I'm going to be picky for a second. Pop open any Calculus book and you'll find a table of integrals:

Stewart 7th Edition Single Variable Calculus

Stewart 7th Edition Single Variable Calculus

For whatever reason, the natural base is always called out as unique and special. If you look at the equivalent derivative table you'll see it gets a call out from a generic exponential. But why do this? The natural base e is just a number, albeit irrational. If you integrate a base e exponential using the generic rule, you get e^x / ln e which reduces to e^x.

Calling it out as something special with a different rule loses site of base e being one of many possible bases for exponential functions. That somehow e isn't just a number. The same hang up exists with π whether you've noticed it or not. And it furthers the love of special cases: specific steps must be remembered for almost everything, when really you've missed an opportunity to show algebra in action.

AuthorJonathan Claydon
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An article made the rounds a while ago: Colleges Reinvent Classes to Keep More Students in Science

A lot of smart twitter folks had the same reaction I had: "duh."

Now, to be fair, at no point in my career have my techniques met with strong critique. No one has accused me of doing harm or doing less than what's required. But there's a stigma. You can feel it in professional development sessions. You're showing some veterans about how great it is to have kids talking to each other and figuring stuff out and they kind of scowl or find a way to tune out. They play along to be polite but you know 0% of what you discussed will make it back to their classroom.

And in these instances it comes back to the eternal straw man: we've GOT to prepare them for college.

No one stops to consider that college teaching is sub-par. Or that the environment is completely foreign to high school. Other than desks and books it's not the same thing. Around the time I saw this article a teacher from another campus was remarking about some college professors who were blown away by Bloom's Taxonomy, something engrained in high school PD for like, ever.

Why do most people lecture eternally? Because they do it in college. Why do most people give mountains of homework? Because they do it in college.

You know what else happens in college? Class sizes of 150. Three hours of "face" time a week. Learning for the sake of learning disappears. GPA becomes the only thing that matters. You give a student two opportunities to perform in 18 weeks and they're going to get desperate. You make a student a number on a list and they won't care about deceiving you. You find the right kids on a college campus and you can earn a master's degree in cheating. No one likes talking about that either. Now, the students I knew who relied on that sort of thing didn't last long, but essay mills are a thing.

You all had the bad professor. Sixth semester fluid mechanics for me. Fellow walks in and says what are your questions. None? Ok. Ninety minutes of reading overheads. Scolding us for not asking questions. Threatening to go faster. Whipping overheads every 15 seconds. Getting more disgruntled each time. A week later he told us that merely sitting there complacently for an entire semester will earn us a C. I got a B in the class, and I know nothing about fluid mechanics (my exam grades were all less than 50%). Nothing he did inspired me to care one iota about his material. Said professor had several prestigious teaching awards to his name because, of course.

It was anomaly, and I had plenty of good instructors. But c'mon, THAT's allowed to happen? That's what everyone was trying to prepare me for?

Why in the world should I do anything like a college professor? I know all of my students by name. I see them five hours a week, every day of the week. I ask them to perform dozens of times and watch them grow from feedback. They produce something every day. There's no need to offload the instruction into a daily pile of homework. Final exams are meaningless to me, because I've watched a student every day for months, I don't need some high stakes 80 question test to tell me if they learned anything.

If a university lecture hall forms the basis for your high school classroom culture, go visit a lecture hall and update your memories.

AuthorJonathan Claydon

Time for another rant. Some topics in math are ripe for abuse. Previously, I looked at the sorry state of rational functions and exponents. Another has drawn my gaze and that's logarithms. I admit that in the two previous scenarios and this one, I am totally guilty of each bad practice.

Logs are usually in an unfortunate position. Often taught towards the end of the year, and tough to demonstrate as practical. Should you go on to higher math and science there are plenty of reasons to love logs, but at the high school level it's pretty rough.

Why? Well, log laws. Logs have a handful of neat properties, and those properties were a way to simplify hard multiplication prior to calculators, and they power the slide rules that sent us to the moon. Log tables were an important part of textbooks. With computers better equipped to handle nasty numbers, the properties get turned into a "nice to know" sideshow.

Case in point:

Pre-Calculus with Limits 2005

Pre-Calculus with Limits 2005

The student is asked to extract these single logarithms into several to demonstrate their knowledge of log laws. The glaring problem is why? Why does this matter? Most of these can't be graphed, but let's examine #37.

Screen Shot 2013-12-30 at 3.44.34 PM.png

C'mon. This sort of stuff right here, the demonstration of properties with no context is what undermines math education. Sure, they'll diligently expand the log for you, but what has that accomplished? What better understanding of logarithms do they have now that they can expand some unwieldy function into a string of unwieldy functions? Why is it a problem worth solving?

