Thursday, October 30, 2014

Boot Camp Algebra?


Is Algebra a problem or an opportunity? 

The Algebra failure rates for high school and community college students are so high that many including the pre-eminent two-year college math group AMATYC are considering eliminating the requirement, with some substituting, eg, Carnegie Foundation's Statways/Quantways.

I raise the possibility that in fact Algebra presents not a problem but rather the perfect and even indispensable opportunity for us to turn American education around, a tightly defined battleground where we can decide once and for all whether American students can once again work hard, persevere, think rigorously, and work meticulously, not just in math, but in all intellectual endeavor academic or professional.

Continuing the unfortunate military analogy, Algebra can be an academic boot camp akin to the military's six to twelve week programs in which undisciplined, out of shape civilians quickly learn the discipline and basic skills required by the services, aided in their efforts by a well-constructed program delivered by dedicated professionals, and by a clear understanding up front that they are in for a challenging experience.

Where We Stand

A lot of folks want to drop the Algebra requirement for college. Why? Because so many cannot pass it. But is Algebra the canary in the coal mine? Algebra is not that hard. If students fail Algebra then we need badly to fix something, but that something is not Algebra. Note that I am worried neither about the canary nor Algebra: I am worried about the atmosphere being fatal to all learning.

Will dropping the Algebra requirement help? The skills required by Algebra -- learning a set of rules, knowing which rules can be applied when, then applying them accurately and meticulously -- are skills required in any job requiring a degree, STEM or not (see next), and many technical jobs not requiring a degree.

Why not train kids up on a generic meta-skill required for any job (or hobby) that is at all interesting?

(See Next)

My most recent computing gig in tall buildings was for a dental insurance provider. Claims adjudicators might have to cover three or more states, each with its own rules and regulations such as compliance windows (eg, claims might have to be settled within thirty days, twenty if delivered electronically). Within each state, different coverage plans were supported, each with different rules. A single plan would have one or two hundred rules covering different procedures and classes of patient and provider. A single rule might be expressed in a technical paragraph twenty to one hundred words long. Rules interacted, with one rule having precedence over another where both applied.

If someone cannot pass Algebra but we Oz-like hand them a degree, how well are they going to do at such a job?

Do not sell students short

Kids vastly prefer succeeding at academic challenges over getting free passes on them. What is hard for many is in fact succeeding, but if Algebra is easy (it is) why not use it as an opportunity to show them how to succeed with a logical set of rules before they tackle their intended profession? The profession itself will bring its own added complexity such as the dental procedures above -- better to get the meta-skills of study, perseverance, and meticulousness out of the way in an isolated context already mastered (arithmetic (I know, see next)) so they can concentrate on the logical system? They can walk before they run..

Next. Yes, I know, one of the big problems is that they have not in fact mastered arithmetic. That too is easy and curable with a bit of work and concentration. See where this is going?

Algebra is perfect because it presents all our learning ills in a nutshell: lack of basic number skills, meta-lack of accepting ownership for learning, in turn reflected in the meta-lack of perseverance. Algebra is a subject challenging enough to require the perseverance and self-direction we wish to cultivate, and a subject which, when mastered, will give students pride and confidence to reach higher. Earlier I said it might be indispensable: what other subject presents such a pure training ground?

Who knows? Instead of failing Algebra, many may be encouraged to aim a little higher at more rewarding (financially and otherwise) STEM jobs. Dropping Algebra will inspire no one, instead merely reinforce for students that they just need to fail often enough and we educators will move the curve.

They do not get much accountability in high school, but will respond well to accountability if we make clear to them that they are in college now and the rules are different. Isn;t that prospect more exciting than continuing the accountability forgiveness by dropping Algebra?

The problem colleges and students have is not that they cannot teach/pass algebra. It is a meta-problem of not knowing how to work, of expecting passing grades to be handed out in return for a minimum of effort. Google “grade inflation" (OK, here it is and here is another) look for the spike (in a chart on the first link) beginning mid-sixties.) And here is an anecdotal but stunning example of how we got soft. The natural consequences of lowering the bar were well documented a decade later: Nation At Risk.

The nice thing about trying against the apparent odds to succeed in teaching algebra: at least we will believe in what we are doing. No one will take satisfaction from abandoning Algebra, except those students who consider it an unreasonable imposition on them. But how are those people going to respond to the demands of their profession? Their resentment of reasonable demands is the natural outcome of decades of apologizing to students for asking them to work. We should jawbone them into understanding that the first thing a TYC or CC will do is disabuse them of attitudes that will make them workplace failures.

