Which Did You Prefer: Algebra or Geometry?

When people find out I am a math teacher, if they don’t tell me how much they hate math and how they were never “good at it” (this comment, BTW, usually culminates with them vomiting on my shoes), they invariably offer that they were good in Geometry and bad in Algebra, or vice versa.

I never ask  people which they preferred in school; for some reason they almost always volunteer their preference.  If you happen to be of this same mind – that you preferred Algebra to Geometry in school or vice versa, consider the following reason why this may be so.

Geometry is very spatial.  Nary a Geometry problem is not accompanied by a figure, usually involving angles, triangles, quadrilaterals, or circles with chords and tangent lines, all chock full of capital letters denoting vertices and such.  True, there may be some algebra involved in the solution of the geometry problem (which, incidentally, is why Algebra generally precedes Geometry in most high school coursework), but  diagrams and figures reign dominant in Geometry.

Algebra, on the the hand, is typically void of diagrams and figures and relies instead on a student’s ability to manipulate symbolic expressions in solving problems.

Therein lies the distinction.  The left hemisphere of the brain is dominant in the cognitive processes using LOGIC, SYMBOLS and LANGUAGE.  The right hemisphere of the brain deals with SPATIAL, CREATIVE, and MUSICAL processing.  That is to say, Algebra and Geometry are processed in different hemispheres of the brain:  Algebra is a left-brain pursuit while Geometry is right-brained.

Of course, some of Geometry requires left-brain logic and some of Algebra requires right-brain spatial/linguistic processing (as in the case of solving word problems).  But on whole, doing Algebra has more in common with writing an English paper than with doing an Geometry problem – at least cognitively.

I myself am a case in point.  I am much more an Algebraist than a Geometer by trade and I am also a writer (not only in the case of this blog, but also for a forthcoming book I am writing that will published by Corwin Press).

I am willing to bet that those folks who liked Geometry tended to pursue artistic careers (sculptures, painters, photographers, architects and musicians) and those who liked algebra tend to pursue left-brain careers (scientists, economists, writers and lawyers).

If you had a preference, which did you prefer in school:  Algebra or Geometry?    dven.

Oh Christmath Tree

So I’m driving down a country road margined with make-shift lots selling Christmas trees every quarter mile and I’m struck by a thought.  The thought is motivated by the all too common experience of dressing a newly purchased tree with a string of lights and realizing that the string of lights runs out before the branches do, leaving either an obvious bare rung of base branches or me having to start all over wrapping the lights. My guess is that we’ve all done this – miscalculated the length of stringed lights with the amount of branches to cover.

What a great query for a secondary math class!  I think to myself.  If I were still in the classroom I would pose this holiday problem to my kids.  It would go like this:

Suppose you had a Christmas tree that was 7′ tall and 5′ wide at the bottom row of branches.  If you had two equal strings of lights with which to adorn the tree, how far down from the top should the first string of lights go so that you had enough left with the second string to cover the rest of the tree?

This is not a simple problem.  It involves surface area of a cone, the Pythagorean Theorem, similar triangles, proportions, and solving a simple quadratic.  That’s what makes it such a nice problem to ponder with kids.  Not only is it seasonal but, like all real-world problems solvable with math, there are many different mathematical concepts and skills embedded in it.  I should point out that I would not actually do this kind of math when dressing a tree, but it is a curiosity that can be solved quite accurately using math.  [For those brave enough to try it, the answer appears below.]

I suppose what drives me to write this blog entry has also to do with my concern that, in this day of state curricula and district pacing guides, math teachers just don’t get to ponder real-life math problems or use relevant, timely quandaries to teach this subject.  I suspect many math teachers would not even notice a real-life application unless it was a textbook word problem or  specifically suggested by the district pacing guide.  In our desperate attempt to unify instructional materials (to make them “teacher proof”), we have removed original thinking from our teachers’ lesson designs.  What also gets removed is the spark that can ignite a teachable moment or a higher-order educational experience for our students.

If you’re a secondary math teacher, I encourage you to try this problem with your kids.  I’d love to hear how it goes.  dven.

Answer.  The first string of lights should last roughly 5′ 1″ down the tree, measured vertically from the top.

The S-Word

Having been in education now for depressingly close to three decades, I’ve noticed that educators are very careful when the opportunity presents itself to describe a student as “smart”.   When I changed teaching jobs and moved from a Connecticut high school to a South Carolina high school in the mid-eighties, I was struck by how teachers used the S-word freely but almost always in a whisper, as in He’s really smart or Oh, she’s smart. I had never heard any teachers in Connecticut ever label a student as smart.  But in South Carolina I heard it fairly often, as if a news flash or notice of some special case.  I won’t attempt to theorize why – it’s just something I noticed.

Then there are those teachers who call everybody smart.  In addition to cheapening the attribute, it’s just not true.  This isn’t Lake Wobegon, after all, where “all the children are above average.”  No one in education seems to want to admit the obvious:  All kids are not smart.  It may be politically correct to say they are, but it’s a lie.  I guess I’d rather be honest than politically correct.

