Wednesday, 23 December 2015

Drawing my Math Webcomic

Almost three years ago, I wrote a post I called "Drawing for a Webcomic". Back then, what I was really doing was illustrating a serial. Now, it's a legit comic. So I thought I would update the post, in part because I feel like I'm terribly inefficient. Let me know if you agree.
I'd ask Koch, but he only speaks in riddles.

STEP ONE: The script

I write the script in TextEdit on my Mac. Usually about four comics worth at once (roughly a mini-arc), which tides me through a month (I publish every Monday). That said, episode titles and mouse rollover comments tend to be put in the week I thumbnail, unless a really good idea strikes me sooner. The script is also extremely flexible, as you'll see.

STEP TWO: Refining the mental image

This step involves thumbnails, effectively a tiny version of the finished page. I learned about the technique at Anime North 2015, and it makes lots of sense. My thumbnails are set up in columns, even though my final panels are a 2x2 square, in part because I hadn't decided on that format when I started, and in part because I find it easier to fit 8 on a page this way.

You can see strip #22's thumbnail pictured on the right. After drawing it out, I realized I'd put Expona on the wrong end of the bar - she'd be looking off the page in Panel 4, hence my arrows and remark "look left" (to have her face the prior panel). This is where I try to vary the character sizes between close ups and long shots.

STEP THREE: Refining the actual image

I jot down the script (more or less), then fit in word balloons before sketching. This step usually takes a couple of passes, one for the overall setting/character location and one for the detail. Sometimes I will do that on a panel-by-panel basis. Here, still with strip #22, I made two overall passes so you can see the difference.
First pass - words and positions

Backgrounds are a pain in the ass. First, because although this step alone takes about two hours (half an hour per panel), it's somewhat mindless work. Unless I have to account for the position of things in the background - that needs thinking! And I cannot simply drop in a prearranged background, because of how my perspectives zoom in and out.

I had no idea how annoying backgrounds would be when I wrote my "Xeno: Paradox Princess" parody, which required objects for the protractor/chakram to hit. I now hope to avoid objects wherever possible. There's also the fact that I will usually need to do a web image search for a frame of reference with a new item, like the exterior of the "Bowditch" here.
Second pass - Faces and detail

Above, you'll see the second pass. I added a shot glass onto the bar, to try and fill the otherwise empty space. The 'Hmm' in Expona's thought bubble (Panel 1) was an addition for the same reason... and while that word will be at the start of the sentence, the lettering itself is a full step later on.


Essentially a third pass, using black ink. Sometimes I catch minor mistakes here, like the fact that I missed drawing Expona's watch in Panel 2. Basically, this step is exactly the same as it was three years ago. There's merely more ink needed.
Third pass - Refinements

You may notice, I don't ink the panel boxes themselves. That's because they're merely guidelines for me when drawing. Ditto for the words, in fact they'll get erased out (see lettering, below). This is the point when I would ACTUALLY scan the thing in. On the same printer I had three years ago, which only my old computer has drivers for.
Pencils erased, scanned in now

STEP FIVE: Panel Template

At this point, I take the scanned image and grab it, panel by panel, dropping it into my panel template file. The "Any Qs" and copyright are already on there. Aside from those, the template is a 2x2 grid, all panels the same size, and it is layered such that I can put the image under the lines. I grab my squares (as square as possible), then shrink the panel layer down to about 300 by 300 pixels (so roughly 25%). The title also gets grabbed. This is the simplest step, but if I drew outside the guidelines, sometimes I'll need to resize and regrab.
Dropped into the template

STEP SIX: Lettering and Cleanup

I'm getting better at judging the spacing for word bubbles. But sometimes I misjudge, or need to rearrange words in a sentence so they don't escape. This time, I decided to have Expona say 'Trigonometry' rather than 'Trig' in the last panel, thus had to pull her word bubble further apart (rather than pushing it closer together). The text is inserted line by line (line breaks are never positioned properly).

The font I decided on back in August was Candara, size 18. Why? Well, it's free. Also, it has a boldface option that's basically the same size as regular text. (When everything is in all caps, you need boldfacing to imply capitals. With this font, I can simply type over an old word with a boldface version.) It doesn't do "Crossbar I's" though, so I have to adjust those manually.

If you don't know what those are, have a look at Expona's last dialogue bubble: "I MEANT IS THERE..." The first 'I' is on it's own, and thus needs top/bottom bars, whereas the 'I' in 'IS' is merely a stick. Incidentally, the 'LISSA' tag in Panel 3 is a different font, Optima. I don't recall why I decided to change it for character intros.

Once the words are in (and script readjusted), with their bubbles being digitally fixed, I tidy up any other pencil markings I see; areas where I extended a line too long, etc. Then I flatten the whole thing (removing layers), resave it as a PNG, and get ready for the next step.

STEP SEVEN: Colouring

I do this the same way I did three years ago as well - basically reach into the prior file, grab the last image of the character, put it (temporarily) on a new layer, grab the colours with the eyedrop tool, and use them on the new strip. If there's objects being introduced (like the Bowditch here) I mix colours up a bit and see how they work out. The pinkish (on the sign and the stool) was to coordinate with Lissa's hair.
You can read this comic in context too

Sometimes final touch-ups are needed. For instance, I noticed that the shelves looked a bit like sequential bars stretching backwards, so cut and pasted a new line for more dimensionality.


I now transfer everything to my current computer. I have the "Next/Previous Page" table in a file that I can cut and paste into the HTML (exactly like my time travel serial). The comic script can then be pasted underneath that, though sometimes I'll simply retype it. The mouse rollover comment is added through the HTML. After the post has gone live, I can finally add it to the table of contents.

And that's it! The whole process likely takes between 4 and 5 hours, but it's rarely done continuously; I take breaks before the image is inked, then sometimes after scanning and/or before colouring. Three years ago, it apparently took less time, but I was doing less of it.

Now, is that process terribly inefficient? Is it even worth it? I suppose that's for you to answer. Let me know below. Thanks for reading!

Wednesday, 9 December 2015

TANDQ 15: Animated Discussion

In 2014-2015 I wrote an education column called "There Are No Dumb Questions" for the website "MuseHack". As that site has evolved, I have decided to republish those columns here (updating the index page as I go) every Wednesday. This fifteenth column marks the last in the series. It originally appeared on Thursday, June 2, 2015.

Should TANDQ be resurrected in 2016? Well, if so, what would you want to read about?

What animation is acceptable at your workplace?

