Category: Algebra 2

A Simple Switch Creates Curiosity

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.

The Struggle of Letting Them Struggle

Today I was teaching an honors level class how to add and subtract rational expressions.  These are the students that have pushed themselves to take Algebra 2 over the summer to advance their potential opportunities as they become upperclassmen.  We’re talking serious students.

To begin the unit, I modeled my teaching after a Dan Meyer inspired idea and, although, not the most exciting thing, it helped students gain a little buy-in.  The previous day we had actually looked at adding fractions, discussed the similarities and differences of adding rational numbers and made significant progress.  So I thought.

We started today by checking over the homework and there were a lot of questions.  THERE SHOULD BE  A LOT OF QUESTIONS.  This is hard stuff and I was fortunate to have curious students, anxious to learn the content at a deep level and ask about their confusion.  If there aren’t any questions after the first day please don’t assume they’re good.  In fact, I could sense the frustration and kept telling them:

This is hard, but that’s why it’s better than many of the other things we look at.  It’s a chance for you to struggle and flex your creativity in solving problems.  Keep trying, keep failing, and keep asking questions.

Instead of moving on, I asked them if they would prefer to practice this a little more; they said yes.  I started by giving them a more simple problem.  Then, I increased the level of difficulty, and finally, gave them this harder problem:

__5__  +         4         –      2  
x(x+2)         -x – 2             5x

The students were able to handle the first two problems fairly well with asking only a few questions.  The final question pushed many of them outside of their comfort zone.  I let the students begin on their own, then as they began to ask questions I went in to help clarify some confusion.  But….it was rough.  I’m talking seriously rough.  Like, why would you do that, how can you even think that after the other things we looked at rough.

What I found myself doing as they asked me questions was becoming overwhelmed.  Student after student asked me questions, and I was having the most difficult time thinking about how to steer them in the right direction without showing them exactly how to do it.  At one point, I told a student:

Just hang on for a minute; I’m going to go through it with the class in a little bit.

It was at this point I realized the struggle of letting kids struggle.  Learning is messy, but when we have a quality problem of difficulty and students ambitious enough to struggle through it, we must ask ourselves whether we are ambitious enough to help them through the struggle rather than re-gain control and show them how it is done step-by-step.

As teachers, we need ways to encourage kids to take risks, while demonstrating that it is okay to do so.  We need to allow kids to make mistakes and fail, but we cannot be there to catch them as soon as it gets a little difficult.  Instead, we need to foster a classroom where creativity is encouraged and wrong answers are explored and shared.  In that moment, I wonder how the learning in my classroom would have changed if I grabbed three students’ notebooks, threw them under the document camera and, as a class, we discussed the math (wrong or right) that the student displayed.

I, like many other teachers, get caught up on the right answer rather than the process of getting there.  Because of this, I get frustrated when students are nowhere near the correct answer.  Instead, we need to embrace the messy process and the learning that is held within.  Maybe the student that struggled and got a problem wrong three times actually ends up learning and understanding the process at a deeper level than the student that got it right on the first attempt.

I ask you to be careful next time you get frustrated because no one in your class is finding the right answer.  Use the opportunity to talk about mistakes and continue to give them chances to flex their creativity and make mistakes.  It’s a struggle.