Random Musings

TYCs are in this mess because HSs lowered their standards. Dropping Algebra simply joins in on that party and passes the problem along to businesses. I understand the political pressure on TYC math departments to lower the bar, but will that achieve anything? Will TYC degrees become as meaningless as HS degrees, a mere certification of attendance?

If students sit back and think that just by showing up in class and doing assigned work (more or less well) they have done their part and deserve to pass Algebra -- well, that is a fatal flaw. No one can learn with that attitude. We need the student involved and active and owning the responsibility to meet an unbending standard.

The folks behind DragonBox Algebra came up with some great numbers with their huge “challenges”. Hundreds of thousands participated. 95% mastered the game, but some required six times more problems and ten times more time. Very encouraging, and one clear indicator of what we can fix about how we teach Algebra: employ mastery (aka competency) based learning.

Colleges have an opportunity -- one that high schools do not have -- to turn things around. They have
a self-selected, more mature population;
who turned to college by choice to move up in the world;
who is paying good money to do so; and
who is no longer backed up by nagging affluent parents or the school boards beholden to those parents.

If we hold this promising audience accountable and make it possible for them to succeed with mastery-based learning facilitated by technology, students will be thrilled. Listen to the Statways/Quantways videos -- successful graduates have so much pride, and some even consider STEM fields! Imagine if they had succeeded with Algebra.

Kids are quite smart. They will admit they have gotten a free ride. Welcome them to college and say, yeah, college is the big leagues, are you ready to step up your game? Then provide support, and watch what happens. No one really enjoys a free ride, everyone enjoys honest achievement.

There is a bit of a win there in that employers expect anyway to train new employees on their systems and processes, so why not move the kids along and Oz-like give them the diplomas they need? Actually, as long as we cannot do any better, that is fine. The employers will work things out or let the employee go. They can even test as part of the hiring process to avoid the substantial cost of a failed hire.

But what if we can do better? What if we can take students who have been passed along by their high schools and teach them how to work, how to persevere, how to concentrate? How much better will they turn out, and how much better will we and they feel about the work we are doing?

An Exemplary Anecdote (and the martial theme continues)

I once took a crash course in kickboxing to get ready for a full-contact tournament. Why the crash course? I had been boxing exclusively. My boxing teacher was right, I would be fine just with my hands, but when I found out my bartender Carlos had had his own martial arts school back in Portugal I asked him to show me a few things.

Carlos turned out to be a wonderful teacher who threw himself into the task. Somewhere around our third session he had me hold a striking mitt head high while he demonstrated a spinning back kick, looking at me over his left shoulder before instantaneously spinning 360 degrees to the right and delivering a tremendous roundhouse thwack to the mitt with his back foot.

I pointed out to Carlos that the fight was in two weeks and I was never going to do something like that in the middle of a real fight, but he insisted we work on it a bit. I like to do what my teachers say, so I made a couple of awful attempts and that was it. For one, I did not have the flexibility to get my foot that high. For another, it had only been a few lessons, my basic kicking skills were not there. Sound familiar?

I was training quite a bit so had plenty of time to work on the kick, and I respected the challenge. I had to work out how to lean my body away so my leg could get that high, then I had to work out how to maintain my balance while making that contortion. To my delight and astonishment, after fifteen minutes of exploration I was actually cracking the heavy bag high and hard with single-motion spinning back kicks. It was a blast.

The next time we worked out I think I had forgotten about it, but sure enough Carlos took out the mitt and asked to see my spinning back kick. I nailed it pretty well and Carlos was impressed. Then it was he who pointed out that I would never throw that in my upcoming fight, asking if I knew why he had me learn it.

No, I said.

For your confidence, he explained. So you would know what you can do.

And he was right. I was now excited about the prospect of adding kicks to my repertoire. No, there was not time enough before the tournament to internalize the technique, but the future was wide open. By taking me beyond my comfort zone into a space I wrongly considered well beyond me, he had opened up a whole new frontier of martial skill for me.

All in the context of one solitary technique.


Can we do with Algebra what my martial arts teacher did with the spinning back kick? Note that I am not talking about making Algebra relevant, I am talking about Algebra as a pure exercise in which students will learn how to learn and discover that the seemingly impossible can with perseverance be performed more or less well.

When we do that, we will not need to “sell” Algebra as relevant. Students will enjoy doing Algebra and take pride in their newfound ability.