There’s my question to you. As I’m not at your workplace, YOU tell ME (in the comments, or email if you prefer) - what is your workplace, and what is the threshold of acceptance there? Would you be allowed to have, say, “Despicable Me” minion figurines at your desk? A “My Little Pony” poster? Could you wear a “Sailor Moon” brooch on the job? Yes, I’m trying something a bit different here, turning this last question back to you. I’ll explain why the subject came up, then provide my own answer too.

This particular idea came out of a panel at an anime convention regarding “Fans in the Professional Workplace”. On the panel were a lawyer, a banker, a teacher, a graphic designer and a librarian. (I’m simplifying the job descriptions slightly.) Now, obviously there are lots of fandoms out there… and on a related note, I recommend checking out the recent “Fan I Am” series of postings by Steve Savage. (Have you considered politics as a fandom?) But I’ve decided to target animation. Partly because it was the primary topic at the panel, but also because there’s a lot of conflicting information about it on the internet, and out in society.

There’s this notion, particularly in America, that animated films are “children’s entertainment”. But are those attitudes changing? Or are we merely expanding the demographic? What about outside the US? The more I look, the more I feel like animation has lots of misconceptions tied to it, perhaps because it bridges so many different genres. Even Ben Zauzmer, who used data and statistics to predict almost all of the 2015 Oscar winners, missed out on his prediction for “Animated Feature”. (For that matter, why didn’t “Lego Movie” even qualify?) So when an audience member in that panel I attended indicated that she was soon graduating from post secondary, and essentially asked how much one could or should put anime out there, I was intrigued. It felt like something relevant for this website [MuseHack]. But there was no clear answer - as often happens with good questions.


To be clear, I’m not talking about whether it’s proper to engage in your hobbies while on the clock. Nor am I suggesting fandom should be completely excluded from a resume, given the lessons it provides that may be relevant to your career. What I’m looking for is the middle ground. Can one person name drop “My Little Pony” in a workplace with the same ease as someone else drops “Game of Thrones”? Remember, one of those shows involves rather more violence and death! Well, it turns out the answer is likely no, as “FiredBrony” found out in 2013. (See my “You Can Be Fired for This” link below. Though let’s not pretend the issue is specific to gender.) So I ask again, in an age where new graduates are worried about simply getting entry level jobs, which workplaces are more accepting of one’s interest in things like anime and animation? My suspicion is “Your Mileage May Vary” (YMMV).

I’ve done some online searching, but figure it’s better to hear from those with actual experience. To that end, let’s first consider the people on that panel. The lawyer had to be totally undercover. The banker was mostly undercover, in part because when she was a summer student, she’d had complaints from older people at work about her reading manga on her lunch break (seen as “unprofessional”). The graphic designer could get away with subtle decorations (like an anime belt on casual Friday), but wouldn’t comment about it randomly. The teacher could potentially say something depending on the audience - for instance, recognizing a Totoro shirt is a safer move than recognizing “Kill la Kill”. And towards the other end of the spectrum, the librarian could discuss anime and manga with coworkers, and even use it to connect with people when done correctly.

While we’re on the subject, some other thoughts that came out of the panel was how the mention of anime (like volunteering at a convention) could make your resume more memorable - but you may also want to look up the hiring manager on LinkedIn to see what the company expects. There was also mention of the fact that, if you type in “The Manga Guide to” into a search engine, you’ll find they’re being used as a learning tool beyond the high school level. Someone’s even doing their PhD as a graphic novel. Is there a generational divide forming? At minimum, I suspect the definition of “professionalism” may be evolving.

Anyway, here’s my answer: I’m a high school mathematics teacher. As the teacher advisor for the anime club at our high school, I have some anime CDs on my desk, and occasionally discuss it with students. I could discuss it with coworkers, but there isn’t much interest. I haven’t thought much about American style animation; I’m no fan of the Simpsons, but I don’t get hassled for that. How about you?

For further viewing:

1. Yes, for the Millionth Time: You Can Be Fired for This

2. Why animated films are the UK’s favourite - and why that’s not likely to change

3. History of Early Animation (video)

Gregory Taylor is a high school mathematics teacher in Ontario, Canada, who does serial writing in his spare time. He can be found on twitter @mathtans, and runs a few blogs including "Mathie x Pensive".

Saturday, 5 December 2015

Know Any Support Groups?

Back at the end of September, I felt like I was in a “Good Creative Place”, particularly related to my personified mathematics webcomic. Well, that’s deteriorated.

Part of it is some other personal difficulties I’m going through, which has me thinking more about my hobbies. Part of it is the fact that the recap post I wrote here last weekend got two Twitter endorsements and six likes with little effort, which is miles away from any reaction to to the actual CREATIVE stuff I do regularly. Part of it is fatigue.

Someone recently mentioned online support groups in connection to a completely different matter. I’m wondering if maybe that could be a thing I need, and if anyone knows of support groups out there for the following things:


I personify mathematics. Still. That serial began in July 2011. I learned it was a serial in March 2013. I gave it up in May 2014. I brought it back as a webcomic in August 2015. Strip #20 posts on Monday. It’s the most important thing I do after family and work.

...Or not.
Back in August & September, I used to be getting 100 hits. Since mid-October, that’s dropped; the average is now about 70 hits (lowest is 59). I had no comments on it through the month of November.

Why I Sigh: I know I’m no XKCD, Ben Orlin, or (x, why?). But when things like the @solvemymaths Mr. Men regularly seem to get 4 or 5 likes and RTs and I can’t even manage THAT after FOUR DAMN YEARS... is my drawing THAT bad? I like to think I’m improving. I know, I’m probably deluding myself.

What Keeps Me Going: Scott RTs me. I did actually get comments from him, Chris Burke and Ashtar Balinestyar prior to November. I had one person ask me about the strips last month - at work, at a time when teachers are pretty busy. And sometimes I reread Audrey McLaren’s old comments from April 2014.

There’s also the fact that I post the images separately to Tumblr, so any hits there wouldn’t show up in my count. But I think only John Golden is tracking me there.


I have been publishing (after editing and drawing) a time travel serial since April 2015. How’s that going?
-Part 1: 129 Hits. Last hit: Nov 28th.
-Part 2: 30 Hits. Last hit: Nov 25th.
-Part 3: 25 Hits. Last hit: Nov 22nd.
-Part 4: 27 Hits. Last hit: Oct 20th.
-Part 13 (Arc 3, my BTTF tribute): 11 Hits. Last hit: Oct 25th.
-Part 24 (End Book 1): 10 Hits. Last hit: Sept 20th.
-Part 32 (published a month ago): 3 Hits. Last hit: Day after publishing.

I have had ONE SINGLE POSTED COMMENT through 36 entries in the story, and it was more an observation. (Granted I’m not including the couple comments regarding the blog itself, or remarks outside the blog.) Two posts out of the 36 got a WordPress “like”.

