tell one narrow part of a story.

Grades aren’t going anywhere anytime soon. However, how we come up with grades can be greatly improved.

What do you think about grades? Should we can them? Do we need to standardize grading? I’d love to hear your thoughts and experiences.

Three Observations from Implementing a Bring-Your-Own-Device Approach in my Classroom


I must admit – I’m a tech-hoarder. Here’s a picture of all the devices that I have been able to collect or purchase so far to help implement a bring-your-own-device (BYOD) environment with my students. I’ve dabbled over the past little while with browser-based apps like Desmos, Kahoot! and Socrative that work with any type of device (Android, iOS, Chrome, Windows) and definitely see how they augment the learning of complex ideas or allow more effective assessment of student understanding. I wanted to write not about any tools that could be used, but rather three interesting things that I’ve observed first-hand while implementing BYOD.

1. Students look forward to using their devices for learning, unless they’re at low battery

As a whole, students are totally into using their phones and iPods for class activities. However, I’ve had times when one or two students say they don’t want to participate because their phones are at 8% power. Nowadays, if the BYOD portion of a lesson is later in the period, I give students a heads-up and allow them to charge their phones in anticipation. I carry some extra USB plugs, micro-USB cords and the odd Apple 30-pin or Lightning cable in my drawer in case anyone needs them. In the future, I’m hoping to make a DIY charging station, like these beauties, for my classroom (I’ll have to channel my inner Martha Stewart).

2. In an BYOD setting, understand that students will do others things on their devices once in a while, but minimize it whenever possible and don’t be mean about it

With lots of devices comes lots of responsibility. Devices can obviously act as distractions for students, even when they’re in the middle of using it for educational purposes. In my class, I see students on YouTube, texting, or on social media all the time. Rather than making things punitive and nasty, I simply nudge them back on task, or start reading out loud what they’re messaging, which usually gets them to stop quite quickly. It’s near-impossible to monitor the use of thirty devices by thirty students, so vigilance is key, but I also accept that students will from time to time do something else on the devices, and it’s okay, as long as it’s not a huge distraction. Even pencils and paper can be a distraction if people start doodling; that doesn’t mean we don’t allow students to use pencils and paper, right?

3. Equity of access to mobile devices is an issue, but not THAT big an issue

Just because many of my students come from lower-income families doesn’t mean they don’t have mobile devices. From my data collection, roughly 60-80% of students will have a device that they can use. Perhaps it’s part of the teenage culture of conspicuous consumption, or pressure that students exert on their parents/guardians, but the majority of students in any given class will typically own a device. For those that don’t have one, I allow them to borrow for the period the aforementioned top-up devices that I’ve collected over the years. Without top-up devices, students who don’t own devices will definitely feel excluded; therefore, they are a must in any BYOD environment, in my opinion.

As school budgets start to get tight, BYOD will be the only viable strategy to involve technology use in classrooms. The Toronto District School Board is behind other boards like Peel DSB that have official policies and supports around BYOD implementation. Students definitely enjoy using their own devices, but some challenges, like charging and distractions, come with the territory. We shouldn’t let these issues get in the way of integrating technology for the benefit of student learning.

P.S. For more info on BYOD in education, here’s a great site.

Making EQAO review a bit more tolerable using Kahoot!, a web-based learning game

IMG_3077January is here, which means it’s Grade 9 EQAO assessment of mathematics season (sigh). I’m not a huge advocate of standardized testing, but I’ve spent a large chunk of my teaching career working on resources and ways to help my students succeed on them. Why? Because achievement on these tests, whether we like it or not, is arguably the most heavily scrutinized benchmark for student learning, teacher competency, and school effectiveness. Politically, the results of these assessments either tell the world that the kids are alright, or that we have a full-blown educational crisis on our hands and we need to scream, flail our arms and run in circles.

For students (as you all might remember from your days of schooling), the greatest challenge of an end-of-semester test is remembering everything that they have learned over the last five months. I devote about a week to try to go over all the main concepts that we’ve done as a class over the course of the semester. In previous years, these review sessions would consist of some teacher-led examples, students working on questions to try to jolt their memory, and some form of summary. It was a bit dull, but I felt that it was the most effective use of time. This year, I’ve decided to appeal to students’ desires to have fun while learning. That’s where the gifts come to play.