Everyone knows the importance of math. The problem is just that they cannot do it. If we can change that, students will feel pride and aim higher and we educators will be able to hold our heads high, having done our jobs. If we drop the Algebra requirement, do you have any idea how hard it will be to get it back in? Let’s just get to work on teaching Algebra better, including first making clear to students that learning Algebra, with our copious help, is their responsibility.

Is there another subject we could use for this intellectual boot camp? Perhaps. I gather Latin has quite an intricate grammar. But is some other subject also the doorstep to mathematics, science, technology and so many 21st century careers? Will acing this other boot camp also lift the aspirations of kids to more rewarding careers?

Again, students questioning Algebra do so because they are failing it. That is the real problem, and that is on what we should be working.

The Programme

Along the way above I touched on the elements of a successful TYC (or HS) Algebra programme in which the failure rates would be closer to 10% than 50%:
  • Attitude. An explicit discussion with students of their accountability, and of their responsibility for their own mastery. We will tell them what they need to do, and it is not all that hard, but they are the ones that have to do the things we recommend; beginning with...
  • ...number facts. Learn them cold.
  • Practice, practice, practice until mastery is achieved, and do not continue until mastery is achieved. Bloom identified this long ago as a big win for learning. The DragonBox experience makes clear different students will get there at different rates, but that a very high number will get there as long as they have an easy way to practice indefinitely with immediate feedback.
  • The prior element dictates that some form of automated practice and self-assessment be involved.


  1. Is all the above self-serving because I am developing an algebra learning web site? Sure, but (a) algebra is not going anywhere, so I will do OK either way; and (b) I am not expecting anyone to accept me as an authority. Any credibility derives from the reader’s own experience of what I am merely reminding them, if I am.
  2. Well, OK, here is my experience: I taught eighth grade science/math for four years, three in the inner city, one in a small working class town. I tutored Algebra privately. I sold a similar desktop app back in the 90s and learned a lot from the feedback. I have had TYC professors speak to me of the problems they face, including students who think they should pass because they attend class reliably. Another spoke wryly of dealing with discipline problems, and this was almost twenty-five years ago. 
  3. I have playfully teased folks on-line for considering dropping Algebra. That is just my style. I understand well (a) the tragedy of kids trying to better themselves by enrolling in a TYC only to come to grief in algebra class (with plenty of student loan debt to pay off) and (b) the dismay of college professors forced to teach eighth grade math. That’s what we need to fix.

Wednesday, October 1, 2014

One Half Is Not Two Quarters? Really, CCSS?

Clicked on a tweet hawking great lessons for the toughest parts of math CCSS standards (as defined by some research or other).

There was nothing at my level of interest (Algebra) but I spotted a lesson on equivalent fractions and went for it because fraction skills matter big time in Algebra. OK, the lesson I spotted was for the third grade (remember that) but I guessed it would still get to the crux of fractions.

Not too far along I tripped over something. Looking down at the page, I thought I saw the assertion that 1/2 and 2/4 were not equivalent. In a lesson on equivalent fractions, it was quite exciting to discover 1/2 and 2/4 were not equivalent.

A, here it is, the section had a subtitle: "To find equivalent fractions, the size of the wholes must be the same." Note that this little paragraph pops out of nowhere in the middle of a lesson on 1/2 being equivalent to 1/4. If I got lost, a third grader would...?

Of course, the subtitle is wrong. The size of the whole has no bearing on the equivalence of the fractions. A fraction of a whole is not the fraction, it is the product of the fraction and the whole. Hey, let's use algebra:

Let us call two wholes x and y. If (x > y), then (1/2x > 2/4y), but the fractions 1/2 and 2/4 are still equivalent. QED.

Oh, wait, you want to call the compound product of a fraction and some whole the fraction? Let us convene a council of mathematicians to decide if .. wait. We are springing this on third-graders right in the middle of a lesson on the equivalence of fractions?

One might want to follow the lesson on equivalence by having students contemplate why a half dollar is less than a 2/4 million dollars, just in case the equivalence lesson causes confusion by misapplication, but... is that a problem?

Every time I look at the Emperor's new CCSS clothes I see lessons more confusing than inspiring, and they are often downright incorrect if one wants to look at them rigorously.

So we are in for five years of refinement of these CCSS lessons to get them right. That is understandable -- it is hard to get anything new right on the first try -- but then should not CCSS be off somewhere in an incubator being refined and tested so it can eventually win on the merits?

CCSS principles may get an "A", but its implementation is still in the first grade and those mandating it now get an "F".