July 19, 2012

What High School Math Teachers can do in Today's Classroom

My past systemic criticism notwithstanding, and desire for change very reasonably postponed, if I had a Math Classroom again, I'd do my absolute best to educate young students, in all sorts of ways. Many positive outcomes are entirely possible, especially if the objectives, methods and goals are presented honestly and straightforwardly, between Administrators and Teachers. For a teacher in such an arrangement, the only tangible questions are: Where's the target? - and - How do you want me to hit it?

In all educational honesty, Algebra is like any other field of study. It presents different aspects of itself to different classrooms, and offers different lessons at different levels. When the rich American in Italy asked his proverbial tour guide, "How long do I need to gain a real appreciation of Rome's culture and history?" the Italian responded, "Un'ora, Una vita." There are layers of brilliance to Algebra that many students will (and ought) never encounter, but there are aspects of brilliance to Algebra that all students can encounter and be 

So, although I strongly believe the US Educational System will eventually diversify, and that this is needed nowhere more so (Godspeed) than in Math classes, the unfortunate reality is that today's Math teachers still have to play with the cards they've been dealt. Better ideas may crack the system someday, but they aren't currently helping those who are currently on the inside. 

So, what can be done, at the moment?

The major trick, I believe, is to teach more than Algebra. The major opportunity is to teach them to learn, to collaborate, to apprehend, to master, and to re-present for their peers. Make no mistake, we CAN get students to master the content, and we CAN get students to conquer the 'State Test', but we'll be MORE effective at that AND more productive in general, if we concern ourselves less with particular standards of how "True Algebra" ought to be taught, and more with how STUDENTS ought to be taught. Sorry for shouting. I get so excited sometimes.

Without further ado, then, here are just a few personal and (hopefully) practical suggestions for anyone with the good fortune to receive a contract this year:

* Ask permission to "flip" the Math classroom. Make one good video lesson per day, have it available on at least one playback screen in the room, playable over and over. Play it to start class and then allow groups to split up. 

* Set up a flexible system to facilitate peer tutoring and Those who finish early can earn bonus points and/or extra rewards for re-teaching the lesson, either in small groups or at the front of the board. Any students who prefer isolation can watch the lesson again as many times as they like, with headphones. 

* Even better, ask permission to post the lessons online, in advance. Aggressively organized kids can do the work before coming to class, and then (again) earn rewards by teaching the lesson to others.

* As much as possible, make the students talk more than you do. Most teachers ask, "Any questions?" And a five second question is followed by more Charlie Brown style Wah-wah-wah. Dynamic teachers ask, "Who can explain this?" Or at least, "Who can tell me what I just said?" Then they listen.

* Require pens, and forbid pencils. The old tradition was born from a scarcity of paper, and modern teachers support it for convenience in grading, but erasers are terrible things. Erasers enable cheaters to change answers, and - what is far worse - they cover over helpful evidence of actual thinking. 

* Give partial credit for observable reasoning. Right thinking, wrong thinking, or absent thinking, the scratch work that gets erased in Math classrooms today amounts to a ginormous waste of valuable information, which a good teacher should use as a gold mine for planning individualized remediation.

* Give 100% make-up credit for anyone who can explain, using any functionally communicative words, why their first problem was wrong, and/or also why their corrected work was more correct. Explanations during class must be offered in writing. Verbal explanations become interviews, and therefore must wait until tutoring time is available. (Point variations apply. If the first assignment was not attempted, 50% make-up credit might be more appropriate, naturally. Etc.)

* Allow calculators on every problem assigned, classwork, homework, quiz or test. Yes, because the Big Test allows them, but far more importantly because of this reason: For math students, calculators are both fearsome and ubiquitous tools, and any dangerous power that is also omnipresent should be something children are taught how to handle, not whether (or when) they're permitted.

* Train students in how to interpret and re-present what the calculator produces. The future of math-related fields is less in need of people who can crunch all the formulas, and more in need of those who can apprehend complex information and re-present it to others in more easily understood terms.

There are more, but those few suggestions best get into the heart of what I have to say about teaching Algebra in today's (US) high school classroom.

And now, if I may elaborate...

Traditionally - no, to be more precise - Historically, there are two reasons why High School Algebra has stood as the principal gateway to collegiate entrance. One reason is fundamental, and the other is practical. Fundamentally, those who master basic Algebra (especially in its more traditional renderings) have demonstrated the ability to reason abstractly. In other words, Math pre-dates IQ tests, and - again, speaking historically - Math was a method both simple and thorough for determining which minds were at the top levels of aptitude, and most suited for college. But today, "everyone" can go to college. No, truly. Everyone can go to some sort of college or other.