Why I Sigh: Whenever I go to someone else’s serial, and I see a single, solitary comment. Like, WOW. To me, that’s epic. Oh sure, sometimes people remark on my site itself, but on November 26th it was back to 0 Hits for me! Heaven forbid I learn anything about my actual story!

What Keeps Me Going: I did get a 3.5 star review in “Web Fiction Guide” by Billy Higgins way back in May. John Golden tweets at me sometimes. Scott Delahunt (my beta) reblogs the whole thing on Facebook, possibly much to the chagrin of his followers. And I had a really nice remark about my writing in general when I did a guest post over at “Legion of Nothing” for April Fool.

There are also 5 “followers” who may be reading through email. Silently.


I’ve written almost 30 of the damn things. The students seem to like them. But it’s like every PD I go to these days is telling me that “singing songs for students may engage them, but they’re not learning” so fine, I suck as a teacher, thank you. I actually couldn’t watch the TMC15 closing video - I didn’t want to see parody joy in others when (outside of the classroom) the vibes I get are silence trending to “what are you doing with your life”.

I really need to start putting together my annual Christmas parody. Did you even know I’ve done those annually for four years? Well, whatever. Last one only had 40 views on YouTube. I can’t even suggest what keeps me going here, because I haven’t written a song parody in months.

I’m a teacher. I pride myself on an ability to give feedback to others. You’d think I’d know where to get some myself. But no - I feel like the only feedback I ever get is the stuff related to my job itself. Which, granted, is appreciated. And the thought is an exaggeration. But December is traditionally lower for views in general, so I guess I need more to keep me going these days.

So, anyone know of any support groups for that stuff?

All I can do is sigh.
All I can do is wonder why.
All I can do is try, try, try...

Sorry, I think that got a bit bitter. Thanks for reading to the end. Namaste.

Wednesday, 2 December 2015

TANDQ 14: Vote of No Confidence

In 2014-2015 I wrote an education column called "There Are No Dumb Questions" for the website "MuseHack". As that site has evolved, I have decided to republish those columns here (updating the index page as I go) every Wednesday. This fourteenth column originally appeared on Thursday, April 30, 2015.

How can I influence a democratic election?

Vote. That’s your best shot, particularly in light of voter apathy. And when you do vote, make it an informed decision, particularly if you’re getting others around you to vote too. I’ve noticed lots of voting talk lately, from the United Kingdom’s general election this May, to the Canadian federal election later this year, to the Hugo Awards ceremony in August. Oh - if you haven’t heard, that last is getting more press now due to the “Rabid Puppies” group (sometimes confused with the “Sad Puppies” group, but they’re in their third year). But take care, as not all democratic voting systems are the same! Let’s delve deeper.

Many countries that are (or were) British colonies will be familiar with the “first past the post” (FPTP) system. It’s more formally known as the “single member district plurality” system, and basically involves marking an “X” next to your candidate of choice - the candidates having been previously chosen by their political parties. When the votes are finally tabulated, whoever has the most “X”s wins - even if that individual did NOT collect an overall majority. To illustrate, let’s say we have three candidates: A, B and C. The election results are A: 25%, B: 35% and C: 40%, so C is put in charge of 100% of the district, despite the fact that 60% of people chose someone else. We can see from this that there are problems with FPTP - not the least of which is that in the long term it encourages a two party system (see the video link at the end). But it gets worse.

Mathematician Donald Saari has shown that the results of a plurality voting system can produce an outcome that is the exact reverse of actual voter preference. See his website for some introductory lectures, or search the internet for the famed “milk-beer-wine” example. As an abbreviated version: Given my election results above, consider a case where all those voting A and B would have ranked C dead last (meaning 60% of voters now really dislike the result) while all those who did vote for C would have picked A as their second best choice (meaning NO voters would rank A in last place - despite A being listed as “least popular”!). Obviously, ignoring those rankings is a big issue! Can including them fix the problem? Well, not entirely. But a ranking system is what they presently use for the final ballot of the Hugo Awards.

Rank and File

What ranking does fix is the problem of “strategic voting”. For instance, if a person really doesn’t want candidate C to win, then even though they like A best, they may choose to throw their support behind B, who has the better chance of beating C. This “spoiler effect” is what can lead to recent UK news articles like “Vote Conservative in seats Ukip can’t win”. I suspect it’s also behind the latest trend in overall election campaigns, which don’t seem to say “Vote for me, B!”, so much as “Don’t let C win!”. Fortunately, this issue evaporates with what we call “Instant Runoff Voting” (or in Australia, “The Alternative Vote”). It’s still a plurality/majority system, but you don’t mark a single “X”: you rank your candidates by preference. This means that, given the same election results above - we’d have no clear majority. So we drop out “A”, and the SECOND choice on their ballot is redistributed. If they all hated C, this results in B: 60% and C: 40%, meaning candidate B is declared the winner!

Of course, this preferential voting system doesn’t eliminate issues like gerrymandering, or address the problem that all voters could have been satisfied with “A”, and the system is still susceptible to mathematical paradoxes. (Not all relationships are transitive in nature: If A beats B and B beats C, it’s possible that C beats A.) But this system does mean that you don’t need to worry about how the other people in your district are going to vote! Which implies that candidates have to campaign based on their platforms, not against someone else’s. The final voting for the Hugo Award gets even more interesting here, in that “No Award” is a valid ranked choice. So to win, the first nominee who ends up with over 50% of votes must be subsequently tested to ensure that the “No Award” choice was not ranked higher than them on relevant ballots. To read more about the Hugo voting system (and to see how they pick their runners-up), go to their website here. Note also that their award nomination process is not the same procedure.

At this point, having torn down Plurality systems, what’s left? Why, the Proportional Representation systems, used by many European democracies. The idea here is that you get two votes - one for the governing party, and the other for a party candidate. Ergo, if the final results show a preference of 50% for Party 1, 45% for Party 2, and 5% for Party 3, the seats are filled to that proportion with the top candidates, as chosen by the second (simultaneous) vote. It’s more mathematically fair... but it’s not perfect either. Proportions cannot be exact, and there is a need for larger districts, meaning less local representation. Of note, in 2007, the Canadian province of Ontario held a referendum on whether to adopt the “Mixed Member Proportional” variation on this system - a system with double the seats compared to the regions - so feel free to watch Rick Mercer’s analysis for more. (The proposal was defeated by 63% of Ontarians, and “First Past the Post” remains in effect.)

