Sunday, February 21, 2010

Another Algebra Problem for the "Ages!"

Forgive the lame attempt at "punnery" in the title but I hope you will see the many layers in the following discussion about a 'typical' word problem involving people's ages. Perhaps, not so typical but I hope it will be topical!

Please bear in mind that this blog is geared essentially toward the K-12 math teacher. Most of the problems I write are not intended to stump people. Rather they are vehicles to engage a discussion of instructional practice as we move toward a leaner, deeper and more coherent curriculum in this country. 

To reiterate a point made many times on this blog:

Most common teacher reaction to these problems and investigations:
"Dave, These are nice but you know we don't have time for this, particularly at the secondary level. What exactly do you want me to delete from the curriculum!"

My Response: "That is the point of why we need to reduce the number of topics covered and alter our paradigms of math education.  The underlying ideas in these problems and explorations include some of the essential ideas of mathematics, algebra in particular. Who am I to make such a pretentious claim? Well, it isn't pretentious at all! Everyone knows that most students go through the motions of learning, concentrating solely on what they believe will be tested. They have little interest in anything else (yes, there are exceptions!). If assessments focus on procedures and superficial ideas, then there is no reason for students to make the effort to actually use their 'reasoning brain'!"

As teachers of mathematics we need to know and understand what the "big ideas" of math are before we can hope to convey this to our students. In the problem below, the essential idea is NOT how to solve a contrived word problem. 

I.  One of the big ideas here is of course to develop a problem-solving approach which can be applied to many problems. 

II. Secondly, the important notion of comparing the number of unknowns (variables) to the number of relationships (equations in this case). If the number of variables exceeds the number of relationships, then we should NOT expect a unique solution to the problem. If the number of equations exceeds the number of variables then there might not be solution at all because of inconsistent conditions. These are huge themes in using algebra to model the real world. How many variables are involved in weather forecasting!

Mr. Oldman is currently five times as old as his great-grandson. In how many years will he be four times as old as his great-grandson will be? Assume ages are positive integers.

Fairly straightforward algebra problem? Easy to guess the "answer" like Mr. Oldman is currently 75 and the young man is 15. In five years, the ages will 80 and 20. Voila! You know in upper elementary or middle or high school there will always be some student who "guesses" these values and sits there complacently with the "I'm done!" look on her face.

But there is so much more here for us to discuss with the students. A much bigger picture with deeper concepts than guessing an answer or setting up algebraic equations.

I will get the ball rolling and leave the rest to my astute readers whose insights always surpass mine...

Why did my hs algebra teacher, Mrs. Hill, always require a chart or table for these problems? Even when I didn't feel I needed it, I had to use it. Decades later, I still do out of force of habit and I've been carrying on her legacy with my own students ever since! Rather than import some html table codes, I will resort to brute force for displaying the table. Pls forgive any problems with the display on your screen.

............................NOW .............. FUTURE (y yrs from now)

MR. OLDMAN.........  5x ................  5x + y

YOUNG MAN.........  x ...................  x + y

Equation:  5x + y =   4(x + y) → x = 3y

Of course, I could have used different variables or more variables and produced more equations but there is an essential truth underlying all of this. As math people and from our experience we could sense from the beginning that there would be no unique solution to this problem. But how do we develop this conceptual understanding in our students?   Some students I believe would intuit this, but many might not.

There's no earth-shattering revelation here! But IMO every student of algebra should experience this and engage in discussion about it.

Ok, teachers of algebra in middle or secondary grades -- how would you develop this into a meaningful structured lesson and, at the end, assess the learning which hopefully took place?

  • Simply show them how to make the chart, set up the equations, solve and note that there is more than one solution? 
  • OR would you allow the students to discover this for themselves? Would you begin by asking them to intuitively guess if there is exactly one solution in positive integers, more than one or none? 
  • What questions can you compose ahead of time? What questions can you anticipate coming from the students? 
  • Can we make the problem into more a puzzle-solving experience. Do you believe the word "puzzle" evokes a different visceral reaction from "word problem"?

