The Real Basic Skills in Math and Arithmetic

Long story short: The writer of a Letter to the Editor recently took me to task over a recent column.  She strongly disagreed with my opinion that arithmetic skills are no longer as important as they used to be, and the view that math and arithmetic are not identical.

Part of the problem seems to be that I was a math educator. (Don’t tell my mom. She still thinks I was a saloon bartender).   Apparently, this means I am unprofessional, don’t use math in the real world, and disqualifies me from any meaningful discussions about basic skills and how students learn them.  

Aside from the Letter’s tone, I really have no quarrel with the writer. She’s entitled to her opinion after all, and a couple of her views aren’t terribly unusual.  But I do think some observations about her points are in order: 

  • This always-present objection about cashiers who can’t calculate change quickly is simply NOT a technology problem. (Nor is being late, phone alarm or not).  Some of us can remember such cashiers (and late arrivers) well before calculators were invented!
  • I will gladly agree that the ability to tackle and solve real world problems is the goal of mathematics!!! It is partly why I believe we need to spend MORE time on that skill these days, and LESS on ‘adding and subtracting large numbers’.
  • I would quickly agree that math anxiety can start when students fall behind in arithmetic skills. Part of my point exactly! (Especially timed arithmetic practice!!  Students begin to think they’re ‘dumb’, when they aren’t, simply because they can’t do something quickly. )
  • The writer seems highly enamored with engineers. I would ask her to show me an engineering firm that still calculates with paper and pencil, or ‘long multiplication and division’.

Digging deeper, I believe the subtle crux of the disagreements here lies in the terms ‘basic skills’ and ‘fluency’.  The writer makes a big deal that basic skills are needed for ‘higher math’ and problem solving.  It sounds reasonable, and I partially agree.  SO, let’s stop and ask: “Just what are basic skills in 2017?  What were they ever?”

Consider Student A who can easily and accurately divide 396 by 52 using long division, but who is stumped by the question “If John donates $396 to a cause in one year, what is his average weekly donation?” Which would we prefer Student A be able to do? Which is the ‘basic skill’?!  Would we say Student A is fluent in division?  Moreover, if Student B knows that division is called for here, does it really matter how he/she gets that answer – mentally, abacus, pencil, or calculator?

This has been our confusion for decades – even before technology.  Basic skills involve knowing when an operation is called for, knowing how to apply that operation (with technique of choice), and then knowing how to interpret the answer in the real world.    Being able to learn and flawlessly perform some rote technique to get ‘an answer’ isn’t a necessary basic skill any longer.  It used to be our only route to getting those answers, but no more.   If we are in trouble, then it is because we have mistakenly focused (even before calculators!) on techniques rather than recognition and application of skills (part of problem solving).

We ALL want our students to be able to efficiently tackle and solve ‘real world’ problems, using whatever tools/techniques are at their disposal.   We should continue to focus on that common goal.  And in the process, using a calculator to get past the tedious calculations does not hurt – indeed, helps with – the real basic skills.

Back to the Future

In this space last time, I related an incident that occurred ‘back in the day’:  I had made a careless arithmetic error that affected my test score and, while I had worked the problems with the ‘correct’ procedures, my teacher had told me that ‘in real life, you don’t get partial credit for a building that falls down.’

Let’s leave that incident itself (and the participants) back in the ‘60s, but move that thought into the present. With an eye on the future, let’s explore the broader issues in play, along with some others.  For those broader issues and dilemmas are still very much alive.

As an educator it seems that the basic underlying question is this:  How do we prepare our students to learn the skills they need for their future professions, and still allow them to make the necessary mistakes that allow them to learn those skills in the process? Where do we draw the line?

Clearly, we want to prepare our students for ‘real life’ situations, and the ‘perfection’ those situations require. (If one builds a bridge, it must stand up to decades of brutal treatment.) Yet our students are NOT in ‘real life’ yet, and they are often far from having developed the skills they will need when they arrive.  Is that not what ‘school’ is for, after all?

These important questions are not restricted to any one discipline, of course.  How do we teach youngsters to ride bicycles without knowing they will spend some time losing their balance, and skinning their knees?  How do we hone the skills of a young future Hemingway without putting up with – and correcting – some pretty boring and pedantic writing in the process?  How do we encourage a future Yo Yo Ma who has just picked up his/her first cello, and who will make some interesting sounds in the beginning?  The analogies are abundant.