Wait, doesn’t that referendum mean that Ontario held a vote on how voting will take place? How very meta. And yet, it’s only by proposing such things that any (necessary?) democratic reforms can occur. Which brings us back to the introduction: Vote, and try to do it in an informed way. It’s probably your best shot at influencing things... short of becoming a candidate yourself.

For further viewing:

1. Why Democracy is Always Unfair

2. Some Sad Puppy Data Analysis

3. The Problems with First Past the Post Voting Explained (Video)

Got an idea or a question for a future TANDQ column? Let me know in the comments, or through email!

Sunday, 29 November 2015

PD: Math in Context

My other two blogs (math webcomic and time travel serial) are queued up early this week, so time for another “Good Professional Development but Had No Time To Blog” post. Today’s topic: COMA Social 2015, from October 1st, featuring Kyle Pearce (aka @MathletePearce). See his presentation, “Making Math Contextual, Visual, and Concrete!” at and/or read below.


While teachers may get excited about the correlation coefficient, students are often less enthusiastic. How can something like that be taught so that students won’t freak out or feel overwhelmed? Kyle says “I thought this is what good teaching looked like - very organized, structured notes”. Yet students were bored.

Maybe technology will change everything? SmartBoards? Well, students were behaviourally better (not talking while he wrote on the chalk board) but still not intellectually engaged. This because it was more of a substitution, not an augmentation... the latter is needed for functional improvement.

Here’s how we learned math: 1. Take up HW; 2. Definitions, formulae, procedures (to set student up for success); 3. Examples; 4. Homework. Yet the examples were abstract, removed of context, and then asking a week later, often the concept hadn’t stuck/was forgotten. This model creates two groups of students: “Good at Math” & “Not Good at Math”. Except “Good at Math” meant understanding terminology, and following procedures, aka “Good at Memorization”. Meanwhile, those “Not Good at Math” might not memorize, or might be capable of memorizing but were unwilling to play that game.

Even students “Good at Memorization” run into problems. There is a larger “Struggle with Unfamiliar Problems” group. (If a test question isn’t like the examples, it is seen as “unfair”.) When we take in new knowledge, we have to tie it to prior knowledge, but traditional methods (and textbooks in particular) will silo concepts into tiny blocks, removing the chance to see those connections.


Showing the connections (how math concepts in one chapter exist elsewhere), and making them more contextual, visual, and concrete, leads to more confidence. Student success and understanding can follow from that. Avoid the natural immunity to change (yes the algorithm is beautiful, yet we need to know when it applies), and a desire to aim for entertainment (showing “pi” as “pie” or inventing a “rap”) - put the student engagement where we need it.

Kyle showed a 7 step process posted next to an “instant brewer” coffee machine. (“Actually I just press this button.”) Do we need an algorithm to use one of the easiest objects in the universe? Must we be told to plug it in? He notes that process wasn’t created for the USER of the machine, but for the OWNER of the machine. Now the user has no reason to ask the owner anything! But what if there’s an upgrade? Such algorithms don’t equal engagement, or understanding. 

While technology can functionally improve a classroom, it’s the task that’s going to redefine the class. (Kyle showed a few apps quickly; don’t find an app specific to fractions, otherwise we’ll be switching from app to app for problems. And “Evergreen” apps can be used but they’re not math specific.) The fear is that people will ignore the effective teaching aspect, which has to happen before the transformational technology.

Kyle posed a question about stacking paper up a wall - and showed this with a photo. (Clipart doesn’t do much for him.) Ideally get a number of questions and settle on one in particular. Here if we ask “How many pack of paper to the ceiling?” we can now get predictions. Use “High/Low” strategy (Dan Meyer 3 acts) - what number would be too low? Too high? With a padlet for Google Docs, predictions can be put online - or jot on a whiteboard and take a picture. (Kyle got a class set of iPads funded.)

Don’t give the number yet. Request what other information is needed: How many papers in a stack? Is that an 8 foot or 10 foot ceiling? Figure out what is useful. Refine the predictions - and Kyle had us log into the playkh website with a PIN to play along like a game show.

Students can upload their own solutions. If Kyle wanted to look at exploring proportions (versus unit rates), he could decide which solution to show the class first. Also look for the best incorrect responses. Students can realize that by not showing work, it’s leading to simple mistakes - and you don’t have to reveal whose solution it is. (You can also rotate on the fly!)


Do teacher solutions look like student solutions? Sometimes it’s unnatural to do things the way we’re asking students to do it. The “game show” can then offer extension problems to check “what do students know, what do students not know” rather than asking. For instance, we may not have talked much about variables yet, but can mention it here to check prior knowledge.

Rather than looking up the solution, actually show it (Kyle enlisted the help of a custodian). His answer doesn’t match the math - students don’t think about this. Weight is compressing the stacks? Pushed up the ceiling a bit? Tower is leaning? Floor is slanted? We have made this more visual. Maybe not concrete in terms of physically holding the paper, but might not be needed here. (Will we remember doing math or merely baking cookies?)

Another extension: What would stacks on a table look like? (Linear?) Can a student identify that there is a relationship here between two variables, not merely a division and done? Which variable impacts the other? Proportional reasoning made more explicitly linear (direct/partial variations). Kyle noted that this wouldn’t all be addressed in a single day.

We might want to bump into the algebra, rather than make it explicit. What goal are we aiming for when we include a table? (Solving an equation.. first differences.. the y-intercept!) What we have as given information is slope and a point. Pull y=mx+b from students rather than have them copy an algorithm (as in “a note”). The only way students can do it if we strip away all this context and use simple numbers is if they mimic the teacher/process.

Move the “algorithm” to CONSOLIDATION after the activity. “What does the point (1,5) represent in the context of the stacking paper task?” (1 package gives us total height of 5 - including table) “What does slope/unit rate represent?” (height of a pack) “How tall would this table be with these numbers?” Scaffold toward tasks, moving towards the abstract, rather than the other way around. Giving procedures and then hoping for problem solving at the end won’t work - they’ll be scared by the end.


Executing things this way can shrink the “Not Good at Math” group down, but MORE, it will shrink the “Struggle with Unfamiliar Problems” group down. Continue digging deeper, when you’re ready to tackle another concept. If there are two package stacks of different heights, can we figure out the table underneath? What is the learning goal for students to bump into here? (Solving given two points. Intuitively uses the slope formula, in context.)

Scaffold out to: What happened here? Does it work all the time? “Our students are capable of doing all these things we want them to - we need them to find creative ways of getting there.” Then simplify what they’re doing.

“We don’t do math because it’s harder, we do math because it’s easier.”

Thanks for reading!
-For last year’s COMA Social recap, see "Public Math Relations" with Marian Small.
-For how I set up my classroom lately, see "Grouping Tagline".