Anxiously awaiting your thoughts which is really the point of having a blog and not simply a web site. Don't leave me twisting in the wind...

"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

"You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught."
--from South Pacific

Friday, February 19, 2010

Does Learning/Playing Chess Develop The 'Mathematical Mind'?

Nothing revolutionary about this question but it's been on my mind for some time and, now, that I have a six-year old grandson who seems to be analytical like me (we speak the same language!), I've decided to introduce him to chess via chess puzzles.

I'm not writing this because I believe one should develop a math curriculum around chess! I'm simply suggesting that any activity which requires problem-solving, analysis and creativity develops the mind. Chess is only one of many such vehicles which drive the student toward deeper thought processes. I could make a similar argument for writing and other artistic activities, but I chose chess because it's a game I love even though I'm still a novice.

Perhaps the most important aspect of all of this for children is that learning chess can be fun provided it is introduced properly. I believe many young children would quickly become frustrated and bored if they begin by learning proper openings and the concepts of positional play in the beginning stages.

There are well-known analogies between playing chess and mathematical problem-solving so I will only mention a few that occurred to me a priori, some of which have been validated by the writings of the brilliant mathematician, Paul Halmos:

  • Chess has an elaborate technical language and is completely deterministic.
  • Chess - like mathematics - requires problem solving, evaluation, critical thinking, intuition and planning

Halmos also believed, as I do, that mathematics is a creative art
"because mathematicians create beautiful new concepts; it is a creative art because mathematicians live, act, and think like artists; and it is a creative art because mathematicians regard it so.”

A New Jersey senator introduced and helped to pass bills in the Senate regarding chess instruction in schools, declaring:
"Chess increase strategic thinking skills, stimulates intellectual activity and improves problem-solving ability."

For learning how individual pieces moves and for experiencing more immediate feedback and satisfaction, I recommend Chess Puzzles, which involve contrived board positions requiring the solver to find a mate in one or more moves. Starting with Mate in One Puzzles seems developmentally appropriate in the Piagetian sense, although I would introduce Mate in Two early on as an extra challenge. That is, I wouldn't spend two weeks on Mate in One to the exclusion of other puzzles. For me, the best online site for doing these puzzles on a daily basis is the Mate in Two Chess Puzzles website developed by John Bain, MS Education, a special education teacher, certified USCF tournament director, and a specialist in textbook and materials modification. John is a teacher in the truest sense of the word!

 I'm not suggesting that children should not also be allowed to try a full game with a parent, instructor or with peers. I would allow the child free exploratory play when starting a game, correcting only invalid moves. This is not a purely constructivist approach however -- I do not expect students to invent the the theory of positional play and the deeper aspects of the game! Guided instruction must be introduced strategically just as it is with mathematics instruction, and, as with math, there is never a substitute for repetition and practice!

I'm opening the "board" for you to jump in. Do you agree with some of my thoughts about chess and math? Do you see other analogies I've missed? Have you personally used chess as a way to develop thinking in elementary, middle or secondary schools? Please share!!

"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

Wednesday, February 17, 2010

Odds and Ends -- Week of 2-15-10

Catchup time...