Today, I centered the (re)learning of linear relations around Kahoot!, a web-based game that students play using mobile devices. In a nutshell, Kahoot! allows teachers to create a multiple-choice quiz and students answer the questions using their devices. For each question, a correct answer earns points, and the faster students answer, the more points they receive. Whoever has the most points at the end wins the right to choose from the wrapped prizes. I created a quiz that contained questions regarding linear relations, with the hope that students were more motivated to learn during class in order to be prepared for the game at the end of the period. To some degree it was a success; I’m trying it for the rest of the week to see if the motivational factor increases or decreases. The game itself was a success – students definitely enjoyed themselves, and I feel that many of them consolidated their learning through engaging with the questions.

Is this a game-changing approach? Probably not, but at least it’s way more fun and positive compared to the alternative. After all, it’s not just about the content we teach, but the positive attitudes that we should try to instill upon our students. If you can have a good time and prepare for a standardized test all at once, then why not do it, right? You can’t spell fundamentals without “fun.”

Using Twitter to Show Student Work During Instruction

Screenshot 2015-01-05 at 9.21.56 PM

Now that school’s back in session, I can finally start blogging about some really educational stuff (not that meeting Paleo authors wasn’t cool or anything).

For the past semester, I’ve put an emphasis on showing student working during instruction. Since I don’t have the luxury of a document camera in my classroom, which would make projecting student work a cinch, I thought of using Twitter and my own smartphone instead, along with my computer and digital projector, to project student solutions for everyone to see. Basically, when students are working on solving a problem that I’ve posed, I circulate and identify one or two solutions that I feel would be worth sharing to the whole group. Using my phone, I take a picture of the solutions, upload them to a Twitter account that I use for this kind of thing, and then display the tweet and photo using my computer.

There are many advantages to displaying student thinking for all to see – it doesn’t have to involve Twitter: (1) often there are a variety of ways to approach a math problem, so showing some off and discussing them sends the message that there is never one correct way to go about solving a problem; (2) I have found that students are more likely to look at someone else’s work than the work that a teacher puts up; (3) students, especially those who haven’t historically felt much success in math class, are eager to have their work shown off and are proud when their work has been chosen as the focus of discussion; (4) it gives students more of a voice during a lesson; (5) students can always look at the Twitter account from their own devices/computers at a later time to look at some sample work.

I envision this idea of using Twitter to show student work expanding in a 1:1 iPad or BYOD (bring-your-own-device) environment where all students have some kind of a mobile device. Each student can create a class-specific Twitter account and can then use their own devices to take photos and upload their work to share, rather than having the teacher do it. A hashtag could be used to make finding student tweets easier.

From my 5 months of experience using Twitter to show student work, I have found it to be very effective in hooking students and also sharing quality work amongst everyone. I would recommend others to try this out, especially if you don’t have a document camera. I think there’s also a mild cool factor that attracts the students because it involves Twitter, something that’s pretty hip with the young ones these days. If anyone can think of a way to show student work using SnapChat, send me an email,

The Paleo Diet: Meeting Stacy & Sarah and my Paleo Experience

Today, I ventured to my first-ever book signing at the Bay and Bloor Indigo to meet Stacy Toth and Sarah Ballantyne, two best-selling authors and voices in the ever-growing Paleo diet movement. Now, to be honest, I only ever listen to their Paleo View podcast when my partner, who’s the real fan, plays it on her phone for everyone to hear, but I enjoy it nonetheless (speaking of my partner, she just worked 26 hours at Sick Kids Hospital and was recovering in bed, hence her absence in the photo). I wanted to buy a couple of Paleo cookbooks and have them signed just as a little present for my slumbering doctor back home, so I arrived an hour early to beat the rush and get in line. But, after my purchase, I noticed that Stacy and Sarah were already sitting at their signing table with nobody else around. Seeing a golden opportunity before me, I made a beeline to the table, introduced myself, got my books signed just like that, and had a nice 10-minute chat about their latest travels (they just got off a plane from Minneapolis), Sarah’s post-doctoral work in Toronto (she did a fellowship at St. Michael’s Hospital), falling glass from downtown buildings, what a great city Toronto is, and miscellaneous Canadiana. It was a great experience and I was really glad to meet the owners of the podcast voices that regularly fill my home.

Am I a devoted follower of the Paleo diet? I’m off-and-on, and not nearly as much as I’d like to be (see my McDonalds post for proof). There’s definitely credible evidence around in favour of the Paleo diet. Eating good meat, lots and lots of vegetables, reducing/eliminating grains, and getting rid of processed foods, to me, just makes a lot of sense. You really can’t argue against eating REAL food. Also, and more to the point, any diet that says that it’s good to eat saturated fat, butter, and bacon will always be viewed favourably in my books. For the two months that I was a really good boy, I actually did lose weight. However, lately – and especially during the holidays – it’s been tough to not overeat or resist the yumminess of cookies and sandwiches. Thirty years of eating whatever without much regard for anything inevitably takes some time to undo. I’m working on it. It doesn’t help that McDonalds keeps sending me coupons.