Wednesday, 25 November 2015

TANDQ 13: Pass It On

In 2014-2015 I wrote an education column called "There Are No Dumb Questions" for the website "MuseHack". As that site has evolved, I have decided to republish those columns here (updating the index page as I go) every Wednesday. This thirteenth column originally appeared on Thursday, March 26, 2015.

Why must my password include a capital letter?

Because the needs of the many outweigh the needs of the few. This being the one year anniversary of my column, I’ve decided to take a look at some rather simple mathematics that is often taken for granted: that of passwords. Then again, as geeks, writers, et cetera, maybe you have a very good grasp of the subject, along with how long it takes your computer hacker character to crack a code. (Maybe you can even teach me a thing or two in the comments!) I’ll endeavour to be entertaining regardless.

Here’s the basics. There are 26 letters on an English keyboard (I don’t know enough to comment on, say, Japanese). Let’s say that your password has to be exactly 10 characters long (makes it easy!). With 26 choices for each entry, the result is 26^10 or 141,167,095,653,376 possible passwords. Now, what if we include capital letters as options? This doubles the total character set, so 52 choices for each entry, resulting in 52^10 or... well, over 144 quadrillion possible passwords. We’ve increased our previous answer by 2^10. But wait, what if we FORCED at least one capital letter instead (required, no option)? Well, this is going to reduce the total. It only makes sense. When you add a restriction to something, the total will decrease.

In this case, with one (or more) of the characters having ONLY the 26 uppercase options, we can effectively remove every password that is all lowercase - in other words, the 26^10 options we had to start. They’re no longer valid. Granted, when you remove 141 trillion from over 144 quadrillion, you still have 144 quadrillion… but the restriction DID make your password a bit easier to guess. What if your password can be any number of characters? That’s harder. What if it must be at least 8 characters? Somewhat easier again - don’t try guessing a shorter password. (What if it’s a maximum number of characters instead? Then it could be that you’re watching Sherlock, Season 2.) The natural question at this point is: Why force conditions that ultimately decrease total options? It’s a pretty good question.

Predictable Entropy

Before we get into that, a word about password entropy. (I am now contractually obligated to point out this XKCD comic. There’s an in-depth analysis of the mathematics behind it in my ‘further viewing’ links below.) The short version: Entropy is defined as the total number of possible resultant states. In terms of a string of characters, this gives: (total_characters)^(length), the way we had 26^10, above. Computers work in binary, so take log base 2, giving: (length)*log_2 (total_characters) as the binary size of the message, aka bits of entropy. You’ll notice that length is the big multiplier. Yes, log base 2 of 26 is less than log base 2 of 52, but adding two more (lowercase) characters is almost equivalent. (12*log_2(26) and 10*log_2(52) are both about 57.)

So, how many bits do we need for a good password? Well, this website link says 72 bits of entropy/security is strong for short term, but 80 is better for long term use (supported elsewhere, as it means 2^80 passwords would need to be tried). How do we get there? With about 94 characters on the keyboard, we’ll need 80 = (length)*log_2 (94), so a length of 13 characters. (PIN numbers, I’m looking at you.) Here’s the interesting thing. This entropy can be similarly achieved by selecting a sequence of random words, known to many as a “passphrase”. Instead of a keyboard, let’s assume a dictionary/vocabulary of 1,000 words. Solving 80 = length*log_2 (1000) means a length of 8 words (repetition allowed). If this doesn’t seem to buy us much, try plugging in the ACTUAL size of your vocabulary to the equation - the number of necessary words will only decrease. (Unless you know less than 1,000 words.)

The caveat to using a “passphrase” is that it does need to be RANDOM. The second word shouldn’t be in any way be determined by the first. Humans are not good with random - we will pick our birthdate, our mother’s maiden name, and something off The List of 2014s Worst Passwords... all in lowercase. Unless, that is, we are forced away from that inclination using (surprise!) some sort of restriction. So even though there are a few of us who can follow the logic of “length over character use”, for the good of the many who would use their password length to expand on 123456, we must succumb to including at least one upper case character, et cetera, et cetera. It’s not all bad - throwing in a symbol does increase the complexity of a passphrase too.

Of course, all of this assumes your hacker is running some brute force algorithm, rather than being a bit more ambitious, and attempting to steal an entire password file off your network. There’s not much an individual user can do there (aside from constantly change their password, and I pretend that’s why my work account forces me to do this) but logically the system itself has security measures in place. For instance, cryptographic hash functions (a nice little application of high school mathematics). Good enough - until we hit something like 2014’s Heartbleed bug, also an XKCD comic. Or until the character in your novel decides to use telekinesis to figure out people’s passwords. But at that point, you might as well call in Sherlock to get his opinion.

For further viewing:

1. Strength/Entropy: Characters vs. Words

2. The math behind passwords

3. TeachNomination: Password Math (Video)

Got an idea or a question for a future TANDQ column? Let me know in the comments, or through email!

Wednesday, 18 November 2015

TANDQ 12: Text Me Never

In 2014-2015 I wrote an education column called "There Are No Dumb Questions" for the website "MuseHack". As that site has evolved, I have decided to republish those columns here (updating the index page as I go) every Wednesday. This twelfth column originally appeared on Thursday, February 26, 2015.

When will paper textbooks go away?

Never. Yes, I say this despite the president of McGraw-Hill Higher Education stating “Textbooks are dead” last October (2014). In my defence, I can point to South Korea, who (back in 2011) declared they would go fully digital on texts by 2015 - only to back off, in part due to concerns over research about how screen time might affect brain development. And it HAS been shown (in an Israeli study) that those reading on a screen (versus from print) will perform worse in a scenario of timed comprehension - even though they thought they performed better. But wait. Notice I didn’t say paper textbooks would remain dominant. The textbook industry does need to adapt. Let’s have a look at that.

Since 1978, the price of college textbooks has risen more than 800 percent (and DO see that link for the comparison graph). In other words, a text that cost $25 over thirty years ago would now cost more than $225 (new). How can the industry get away with this? Partly because, owing to consolidation, 5 textbook companies now own more than 80 percent of the publishing market. So there isn’t a lot of competition. It also helps that this is a market where the consumers (the students) don’t get a say in the product they have to purchase. (Or do they? More on this later.) But here’s the thing, NO ONE has money for textbooks right now. Even in public education, school budgets are being slashed to lower your taxes, meaning older textbooks cannot be replaced (see also: street potholes). It’s probably even worse than you think - for instance, schools can supplement income with cafeteria sales, but now that all choices are (mandated to be) healthy, students are crossing the road to eat at McDonalds instead. It’s 2015, and I teach a Grade 12 course out of a textbook published in 2003 because THAT is REALITY.