A better approach might be the algebra that proves this scenario.

Screen Shot 2013-12-30 at 3.48.48 PM.png

Lacking real world context? Yep. Contrived? Yep. Unwieldy? Yep. But what's different? Instead of demonstrating log laws for the sake of demonstrating log laws, you can see where they expand the algebra skill set. That hey, there's two answers, so an embedded quadratic shares the properties of a normal one?

Curriculum needs to flow. The work done in September should connect to that taught in March. A series of random diversions dilutes the big picture. Log laws have a place, but they should not be trotted out to the center ring and forgotten a week later.

AuthorJonathan Claydon

Which method of solving a problem would you prefer to explain to a group of novice mathematicians?

What purpose does it serve to let math technology sit still? Will the class of 2020 still take the SAT on a calculator from 1996?

Stop making students rediscover fire.

AuthorJonathan Claydon
2 CommentsPost a comment

Teaching is not a rich person's game. Duh. But marketing materials have a way of displaying all of us as extreme penny pinchers. Take, for example, a couple excerpts from teachers featured on the restyled Apple and Education landing page (plenty on this in the future). The spotlights on iPad and Mac deployments have some good things to showcase, but there's a lot of the same old marketing talk. Then there's this:

Brandi McWilliams:

Every teacher is on a budget. If you see an app that costs money, keep looking. A lot of times there is something similar for free.

And a second.

Amy Heimerl:

I always search the free apps first. There are so many great free apps out there in every subject.

I mean no disrespect to these educators. Like of all us, you make the best of the tools at your disposal. If a district is willing to fund an iPad program, good staff find ways to make use of the investment. But these quotes are dangerous stuff.

I have no problem with free. Many good tasks can be accomplished on an iPad using the built-in apps or ones that are free. These quotes suggest that you have no business paying for anything, that free always wins. 

A lot of students are looking to become software developers. It is an industry hurting for native talent. Computer science is a degree that can take you anywhere (just ask crazy engineers in education), and the kind of students who would make great programmers don't always have access to learn about it in school. To say free apps are the only way to go really devalues the work of a computer programmer. What happened to a business model of selling a product for more than it cost to make? What kind of support do you expect from a developer who makes little to no money on their app? What are the odds that free app still works when it's time for iOS 8? What is the impact of going with free apps that are supported by ads? What are you telling students about the value of hard work?

There isn't much more of an App Store gold rush anymore. These guys are not all millionaires.  Some are lucky to get sales that cover the cost of development. To devalue their product in such a way that free is the only acceptable option does a disservice to their profession.

Aren't teachers the perfect demographic to understand what it feels like to have your craft devalued?

Let's assume the school bought all the iPads in your room, so you never had to make the $300-500 investment. A $5 app (1-1.6% of the purchase price!) does not have to be purchased for each and every one of your classroom devices. Signing in to the App Store on each iPad will allow a retrieval of purchases: 


I have 15 iPads in my room. A lot of what I use is free. Some of it was paid ($5-15). Anything I paid for was purchased once. Plunking down $5 for an app that can be put on 15 iPads quickly sounds like a bargain to me. Why restrict myself to free? 

What did you spend at Office Depot this month to get ready for school? 

If we're starting to say apps are just as important as pencils, where's the demand for free pencils? 

Though we did solve that free calculator problem. 

AuthorJonathan Claydon
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Interactive whiteboards haven't been in my school long. We're in year two or three where you can expect most of the classrooms in the building to have one. A student is used to staring at one six or seven times a day. What's strange is the built-in respect the technology has earned in so little time.

The piece that garners the most attention is the large plastic square hung on the wall. It is the One True Source™ for all interactive classroom goodness. What is it really? A big mouse. But it SEEMS like it's doing all the work because the magic pen will let you manipulate items on the screen. 

As long as an image appears on the big plastic square, the board is "in use" whether I'm standing in front of it with the pen or not.

Case in point, a few years ago I stopped standing in front of the board

The projector was still on. An image appears on the plastic slab. I wasn't using the software that comes with the board. I didn't touch the pen for months and months. But as far as the students were concerned, it was in use. 

Then late last school year, we went from a dim projector to something a little clearer at the front of the room:  


The comments were instantaneous. "Wait, you aren't using the ActivBoard?" "Why aren't you using the ActivBoard?" "What happened to the ActivBoard?" They do refer to it by name, so success for the marketing department there.

And I stand there puzzled. I never touch the pen. I never use the official software. I never bring any of them up the board to "engage." Every advantage to teaching off a computer is still present, but roll something in front of the slab and the room goes crazy.