A few years ago, I shared with my Grade 9 math students this blog post that contains a scatter plot showing the lack of association between saturated fat intake and heart disease. They were surprised to see that the negative messaging that they had received regarding saturated fat was actually not backed up by scientific evidence. But, as I am not a health professional, I didn’t tell my students to head to the nearest fridge and start downing butter sticks. I let them think about what the data might suggest and left it at that. Now, with some mainstream media writing about how doctors got saturated fat all wrong, I would direct students to read up on the topic, and also read some counter-arguments, and make up their own minds.

All in all, today was a fun day. My partner loved the cookbooks, and now I have no excuse to ever eat McDonalds at home, ever.

Unless more coupons come my way…

Critical Thinking and Gas Prices


Today, I did something that I haven’t done since my family got our first car four years ago – I filled up the tank with sub-90-cent gas. Oddly, in addition to feeling the obvious joy for saving money, I felt a momentary sense of worry about turning into a liberated gas-guzzler. Like many others, my driving habits reflected the tough times of $1.30/L fuel – I drove less, travelled to more local options, and conserved gas however I could (e.g. opening the windows instead of using the air conditioning). Now that gas feels like a bargain, I fear that I and others like me will start taking long joy rides with the AC blasting for no good reason (even in the dead of winter), and in so doing, contribute even more of those nasty greenhouse gases I’ve heard a bit about to Mother Earth. After all, the whole reason fuel-efficient, hybrid and electric cars came to be is due to the economic pressures that came with high fuel prices, and not for any environmental considerations. Money drives behaviour, unfortunately. When it comes to low gas prices, it’s not all roses and rainbows.

So what’s the educational spin in all this? Students should be encouraged to explore all sides of any issue. Role playing and debating are some simple ways to promote critical thinking. This coming semester, I’ll be incorporating social justice issues into math, and no doubt there’ll be some serious debating happening.

In the meantime, I’ll try to enjoy the low gas prices and, well, walk, We’ll see how that goes, as I stare at a snowstorm out the window.

Inspiration from McDonalds



Alright, so maybe I shouldn’t be outing myself as a more-frequent-than-I-should-be customer of McDonalds, but today’s visit was interesting for several reasons. Firstly, my order was taken by an old CW Jefferys student of mine who’s doing a paid co-op placement, so we got to catch up and talk about her plans after high school. Also, I learned that my Filet-o-Fish (pictured above) was made with MSC-certified sustainably-sourced wild Alaskan Pollock, which basically means that it’s a bit better for me than farmed fish, the fish lived decent lives and weren’t fed junk like GMO corn or soy. Also, all sorts of math can be done using the McD’s own nutritional information, not only to learn math concepts, but also to use the math to raise awareness of making informed decisions about the food students choose to eat.

For instance, have students create three meals (breakfast, lunch and dinner) that they would enjoy eating, and then calculate the percentage of daily intake for calories, fat, etc. according to Canada’s Food Guide. Then, have students create three healthier (and I do use that term loosely) meals from the list of McDonalds food items and see the difference in the percentage of caloric and nutritional intake. I’m quite sure that plenty of questions about what is/isn’t healthy will abound in this exercise, which is an important lesson in and of itself. I’ve always thought about trying something like this with my Grade 9s. Now that I’ve thought it through a bit more, sounds like it’s definitely worth doing next time around.

I knew going to McDonalds today was a good idea.

Math with Mewtwo – Learning math using Pokemon TCG Cards

Pokemon cards

If you are like me and a proud parent of a 7-ish year-old boy, you’ve no doubt encountered Pokemon cards – you may have even been completely into them yourself as a youth (personally, I was more into Power Rangers). These cards with pictures of strange animal-like creatures are part of a trading card game similar to Magic, and my son LOVES them. So, why not try to make the most of this often expensive interest and make it educational? For the past few months, rather than playing the real rules of the card game, we play with them by taking turns “battling” each other’s Pokemon with their attacks and associated amount of damage until the opposing Pokemon’s health points (HP) are reduced to zero. Needless to say, this involves plenty of subtraction using mental math. I’ve noticed that after several months of playing, my son’s subtraction skills are getting stronger – he even explains what he’s doing in his head after each calculation. Plus, he’s willing to do this for, like, an hour or two. It’s amazing how much learning kids will do if the activity is a game or using something that they’re totally interested in. That’s what I’ll be bringing back to my classroom after the winter break. Whether it’s basketball, dancing, or Pokemon cards (my Grade 9s LOVE Pokemon, too), make math fun and relevant.