Category Archives: Technology

Maybe we need to think of it as teachnology rather than technology.

The time with our learners is limited.  We have to make some very important decisions about how to use this time.  We must consider the economics of our decisions based on the resources we have.  Is it cost effective, cognitively, to spend multiple days on a learning target to master something that a machine will do for us?   

 Is what we label as problem-solving and critical thinking really problem-solving and critical thinking or is it just harder stuff to deal with?  Can we teach problem-solving and critical thinking in the absence of context?

Do we have a common understanding of what good problem solvers and critical thinkers look like, sound like, and think like?  If we are teaching problem-solving, critical thinking, and creativity, shouldn’t we know what that means to us?  Shouldn’t we be able to describe it?

Does technology hamper or enhance a learner’s ability to problem solve and think critically?  I think I might be back to the struggle of using calculators to compute and a spell checker to write.  Do we even know enough to make a decision about technology until we experiment and learn by doing? 

If you have not read Can Texting Help Teens with Writing and Spelling? by Bill Ferriter, stop reading this right now to read Bill’s post.  It is a great example of leveraging technology to promote creativity and critical thinking using technology.  Read about having students write 25 word stories.  This is teachnology, not technology.  Tweet, text, type, write on paper – it doesn’t matter – unless you want to publish your work.  The technology, Twitter in this case, aids in the critical thinking; you are restricted to 140 characters.  The technology offers the learner a way to publish and see other published work. 

My ability to transport myself from place to place is actually enhanced and improved because of my truck.  I have no idea how my truck works other than gas goes in, step on the brake to stop, R means we are going to go in reverse, etc.  I do not need to understand the mechanics; I can have that done.  

I do not need to understand the mechanics; I can have that done.   I don’t need to know how to change the oil in my car.  I need to know that I need to have the oil changed in my car.  And, very important, I don’t need to learn this lesson by experience.  It is too expensive to learn experientially why I must have the oil changed in my car.

Isn’t it too expensive to spend 2-3 days on some topics that we traditionally teach?  Are we getting the biggest bang for our cognitive buck?  Often our learners can’t see the forest for the trees.  They never get to the why because of the how.  Don’t we need to learn when and how to use technology not only to engage our learners, but to increase our cognitive capital?

How can we learn to ask

  • Why are we learning this?  Is this essential?
  • Will technology do this for us so that we can learn more, deeper?
  • Does this have endurance, leverage, and relevance?
  • Shouldn’t we use technology to grapple with the mechanics so the learner shifts focus to the application, the why, the meaning?

Is Efficiency Without Understanding Efficient?

Time and meaning.

How often do we hear the following? “I don’t have time to _____.”  “I can’t take the time to _____.” “If I do this, I won’t have time to _____.”  “I can’t believe they don’t understand _____.”  “It’s like they’ve never seen or heard _____.”

Time and meaning?

Time and meaning.  How often do we hear the following? “I don’t have time to spend 3 days on this section.”  “I can’t take the time to do a project.” “If I do this, I won’t have time to teach them everything they have to learn.”  “I can’t believe they don’t understand _____.”  “It’s like they’ve never seen or heard _____.”

Time and meaning…

How often do we hear the following? “I don’t have time to spend 3 days on exponential growth (slope, poetry, reconstruction).”  “I can’t take the time to do a project.” “If I do this, I won’t have time to teach them everything they have to learn.”  “I can’t believe they don’t understand exponential growth (slope, poetry, reconstruction).”  “It’s like they’ve never seen or heard of exponential growth (slope, poetry, reconstruction).”

Time and meaning!!!

We want our learners to be efficient.  We teach shortcuts, right?  I’ve been wondering about shortcuts for a while.  Is it a shortcut if I don’t know the long way?  We teach King Henry Died Monday Drinking Chocolate Milk to help learners become efficient about the order of prefixes in the metric system.  We teach Please Excuse My Dear Aunt Sally to help learners remember the order of operations.

Now, don’t get me wrong.  I LOVE mnemonic devices!  In How the Brain Learns Mathematics, How the Gifted Brain Learns, and How the Special-Needs brain learns Dr. Sousa gives evidence that process mnemonic devices are powerful for learners, particularly those with dyscalculia.

