Monthly Archives: September 2011

The Demise of Graphing Calculators?

Here’s an interesting question posed on a graphing calculator discussion group:

At Hypothetical High School, all students have laptop computers 24/7, and fast and open Internet access both at school and at home. Students will already have free access to Geogebra, Excel, Microsoft Maths, Wolfram Alpha, and heavens knows what else.  If more Maths power is needed, it can be bought relatively cheaply….

For such a school, is there any justification in asking parents to pay an extra AUS $190 for a graphics calculator?

Just so I don’t confuse anyone, remember that I am a firm believer in the power of technology for enhancing the relevance and power of mathematics teaching.  I have seen students explore and discover amazing mathematics because they had the instant feedback abilities of mathematics software (handheld and computer-based) that answered their “what if” questions whenever they occurred whether or not an official teacher was present.  It levels the “playing field” for all students of mathematics and grants them access to understandable answers that sometimes result from intervening mathematics beyond the current reach of the explorers.  It is one more tool that can be used to lure the curious into the beautiful worlds of mathematics and patterns.

But if all students and teachers have access to calculation/graphing/manipulation tools that are far faster than any handheld calculator, how can we possibly justify charging (or asking) families to pay even more?  The only argument posted in the discussion group in response concerns high stakes testing.  Real or not, that strikes me as an anemic response for many reasons.

  • We already ask families (or all tax-payers via school-funded testing) to shell out huge sums for annual testing.
  • Testing already occupies a disproportionately (and dis-appropriately?) large amount of the focus of many schools.
  • If math and science “explorers” already have laptops, won’t requiring them to learn the specific workings of handhelds just take up more classroom time that should be spent on content?
  • Are we really so wedded to testing that we are willing to spend extra time, money, and other resources to keep it in place?

As much as my students and I have grown from the presence of graphing calculators in their hands over my years as a math teacher, is it time to say goodbye to my old friend, the handheld calculator?

Quadratic Explorations

I first encountered this problem about a decade ago when I attended one of the first USACAS conferences sponsored by MEECAS in the Chicago area.  I’ve gotten lots of mileage from it with my own students and in professional circles.

For a standard form quadratic, y=a*x^2+b*x+c, you probably know how the values of a and c change the graph of the parabola, but what does b do?  Be specific and prove your claim.

Elementary Multiplication

One of my daughters is now in 2nd grade and I’ve always been interested in keeping her curiosity piqued–whether in math or any other discipline. I never want to push her to memorize anything or accelerate her learning beyond what she’s ready to engage.  But she has always enjoyed games and has been intensely interested in art.  Following are some ideas I’ve been playing with my daughter during our recent conversations.  Perhaps some of parents out there can benefit from my ideas or others can give me some additional leads on other good ideas

I always play number games with my daughter.  A few years ago I asked her how many apples (or dolls, or crackers, or whatever was in front of her at the time) she would have if she had 2 and I gave her 2 more.  There were many variation on this theme.  Eventually the numbers grew larger and then I asked her how many I would need to give her if she had 2 apples now and would have 5 after my donation.  It was my attempt at introducing subtraction without needing to name a new concept.  From my end, this has worked well.  My daughter likes playing with numbers and I keep pushing the window of what she can handle.  I make it clear that she can always ask for hints and that I’m never disappointed if she can’t handle a question I give her so long as she tries.  It’s a delicate balancing act, reading my daughter’s readiness and trying not to overburden her.  When I misjudge, her blank face tells me to go in another direction.

I’ve been seeding the idea of commutativity lately.  When I ask her something like 10+2, I always follow with a 2+10 and ask her if she notices anything about her last two answers.  At first she didn’t notice, then she saw that the answers were the same, and recently she has been been telling me that you can “flop the numbers” in addition and get the same answer.  I knew the idea had begun to sink in when I asked her 4+8 and she asked if it was OK by me if she added 8+4–it was easier for her to add on 4 to 8 than 8 to 4.

Today, she mentioned negative numbers and I jumped on her exploration of commutativity.  She told me that she knew “somehow” that a subtraction gave a negative result if the “second number was bigger.”  I told her that the only difference between 10-13 and the 13-10 that she already knew was that 10-13 gave a negative answer.  Further details can happen later, but for now, I jumped on a moment of interest and continued a game that we’ve been playing for months. Her face lit up when she realized that negative numbers really aren’t that hard!  It’s never about memorizing facts and I’m always ready to back off.  My mantra:  Keep reading your audience and keep it fun.

Here’s another set-up I started a week ago.  I’ve never seen multiplication started from this angle (but I’ve not been trained as an elementary teacher either).  Nevertheless, I was thinking about how to introduce the concept of multiplication without making it a chore or making it a new idea, so I tried tapping into her art interest.  Two weeks ago, I asked her how many ways she could arrange 6 dots into rectangles.  Grabbing some paper, she quickly made an arrangement of 2 rows and 3 columns and a short time later, 3 rows and 2 columns.  It took some prompting to get her to see a line of dots as a rectangle 1 unit high (or 1 unit wide), but the hook was set.  What follows is a sampling from a journal she keeps for playing around with shapes or math ideas.  I had asked her to try this rectangle arrangement of dots for every number from 1 to 20 using what she had learned from arranging 6 dots.  I asked her to list beside each arrangement the dimensions of the rectangles she could find.  She missed a few, but I’m a pretty proud dad right now.

While she’s just scratching the surface of multiplication right now, I’m pretty psyched that she has written multiplication while thinking that she was just describing the dimensions of rectangles.  What I really think is cool is that she is learning multiplication in reverse–starting with the products and learning the factors.  Eventually, we’ll do this in the “normal” order, but for  now, the art connection has her totally hooked.

Also, she already knows about even numbers and odd numbers.  For now, I have future plans to introduce prime numbers as those that can form exactly 2 rectangles.  We’ll also explore commutativity further once she gets more comfortable with multiplication.  Down the road, I see showing her that 1 is a special number because it is the only number that has only 1 rectangle.  Also, perfect squares (the square numbers that we’ve also discussed) are the only numbers with an odd number of possible rectangles–another consequence of commutativity.

I’d love any feedback on these rambling musings.