MathNotations

First a humorous aside from one of my friends on another message board. A friend emailed it to him so it's probably making the rounds of the web. In case you haven't seen it, here it is...

A recent study found that the average American walks about 900 miles a year.

Another study found that Americans drink, on average, 22 gallons of alcohol a year.

That means, on average, Americans get about 41 miles to the gallon.

Kind of makes you proud!



One of our new and devoted readers, Florian, contributed the following unusual digits by algorithmic construction problem. This is a wonderful example of a different type of solution, since a standard algebraic approach should prove fruitless. Florian is our resident computer scientist. That should help you understand how he devised this question.

Suppose a₁a₂a₃...a_n-16 represents an n-digit positive integer whose units' digit is 6. Find the least such positive integer satisfying the property that when the number is multiplied by 2, the result is 6a₁a₂a₃...a_n-1 , the n-digit number whose digits are the same as the original number except that each digit is shifted one position to the right and the rightmost digit '6' rotates to the leftmost position.

Have fun looking for this 18-digit number! Would a calculator be useful here?

Variations and Extensions:

Here is how one could modify this for middle schoolers:
(i) Give them the 18-digit number to start with (sorry, I'm not giving this away yet), have them multiply it by 2 using paper and pencil and see how long it takes for various students to see the surprising result. (Yes, Steve, they actually are expected to multiply with accuracy!)
I guarantee they will express surprise!
(ii) Now ask them to figure out how they could construct the digits of the mystery number, one digit at a time. Some will catch on quickly, others will need guidance.
(iii) What questions should occur to students as they are building this number? You may need to ask them if they believe this process eventually has to terminate.

Extension for the Very Highly Motivated (or for people like me who need to get a life!):

Construct the 42-digit number a₁a₂a₃...a₄₁5 (ending in the digit '5'), which when multiplied by 5 is of the form: 5a₁a₂a₃...a₄₁, in which the result has the same digits as the original number with each digit shifted one position to the right and the rightmost digit rotated to the leftmost position.

Note: Check my accuracy on this!

Posted by Dave Marain at 5:36 PM 5 comments

Labels: algorithms, math challenge, math humor

I'm posting this again (and I will probably be posting frequent updates over the next 30 days) to make sure everyone knows there is now a poll in the sidebar! The original post explaining this is here. Please choose the option that most closely matches your feelings about which multiplication algorithm(s) should be taught in Grades 3-5 as well as the issue of mastery.

I believe you're only allowed to vote once. After you submit the vote, the current tally is updated and the results appear in the sidebar in place of the survey options (only snippets may appear). Pls let me know if you're having difficulty reading the 4 options before voting.

This poll is an opportunity for your voice to be heard regarding a critical issue in mathematics education. I hope you will participate.

Posted by Dave Marain at 8:51 AM 0 comments

Labels: algorithms, multidigt multiplication, poll, survey

Update:
A poll has now been created in the sidebar! Please express your preference. (Thank you, Jonathan, for making this suggestion).

As if the educational fate of our children could be determined by a poll of our readers...
Hey, you never know.

What's even more incredible is that individual teachers, schools, districts, and states still feel that this is a 'local' decision. After over two decades of this debate, children and their parents still have little to say about these kinds of curricular decisions that will impact on the mathematical futures of another generation. After all, the experts know what is best, right? Well, there are many many experts who all feel sure they know the answer to this question. Trouble is, they do not all draw similar conclusions and spend most of their time defending their choices or their particular agenda or favorite set of materials.

Finally, MathNotations will settle this debate once and for all. The results of this survey (assuming a minimum of 3 responses) will be used to influence national policy for years to come!

Here are your options regarding your preference for how multidigit multiplication should be taught in Grades 3-5 :

(A) Teach only the traditional algorithm and expect mastery
(B) Teach the 'partial products' method to develop understanding of place value and the traditional algorithm; teach the traditional algorithm as a more efficient method and require it; expect mastery
(C) Teach the 'partial products' method to develop understanding of the traditional algorithm; teach the traditional algorithm as a more efficient method; give students a choice of methods; expect mastery of at least one method
(D) Model other methods (e.g., 'lattice method') and encourage students to invent their own method; do not require any particular method or mastery

Now, don't miss this opportunity to be part of an historic decision. Your vote does count...

Posted by Dave Marain at 2:49 PM 28 comments

Labels: algorithms, curriculum, multidigt multiplication, national math curriculum

If you were looking for a challenge here in higher math using combinations and permutations, sorry to disappoint you! I felt compelled to write this essay after watching my wife patiently attempting to teach one of my children how to open a combination lock. She doesn't think of herself as a teacher, but, she is, and, in many ways, far more skilled than I ever was.

One of the rites of passage for many middle schoolers is mastering the intricacies of the combination lock for their lockers, somewhat akin to elementary schoolers learning how to tie their shoes. Do you remember the frustration you felt the first few times you tried to solve the puzzle of these locks? Do you recall your euphoria when it magically opened? Consider all of the 'skills' involved and think of the parallels to mastering the algorithms of mathematics:

(1) Fine motor skills required to precisely turn the dial and stop at the correct number
(2) Memorizing the 3 numbers in sequence
(3) Understanding the difference between Right and Left when rotating the dial and retaining the R-L-R sequence
(4) The absolute discipline and precision required - close is not good enough
(5) The dreaded second step of the process needed in going 'past zero'
(6) The extreme feelings of frustration from failing repeatedly and the inclination to give up, yet driven to continue
(7) The elation felt in getting it the first time all by yourself, only to be followed by despair when you can't seem to duplicate the feat!
(8) The feeling of accomplishment when you can do it almost every time without anyone helping you
(9) Is there any substitute for independent practice in achieving mastery here?
(10) How important is motivation here in driving the child to continue in the face of adversity?

What about the challenges faced by the 'instructor' here? If you were the one who helped someone succeed, did you find it frustrating or did you have 'unlimited' patience? Did you have to practice it yourself first and think about breaking this 'automatic' process into simple discrete steps? Did you have to try different verbal instructions (for example, using 'down' and 'up' vs. 'left' and 'right') or different techniques of one approach failed? Did repeated demonstrations in front of the child suffice? Did the child say, "Let me do it by myself?" If you've helped several children learn to 'unlock' the combination, did you use the same approach successfully with each child? Are some youngsters simply unable to 'solve the problem' at that time and need to be given a key lock instead as an accommodation? Is making this concession detrimental to their self-esteem and eventual development or is it reasonable at that time? Will some of these youngsters be able to succeed later if given the opportunity to try again (when developmentally ready)?

Is there a metaphor here for teaching children mathematical algorithms? By the way, can you think of others skills or concepts involved in opening the lock that I overlooked? Pls share!

Now, parents, extrapolate this 'teaching' process to dozens of unique math students every day with a myriad of different algorithms over the course of a school year? Anyone can teach, right?

I realize some of you will see the flaws in this metaphor and will point out all the differences between opening the lock and solving a mathematical problem? I know the parallel is far from perfect but this is something that just struck me and I had to put my thoughts down. You know, like a journal, a diary, a blog... Your thoughts?

Posted by Dave Marain at 6:11 AM 0 comments

Labels: algorithms, learning, teaching