Teaching Creativity in Mathematics

This will be the first of two ‘blog posts on an activity that could promote creativity for elementary, middle school, and high school students.  A suggestion for parents and teachers is in the middle of this post.

ABOUT A DECADE AGO, I first discovered what I call the Four 4s activity.  In brief, the game says that using exactly four 4s (no more, no less, and no other digits) and any mathematical operation you want, you can create every integer from 1 to 100.  Two quick simple examples are \displaystyle 3= \frac{4+4+4}{4} and \displaystyle 16= 4\cdot 4+4-4.

As for mathematical operations, anything goes!  The basic +, -, *, / along with exponents, roots, decimals (4.4 or .4), concatenation (44), percentages, repeating decimals (.\overline{4}), and many more are legal.

At the time, I was teaching a 7th grade prealgebra course with several students who were struggling to master order of operations–that pesky, but critical mathematical grammar topic that bedevils some students through high school and beyond.  I thought it would be a good way to motivate some of my students to 1) be creative, and 2) improve their order of operations abilities to find numbers others hadn’t found or to find unique approaches to some numbers.

My students learned that even within the strict rules of mathematical grammar, there is lots of room for creativity.  Sometimes (often? usually?) there are multiple ways of thinking about a problem, some clever and some blunt but effective.  People deserve respect and congratulations for clever, simple, and elegant solutions.  Seeing how others solve one problem (or number) can often grant insights into how to find other nearby solutions.  Perhaps most importantly, they learned to a small degree how to deal with frustration and to not give up just because an answer didn’t immediately reveal itself.  It took us a few weeks, but we eventually completed with great communal satisfaction our 1-100 integer list.

PARENTS and TEACHERS:  Try this game with your young ones or pursue it just for the fun of a mental challenge.  See what variations you can create.  Compare your solutions with your child, children, or student(s).  From my experiences, this activity has led many younger students to ask how repeating decimals, factorials, and other mathematical operations work.  After all, now there’s a clear purpose to learning, even if only for a “game.”

I’ve created an easy page for you to record your solutions.

A FEW WEEKS AGO, I read a recent post from the always great MathMunch about the IntegerMania site and its additional restriction on the activity–an exquisiteness scale.  My interpretation of “exquisiteness” is that a ‘premium’ is awarded to solutions that express an integer in the simplest, cleanest way possible.  Just like a simple, elegant explanation that gets to the heart of a problem is often considered “better”, the exquisiteness scale rewards simple, elegant formulations of integers over more complex forms.  The scale also includes surcharges for functions which presume the presence of other numbers not required to be explicitly written in common notation (like the 1, 2, & 3 in 4!, the 0 in front of .4, and the infinite 4s in .\overline{4}.

In the past, I simply asked students to create solutions of any kind.  I recorded their variations on a class Web site.  Over the past three weeks, I renamed exquisiteness to “complexity” and re-ran Four 4s across all of my high school junior and senior classes, always accepting new formulations of numbers that hadn’t been found yet, and (paralleling Integermania’s example) allowed a maximum 3 submissions per student per week to prevent a few super-active students from dominating the board.  Also following Integermania’s lead, I allowed any new submission to remain on the board for at least a week before it could be “sniped” by a “less complex” formulation.  I used differently colored index cards to indicate the base level of each submission.

Here are a few images of my students’ progress.  I opted for the physical bulletin board to force the game and advancements visible.  In the latter two images, you can see that, unlike Integermania, I layered later snipes of numbers so that the names of earlier submissions were still on the board, preserving the “first found” credit of the earliest formulations.  The boxed number in the upper left of each card is the complexity rating.

4s_1

4s_3

4s_2

The creativity output was strong, with contributions even from some who weren’t in my classes–friends of students curious about what their friends were so animatedly discussing.  Even my 3rd grade daughter offered some contributions, including a level 1.0 snipe, \displaystyle 5=\frac{4\cdot 4+4}{4} of a senior’s level 3.0 \displaystyle 5=4+\left( \frac{4}{4} \right)^4.  The 4th grade son of a colleague added several other formulations.

When obviously complicated solutions were posted early in a week, I heard several discussing ways to snipe in less complex solutions.  Occasionally, students would find an integer using only three 4s and had to find ways to cleverly dispose of the extra digit.  One of my sometimes struggling regular calculus students did this by adding 4′, the derivative of a constant. Another had already used a repeating decimal ( . \overline{4}), and realized she could just bury the extra 4 there ( .\overline{44}).  Two juniors dove into the complexity scale and learned more mathematics so they could deliberately create some of the most complicated solutions possible, even if just for a week before they were sniped.  Their ventures are the topic of my next post.

AFTERTHOUGHTS:  When I next use Four 4s with elementary or middle school students, I’m not sure I’d want to use the complexity scale.  I think getting lots of solutions visible and discussing the pros, cons, and insights of different approaches for those learning the grammar of mathematical operations would be far more valuable for that age.

The addition of the complexity scale definitely changed the game for my high school students.  Mine is a pretty academically competitive school, so most of the early energy went into finding snipes rather than new numbers.  I also liked how this game drove several conversations about mathematical elegance.

One conversation was particularly insightful.  My colleague’s 4th grade son proposed \displaystyle 1=\frac{44}{44} and argued that from his perspective, it was simpler than the level 1.0 \displaystyle \frac{4+4}{4+4} already on the board because his solution required two fewer operations.    From the complexity scale established at the start of the activity, his solution was a level 2.0 because it used concatenated 4s, but his larger point is definitely hard to refute and taught me that the next time I use this activity, I should engage my students in defining the complexity levels.

ADDENDA:

1) IntegerMania’s collection has extended the Four 4s list from 1 to well past 2000.  I wouldn’t have thought it possible to extend the streak so far, but the collection there shows a potential arrangement of Four 4s for every single integer from 1 to up to 1137 before breaking.  Impressive.  Click here to see the list, but don’t look quite yet if you want to explore for yourself.

As a colleague noted, it would be cool for those involved in the contest to see how their potential solutions stacked up against those submitted from around the world.  Can you create solutions to rival those already posted?

2) IntegerMania has several other ongoing and semi-retired competitions along the same lines including one using Four 1s, Four 9s, and another using Ramanujan’s ‘famous’ taxi cab number, 1729.  I’ve convinced some of my students to make contributions.

Play these yourself or with colleagues, students, and/or your children.  Above all, have fun, be creative, and learn something new.

It’s amazing what can be built from the simplest of assumptions.  That, after all, is what mathematics is all about.

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7 responses to “Teaching Creativity in Mathematics

  1. What do you say/do to dissuade them from googling answers?

    • To some extent, it’s about the game and being creative and noting that all answers are different and valuable, even if they aren’t the “most efficient”. As an example from the ‘blog post, the boy who initially submitted 5=4+\left( \frac{4}{4} \right) ^4 didn’t end up with the most efficient solution, but his use of the \frac{4}{4} fraction and exponents were good clues for how to lose unneeded 4s and how to ramp up smaller values using exponents — valuable lessons for many larger integers.

      Some people always look at the answers in the back of the book. If you are worried about that aspect, I might vote to not use the “sniping” aspect of Four 4s. I was really successful the last time I used Four 4s without sniping with middle school prealgebra classes keeping lots of blanks (5+) open for each number to encourage different solutions. In the end, I was trying to celebrate individual thinking and creativity and creating new solutions from what others had done previously.

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  4. Hi Chris!

    Thank you for this! It looks like a lot of fun. I’ve recommended this to my daughter’s teachers.

    Kay

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  6. Pingback: One of my favorite number activities is the 4 4s | 17GoldenFish

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