Staring at a technology is not equivalent to understanding how an 84" mouse works.

AuthorJonathan Claydon

I brought this up once before  and it triggered a reminder about another subject we put too much importance on in Algebra II: exponent rules. Again, they are important and they have a place, but sometimes it just goes a little too far:

Holt Algebra 2 with Trigonometry 1986

Holt Algebra 2 with Trigonometry 1986

Similar to the other exercises I pointed out, these appear towards the end of a problem set. But what do these represent? Can you name something that is modeled by a ninth power? How often are scenarios controlled by 3 variables of incredibly high powers like this? 

I know what you're saying and I agree, the ability to simplify expressions like these serves to demonstrate a high level of comfort with mathematics. Too often I've taught this unit and the exponent rules only serve to confuse. Students are very good at subtracting or adding the powers of like variables, but they like to apply it to the constant as well. In #32 for example, a good number of my students would put an 11 in the denominator of the simplified answer. 

Is that a failure on my part? Sure. Should we spend more time on the underlying misconceptions? Sure. Do we have the time to do that?

That last question is the tricky part. The length of Algebra II curriculum is so long that there is pressure to keep the foot on the gas. With a long parade of functions to get through, we've 3 days to master these exponent rules kids and that's it. The problem is they never get mastered because there's no other areas of Algebra II where it's necessary to reduce 8th, 9th, and 10th powers. It'll appear in Calculus, but the context of finding the derivative of an 8th power is just as contrived. 

I also hear you say "but students are being forced into remedial math now more than ever!" Will continuing to power through an incredibly long curriculum help this? Or did the curriculum creep cause this in the first place? Would a more focused topic list in high school drive home low-level skills that would improve scores on college math placement tests? 

And speaking of contrived context: 


Not an authentic textbook problem, but we've all seen something like this in an attempt to work in a geometry crossover.

AuthorJonathan Claydon
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I might draw some ire for this one. I'm not much of a pure mathematician. I don't care much for making elegant proofs, the properties of oddball trig functions, or generalizing the sequence for making inscribed polygons (although they are pretty).

Clearly, because I don't care should mean nothing to the people that enjoy that sort of thing. Go nuts, ignore me. 

Where these highly abstract tasks go awry is when you try to put them in your high school textbook. Is this an implication that high school students should be capable of these? Am I bad teacher for skipping them? What are these problems trying to accomplish? 

There are many offenders, but rational functions, you by far can be the worst. 

Exhibit A: 

Holt Algebra 2 with Trigonometry, 1986

Holt Algebra 2 with Trigonometry, 1986

Granted, these were towards the end of a rather lengthy problem set. But what are these then just a way to find a problem that will take 10 minutes or so to simplify? Assuming the student doesn't trip over the many many algebra hurdles lurking in here. 

The last one is interested in the roots of a random rational monster.

Screen Shot 2013-07-09 at 1.43.51 AM.png

Nasty asymptotes abound, and there appears to be only one solution. I could see a more advanced math class tackling this function. Certainly I made one of you reach for a pencil to see if you could find the answer numerically (honestly, I almost did too). But why is this in high school math? 

Exhibit B:


At least the headings are up front about it. My college Calc I professor had us tackle something like the first ones. I have zero interest in messing with that bottom one. 

These kind of exercises have a place, but I don't think it's in high school. We just aren't given the time to make the foundational skills solid enough. My main criticism of curriculums that are a mile long is that there is no time to stop and really spend the hours working towards mastery of a skill. If your academic Algebra II sections are anything like mine, all the algebra hurdles from the book exercises would completely obscure whatever I was trying to teach about a rational function. It's a problem in Pre-Cal and Calculus. Strong algebra skills get sacrificed in the name of these random special cases because it's on the semester sheet.

Anecdotally, rational functions are big targets for the "when will we use this in life" crowd, in which case I agree with the frustrated student. 

AuthorJonathan Claydon
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Google had a keynote of epic length earlier in the week where they outlined plans for their platforms over the next 6-12 months. Tons of bullet points, lots of promise, though if last year is any indication, most of their hopes and dreams won't play out like they wanted. Deep within the 3h 25min feature film was 6 minutes about captial E Education. If you cue up the full keynote, the education part runs from 1:14:00-1:20:00.

In the brief section they outline a problem they encountered: teachers find there is a gap between what technology is capable of and what is practical for school environments. Google sees this as an opportunity to help them out. Their solution has two main ideas: fast deployment, and curated content.

The Promise

From Google Play for Education:

After finding apps they want to use, educators can push them instantly to student devices over the air. They can send the apps to individuals or groups of any size, across classrooms, schools, or even districts.

