Pedagogical Pickle

The options are many, the outcomes are uncertain, the needs are tremendous, and the urgency is high.  Are you with me?  If you follow my blog, and its purpose, perhaps.  If not, I speak of the current plight of secondary math education in the U.S.  As presently defined, the math competencies of our nation’s youth, on average, are falling off the proverbially cliff; they started on this downward trajectory a few decades ago – why exactly is not known, or if so, not widely shared.  Nothing has been able to slow its decline, and it is not clear if it is even possible, regardless of claims from both ends of the educational spectrum.

Conservative stalwarts to progressive reformers, public officials to private foundations, seasoned teachers to fresh-out Teach for America volunteers, all offer their laundry list of “do’s & don’ts” that will reverse the national slide.  However, none have shown that their approach is repeatable or scalable, not to mention effective across our great nation with its diverse student populations, many with unique needs that tax even the most patient people we can imagine.  What is one to do?  I am not sure.

I do know that I plan to do my best to do my part in reversing the slide.  My approach comes from a strong desire to reveal the multiple ways math can be approached, that there is no one right way, that no one is incapable of learning how to master math, and giving up on yourself is not an option.  I plan to hold students to high standards with high expectations, not for my edification, but for theirs, and if I benefit in the process, so be it, so I can reinvest that improvement back into my students.

Which brings me to my current dilemma.  Is it best to show students procedures that are easy to remember in hopes they recall the procedure, apply it correctly, and hopefully, more often than not, respond correctly on the various assessments placed before them?  Even if they may not know how to apply what they’ve learned in a problem that deviates slightly in context, content or construction from what they may have grown accustomed to seeing?  Is it OK if they simply show rudimentary facility or is comprehensive transference required?  What is best for the future of those students and our nation?

And what about those who struggle mightily to recall the simplest facts and procedures?  Do we allow them to continue to fail assessments that require proficiency in both facilities?  And if we do, what harm, exactly, does that cause our society?  Are they less capable of benefiting our local communities because they struggle with algebra, geometry or other “required” mathematical subjects, theorems, rules, etc?  Are the plethora of standards truly necessary for all?  Is a hybrid approach possible?  What does equitable learning look like for students who struggle so?  Does our nation have what it takes to get us back to where we were a few decades ago.  Or are we doomed to be ranked 12th or worse in the world going forward?

As a specific example of pedagogical dilemma currently rattling around in my mind, here is a simple algebra problem dealing with proportions:  13/6 = 52/z.  There are a few ways to approach this problem.  The traditional approach taught uses “cross multiplication” to yield (13)(z) = (6)(52), followed by multiplying 6 & 52 yielding 312 which is then divided by 13 to yield 24.  This is the popular method to teach struggling learners since it minimizes deeper thinking, by simply following a procedure independent of the numbers themselves, especially if one can use a calculator.

When I teach the same problem, I never multiply the 6 & 52, I leave them and place the 13 in the denominator yielding (6)(52)/13 whereby I then see that 52 divided by 13 is 4 which multiplied by 6 equals 24.  This approach is simplest to me, since multiplication of smaller numbers are involved, often just requiring recall of the “tens times tables.”  There is no messy multiplication or division that often results in minor mistakes and ultimately incorrect answers; and on multiple choice tests which are so often given at this level, students obtain scores that do not reflect their level of understanding.

So, while I might be limited in revealing different approaches like this when I am a student teacher, I fully intend to explore whether my approach is easier to understand with my future students.  Realistically, I may not know which is easier unless I conduct an experiment where randomly selected students are separated into two groups where one receives instruction in one approach and vice versa.  However, even this experiment may not reveal whether it is better or not across other classrooms, teachers, etc.  So I am left with the dilemma I started this posting with:  what is best for the student?  our local community?  an ethnic group?  our city, state, country?  If anyone knows, I am all ears.


About Dave aka Mr. Math Teacher

Secondary math teacher teaching math intervention, algebra 1, honors precalculus, and AP Calculus AB. I spent 25 years in high tech in engineering, marketing, sales and business development roles in the satellite communications, GPS, semiconductor, and wireless industries. I am awed by the potential in our nation's youth and I hope to instill in them the passion to improve our world at local, state, national, and global levels.
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10 Responses to Pedagogical Pickle

  1. Reblogged this on Reflections of a Second-career Math Teacher and commented:

    I’m still wrestling with many of the questions I posed as a student teacher in this blog post from four plus years ago…


  2. pusdcitizens says:

    Do you have a reference for the precipitous decline in math scores?


    • Nothing specific as I was speaking to the rhetoric bantered about at the time re: our performance on the PISA, primarily, but also on a variety of other ill-defined measures….I meant to add a caveat in my re-blogged post that I question whether there has truly been a decline per se, as specific measures of performance over time are few and far between but decided against it for brevity’s sake.

      The gist of what I wanted to convey in the original post is that there is a strong belief (correct or not) that mathematics performance has declined, on average, as much focus in education centers on the so-called “achievement gap” and other declines in college STEM enrollment, etc.