I should point out that I use the term “smart” not in a Howard Gardner sense, but more in a Martin Gardner sense.

I should also point out that my premise that not all kids are smart does not in anyway lead to the conclusion that they cannot learn or that they should not be taught.  Quite the opposite:  All kids can learn, and it is our job to find a way and a pace that accommodates them wherever they may be on this continuum of smartness.  If they’re on the low end (i.e., not smart (there, I said it)) we must work even harder to reach them so that they do learn and are successful.

Why do we have such a problem saying some kids are smart and some are, well, not smart.  We immediately acknowledge without pause that some kids are athletic and some are not athletic.  Or that some are creative and others are not creative.  Why is smart any different?  Is it that, as a society, we place more value in being smart than, say, being athletic?  Not really.  Look at the mean salary of professional athletes compared to the mean salary of rocket scientists.

So what’s the deal?   dven.

Their Best Learning

I just spent three days consulting in a district in a neighboring state.  At the end of the second day I met with an administrator , as I typically do, to debrief the first two days of my visit.  During this meeting, I was struck by a comment she made.  She said, “It’s not about our best teaching, it’s about how our kids best learn.”  It wasn’t just a clever, pithy remark; to me, it signifies a colossal shift in thinking about teaching and learning.  It’s not about our best teaching and this age-old focus is beginning to change in districts everywhere.  It’s about what they are (or are not) learning.

Embedded in her comment is the notion that our kids don’t all ‘learn best’ in the same way.  That’s one of the shortcomings of the ‘best teaching’ mentality:  unless we design lessons that accomodate diverse learners and put into practice the oft-spoken language of differentiated instruction, our ‘best teaching’ – good as it may be for some kids – may sorely miss the mark for many other kids.

When we all as a community of educators shift our emphasis from good teaching to good learning, we begin to design different kinds of lessons, lessons that reflect “how our kids best learn.”  dven.

Poor Kids

The widespread incident of neighborhood violence and drug usage in America’s inner cities has had and continues to have a devastating impact on her urban schools.  There’s no denying it.  The district in which I work is the nation’s 18th largest district.  Our city has the state’s highest crime rate and the neighborhood in our city with the highest crime rate includes a middle school in which I was recently stationed for 6 intensive weeks as part of LEA Improvement (since our entire district of 180 schools is currently in Corrective Action, as per NCLB legislation).  That is all to say, I get to see the impact of neighborhood crime, violence, and drug use first hand.

But these are all corollary to a more fundamental problem: poverty.  The number of children in the US living in poverty – both urban and rural poverty -  is staggering.  The graph below depicts America’s children living in as compare to the poverty rate of other  countries.

How can the most economically advantaged country in the world permit so many economically disadvantaged children?
I first saw this graph as I prepared a workshop for teachers on Urban Education.  The more I researched this topic, and learned about the incident of students living in poverty, the more I became outraged.  How could this state of affairs have happened in our America?
I don’t have a solution – there is no quick and easy solution.  But as a citizen and especially as an educator who is routinely entrenched in the mess it has caused, I am outraged and think every citizen should be likewise.

The impact of poverty trumps the impact of drugs, crime, and violence.  Indeed, the latter follow quite predictably from the former.  dven.

Thanksgiving & Year-round Schooling

When I was in the classroom, just a few years ago, I always looked forward to Thanksgiving. Sure, it represents the first significant (and well-deserved) holiday break from school, but that wasn’t the only reason. For me, Thanksgiving represented the psychological half-way point of the school year.  Once Thanksgiving passes, it seemed to me, the year starts to really fly by and before I knew it, Spring was upon me.  I realize, of course, that Thanksgiving is a fair bit shy of the actual, chronological midpoint, but it always seemed like half the year was over upon its colorful and self-indulgent arrival.

But it occurs to me that this feeling could change in the event that the resurgent talk of year-round schooling becomes a reality.  What gets me about all this talk is that the proponents of year-round schooling – most often people not in education – act as though extending the school year will, by itself, increase student learning and improve sagging student achievement.

I’ve read the recent NAEP report and I know we’re not doing so well, overall, as a nation in an increasingly flat world.  But more of a bad thing is not a good thing.  It’s just…well….more of a bad thing.  That’s like going to a really bad restaurant which serves really bad food and somehow feeling good about the experience because the portions were really large.

I would like to know from my readers what you think about year-round schooling.  And while you’re responding, tell me:  Does Thanksgiving feel like the psychological half-way point to you?  dven.

Sound off: Teacher Salaries

I recently learned that a typical Supreme Court Justice earns about $200,000 a year.  Judge Judy, on the other hand, makes about $25,000,000 a year.  What is wrong with this picture?!!

The average teacher in public K-12 education earns $54,000 a year.  I won’t even venture a guess as to what the average non-college educated, professional athlete makes in a year.