In the keynote video, there is a short sequence where an app is located via desktop browser, a Google account group is specified, funds are deducted as necessary from a district account, and the app is marked for deployment to the specified account group. Details are fuzzy on how this will look in practice. The implication is that student devices signed in to a privledged account will be pushed new content all at once. Should this work out, it would definitely beat the current way to get an iPad up and running. Apple has tools for mass deployment, but it's really intended for corporations, you have to have a business level iTunes account. It is not friendly to a situation where the teacher is seen as the primary maintenance person.

Second, they hint at a special section of the Google Play store with curated education content:

With Google Play for Education, teachers and administrators will be able to browse content by curriculum, grade, and standard — discovering the right content at the right time for their students. If your app offers an exciting new way to learn sixth grade algebra, we'll make it easy for math educators to find, purchase, and distribute your app to their classes.

I'm not sure if this is going to have the desired effect. Curated lists of "great education apps" are numerous. A search for "education iPad apps" yields a ton of "Top 12 Must Have..." or "20 Apps Teachers Can't Live Without" articles. Plus, Apple even carved out a chunk of their store for iTunes U to give us...hundreds of videos of lectures?

The Practicality

Google's first issue is one I hope they solve and Apple notices. Distribution and deployment should not be as difficult as it is currently. iPads are not what I would consider difficult to get up and running, but I'm a guy with an engineering degree who has played with computers since I was 8. Half the problem is absolute techno-phobia/indifference in your average classroom teacher. It's no different than a problem math student. A solution with more than one step is an instant "no way, that's too much mister." Should Google succeed in making it easier for an individual teacher to get something up and running, good for them.

The huge huge elephant in the room here is that at no point during the 6 minute presentation was there word one about monitoring the quality of technology based instruction. If you look under the Get Started section of their Educator Program, there are no guidelines about what teachers will be reviewing these things, how quality is determined, or provide a single sample use case for an aspiring developer. There seemed to be no focus on use cases in the Education Store screenshots either, just showing it as a nicely designed app directory.

The Problem

This scenario seems to be continued lip service to education without providing a single example of what they think a good education scenario looks like. Unfortunately, educators/administrators will buy what they're selling because a) technology stuff is hard b) Google and Apple are smart, they must know what they're doing.

But they don't!

Google's scenario will continue the idea that magical, wonderful education technology can be found with a magic bullet app. In their keynote, the presenter mentioned 550 different apps were used in their two pilot schools. 550! How do you build lasting routines like that? How do you expect a strapped school staff to manage that? What in the world is significant about 550 apps other than being a really big number that you can impress people with during a keynote?

Apple's official party line isn't any better. iBooks Author supports the notion that digital textbooks with embedded video are the answer. iTunes U supports the notion that there's nothing wrong with the current structure of college lectures, and that the world will be better served by getting to sit in a boring class for free. In March of this year, Apple published a very well made video about a high school in Boston that went 1:1 iPad. Take the time to watch it and see if you can find what is revolutionary about any of the activites done by those students. They are able to create some movies, ok. But...they take notes...digitally. They read a textbook...digitally. In one scene, a teacher projects something from a 90s era overhead and a student takes a picture of it.

Google set out on this mission because "when I go visit my kids' classroom it looks pretty much like it did when I went to school." But what are they really doing about it that isn't just a new coat of paint?


AuthorJonathan Claydon

I recently closed out the academic portion of Pre-Cal. We're in the middle of producing a short film and then tackling some small projects that summarize everything we've done this year. The final act of their tested curriculum involved a little bit of calculus. We went through finding limits by direct substitution, determing the value of limits that seem to be undefined, limits at infinity, and derivatives.

Every year I play a dirty trick on them in the section on limits at infinity:

Photo May 07, 9 52 15 PM.jpg

It's an exercise that draws inspiration from this test question by Sam Shah which was in turn inspired by a post from Bowman Dickson. Basically, what happens when you present a student with the unexpected? In the 90 or so papers I have, there are dozens of students that made the same mistake on this question. There were no in class examples like it. Though I did mention that the largest exponent has control of behavior throughout the course of the material. This is the kind of thing that makes it clear who processed the full concept and who was pattern matching to examples.

I would fall prey to these oversights in school all the time. A particularly nasty Thermodynamics exam I took was full of things like this.

Now the conundrum: is this a good test question because it showcases who internalized the concept completely? Or a bad test question because the answer is so obvious to me, the seasoned veteran, and hahaha can you believe these inexperienced teenagers are no match for me?

AuthorJonathan Claydon