Knowing without understanding

This is a great article from K-12 education about teaching for understanding, http://www.mathsolutions.com/documents/NumberTalks_SParrish.pdf.  I am guilty of teaching "tricks" thinking that it helps learners be able to do math better.  However, my focus now is teaching for understanding and encourage my learners to think beyond the "trick".  

My favorite trick to teach learners was "Keep Change Flip" when dividing Fractions, learners would just do this procedure but not one really knew why we did it. 

What are some of your math tricks that you teach but would be interested in finding a different conceptual approach for?  Let's see if we can come up with a list of great alternatives to use.

Brooke Istas
SME, Math and Numeracy

Comments

   I've got some folks in the MTBOS ("Math twitter blogosphere") who think tricks are anathema ... but they're working with younger students with fewer misconceptions, and a whole lot fewer "pieces of pie" in ALEKS to plow through.   So last week I was telling a ton of people confronted with problem after problem after problem that hey, the first number is your starting point... so you don't change that... but we change the division to multiplication and the next fraction to its reciprocal.,.. and yes, "Copy, change, flip" is a good way to remember that procedure. 

    I try to spend a bit of time connecting their idea of "half of something," which many of them already have, with what it looks like :  40 x 1/2.   The whole language thing gets a little hairy -- we're multiplying by a half.   Oops, except we're actually dividing by two. 

    I'm going to figure out a way to explicitly explain that little language thing... it's almost big enough to be a "big idea."   After all, with negative integers we can "add the opposite" and if they get further into algebra, they have the fun of taking roots of powers.   The idea that things can be done and undone is an important one.   

  

It's challenging when a student is trying to do a five or six step problem... and she thinks 7 - 5 might be 6... but I can hope that tone and language are helpful with the fraction parts.   No, I don't have time to build up the concept with the "denominator -- it means name -- the bottom number isn't the amount, it's the size" -- this is the tutoring lab and ... that part was supposed to have happened weeks ago... so now these are much more advanced problems.   

     However, I can at least duly note that "hey, it's adding and subtracting... you have to add the same things to the same things, and fourths aren't the same as fifths."   And I emphasize the "number part" of the fraction with my tone so that 3/5 + 3/4 is "three of these and three of those..."  

At least, she's seen the process so when I get it started she can finish it.  She's been depending pretty entirely on the calculator (and may have been shown how to do the fractions with it), and since they don't get to use the calculator for lots of things, that's not going to be good... 

... but it also reminds me that what are perceived as mnemonics or "tricks" can, in fact, be conceptual.   I want a student to see a plus sign and, perhaps, hear my annoyingly repetitive quip:   "It's same to same in the addition game,"  whether it's common denominators or like terms or...   

Susan,

Thank you for your response!  I am sure others of us in this community have thoughts on this topic, too.  Anyone else care to chime in?  I am curious if we are truly helping students with just teaching tricks or if a deeper understanding does eventually form?  Are we truly reaching for that deeper, richer, experience that will last a learner for the duration of their life?  

Thoughts?  

Brooke