“Process mnemonics are so effective with students who have trouble with mathematics difficulties because they are powerful memory devices that actively engage the brain in processes fundamental to learning and memory.  They incorporate meaning through metaphors that are relevant to today’s students, they are attention-getting and motivating, and they use visualization techniques that help student link concrete associations with abstract symbols.”
How the Brain Learns Mathematics, David Sousa

How am I, how are we, helping students link concrete associations with abstract symbols?

Our current learning target in Algebra I involves exponential functions – exponential growth and decay.  We can just teach them the formula, but are we really teaching them if we do that?  Haven’t they been given the formula before?  How do we link concrete meaning to the abstract symbols in the formula?

My teammate, @bcgymdad, taught me how to do this a couple of years ago.  It takes more time to teach it – several days.  I’d like to describe it to you; you can decide about time and meaning and efficiency.  I’d LOVE to know what you think! Oh, and sorry for the pseudo-context. We had a sense of play; we had fun, and we learned. 

Question 1:

I need to hire two of you. You can pick up some quick spending money.  Volunteers?  Great!  The job is to clean windows for 20 days.  DG, I want to hire you and I’ll pay you $40 per day; does that sound fair? <Yes, ma’am>  GW, I also want to hire you, but your payment plan is different, okay?  I will pay you $0.01 today, tomorrow $0.02, $0.04 the next day, and so on. Not a new problem to me, but apparently a new problem for the learners.  I took a quick poll of the class.  Whose payment plan would be best for your if you intend to complete all 20 days?  The vote was great; it split right down the middle.

Big questions:  If both workers complete the 20 days, how much will they each be paid?  Instantly, everyone knew DG would be paid $800.  <Yeah, baby!>  How much would GW be paid?  They just sat there.  Really!  Waiting for me to tell them; they are so conditioned that even after 4+ months with me, they waited.  I had to say “Don’t you have a calculator? Figure it out?”  Then my favorite question “is it okay if we work together?”  AHHRRRGGGG!!!!!!!  Are you kidding me?  YES!

Not one learner, not one, thought to use a spreadsheet.  Never occurred to them; they didn’t know how.  We stopped; we took the time to learn. 

Stage 1:  Simple spreadsheet formulas – make the spreadsheet work and why to use a spreadsheet.

   

 DG is feeling “ripped-off”.  So let’s change his daily rate to $100.  BOOM! The power of spreadsheets.

   

The question…will the time taken to do this work numerically connect meaning to the abstract symbols?  The first meaningful connection popped up immediately.  Again the question…How much was GW paid for her 20 days of window washing? 

My learners who speak before they think belted out “$5242.88!”  KC, profoundly quiet reserved KC, said loudly with a great frustrated voice: “No she did not!  That’s how much she was paid on day 20!”  Meaning!  This led them to ask me how to find the total.  I love it, love, love, love it when they ask me to teach them something. 

   

We graphed the data.  Look how much can be learned graphically. Now we can visualize the difference in constant rate and exponential rate.  Then we wrote equations.  It made sense to them that the equation for GW was y = 0.01(2)^x.  Interestingly, they had a little trouble getting to DG’s equation y = 40.  Sigh…so much work to do to connect ideas.   

   

While my learners could not solve for the day DG and GW would be paid the same wage algebraically, they can all tell me when looking at the graph.  Are we letting the analytic algebra, the efficient way, hamper learning and understanding?

Question 2:

ES has $1500 and invests it at 8.5% interest compounded yearly.  In 10 years, he will be 24 and, hopefully, graduating with his masters degree. How much money will he have at the end of 10 years if he just makes this initial deposit?

Can you apply what we just did with spreadsheets to answer this question? Oh, if you know the formula, just use it.  It is more efficient.  Does anyone know the formula?  Nope.  They know there is a formula, but they don’t know it.  And that is OK. 

Without direct instruction from the adult in the room, one learner realized that you had to have a year zero.  This rumor then spread throughout the community very quickly.  Oh sure, there were questions about getting the spreadsheet to work, but they were confident about their math/arithmetic.  Well, oops, some had to remember that 8.5% is not 8.5; it is 0.085.  But they learned it experientially and from each other; they were not told.  They learned from the data; it did not make sense. 

   

Again, the power of the spreadsheet.  8.5% is not at all realistic for 2011.  What happens if we change the interest rate to 1.5%? 

   

How long will it take ES’s money to double?  The spreadsheet is not efficient.  Using a graph is much more efficient.  This is why we need to understand the formula, but not before we understand the problem.

