Choosing Whichever Comes Last

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We’ve all heard the commercial.  Buy a new vehicle and get some kind of 10 year/ 100,000 mile warranty.  When I was younger, I remember being curious about how it can be both 10 years and 100,000 miles under warranty only to find out that the warranty is covered until whichever happens first.  This makes sense for a business model, but I find this idea of “whichever comes first” creeps into our daily decision making and causes many of us to stop short of our true potential.

These thoughts came to me during a run.  I was out for a quick run and I told myself I was going to run for three songs then turn back around.  As the third song ended, I looked ahead and saw a side road about a quarter mile ahead.  I had the option to reach my goal of running for three songs or to push on for just a bit more.  I chose the latter.

The first issue many people face in reaching their potential is not setting a goal.  It is very unlikely to grow or even feel accomplished if you don’t have a goal in mind.  The second issue is that often people are satisfied with reaching their goal and, once they reach it, stay stagnant.

When given the option of 10 years or 100,000 miles I challenge you to choose whichever comes last.  Run to the next street, then one hill more, then finish that song.  Get your degree, get the job that you’ve dreamed of, but don’t ever stop striving for the next step.  You are bound to fail or come up short sometimes, but its the only way you know you gave it your all.  Learn from it, and get after it again.  Goals are great motivators, but they are just the beginning.  Greatness in school, in work, in relationships, and in life happen when you’re given two options and you choose whichever comes last and whichever takes the most work, reflecting on your journey and constantly preparing for the next challenge.

A Simple Switch Creates Curiosity

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How often have you heard this in a meeting with your department, a parent, or even in your own head:

“Kids are too dependent on calculators!  They can’t do anything in their head any more!

There are countless examples of 17 year old kids not being able to complete simple arithmetic which is distressing, but it is not the argument I am here to make.

Instead, I am concerned the fear of calculator dependence is negatively affecting the way teachers design lessons, structure discussion, and assess their students.  When that fear is in the forefront of our minds, we ask students to put the calculators away.  “We need to understand how to do this by hand before we use the calculators”, you might say.  I ask WHY?

Living in the 21st century, we have technology all around us and if I want to know the answer to something I am going to google it.  If the calculator can do the problem for us, why are we wasting our time?  Maybe you argue you have to understand the process to really get what’s going on.  Okay.  My argument becomes: make a question that demands I understand the process.

I am not arguing that understanding how to complete a problem by hand is a bad thing; instead, I’m arguing it’s a great thing!  But we need to make students feel the need to understand.  We need to show that the calculator can calculate, but only humans can think, dig deep, and discover connections.  We need to design problems where the calculator can’t solve it in one step or at all so it once again becomes a tool in the learning process rather than the process itself.


 

To make my point, I will use the example of a lesson devoted to adding and multiplying 2×2 matrices.

Method 1

  • Give students an example of two matrices adding together, then work on the problem with them and show them how it is done.  Leave time for questions.  Then give them a few problems to try before moving on.  Pause for questions.
  • Now give students an example of two matrices multiplying together.  Don’t forget to warn them that this one is tricky!  Then, go through the process with them and take questions.  Then give them a few problems to try and walk around to help answer questions.
  • Two days later, show them how to complete it on the calculator.

Method 2

  • Show students how to add two matrices together using the calculator.  Have them figure out the pattern. Takes about 1 minute.
  • Show students how to multiply two matrices together using the calculator.  Have them figure out the pattern.  Takes very long.  Eventually after giving some hints and gentle nudging, students (maybe not all of them) figure out the pattern and share it.  Without discussing right or wrong, put up examples for students to try by hand and then check them with the calculators.

 

An amazing thing happens in method 2.  Students begin to view the challenge as a puzzle to figure out rather than an “enter” button to be pressed for an answer.  If you pause long enough after kids first type the multiplication into the calculator someone will ask, “why does that work?”. THAT’S LIKE NEVER ASKED!! YOU WANT TO KNOW WHY?!?  It’s a cool feeling.

Even though we allowed the students an opportunity to struggle, we have to wonder what motivation students have to remember the meaning of a topic and how to complete it by hand.  Why can’t they just go back to using their calculator?  The truth is, they can….if you design problems with simple answers. (Assume these are 2 x 2 matrices below being multiplied)

Easy With Calculator

[  1    2 ]      [ -2   4 ]
[  7   -4 ]      [  0   8 ]         =     ?

Not Easy With Calculator

[ 2     4]       [3      x]                [4      12]
[ 1     1]       [-1    6]        =       [14    9 ]

 

A slight change in thinking renders the calculator powerless or, at most, a guess and check monster that drags out the process.  Instead of fearing the power of a calculator, we need to make kids jealous of the power it has and push them to ask why and how it works!  If we can do that, while creating challenging questions that force students to think deeply, we won’t have to let the fear of calculators cloud our judgement.  A simple switch can lead to more curiosity, discovery, and understanding for students.