As we prepare and work with our students in all their disciplines, it makes sense that we must be patient and allow – even encourage – the missteps that will occur along the way.  It’s all part of a learning process that prepares students for ‘the real world’. Part of what makes a good teacher is knowing when and how rigidly to walk that fine line for the benefit of all.

Interestingly, though, there is another side of the same coin.  There are often places where we are shortsighted in the other direction, and could do a better job of preparing for ‘the real world’ earlier than we do.  In my own discipline, for example, I have often wondered how we prepare students for the ‘real world’ where folks work in teams, use technology (both for communication and computing), and often extend projects over weeks at a time, when we force them to do arithmetic problems by themselves, using paper/pencils only (no technology), and operating within a time limit!  It doesn’t make sense.  Other such examples abound.

As always here, the ongoing task of educating our youth for the future is a long-term process that is a delicate tightrope walk.  We must simultaneously keep our eyes on the future product, while focusing on the present, and working with students as they are when we get them.  It’s a delicate and dangerous balance.  It’s one small part of what makes education both a challenging and a rewarding endeavor.

A Case of Mistaken Identity: Arithmetic and Mathematics

Years ago, a cartoon in the Chronicle of Higher Education jumped out at me.  I ended up using it frequently in workshops with teachers for the rest of my career.

Here’s the picture:  A traditionally stereotyped matronly female teacher sits at her traditionally stereotyped desk (in the front center of the room), watching a traditionally stereotyped Johnny do some arithmetic problems on the blackboard (Remember those?).  She is frustrated and scolds him “Your math skills are horrible!  How do you expect to get a job in you can’t add and subtract?”  Little Johnny answers brightly, “No sweat!  I’m going to be a Congressman!”

Many of us will laugh (and/or cry?) at that cartoon, but, in my opinion, the humor is masking a case of mistaken identity. And it’s one I worry is still prevalent today.

The mistaken identity is this, in six short words:  Math and arithmetic are not identical.  We’re broached this topic before, but it’s always worth another visit from a new angle.  The importance and ramifications of this mistaken identity cannot be understated.

Naturally, of course, arithmetic is a part of mathematics.  But the two subjects are not interchangeable. Any more than punctuation and skillful writing are interchangeable.   For countless years, there grew up this impression that mastering six to eight years of paper/pencil arithmetic (that is often timed!) is what mathematics is about.  Does this impression still linger?

This wasn’t such a terrible mistake ‘back in the day’, when higher math necessarily required great deals of calculating, and our jobs in the workplace often required extensive shop-keeping skills without the benefit of a calculator.  Knowing one’s times-tables was more than handy – it was practically necessary.

Clearly, however the world and the workplace have changed – drastically.  And because of that, the classroom – and the mathematics skills taught there – are necessarily changing too. But not always as fast.  Do we really want, for example, to spend much – if any – time anymore on learning times-tables and other purely arithmetic procedures?  When was the last time those were used in the workplace, especially where time is money?  It’s not unlike continuing to spend class time learning to saddle a horse so that one can travel later.

Not only has the world changed, but so have the basic skills needed to survive in it.  We need to focus on helping our students learn to tackle and solve problems, using the tools they have at their disposal.  This is NOT minimizing the necessity of learning important skills. It is instead to reinforce that nowadays, almost 2 decades into the technologically oriented 21st century, we need to be sure to know what those skills are.

Perhaps I’m out of date to think this mistaken identity still exists.  I hope so.  But every time I hear the careless phrase “do the math!”, I wonder.  Math and arithmetic are related, but one is not the other.

So, let’s return to Johnny, our aspiring Congressman from above.  I don’t believe Johnny will be an effective congressman without having a good working knowledge of things like statistics, estimation skills, problem solving, interpreting graphs/spreadsheets, handling data, and even number sense.  (How many of our politicians really know the difference between a billion and a trillion?)

But I do believe Johnny can learn and use these skills successfully, whether he remembers (or even learns) his 12-times tables, e.g.  And that fact continues to have increasingly pertinent implications for the classrooms of our community and our nation.