   

Do you think the spreadsheet work will help learners understand?  Does taking the time to work with the numbers help students understand the problem?  Will it help students interpret the graph?

Time and meaning…If we take the time to teach multiple representations of the same idea will we increase the opportunities for students to find meaning and understanding?

Learning, liking what we do, bright spots, and literacy

I do not know how to find the square root of a whole number without a calculator. I have number sense; I can estimate that the square root of 21 is between 4 and 5, closer to 5. I wonder if my students’ work with fractions and decimals falls into this category.

From Powerful Learning What We Know About Teaching for Understanding: “Students do not routinely develop the ability to analyze, think critically, write and speak effectively, or solve complex problems from working on constrained tasks that emphasize memorization and elicit responses that merely demonstrate recall or application of simple algorithms (Bransford, Brown, & Cocking, 1999; Bransford & Donovan 2005).”

I worry that our students never get to the solve-complex-problems stage of learning in math. I’m afraid that we assume if they can’t do the “basics” then they are not qualified to attempt sophisticated interesting problems. If we would dare to start with the complex problem, would we interest more students in learning – even learning the “basics”? If we allowed technology to crunch the numbers, would students experience more engagement and attempt more interesting, complex, elegant problems? Would they ask to learn to improve the “basics” that they deem necessary or important? Would they use technology to aid in their learning? Are we brave enough to test this hypothesis?

In Bo’s It’s about Learning blog from June 25, 2010 I read: “We should be recreating more of the moments when things work well, when our strengths are revealed and engaged, when our efforts are at our best.”

Imagine you are sitting in Algebra I looking at one of your papers where every problem is wrong; you do not have one right answer on the entire page. You know that you have done everything the way you were taught. You know and can express that you have used the correct inverse operations to solve the equations? Or, you know the quadratic formula and can correctly interpret the results IF the results are correct? How frustrating!

Think of Gillian from The Element. Gillian did not perform well on tests; her work was difficult to read, and it was often turned in late. She was “a problem” in class. Sound familiar? In Gillian’s case, she needed movement. What if you need technology? What if we could reveal your strengths in algebra by simply allowing you to leverage technology to show your work and effort at its best? Would you find the motivation to work on your deficits if you find your strengths first?

Calculator is to Arithmetic as Spell Checker is to Spelling???

Is using a calculator for math comparable to using a word processor for English? Is the calculator an arithmetic checker like the word processor is a spell or grammar checker?

My friend Jeff makes a good point about technology integration (advances) in English. Students that use a word processor must still proof their writing. Do we worry that kids won’t learn to write because of spell check or grammar check? Or, do we think that because of these tools they are free to concentrate on ideas, organization, voice, word choice, sentence fluency, conventions, and presentation? (Okay, conventions have to do with grammar and spelling, but I’m making a point here.)

The grammar or spell checker does not always catch “there” when I mean “their”. A calculator will not catch that I meant (-2)^2 when I entered -2^2. The calculator does not know that I mean 1+(6+4)/2 when I enter 1+6+4/2.

As Peyton pointed out in the Writing Workshop meeting, MS Word will not alert you to your error in writing “warmest retards” when you meant to write “warmest regards”. Your calculator will not alert you to an entry error; it will not know that you entered -3.75 when you wanted -3.57.

The spell checker automatically corrects some of my incorrect spelling. When I type “recieve” it automatically changes it to “receive”. When I type “calcualtor”, the word automatically changes to “calculator”. When I write “I never here anyone…”, the grammar checker alerts me to check my spelling or word choice. The Nspire calculator will autocorrect a little bit, but it assumes what you mean. For example, if you open parentheses, it will close them. However, you must make sure that it closes where you intend to end the grouping.

This is an interesting place for me in my thinking. I know that it has to do with age appropriate learning. I believe that young children should learn their numbers and arithmetic just like they learn their letters and words. I believe that junior high students should learn how to graph and solve equations by hand, graphically, with tables and spreadsheets, and with technology.

How much more could we learn about algebra, calculus, and statistics if we used technology to accommodate 8th graders that struggle to compute? Don’t our students need to spend more time on data gathering, mathematical modeling, and interpreting graphs and less time on mechanics.

Have you seen Conrad Wolfram’s Ted talk Teaching kids real math with computers?  What do you think?

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Just look at any TED talk by Hans Rosling to see examples of how critical the analysis and synthesis of mathematical information is to our future.