But Seriously... Is Algebra Necessary?

A friend tagged me in a FB post with a link to a New York Times article by Andrew Hacker Is Algebra Necessary? I need to process this, so here goes....

I saw the article as having two distinct parts. Here are my reactions to each part.

Part The First: Math Is a Useless Pain

The first half or so of the article indicates that math causes high dropout rates and is basically useless to everyone not bound for MIT or Caltech. Well, if math causes high dropout rates, then I think there is a reasonable discussion to be had about how we address the issue. What should we do?

1. Prepare students more effectively from the beginning. That is, should we fix our elementary and middle school math programs so that students can have more success later in their math careers.
2. Fix the high school math curriculum so more students can find success and gain the math skills they need for life and for later mathematical study.
3. Give up. Stop requiring so much math.

The eye-grabbing headline and some of the arguments in this first part seem to imply that Hacker is suggesting a focus on option 3, but after reading the second half of the article, I think he is making more of an argument for option 2. I believe that we need to focus on options 1 and 2.

By the way, the Common Core standards should  help with option 1. A coherent curriculum that focuses on important mathematical practices and covering fewer topics, but with more depth is a great place to start. It isn't a panacea, but spending more time on fewer topics can help everyone out. I think the Common Core standards do a pretty good job of mapping out a coherent mathematical progression from elementary school through the end of grade 8. Then, we come to...

Part the Second: The Traditional Algebra 1-Geometry-Algebra 2 High School Math Sequence Is Mostly Useless and Should Be Improved

OK. This is where Hacker and I agree. In an earlier post, I said that we drive too many kids (like cattle) on the pathway that leads to Calculus. As I said in Math Illiteracy, everyone needs to master some math skills and understand some math concepts, but these don't necessarily map well to the cattle drive to calculus that I think is over-enrolled. By pushing just about every kid through the same cattle drive, we are driving kids away from math and putting up barriers to later pursuits that should not be dependent upon relatively esoteric skills such as learning how to factor cubic polynomials.

Maybe most kids should take a (perhaps more applications-oriented) course in Algebra, but maybe fewer kids need to take geometry and Algebra 2. Maybe a course in Practical Math (the actual title of a course on which I am working) could help. This sort of course can help students see how math can be a tool for solving problems. It can still be a pretty rigorous course, though it certainly wouldn't prepare anyone for calculus. But then again, not everyone needs to take calculus.

From my perspective, the tricky part centers around mobility. If we have a math pathway that prepares potential engineers, scientists, and mathematicians with the math skills they need, and we create a separate pathway that helps the general non-mathy population acquire the skills they need, then how do we ensure that a student who starts on a less rigorous pathway can change his or her mind and switch to the mathy pathway? How do we make sure that the less rigorous pathway doesn't become a dumping ground for students who are ill-prepared for the more rigorous pathway? These are very real concerns. Perhaps I'm being a paranoid Black man, but I know that this sort of dumping ground mentality has been wide-spread in the past, and I'd hate to create a two-tiered curriculum that institutionalizes it.

The Bottom Line

As Hacker says:

Mathematics is used as a hoop, a badge, a totem to impress outsiders and elevate a profession’s status.

Here, he is talking about how medical schools (and other post-graduate programs) use calculus as a way to identify students who have the right academic stuff. Honestly, I can't see that changing any time soon. Also, I'd be interested in a study about this. Maybe being able to learn calculus has a high correlation with the ability to survive and thrive in medical school. Maybe this extends to undergraduate studies as well. As the author of the A Mind for Madness blog says in Thoughts on Nicholson Baker’s Case Against Algebra II:

... if a high school diploma is meant to indicate some level of readiness for college, then algebra is probably a good indicator. This does not mean that you will use it, but will just point out that you have some ability to do some abstract things. I’m not saying it is the only way to test this, but it is probably a pretty good one.

Regardless, I think we need to:

• Prepare more students for success in mathematics more effectively. We shouldn't focus solely on the cattle drive to calculus, but then again, we shouldn't ignore it either.
• Do a better job of creating math courses that engage all students. We need to have students wrestle with more authentic real-world applications and see how math can help solve real problems.
• Help students see the beauty in math in addition to the utility it affords in problem-solving contexts.

Paying for Cheap Gas

A few weeks ago, xkcd had the comic Working, which put to mind an app I need to develop or commission.

Whether it makes sense to drive out of your way to save on gas is more complicated than just the economic argument the xkcd comic puts forth in the image. The caption gets at the more interesting thing to me. When you drive farther, you are using up more (probably expensive) gas.

Let's take a simple example. You need 10 gallons of gas and have a gas station right on your way to work. But if you go 2 total miles out of your way (let's say it's a mile in the wrong direction), you could save 5 cents per gallon. Does it make sense to make the detour?