The Tale of the Middle School Assignment

One of my favorite student teaching/learning activities occurred each Fall in a Capstone class for future middle school teachers. It was usually fun for all, and always enlightening for my soon-to-be teachers – and often for me!

Early in the semester, I’d divide the class into three to five small groups.  I’d usually provide 3 varied topics, from which they were to pick one to construct an assignment for a hypothetical class they would teach.  They were to work in a group, create one assignment as if they were passing it out in a class, and have this ready by the next class period. 

I might note that it was fun to wander around the room and listen to some of the conversations as they began to work on these.  It was a good exercise in viewing what they (and others in their group) thought the assignment should include and look like.  There were some interesting dynamics there, as they experienced each other’s ideas.

Usually, one of the 3 topic choices was ‘non-mathy’, asking students to pick a favorite mathematician, learn generally about them, and then share in a short paper. This was often the one the future teachers would pick.  For this reason, as well as to make the thoughts more generally applicable, I’ll focus on that one.

For each assignment, I would later prepare two ‘middle school’ student responses.  In each case, the response of “Sally Square” would be as clearly excellent in content as I could make it, but there would be minor   instructions not followed.  If the requirement was to double space, Sally might forget and single space.  If a 3-page paper was assigned, Sally might not stop until the 4th page, rather than cut material.  You get the idea.

The other response was from “Tommy Triangle”.  Tommy always followed the letter of the law perfectly, but his was clearly a typical-squeak-by submission.  It was often sloppily written, perhaps had grammar mistakes, and had obviously been ‘thrown together’ at the last minute.

In the following class, the students were asked to re-group and give each paper a grade or score.  All groups always agreed that Sally’s assignment was ‘better’ and she had learned more, but they were often in a quandary about how to score the separate papers.  All of them were naturally disappointed in Sally for ‘not following instructions’ – who can blame them? – and they were often astonished to find themselves giving the papers similar grades.

When this happened, their first instinct was to fix the original assignment by establishing more parameters.  Usually they quickly realized that could only make the possible predicaments even worse.  I used to gently mention to them that in these cases, sometimes less is more, especially when the goal assessing learning.  I told them of a special middle school teacher I knew who used to add “turn in something your parents would be proud of”.  This often succeeded better than any rubric!

There was no need to grade this final score-the-papers activity – there were no right answers, after all – but it almost always engendered some great discussions, insights, and reflection.

Primarily, it allowed the students to experience for themselves some truths about assessment that they might not have really accepted from a ‘stuffy college prof’.   1)  Assessing authentic learning is rarely easy, even in a math class.  2)  Good assessment must first involve knowing what you want them to know.   3)   It’s easy to fall into the trap of assessing (or over-assessing) something else, if you’re not careful.

Flying, Learning,Teaching

I’ve always had a love/hate relationship with flying.

On the one hand, of course, flying certainly is time-efficient for getting to destinations.  And it’s often a beautiful, awe-inspiring view from up there, isn’t it?

On the other hand, I’ve always been one of those ‘flying doubters’ who, trusted science notwithstanding, isn’t sure how those huge things get off the ground.  I can’t always make a whole flight without seventeen irrational fears invading my thoughts.  Looking down from a plane’s window is a curious mixture of majesty and discomfort.

The love and the hate butted heads in my 30s.  I had read several of Richard Bach’s books.  I was captured by his love of flying and that sky which is always perfect. (I still love those books.) I think that was the motivating factor, but for whatever reason, I decided to take flying lessons.

My flight instructor was a great guy and an accomplished pilot.  He knew his stuff, and I liked him.  But I had a tough time learning from him.  Perhaps it was his style, perhaps it was my uncertainty – doesn’t matter.  What would happen is that we’d be up there on the downwind leg (before turning twice to land), and he’d start rattling off a list of ‘things to do’.  “OK, bring the speed down, trim the flaps, prepare to turn” and several other instructions which rapidly blended together into one blur of sound.  I’d be doing one thing and miss two of the instructions.  Some students take to flying instantly, but I wasn’t one of them.

It was then that I gained an appreciation of how hard it must be to teach flying to folks that don’t ‘take to it’ instantly.  And it was then that I gained a fresher perspective of the teaching/learning dichotomy.