So the first fix involves those unsustainable prices. The second item is more a need to adjust for the slow pace of the education industry. In a prior column, “Getting Graphic”, I noted that “huge technology upgrades are only possible every six or seven years, if that”. It’s largely due to money. But a slow pace isn’t necessarily a bad thing; these are your children we’re talking about. A new drug needs to undergo rigorous testing before being put on the market, otherwise someone gets sued. Yet (it seems to me) that someone can come up with a new education idea, write a book about it, and try to implement it immediately. If it doesn’t work right away? Okay, sorry about your kid’s education, we’ll try someone else’s idea next year. Seriously? (Incidentally, that is not an attack on things like common core, which involved years of research.) So yes, education is perhaps a couple beats behind the mainstream - that’s not something to attack, merely something to remember. There is still a need for paper texts in education even after a majority of society has “gone digital”... which, granted, is coming up fast, if it’s not already here.

Future of Textbooks

So where are we headed? Let’s take a moment to look at where we’ve been. From a look in the book room at my school, a Grade 9 math textbook from 1986 had 11 chapters, and about 450 pages. The format was a page of explanation, a page of exercises, repeat. It contained an occasional black and white (or red-tinted) image. A Grade 9 math textbook from 1999 had 11 chapters, and about 660 pages. The format was 3-4 pages of explanation and examples, then 3-4 pages of exercises. I would say there is only a 25% chance that you would open the book and NOT immediately see a full colour graphic. Our Grade 9 math textbook from 2008 has 8 chapters, and about 620 pages. The format is 4-6 pages of explanation and examples, then 3-4 pages of exercises. There is a huge margin around the perimeter of each page to drop in pictures, or to highlight key terms (otherwise it’s left blank). What do we conclude? That the trend is towards increased examples and visuals. I do question how seeing a picture of someone skiing is more likely to prompt answering a question about “slope”, but one hopes there’s some science behind it.

Looking forwards, the nice thing about an online/digital version of such a text is that the graphics can be made dynamic. They can allow for self-exploration of concepts, rather than simply accepting them on faith (or believing in them because of the smiling photo in the margin). But here we run into a problem - any company can potentially put something like this together, given the right materials. How do you stand out in a crowd? Well, most of the industry seems to have decided that metrics are the way to go, and wow, does this feel like a bad decision! “We must time how long the student spent reading page 3! How often they attempted problem 1.6!” and so on. No. First, we really don’t. While a generalized study might be good (for instance, to see if screen reading really is inferior to print), such data is meaningless without an individual baseline, or any idea for how to apply it. And I don't see us there yet. Second, educators are swamped with extra work as it is, they don’t have time to pore over the metrics of 90 individual students. Finally, putting more effort here feels like it’s taking away from the more dynamic possibilities mentioned above, turning exploration into more of a “hide and reveal” exercise.

Recently, there’s one more issue at play. Post-secondary students are taking more of a stance with regard to the notion of “having” to buy a textbook. A US study conducted in Fall 2013 reported that 65% of university students decided against buying a textbook - even though 94% were somewhat or significantly concerned that this decision might affect their performance. The same study showed that the high cost of textbooks could even affect student course selection. (Aside: John Oliver has a piece, not about textbooks, but about student debt, which looks at for profit schools. See “no one has money”, above.) But there are alternatives to straight defiance, those being: buying used books, the use of an open textbook (one freely available online), piracy (it does exist) - and textbook rental. An opinion article in Forbes claims that low cost rentals are the real industry disruptor, even ahead of digital. There may be something to that.

Because here’s the last piece of the puzzle: Even now, not everyone can afford the technology to view an online textbook. They have to go with a (rented?) print version - or not at all. There’s also a case to be made for the visually impaired student, who uses a text in braille (incidentally 8 times the size of a regular math book). Or other students with exceptionalities, perhaps who experience trouble with focus when it comes to online reading. THIS is why I do not see the paper textbook vanishing. Ever. If it does, I foresee a backlash once more research has been completed into reading from electronic screens. Yet, even so, the textbook industry needs to adjust. And I fear it’s not adapting as well as others believe. But you don’t have to take my word for it.

For further viewing:

1. 2 Perspectives on the Future of College Textbooks

2. Forget the Future: Here’s the Textbook I Want Now

3. Google Interviews Students: The Future of College Textbooks (video)

Got an idea or a question for a future TANDQ column? Let me know in the comments, or through email!

Wednesday, 11 November 2015

TANDQ 11: Rate This Post

In 2014-2015 I wrote an education column called "There Are No Dumb Questions" for the website "MuseHack". As that site has evolved, I have decided to republish those columns here (updating the index page as I go) every Wednesday. This eleventh column originally appeared on Thursday, January 29, 2015.

Are rating systems skewed?

If responses are voluntary, yes. If they’re not - the ratings are probably still skewed. Despite this fact, people will often check a product’s “rating” before making a purchase. Online reviewers (for movies, video games, etc) will also tend to use a “star” system variant in their regular column/show. Perhaps you’ve even been asked to code up a rating system for someone else to use? Regrettably, while there is more than one type of rating scale out there, the problem of skew - which lends itself to an overestimation of reality - is pervasive. Let’s explore that further.

The first problem is one of averaging. If every review is given equal weight, we can end up with a situation like in this xkcd comic, where the most important review is lost in the other noise. (“You had one job!” comes to mind - though of course that phrase itself isn’t accurate.) In the same vein, an item that has 3 positive reviews out of 4 would get the same mean rating as an item with 75 positive reviews out of 100. But while the percentage is the same, the second item is of lower risk to the consumer, because it’s had 96 more people try it out. There’s also the question of when these reviews were posted - are all the positive reviews recent, perhaps after an update? All of this is useful information, which becomes lost (or is perhaps presented but ignored) once a final average score is made available. That’s not to say that the problem has never been addressed - the reddit system, for instance, tackled the problem mathematically. Randall Munroe (of xkcd, see above) blogged about this back in 2009. But in general, the issue of weighted averaging is not something a typical consumer considers.

Even after all of that, there is a second problem. Who is writing these reviews? Everyone who made the purchase, or who saw the movie? Of course not. Generally, a high emotional response (either good or bad) is needed to motivate us to provide the feedback. This means that a majority of responses will either be at the highest level (5), or the lowest (1). Does anyone reading this remember when YouTube had a five star rating system? It has since become “I like this” (thumbs up) or “I dislike this” (thumbs down), because five years ago, YouTube determined that on their old system, “when it comes to ratings it’s pretty much all or nothing”. Now, given these polar opposite opinions, one might expect a typical “five star” graph to form a “U” shape, with a roughly equal number of high and low rankings, tapering down to nothing in the middle. Interestingly, that’s not the graph we get.