If your car gets 20 mpg in the type of driving you do for this detour, then you are using an extra 1/10 of a gallon to save 50 cents. If gas costs less than $5.00/gallon, then your detour makes sense (ignoring the cost of your time and the increased wear and tear and maintenance for your car).

Now, let's try another example. Let's stick with the 20 mpg because it's an easy number to work with and probably not too far off for many non-hybrid cars.

detour distance: 5 miles (1/4 of a gallon)
save per gallon: $0.10

This scenario makes sense if the cost per gallon is less than $4.00/gallon.

Now, let's look at a decision I get to make. These numbers represent the data for my nearest Costco gas station:

detour distance: 20 miles (1 gallon)
save per gallon: $0.14

This scenario makes sense if the cost per gallon is less than $1.40. Notice that this detour doesn't even make sense if you drive a car that gets 40 mpg.

Anyway, it amuses me that people start going out of their way to save money on gas when the cost of gas goes up, but when the cost of gas is high, the detours make less sense than they do when gas is cheaper. I need to write an app that doesn't just find cheap gas, but also allows you to put in your fuel efficiency and gallons needed to help you determine which cheap gas is worth the trip.

Khaaaaaaaaaan!

A friend at work sent me this link to a 60 Minutes article Khan Academy: The Future of Education?

I haven’t seen the 60 Minutes piece, but have tired of the idea conveyed by the title. I have blogged about Khan before, so I'll keep this short.

• Mathalicious's blog has a nice article Khan Academy: It's Different this Time. 
• On his MAA blog Devlin's Angle, Professor Keith Devlin brings us The Difference Between Teaching and Instruction. 

I could go on and on with links to articles that put Khan in some reasonable context, but I’ll spare you the pain. The basic idea is that I am not alone in thinking that if Khan is really the future of education, then we are in deep doodoo. Frankly, I would not really call Khan Academy’s stuff “education,” but rather would consider it “instruction.” If we turn education into nothing but a series of activities that a microchip can perform, then we are on a very dangerous path.

Don't get me wrong. Khan isn't bad. Using it as a way to support struggling students, or as a way to help flip a classroom and allow teachers to focus on engaging with students, it could have great value. The Daily Riff has The Flipped Class Manifest, which provides some insights into how and when flipping a classroom can work, as well as a nice set of links to articles that provide more depth. 

Regardless of its benefits, my concern is that Khan Academy could be used to replace teaching with instruction, and I think that's bad.

New Math Education Wars: The Wrath of Khan

With their tutorial videos, Khan Academy has made a big splash in the education world. Having Bill Gates sing your praises in a TED Talk is a sure way to get a lot of attention.

Over at Action-Reaction, Frank Noschese has an article Khan Academy: My Final Remarks. I agree with many of his points. Here is a quote from near the end of the post:

Khan Academy is just one tool in a teacher’s arsenal. (If it’s the only tool, that is a HUGE problem.) Khan Academy can be useful for some kids as vehicle (build skills) to help them get to better places (solving complex problems).

On his Quantum Progress blog, John Burk also chimed in with Project Euler vs Khan Academy: The Future of Online Learning. I agree that Project Euler is really, really good, but that is a topic for another post.

Back in 2007 (before I ever heard of Khan Academy), I started developing instructional videos that are somewhat similar to Khan’s work. I think they are great tools, but they are not the be-all and end-all of online education. When placed in the context of a complete course, they can be helpful tools. We need to continue to innovate, but just shouldn't get too carried away by the latest educational fad and think it will solve all our problems. Good teaching isn't about amazing lectures. It's about feedback and engagement and connections.

At their best, instructional videos are pretty sentences. Quality education is a set of engrossing interactive stories.

Brain Calisthenics: Using Perceptual Learning

A friend sent me a link to a New York Times article Brain Calisthenics for Abstract Ideas. The idea is that "playing" with a concept can yield big educational gains. A strictly top-down approach to education (here's the formula, now apply it) is not necessarily the best way to work. This perceptual approach seeks to leverage our ability to build intuition through experience.

Here are some of the games the article references.

Basic Math

Measurements and Graphing: Match the equation to the graph and learn to perceive basic measurement concepts.

Positive and Negative Feedback

Extreme Ball: Time a fan to blow and push a ball attached to rubber bands.

Extreme Population: Help your city reach a population of one million citizens.

Stabilize Ball: Time a fan to blow and stabilize a ball attached to rubber bands.

Stabilize Population: Help your city's population stabilize at 500,000 citizens.

MathEd: New Blog Idea: Real Math Applications

Here is an idea I have had for a few months, but am finally getting around to putting it out there.

How about a new blog that is all about how REAL math solves REAL problems. I am not talking about word problems. I am talking about people documenting ways that math helped them solve real problems. Note that this is not about finding applications for all the math in the course you are teaching or developing right now. This is about finding a way for people who use math to solve problems to share these problems and solutions.