My perspective about learning came from the fact that, if I was going to do this, I wanted to by-gosh learn to fly, and not just ‘pass the test’.  I wanted to know what I was doing.  I wouldn’t have time to encounter a situation for which I would have to think “Ack, what was I supposed to do here?”  (This came perilously close to happening once.)

And my new perspectives about teaching came, of course, from that fact that I taught a subject which – like flying-  not everyone quickly takes to.  (Did anyone compare my rattling-off-instructions description to their math classroom?)

How many of us still believe that teaching consists of providing a ‘here’s what to do’ list, and that all students successfully and naturally learn that way? (And, even if they ‘pass the test’, do they by-gosh know?)

Isn’t it interesting then, that in our system – at any level – a teacher routinely enters a class with some students who instantly take to the subject(s), and handfuls of others who don’t.  Then, we and the system expect all those students to be equally challenged and to all end up at some ‘proficiency level’ at the same time.  And we become worried if they don’t.  This is especially true in my discipline for which the extremes of ‘drawn to’ and ‘repelled by’ are as varied as those of flying.  It’s a huge challenge that we don’t always address too creatively.  Several topics are still begging for more attention here, but I must close.

I eventually learned to fly.  I survived all my solo hours and cross-country trips.  Moreover, I learned lessons beyond ‘flying’.  My experiences in the cockpit provided valuable insights that followed me into the classroom.

The More Things Change . . . in Technology

One [topic that can help prepare students], is one that is rapidly being phased out by technology.  It is the ability for a student to express himself/herself on paper, in the written mode, using thought and creativity.

I’m not sure if the reader/respondent intended to include the words ‘on paper’ or not.  Perhaps it was unconsciously added without thinking.  But those two words make quite the difference in the context.

On the one hand, I could not agree more that everyone needs to be able to express themselves thoughtfully (in any mode), but I wonder if they really need to be able do it on paper.  Are writing skills “being phased out by technology?”

Having experienced both ‘paper/pencil’ and ‘digital’ writing during school and my career, I now consider myself a better writer while using technology than I was before its availability.  That may not be a universal opinion, but even that underscores the point:  The ability to thoughtfully express oneself is not a function of the medium used to do so.

The opening quote reminded me that ‘technology’ has always been changing. It also reminded me of the implications for education amidst times of change (which is to say always?).

Long ago, I discovered a fun page of quotes about change that I used to use with teachers.  It was entitled The More Things Change, and looked like this (with some minor editing):

 1703:  Students today can’t prepare bark to calculate their problems.  They depend upon slates. When their slate is dropped and it breaks, they will be unable to write! Teachers Conference 1703

 1815: Students today depend upon paper too much.  They don’t know how to write on slate without chalk dust all over themselves.  What will they do when they run out of paper? Principal’s Association, 1815

1907:  Students today depend upon ink.  They don’t know how to use a pen knife to sharpen a pencil.  Pen and ink will never replace the pencil.  National Association of Teachers, 1907

1929:  Students today depend upon store-bought ink.   When they run out of ink they will be unable to write words or ciphers until their next trip to the settlement.  A sad commentary on modern education. The Rural American Teacher, 1929

1941: Students today depend upon expensive fountain pens.  They can no longer write with a straight pen and nib (not to mention sharpening their own quills). We parents must not allow them to wallow in such luxury to the detriment of learning how to cope in the real business world, which is not so extravagant.  PTA Gazette, 1941

1950: Ball point pens will be the ruin of education in our country.  Students use these devices and then throw them away.  The America virtues of thrift and frugality are being discarded.  Business and banks will never allow such expensive luxuries!  Federal Teacher, 1950

There’s always the chance these quotes were invented, of course, but that’s not the point.  These things likely were commonly said and felt at the time, and we could easily hear our own version of this list today, involving calculators, word processing, cell phones, social media, etc.)

Things are always changing, as we know.  The point is, we experience that change as an evolution, not a revolution, so it’s hard to keep perspective.  And there’s the rub for our education systems.  In a few years, we will be laughing at today’s list, as we now laugh at the ones above.  The question is:  What will be on the ‘changing’ list then, and are we preparing our students for those things now?