      Personally, I do not believe that the number of students attaining at the highest levels has declined precipitously. However, I do speculate that the percentage of those students may have declined somewhat, and the average could be moving downwards. Also, I do not believe we are developing our best and brightest as rigorously as possible.

      Whether that matters in reality or not is difficult to prove. Nonetheless, I believe it makes sense to strive to improve our abilities over time, which is where my passion resides.

      Thanks for commenting.


      • kentilton says:

        Don’t know about PISA, but college teachers certainly noticed a fall-off requiring them to teach high school math 9and writing). I was just at the 2014 AMATYC conference and learned they now teach not just Algebra but arithmetic, and more than one year of it.

        My guess as to what happened is 60s-70s loss of (parent) will, written up in some detail here:

        FWIW, I am not for a return to “do or die” grading. But what we did was drop that and not replace it with anything. Hence colleges ended up teaching high school.

        My guess is the replacement for threat motivation has to be motivation through the love of learning. That means knocking down the factory model of education altogether and empowering and enticing students to learn for the sheer joy of it. Big project, but it also sounds like fun for us and students.

        Too bad the Gates did not put their money behind that.


      • Thanks for the comment, Ken. It is a shame grades have been hijacked for so long. I was not aware of the extent or duration…

        I resonated with the ACSD article referenced in your post re: killing with kindness, which is the source of the following excerpt.

        “One wonders how many of these [college] dropouts got good grades in high school, only to discover on entering college that their schools, by lavishing them with unrealistically high marks, may have actually been killing them with kindness.”

        I am (perhaps foolishly) trying to ease the transition from HS to college for my advanced students by holding them to high(er) standards than what they may have experienced previously. My belief is that a slight shock to one’s intellectual self-esteem in a supportive high school environment increases a student’s likelihood of success in more advanced coursework in HS or college, an inoculation of sorts for the academic intensity of college.

        I’m not sure my students’ parents agree though…and reading your post and associated references helps me understand why a little more…


  3. Dave your senior ( experience and age only) says:

    Or maybe seeing what multiplier of 13 will produce 52 and applying it to the denominator. Less rote learning ( is it learning ? ) requires more thinking; maybe the brain at certain ages just can’t do it for most. Dependence on calculators ? Are the tests over time testing the same concepts with similar rigor ?


  4. kentilton says:

    I think we have to let the kids own their own learning, mostly because different folks grok the same thing different ways.

    I always began a lesson with a deep motivation/justification for an Algebraic procedure. The strongest students would get it and be able to work from that, the weaker students at least believed that I was not just making up the new procedure out of thin air. And for them I then taught the procedure. The good news is that for some students enough procedural practice needs to them finding on their own a deep unifying justification of a procedure.

    The idea of spotting the divisibility of 52 by 13 i would consider an advanced skill. To help the weakest students i would provide the “can’t miss approach” even if it led to more interesting arithmetic. (I do not think there is a winning strategy if one wants to teach Algebra without them doing arithmetic.) i would then also point out that one can look ahead and avoid multiplying everything out to save some work. Again, that would be put out there for those who would have the ability to spot such shortcuts and the laziness to want to avoid just multiplying everything out.


  5. Dave your senior ( experience and age only) says:

    good thoughts here, thanks


  6. rokballard says:

    Dave —

    I identify with your current frustration and burnout. I feel like I understand your underlying goals; they sound much like my own. One difference is that I’ve been teaching almost 28 years and in a setting far different from yours–small (read “tiny”), rural, sheltered. However, even in my sheltered environment, I have witnessed a change over the past 10 years or so. Students (from top to bottom) with less basic skills, number sense, and will to push through difficulties. Less parental support. It IS hard to carry on in the face of these challenges–I do so because I consider teaching a calling.

    It grows increasingly difficult to do the more that is needed to teach effectively with the plethora of other duties added to what we do. Right now I’m struggling to decide WHAT are the MOST important tasks for each class (I have 6, ranging from grade 7 to Algebra 2) to experience. Tasks/topics that are most important for each student’s 1-quality of life, 2-future math courses, and 3-state testing. Something to provide some focus because every topic I’ve tried the first semester in grades 7-9 has been a battle because the students lack the necessary background skills.

    I’d love for somebody to give me the magic bullet. I don’t even have colleagues with which to argue–I’m the entire 7-12 department! I only know what I’ve known for years–there is no recipe for good teaching. It’s different every year with every different class.

    Keep up the good fight. I choose to believe it is worth it. Even when faced with the occasional ungrateful parent and the more common parent with unrealistic expectations, there remain those students for whom I know I have (via the grace of God) made a difference. To read your student letters, I believe you have some of the same.

    And if you or your readers can tell me the top 3 topics to master in each of 7th grade, 8th grade, algebra 1, geometry, algebra 2, and contemporary mathematics — that’d be great, too!


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