What would happen, do you suppose, if the average teacher salary in the U.S. was, say, $100,000?  How much greater would the pool of new teachers be?  How and in what ways might this transform our educational system?  How much better would our public schools be?  How would this impact our current, abysmal national drop-out rate, or our society as a whole?  How much more competitive would the U.S. be in the global economy, the same one in which we are slowly and steadily losing ground and falling behind?

There are those who would argue that young people who go into teaching do so for the other more important and significant rewards of impacting young lives. I would not disagree.  But anyone who has been in education for as long as I knows that we don’t always attract the best and the brightest to our field.  Could it be that if young people were paid competitive salaries to teach – salaries comparable to those offered by business or even medicine – we would attract more talent, while at the same time preserve the more nobel aspects and motivations for entering our profession?

I know, I know.  This is a pipe dream.  But something must change:  approximately  50% of our present teacher work-force will retire in the next decade.  Who will replace them?  dven.

Math Across the Curriculum

There is a push in many schools and districts to adopt Writing Across the Curriculum initiatives. This is not a terribly new idea; indeed, many schools began pursuing this notion more than a decade ago. But more and more schools, suffering from low ELA scores, have begun to realize that reading and writing are fundamental to student success in most every subject. The research is clear that deficits in reading and writing correlate to deficits observed in student performance in other subjects. In fact, I have recently been contracted by a district in a neighboring state to conduct four full-day workshops to their teachers on the subject of Writing in Math.

In a parallel way (no pun intended), I’m an advocate for what I call Math Across the Curriculum, particularly for grades K – 8. While not quite as ubiquitous as reading and writing, math concepts and usage can be embedded in nearly every subject. Math’s relationship to science is clear – and as such, this relationship often overshadows Math’s connection to other subjects like Art, Music, Social Studies and even sports.

As a math teacher, I would love to team-teach a middle school class with a social studies teacher and apply math to everything from voting to social networking. When kids see connections of math to other subjects in unanticipated ways, they come to appreciate it more and math concepts become useful tools in order to better understand our world – instead of a seemingly irrelevant body of rules and procedures.

Additionally, I’d also love to offer a college course to teachers on Math Across the Curriculum, helping teachers of other subjects understand math’s applications to their subjects and explore ways to embed it in their classes. dven.

<To invite Mr. Venables to your school or district to teach your teachers what math lives silently in the course they teach, contact him directly at danielvenables@mac.com.>

90/90/90 Schools

Doug Reeves’ work with 90/90/90 Schools is compelling stuff. For readers unfamiliar, researcher and educational author Douglas Reeves looked at public schools that were composed of at least 90% minority students, had at least 90% of their students on Free and Reduced Lunch and had a 90% or better pass rate on state End-of-Year assessments. He compiled a staggering battery of data on these schools and sought to find out what else they had in common and what they were doing to produce such impressive student results.

He found five basic tenets common to these schools that were correlated to student performance. They are:

1. strong focus on achievement

2. clear curricular choices

3. frequent and multiple assessments

4. strong emphasis on writing in all subjects

5. external scoring of student work

The point of this blog entry is not to expound on these 5 characteristics. Teachers and schools interested in learning more about each of these attributes should go here for a short summary of the research.

The point here is that there are models of excellence for schools that are most commonly low-performing and in many ways operate “against all odds.”  If your school is a 90/90/<90 school, I urge you to take a hard look at Dr. Reeves’ work. It not only shatters demographic stereotypes, it provides hope to teachers who teach every day against the odds, hope that their students too can excel in spite of the odds. dven.

Tough Kids = Low Standards?

For the three years I was an academic coach at two of our district’s lowest-performing high schools I had the opportunity to visit lots of classes. Lots. I estimate having observed close to 700 – 800 math classes.  In this time, I saw some exemplary teaching and I have shared many of the best strategies and activities that I observed with other districts when I consult throughout the southeast. But I also saw some very ineffective teaching; in fact, the reason I was assigned to these schools was because they were low-performing and because classroom instruction was most often lacking.

What I noticed with the ineffective teachers was they seem to have one thing in common:  they didn’t expect much of their students. It was as if somewhere in their careers they decided that kids in these high-needs schools weren’t capable of producing quality work or high-level thinking. With this self-fulfilling prophecy as the backdrop, they lowered their expectations of their students and of themselves to pathetically low levels. It was as if they sold out, growing complacent with shoddy work and being satisfied with any work their students produced. That the same quality of work (or lack thereof) would never be accepted in the more affluent, “whiter” schools did not seem to phase them or even enter their minds.

Was it that they – the teachers themselves – had little idea what quality work was, or that they simply stopped battling kids to do good work? I don’t know. What I do know is that every time a teacher accepts shoddy work and tells kids it’s good, or good enough, we are increasing the achievement gap. And when we do, we contribute to the educational inequity that plagues our public schools. We become part of the problem, perpetuators of the problem, and not part of the solution.

Love to hear what you think…. dven.

<To invite Mr. Venables to your school or district, contact him directly at danielvenables@mac.com>