The second reason why Algebra has historically stood as the principal arbiter of young people's collegiate potential, as I said, is more practical. In the process of mastering basic Algebra (again, this is more true as you move closer toward more traditional curricula) students are required to process familiar situations that contain endless variety, to process and then to decide! A simple two-step equation presents a surprisingly high number of variations on its own theme. 

Practically, one who memorizes patterns and regurgitates "steps" in responding, honestly, can get by alright without help about 50% of the time. Critically, however, one who captures the internal logic of what balancing equations actually requires - and actually this can be done by any cool headed kid who takes enough time to observe while sincerely engaged (the hard parts there having nothing to do with mental aptitude and all to do with general mental state at the moment of study) - the one who gets what's really going on has a rare opportunity. Such a student can do more than respond in a limited number of situations based on what "steps" he's been programmed and authorized to follow. Such a student can become an autonomous decision maker.

In other words, proper Algebra requires complex analysis and decision making.


Like any other field of study, practical Algebra can still convey benefits to various worthwhile degrees.

"Un'ora. Una vita."

Of course there are limits to what we can teach with our limited resources, but Algebra always contains within it an inherent richness of its own challenging nature... and there are no limits to how creatively we can adapt Algebraic learning objectives (from probably any curriculum) and make their completion serve the increased growth of the student... rather than purely making the student serve the needs of the Math curriculum.

There are ways to achieve what the system requires - today - without bucking the system, but while ALSO providing more diversified levels of engagement with what Algebra does and with what Algebra requites. Yes, requires.

In a lot of Math classrooms, typically, teachers come to believe that they have to pretend everyone's going to college, and yet the most common method of getting them there doesn't necessarily prepare kids to do much more than take orders (and never to issue them). 

There are more educational opportunities in present day High School Education than a lot of Math Teachers often realize. Yes, there are always personal opportunities. You help a kid, you support a kid, you drag them kicking and screaming to graduation, you change a life, you sow a seed. Amen and amen. But I said educational opportunities.

If we're going to teach Math, we may as well teach Math in a way that combines what we all need (kids, teachers, schools and politicians all have particular problems today) with what Math needs. Math needs to be understood. It requires processing and representation. It should not merely be regurgitated.* 

(*Please note: this is somewhat equally true for Literature, History and Science as well... but those subjects can more easily be reduced to regurgitation of facts, which is what a lot of Math gurus have been trying to do with Math now, in high schools, for years. Yet, Math remains unique in it's inevitable demand for decision. This remains a central part of my thesis here, and this remains the best reason to do Math as more than algorithms and facts.)

What Algebra offers - at almost any level of offering - is more than traditional forms of algorithmic troubleshooting and transformation of symbolic equation language - to any particular standard. 

What Algebra offers is a unique opportunity:

* Students can be challenged to embrace the abstract (at any level of abstractness).

* Students can be engaged to observe details of the abstract (and to contrast divergence with patterns).

* Students can be allowed to communicate their incorrectness (and to walk better by stumbling onwards).

* Students can be encouraged to re-process and to re-present ideas and information that is new-to-them.

* Students can be challenged to teach others, and thus discover (and plug) various holes in their own understanding.

* In solving problems (applying the lesson), students can be challenged to perform practical analysis (really just basic and careful observation), and then to decide what's required, and then to implement what's required, and then to test that solution with further analysis (ideally through multiple methods of testing).

To sum up:

Teaching Algebra does not have to be all of these things - or any one of these things - at every possible moment... but teaching Algebra ought to be all of these things - in various degrees and for various students - at some key points during the year.

And so therefore, if you are so fortunate as to receive a teaching contract this year, I will tell you this one more time. It doesn't matter so much what curriculum you're given by Admin, or what key skills need to be tested by the State at year's end, or how closely the methods you're asked to use might approximate "Math" as you yourself once experienced it.

The only questions that matter, for you, my dear Algebra Teacher, are these:

Where's the target? and How do you want me to hit it?

All the big things that Algebra CAN teach CAN be taught with great impact at various points in the year.

Someday US schools will diversify helpfully. Until then, all students will continue being forced to take Algebra, and all districts will continue to "water it down" (comparatively speaking) . Nevertheless, the fundamental aspects of what makes Algebra worthwhile can quite often be utilized to exact more from (and provide more to) your students.

Take it from me. I've been on the inside and the outside. I've talked to teachers in various schools, states and districts.

In the present system as it stands, there are more ways to do Algebra, helpfully, than a lot of teachers [want to] imagine...

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-- Isaac Newton