Connect Abstract to Practical…Meaning to Mechanics

Square-foot gardening, heard of it?


Hold that thought for just a minute…We’ll come back to it I promise. 

I am conflicted about what I am teaching.  I want to say my team is conflicted about what we are teaching, but I think it is really me.  I worry that our teams are on pedagogy auto-pilot; we teach what we’ve been teaching the same way we’ve been teaching it, which would be fine IF they were learning it, but they are not – not everyone.  It is not relevant, and they know we are going to teach it to them again next year.  What is the point in learning it now?  (Totally over-simplified but it makes my point.  Forgive me, please.)

We don’t know how to find balance between by-hand skill and skill with technology.  It is such a hard question and so risky that many choose not to consider it.  We think “I learned it this way; this is the safe way; I understand this.  Since I don’t know what is right concerning technology, I choose to ‘do no harm’.”

I’m supposed to teach a unit on operations with polynomials.  I am searching for context.  Why do we need to learn to add (subtract, multiply, divide) polynomials?  Who cares?  Where is the application of this skill? 

Is the entire history of what can be taught in algebra based on what was possible to learn before technology teachnology?  I can and need to factor polynomials if I want to graph them.  Isn’t that the number one reason to teach factoring?  BUT, don’t I now have a calculator that will do that for me? 

Not only do I have a calculator that will graph any polynomial, I could have a calculator that will do the algebra for me too.  Interesting…Is it important to factor big-hairy polynomials or is it important to know what the factors tell me about the polynomial…to make a connection from the algebraic to the graphical to the numeric? 

Isn’t it a forest and tree thing?  If I can’t factor, I can’t see why I need to factor.  If I don’t see why I need to factor, why do I need to learn to factor? 

My current conflict is about adding polynomials.  If I don’t see why I need to add polynomials, why do I need to learn to add polynomials?  I want my learners to know how and why to add polynomials, but how much is enough by-hand?  How much time should be devoted to the mechanics of adding polynomials?  In the face of technology teachnology, shouldn’t we focus on meaning rather than mechanics?

For example, a great learner question is about adding like terms.  Why can’t you add x and x^2?  Why aren’t they like terms?  Aren’t they both x ’s?  If learners don’t understand the why will the ever care about the how? Do we stop to think about why you can’t add x and x^2?  Can we give an understandable explanation? 

I can use my calculator to show that you can add x ’s together; you can add x^2 ’s together, and you can’t add x ’s to x^2 ’s.  Boom.  Done.  But, this technology does not tell us why.  Why can’t you?  What reason – meaning – prevents these variables from being “like” terms?


This brings us back to square-foot gardening…Doesn’t this picture illustrate why x can’t be added to x^2?  Think about it…


Traditionally, each box in the picture is 4’x4’ and is subdivided into 1’ sections.  You plant in each section.  How many x ’s can you see?  How many x^2 ’s can you find?  Can you see why x can’t be added to x^2?  Can we use this image to give meaning to x and to x^2?  Aren’t units (shout out to all science teachers) critical to this understanding?

If we want art and design, meaning and service, reason and understanding, integrated studies and PBL, and so much more, don’t we need to connect the abstract to the practical?

Why would anyone garden this way?  Is square-foot gardening more or less efficient than gardening in rows?  Can you grow more or less in squares?  How can you efficiently irrigate this type of garden and recycle at the same time?  Would a 4’x4’ garden improve your heath, your lifestyle, feed the hungry?


Mel Bartholomew – Introducing Square Foot Gardening

Frequently Asked Square Foot Gardening Questions


SpinPost – PBL


  1. It might not be the same thing, but in languages, I feel that we need to teach students how to use translation tools and resources (in particular online translators), not simply shy away from using tools that are obviously helpful because “students should know how to write without depending on anything else or anyone else.” That’s my two cents.


  2. Thanks Kristen! I do think that it is the same thing. I want my learners to be confident problem solvers using algebra. I want them to think with technology. I want them to leverage technology to help find relevance, meaning, and endurance. We want learners that embrace problem solving and critical thinking; we do not want math avoidant citizens.


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