J Walking

The graph from the YouTube blog link above is typical, known to some as the “J-shaped distribution” or “J-curve” (not to be confused with the one in economics). It’s so named because there are an overwhelming number of high “five star” reviews on the right, tapering back to almost nothing in the middle - with a small hook on the left, as the “one star” reviews slightly nudge the curve back up. Calculating the mean of a system like this, where both the mode and the median number are equivalent to the maximum, will place the “average” somewhere in the 4’s. In fact, this column came about because of a tweet I saw questioning why an “average” review (3 out of 5) would be considered by some people to be “bad”. Setting aside the fact that some dislike being called “average”... if the J-curve predicts a mean higher than four, the three IS below that. Isn’t that “bad”?

The trouble with comparisons is how useless they are, until you acknowledge what it is you’re comparing yourself against. If you’re comparing a “3” against the rating scale, it’s average - even above average, if the scale is running 0-5, not 1-5! On the other hand, if you’re comparing a “3” against existing ratings for similar products, or on prior data for the same product, the “3” might seem less good... it’s origins may even be called into question. Which actually brings up a third problem, namely that a person may intend to rate something at a “3”... but upon logging in and seeing all the other people who have rated it higher, succumb to “peer pressure”! Giving it a “4” in the heat of the moment! And we haven’t even touched on the problem of illegitimate accounts, created solely to inflate (or lower) the average score of a product. (Of course, what you should probably be doing is comparing the “3” against other scores from that same reviewer. Ideally their scores on your own previous outputs.)

Now, is there a way we can fix this rating system problem? One solution might be to force every user/viewer to provide a review. If all the people with a “meh” opinion were forced to weigh in, it would fill the J-curve. But should their input be given equal weight? After all, being forced to do something you don’t want to do is liable to either lower your satisfaction, or cause you to make stuff up. (Though implementation is not impossible - for instance, requires ratings for a certain number of downloaded videos in order to download more.) Another solution might be to adjust the scale, as YouTube did (or going the other way, IMDb uses 10 stars), but this merely tends to expand or compress the J-curve, rather than actually solve the underlying issue. In fact, I have not come across any foolproof rating system in my research - even the great critic Roger Ebert once said “I curse the Satanic force that dreamed up the four-star scale” in his post: You give out too many stars. (I recommend reading that, as it also points out the problem of having a middle position.)

Meaning it comes down to this: I don’t have a perfect solution. Much like Steven Savage and the issue of franchises from earlier this week, I’m just putting it out there. In particular, should we really trust the ratings we find online? Is achieving an unbiased rating system impossible (short of reading minds) - but the effect something we can ultimately compensate for, the more we understand it? Then again, despite this being the age of social media where everyone's weighing in, reviews might be better left in the hands of the professionals - those people who are paid to assign such ratings for a living. I dunno, do you have an opinion?

For further viewing:

1. A Statistical Analysis of 1.2 Million Amazon Reviews

2. The Problem With Online Ratings

3. Top Food Critics Tell All: The Star Rating System (3 min video)

Got an idea or a question for a future TANDQ column? Let me know in the comments, or through email!

Wednesday, 4 November 2015

TANDQ 10: Around the World: France

In 2014-2015 I wrote an education column called "There Are No Dumb Questions" for the website "MuseHack". As that site has evolved, I have decided to republish those columns here (updating the index page as I go) every Wednesday. This tenth column originally appeared on Tuesday, December 30, 2014.

What is the education system like in... France?

This marks the second of a semi-regular set of columns looking at education systems in different parts of the world. The first looked at England, hence France seems a natural second step. My belief is that this is useful, not merely to learn, but it can help a writer whose fictional characters originate from another country (or world?). Usual geographic caveats apply here, in that when I say “French” I’m discussing France and not, for instance, the province of Quebec in Canada. Which would be somewhat different.

In France, education is free (and compulsory) for children aged 5 through 16. This starts with an “école maternelle” (possibly as early as age 3). Primary school (école primaire) then lasts for 5 years (ages 6-10), middle school (collège, also known as secondary school) lasts for 4 years (ages 11-14), and high school (lycée) lasts for up to 3 years (from age 15 to past the compulsory age). Of note, the French grade numbering system goes backwards compared to North America - the first year of collège is the largest number, year 6 (6ème). It is followed by year 5. The first year of lycée is year 2, then year 1 (première), and then the final year: terminale. The individual years are also grouped into various cycles.

There are 158 days in a typical school year, separated into three reporting terms. Instruction occurs on Mon, Tues, Thurs, Fri, and another half day (traditionally Sat, but in most regions this is now on Wed) to make 26 hours of instruction in a week. The school year begins in early September and runs until early July, during which there are four breaks lasting for two weeks. These holidays begin in: October, December, February and April (where actual dates for the latter two vary based on region - Zones A, B & C). Unlike England, there are no school uniforms - the closest thing they had was already being phased out in 1968.

Testing, Testing

There is no formal testing done at the national level until the end of the 3ème, before lycée. It is at this point that a national exam allows one to obtain the “brevet des collèges” – though one can still attend a lycée without it, as long as their grades are high enough. This exam is one tool used to help determine a path (and lycée) for the last three years of schooling – regular or vocational. Notably, two foreign languages are already needed by this time (selected at the 6ème, and the 4ème).

For the first year of lycée (year 2), courses involve both core subjects and electives, leading to a choice in year 1 of “baccalauréat general” (for one of: Literature/Language, Science/Math, Social Science), or “baccalauréat technologique”. Exams are written at the end of the première, for not only French language and literature, but also for a “minor” area of study, chosen at the start of that year. Then, before graduating, there is another set of exams at terminale. These cover philosophy, and other subjects studied. The final score is a weighted average across all areas, meaning it is impossible to fail a single course - you either pass, achieving at least 10/20, or you must retake the whole year. If you are close (at least 8/10) you may be given the opportunity of an oral exam to make up the difference; an oral is also compulsory for the Literature/Language stream.

Beyond lycée, there is a public university system, but the top schools - “les grandes écoles”, which specialize in engineering, business, etc. - require entrance exams. Napoleon brought this system to Italy, which gives a sense of its history. The intention here is to put emphasis on one’s merit and ability, and not on one’s social or financial status. The exams are given in both written and oral form, where a certain mark must be obtained on the former in order to be considered for the latter. There is no mark threshold here for acceptance - there are limited spaces, and as such you are competing against everyone else who is taking the exam that year. Hence students will typically do an additional two to three years of study (in either a public or private institution) before writing these higher education exams, which can only be repeated once. Once a student is accepted into a post-secondary program, a Bachelor’s Degree takes three years (at either a University or a Grande Ecole). To become a teacher in France, a European candidate needs a three year diploma to be eligible to sit for a competitive examination. Once on the job, they are evaluated by national inspectors.