Here is a statistics example from my office:

The Problem: A Project Manager at my company came to me asking how he could present data about variances from planned budgets in a way so that the big variances (both over and under) from budget wouldn't distort the data. When he used means to display the data, a few outliers over stated the degree to which early budgets varied from the final numbers.

The Solution: I first suggested that he look at a trimmed mean, but the solution that ultimately worked was using medians. Medians are less influenced by outliers than the mean, so using them got the job done.

The goal with a blog like I am proposing is to provide a rich repository of ideas for educators and learners to see how math is really used. Note that the problems don't have to come up in office settings. Some might come up at home or in sports or any other context, but in each case, math needs to be used to solve a real problem.

Any interest? Has it already been done? I think we need a set of contributors for this endeavor. I am fine being a contributor, but couldn't take the entire burden on myself.

Geeking Out With Frink

A friend and former student tweeted a link to Frink, which seems interesting. According to creator Alan Eliasen, Frink is...

... a practical calculating tool and programming language designed to make physical calculations simple, to help ensure that answers come out right, and to make a tool that's really useful in the real world.

So as a math guy, it seems relevant to what I do. I'll check it out, but the fun part is reading the documentation. Check out the Sample Calculations section of the documentation. Feel free to ignore the math. It's all about the narrative he wraps around it. Really amusing stuff. Here is a very small sampling of Alan's comments/quips.

Superman:

Sure, he's a great guy, and, sure, he's the Defender of Truth, Justice, and the American Way, but can't he find a better use for his super-powers than schlepping some shiksa into the stratosphere? Shovel my walk, he could, in 3 seconds--and me with the sciatica.

Flatulence:

Fart jokes. Sheesh. If Frink isn't a huge success, it's not because I didn't pander to the Lowest Common Denominator.

The ramifications of the most famous of Einstein's equations:

Unbelievable. The energy in a teaspoon of water, if we could extract it, is equal to burning more than 3 million gallons of gasoline.

The amount of heat emmited by a human body in a day:

... your average power and/or heat output is slightly less than a 100-watt bulb. (Note that your heat is radiated over a much larger area so the temperature is much lower.) Many days I could be replaced entirely with a 100-watt bulb and have no discernible effect on the universe.

Anyway, check out the documentation. It's fun stuff and did a great job of getting me interested in using the tool for exploring quantities all around me.

Web Stuff Friday: Teacher Blogs Worth Reading

I have found a group of math teachers online who have made a really strong online community. Together, they are questioning, innovating, and making positive change happen and then sharing through blogs and tweets. It's incredibly inspiring. Anyone who is a new math or science teacher should read these blogs and follow these folks on Twitter. Get into the discussion.

Honestly, this is the best use of blogs and Twitter I have seen yet. These folks rock the house.

That said, here is a list of some of my favorite math teacher bloggers/twitterers:

Dan Meyer (@ddmeyer)

Blog: dy/dan

Dan has been featured on TED, so lots of people know who he is.

Kate Nowak (@k8nowak)

Blog: f(t)

Hers is the best subtitle out there. I will make you click the link to read it. Go ahead. I'll wait....

Sam Shah (@samjshah)

Blog: Continuous Everywhere but Differentiable Nowhere

Sam's post on the Blogotwitterversphere is great, and attaining mention on his Favorite Tweets page is a goal for every math teacher tweeter.

Matt Townsley (@mctownsley)

Blog: MeTa Musings and With Students in Mind

Like me, Matt is a former HS math teacher turned curriculum dude. His focus on assessment is a good thing.

Jason Buell (@jybuell)

Blog: Always Formative

Jason's ideas about assessment are important (and by "important" I mean that he and I agree on some key points).

BTW: To get plugged in to the flow of the conversation, check out the mathteachers list maintained by Jackie Ballarini. Great, diverse list of folks.

Note that I have left a ton of people off this list. Check out the blogs and the teacher twitter list and start roaming to find more.

Seeing the Language of the Universe Through GeoGebra

Galileo once said:

The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

Good stuff. Think back on the post Math Illiteracy and this becomes pretty distressing.

In all honesty, I was never much of a geometry guy. I like algebra and math modeling, but have generally avoided geometry, which has always seemed like stuff that was either obvious or magic. I never wanted to teach it because I never felt like I had a good enough intuitive feel for the content that I could convey to anyone else.

In the past several years, some software has helped me see the light of Galileo's wisdom. First, I liked Geometer's Sketchpad, but now I am digging some new (and free) software called GeoGebra. I think it is really cool stuff. If I had GeoGebra at my disposal when I was a teacher, I would have begged to teach Geometry. Software like this really makes the ideas of math come alive. Install it and give it a try. When you actually engage with the content, you can see beyond the onerous two-column proofs and begin to understand the language of which Galileo was speaking.