  • So where are the answers and followup discussion to the previous two problems I posted? They're coming. I plan to make a video of the investigation in the "143" problem. 
  • I think it would be so useful to post videos of actual lessons that demonstrate how some teachers implement explorations in the classroom at all levels from K-12. I'm sure there are some of these already online but they seem to be few and far between and I'm really referring to "best practices" here to serve as models for others. We have NCTM Illuminations for example but I'm really looking for something else here.
  • More important than investigations is to see models of daily lessons which incorporate the best of what we know about effective practice. Lessons which show HOW to blend procedural and conceptual understanding, help students develop skills mastery while engaging in rich problem-solving. Easy to do? Of course not, but if we want the US to be academically competitive, we had better move quickly in this direction and use international models to guide us.
  • How are you, the mathematics teacher, dealing with all of the confusing and overwhelming issues in math education today? OR have you learned to ignore all the "noise", close your door and simply go about the business of teaching? If you're able to, that is! Unfortunately some of the decisions which are being made independently of your input will have significant impact on how well you will be able to do your job today and in the future...  Issues like
    • "Algebra for All?" So where has that experiment gone?
    • States joining consortia to develop a common standards and assessments in math
      • How many consortia should there be? Will most eventually merge into one or two?
      • Will common math standards ultimately lead to more consistency in content  -- i.e., that which is actually taught in the classroom?
      • Are math ed departments in colleges and universities adapting rapidly enough to prepare preservice teachers for the paradigm shifts which are occurring? 
      • Will "methods" courses in ed schools increase focus on actual content, e.g.,

        "You will all prepare a lesson on the effect of changing the parameter "b" in a quadratic function. Your lesson should utilize multiple representations and include a carefully planned series of Socratic questions which develop meaning and conceptual understanding for the algorithms. Specify what actions you took to balance procedural learning with conceptual understanding. Also, be prepared to answer the essential question: "WHAT ACTIONS DID YOU TAKE AND HOW DID YOU ASSESS THAT LEARNING TOOK PLACE?"
      • How can we all use emerging technologies to enhance our teaching repertoire. Regardless of whether you "tweet", "Buzz" or Facebook, or all of the above, how can networking and sharing make us more effective teachers? This is sort of obvious, but how many of us are using these on a daily basis in our planning and in our classrooms? 25% 50% More?
      • What are the most effective online learning tools for mathematics, particularly middle and secondary math? Many excellent bloggers have researched and compiled excellent lists of resources, but specifically, what are the best online video sites (interactive or not)? This is crucial for me and it's something to which I want to dedicate myself.
      • I choose not to get into NCLB or Charter Schools debates at this time...
      • So where is my new website I've alluded to? It's taking forever to set up, but I want to do it right. More to follow...

Ok, so we have to have a little problem for your students to think about. This is part of a well-known genre of "puzzles" which frequently travel across the web and are always intriguing for students and adults alike. You've probably seen it...
For this blog, the essential question is, how we can make this a teachable moment in our math classes?


2% or 98%

This is strange...can you figure it out?

Are you the 2% or 98% of the population?

Follow the instructions! NO PEEKING AHEAD!

* Do the following exercise, guaranteed to raise an eyebrow.

* There's no trick or surprise.

* Just follow these instructions, and answer the questions one at a time
and as quickly as you can!

* Again, as quickly as you can but don't advance until you've done each
of them ..... really.

* Now, scroll down (but not too fast, you might miss something).

Think of a number from 1 to 10

Multiply that number by 9.

If the number is a 2-digit number, add the digits together.

Now subtract 5.

Determine which letter in the alphabet corresponds to the number you
ended up with

(example: 1=a, 2=B,  3=c,etc.)

Think of a country that starts with that letter.

Remember the last letter of the name of that country

Think of the name of an animal that starts with that letter.

Remember the last letter in the name of that  animal.

Think of the name of a fruit that starts with that letter.

Are you thinking of a Kangaroo in   Denmark  eating an Orange  ?

I told you this was FREAKY!! If not, you're among the 2% of the
population whose minds are different enough to think of something else.
98% of people will answer with kangaroos in  Denmark  when given this

Is there some basic number theory here? 
What would you want your middle schoolers to do with this after they play it a few times?

    "All Truth passes through Three Stages: First, it is Ridiculed...
    Second, it is Violently Opposed...
    Third, it is Accepted as being Self-Evident."
    - Arthur Schopenhauer (1778-1860)

    You've got to be taught
    To hate and fear,
    You've got to be taught
    From year to year,
    It's got to be drummed
    In your dear little ear
    You've got to be carefully taught.
    --from South Pacific