Notation

Evelyn Lamb put some interesting thoughts about notation on Twitter last week starting with this tweet:

 

Lamb’s sequence of tweets got me thinking about some of my own battles with notation over the years.  I remember, for example,  struggling mightily with the notation in Hartshorne’s Algebraic Geometry in graduate school.  So, with the thoughts about notation in kicking around in my mind, I had an interesting week watching both of the boys have their own struggles with math notation.  I hope 20 years from now their memories of the Art of Problem Solving books don’t match my memories of Hartshorne!  (note post writing:  I’m surprised that another immediate reaction to Lamb’s post didn’t involve learning Latex!!)

First up – some of my older son’s struggles.  We began a new chapter about exponentials and logarithms this week.  Unfortunately a work trip to London next week is going to mean putting this chapter on hold for a bit, but I was still really excited to dive into this new subject.    We spent the first few days going through several different examples and eventually came to problem 19.4 in  Art of Problem Solving’s Introduction to Algebra book:

Solve the equation:  2^{(16^x)}  =  16^{(2^x)}

The difficulty my son had with this problem came in simplifying the right hand expression.  His instincts told him that  (2^4)^{(2^x)} was equal to 2^{8^x} instead of 2^{2^{(x + 2)}}.  It was a little surprising to me that the introduction of powers in the powers would cause as much difficulty as it did for him.  We ended up spending two full days on this problem – making various charts of numbers and doing several different versions of the problem.  He didn’t have too much difficult with examples which had only numbers, but problems that included variables with powers in the powers were really a struggle for him.   After a few days  he seemed to get a little more comfortable and was  able to work though this problem without too much help:

At essentially the same time I was in the review section of chapter 5 of Art of Problem Solving’s Prealgebra book with my younger son.  We were reviewing some simple algebraic expressions and eventually landed on a problem with an expression similar to this one:

\frac{4x + 3}{2}

The algebraic expressions without fractions hadn’t given him too much trouble, but the introduction of fractions in the expressions really caused him to struggle.    His instincts told him that since the 4 could cancel with the 2, but the 3 could not cancel with the 2, the expression would simplify to be \frac{2x + 3}{2}.  As with the discussion with my older son above, what seemed to me to be a minor change – in this case the inclusion of fractions – caused quite a lot of difficulty.  Working through the difficulty with the fraction notation took a couple of days.  We worked through some examples without variables and then worked on some progressively more difficult examples which included variables.  Eventually we tried out a little “find the error” problem which he seemed to understand pretty well:

 

One of the nice things about working through all of this math with the boys is that we aren’t really in any hurry.   If something isn’t making sense we have the opportunity to take a day or two (or longer) to make sure the concepts are sinking in.   Lamb’s twitter post about difficulties that her college students were having with notation definitely brought back memories of  my own struggles in graduate school.   Maybe she just planted the idea in my head, but it was interesting to observe the notation-related struggles that my kids were having as their own math work.  I wonder if you ever get to a level of math where that struggle goes away?

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