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?

Funerals, Fiddlin’, and the Future

Earlier this spring, I attended the funeral of a former colleague who was a long-time music educator at College of the Ozarks.  I was impressed and inspired by the folks who spoke and shared stories about my colleague.  Most of these speakers were former students now doing impressive things in all realms of the music world.

When a funeral is done right, it seems to me, one leaves the service inspired by the deceased’s life, but also reminded of some important perspectives about life in general, and in this case, education. I know that as I left, I was indelibly reminded of three things.

I was reminded, first, of the value and impact of good teaching and a good teacher.  Students who spoke of their mentor’s influence in touching – and funny – terms were clearly now passing along those same influences in multiple ways to those for whom they have become mentors.  And, of course, this impact will continue for untold generations.

Second, I was reminded of the importance of music in particular – and the arts in general – in our lives.  Which of us does not have our own music (of whatever style) that speaks to us, that moves us, that energizes us, that supports us?

Finally, of course, I was then similarly reminded of the resulting importance of music in particular – and the arts in general – in our educational systems.  Think of that music that moves us, whether classical, rock & roll, or other.  Where do we think those musicians first discovered – and then began to cultivate – their craft?  Do we think they suddenly decided, after graduating from high school, “I want to play the saxophone”?

We must somehow, as a society, find ways to slow the elimination of music in particular – and the arts in general – from our educational curriculum.  Or we must find a (financial?) way to allow schools to continue to provide these crucial elements of education for students to discover.  Or, we must find ways to do it by ourselves.

Along those lines, have you heard of Ozark Mountain Music, Inc?   Their mission is to pass the traditional music of the Ozarks on to other generations of fiddlers.  Ozark Mountain Music runs fiddle camps each summer, and after-school programs during the school year.  The Possum Holler Fiddlers are the showcase fiddlers of the program, making performances at all kinds of events throughout the year.

The program receives some support from the Missouri Arts Council, but it is the brainchild of Bob and Karlene McGill. They are doing wonderful things with the programs – and with the lives of the young fiddlers, some with exceptional talent, previously unknown. Perhaps your (grand)child could be one of them? Discover much more (including information about their summer camps!) at Ozarkmountainmusic.com, There are pictures and contacts at the site.  And why not donate as well?

Traditional Ozark fiddling is not likely to be in the curriculum of many schools, so here is a group doing something about that on its own.  Will we eventually need the same type of groups for the more traditional music and arts programs?

All of which leads us to the future.  How do we manage, in these challenging times for education, to preserve environments which sustain excellent teaching and mentoring for our students?  How do we manage to preserve music in particular – and the arts in general – in our schools?  We must find creative ways to maintain these crucial features of our educational systems.  We must do this for our society, for our students and for their futures.

Counting the ‘Counting Problem’ Answers: Revisiting our Menu Choices

If you’ve just joined us, our last episode ended while we were still sitting in a restaurant, pondering our situation:  We had NINE (9) choices for entrees, from which we could pick ANY 2 for the daily special.  The menu suggested there were 81 combinations from which to choose, and we weren’t sure we agreed.

Responses to the set-up in that last column were both fun and varied.  And, by the way, noticeably free from the oh-god-I’m-supposed-to-know-the-answer feeling often encountered in a classroom.  Part of the point, naturally.

When the dust settled, the majority (but not all) of respondents thought the restaurant was wrong. I received proposed solutions of 36, 45, and 48, as well as the 81.  Surprisingly, I did NOT receive a submission of 72, which I had expected.  See below.

One reader submitted her answer of 36. Here is her (slightly edited) approach:

I used an easier problem of 4 foods so I could find the pattern. I used ABCD and saw that there were 3 choices for A, 2 for choice B, 1 for C, and no new choices for D. So, with 9 choices it would be 8+7+6 etc, or 36 different choices.

Did you get the idea?  This is a nice approach, and I particularly liked the ‘used an easier problem’ as a first step!

Another reader also arrived at 36, and further noted that if you allow two of the same choice (chicken and chicken – remember?), that would add nine meals to the total, bringing the total choices to 45.

A reader who is also a teacher volunteered that “problems like this made good homework problems and got family members involved, as well!”

So. Where did the ‘81’ answer come from, and what’s ‘wrong’ with it?  Maybe nothing! It appears the menu designer multiplied 9 means times 9 (each of 9 entrees can be paired with each of the other nine, allowing repeats?) to get 81.

This is what often happens with solvers.  They usually decide not to allow repeats, then pair each entree with 8 others, giving the 72 (9×8) meals I had anticipated I might get.  Note that, for my money, this is good reasoning!!  Where’s the hidden flaw?

When the above happens, the solver often proudly displays 9 columns of 8 meals each.  One can then often point to a choice (say, AC in column 1) and then find its counterpart elsewhere (CA in column 3) and ask, “is it a different choice if you turn the plate around?”

The response is usually a broad grin (and a forehead slap?) which indicates the light coming on!  At this point, they often add, “OK, divide 72 in half and you get 36!”  Another very good approach!   (One submitter above even noted “I decided chicken and grits is the same as grits and chicken.”)

Interestingly, every so often, the (smart-aleck?) response to that plate-turning question is “Yes, it is different.  I eat my meals from left to right!”  And with that condition, 72 (or 81) becomes a ‘right’ answer! Ah, conditions!

So, in real life, with these variables, we usually agree on 36 or 45 possibilities.

I must add this fun PS:   A teacher once sent me an ad from the St. Louis paper featuring a similar display for the same restaurant chain.  The ad proclaimed: “Here are 3 of the 54 choices possible!”  54?! WHERE did the 54 came from?!  Any thoughts out there?