Outside of the public school board, there are independent private schools, many of them Catholic; there are also five Catholic universities. Religious instruction can be included at these schools, though as long as they also follow the same (national) curriculum as state schools, teachers are still paid by the state. (Of note, they are not paid at the same rate, and their qualifying test, while written to the same standard, is different.) This means that private school fees can be quite low, and compared to the US, a greater percentage of French students attend private schools (though this number is less than 20%). With respect to current events, there aren’t any current reforms in the French system (as compared to the US or England). Time magazine did criticize them in 2010 (in particular for their philosophy requirement), and some feel change is needed to the “grande école” mindset. Do you have any thoughts about the Education System in France? Feel free to comment below!

With thanks to José Piquard, for fact checking. Any remaining errors are my own; please advise, so that I can correct them.

For further viewing:

1. A Typical Day of a French Student (video by students)

2. Education in France

3. France Guide: The French school system

Got an idea or a question for a future TANDQ column? Let me know in the comments, or through email!

Wednesday, 28 October 2015

TANDQ 09: Price Points

In 2014-2015 I wrote an education column called "There Are No Dumb Questions" for the website "MuseHack". As that site has evolved, I have decided to republish those columns here (updating the index page as I go) every Wednesday. This ninth column originally appeared on Thursday, November 27, 2014.

Why don’t more prices end in round numbers?

In large part due to the “Left Digit Effect” - but as a bonus, I’ll also mention “Benford’s Law”, and the pricing of your own items. It seems a sensible topic to tackle as we head into the season of Holiday Shopping, right? Not to mention Black Friday/Cyber Monday for the Americans.

No one (that I’ve been able to discover) knows where the practice of ending a price in a “9” or “99” began. It’s been suggested that doing so would force the cashier to open the till to make change, so that the sale would become a matter of record. But perhaps business owners simply started noticing that pricing at xx.99 was good for sales. Because it is. This has been shown experimentally. Even going back to the 1960s, when a liquor store in Southern California found that pricing their wine UP to 99 cents (from 79 and 89 cents) increased the number sold. How could this be?

One issue is the Western manner of reading, which involves scanning from left to right. So, upon seeing the price $39.99, the left most digit is seen first, and is thus given greater weight - even though by the time we get to the end, the price might as well be $40. (Particularly in countries which no longer mint pennies!) Consider, does it immediately register to you that a $5.99 item is actually double the cost of a $2.99 item? The term “Left Digit Effect” is used to describe how consumers reading $5.xx will interpret “$5 and change”, even if the cents given mean the cost is “Almost $6”. Which, granted, doesn’t quite explain why raising a price would result in better sales, but there’s an element of psychological pricing involved too - if you DO see the price as “Almost $6”, you may get the false impression that the item is somehow on sale. Even if $5.99 is the regular price.

All that said, there’s one other mathematical aspect in play, involving percentages.

Thirty Percent Chance

It turns out that not all leading (“left”) digits are created equal. While truly random numbers (like the lottery) will be evenly spread out across all digits, and truly constrained numbers (like ones which actively eliminate digits) are not subject to the following effect, a set of random measurements (for instance, house addresses) will tend to start with a “1” more often than a “2”, a “2” more often than a “3”, and so on. In fact, the left-most digit in most data sets turns out to be a “1” fully 30% of the time! That’s not even close to one ninth! The mathematics behind it is referred to as Benford’s Law, which describes the probability of the first digit (“d”) using logarithms. This law is even “scale invariant”, meaning it works regardless of whether you measure in metric, in imperial, in dollars or in euros. Why is this useful? Well, for one thing, when the expected first digit pattern is MISSING, we can identify voting anomalies, or catch those committing tax fraud. Yet how does this connect back to shopping?

At first, it seems like a complete contradiction - shouldn’t we see more “1”s, not “9”s? But remember, Benford’s law talks about the leftmost digit. The second digit does not follow the trend to the same extent, and by the time you reach the fifth digit, number choice is fairly uniform from 0-9 (all other things being equal). Why? Let’s consider the percentages. If an item is valued at $10, to move that first digit to $20, you need to double the price - a 100% increase. But for an item at $20, moving it to $30 merely requires a 50% increase - even though both cases involved an additional $10. And moving a $90 item to $100 is trivial - only a bit over a 10% increase in price. (At which point changing the $100 item to the next initial digit, $200, is again fully double.) Such is the nature of logarithms. So why not leave the price at $99? There's not much percentage to be gained by changing it.

Consider also discounts. If there is a 50% discount on any item (under $100), it will end up starting with a “1” so long as the regular price was anywhere between $20 and $40 - yet with the leading digit now being a “1”, it might appear to be an especially good deal. If we increase the discount to 60%, an item at $19.99 would have been less than $50 anyway ($49.98) - yet we may not stop to consider that actual drop in price. We may also perceive a $100 item being marked down to $89.99 as being a much better deal than seeing a $116 item priced down to $105.99, because of the change in place value - even though the price differences are the same.

So, can we relate some of this to pricing your own items for sale? Well, while the “Left Digit Effect” might be in play, a study last year suggested that customers prefer to pay in round numbers. Because really, when was the last time you were at the gas pump, trying to hit a total that ended in .99? In fact, given a “pay-what-you-want” download plan for the video game “The World of Goo”, this study found that 57% of consumers chose to pay round dollar amounts. (I’ve also noticed Kickstarter pledges tend to go in round numbers - is that built into their system?) Some stores will even use round number prices to create the impression of added value or quality. But before we disregard the psychology entirely, there are applications outside of shopping. The link (below) about game prices uses the “Left Digit Effect” as a reason to award 3000 experience points - rather than only 2950 - when you’re coding up your game. After all, the percentage increase from 2950 to 3000 is below 2%.

Will you now pull out your calculator when doing your shopping? Probably not, so your best take away here is to avoid making spur of the moment decisions, particularly when looking at what, on the surface, seems to be a “great deal”. Oh, and you should also check before following any other advice about “secret codes” used in pricing.

For further viewing:

1. Why Game Prices End in .99

2. Benford’s Law (with graphs)

3. GoodQuestion on WCCO News (3.8 min* video)
    * - see what I did there?

Got an idea or a question for a future TANDQ column? Let me know in the comments, or through email!