Math Play and New Beginnings

I’ve been thinking lots lately about the influence parents and teachers have on early numeracy habits in children.  And also about the saddeningly difficult or traumatic experiences far too many adults had in their math classes in school.  Among the many current problems in America’s educational systems, I present here one issue we can all change.  Whether you count yourself mathphobic or a mathophile, please read on for the difference that you can make for yourself and for young people right now, TODAY.

I believe my enthusiasm for what I teach has been one of the strongest, positive factors in whatever effectiveness I’ve had in the classroom.   It is part of my personality and therefore pretty easy for me to tap, but excitement is something everyone can generate, particularly in critical areas–academic or otherwise.  When something is important or interesting, we all get excited.

In a different direction, I’ve often been thoroughly dismayed by the American nonchalance to innumeracy.  I long ago lost count of the number of times in social or professional situations when parents or other other adults upon learning that I was a math teacher proclaimed “I was terrible at math,” or “I can’t even balance my own checkbook.”   I was further crushed by the sad number of times these utterances happened not just within earshot of young people, but by parents sitting around a table with their own children participating in the conversation!

What stuns me about these prideful or apologetic (I’m never sure which) and very public proclamations of innumeracy is that NOT A SINGLE ONE of these adults would ever dare to stand up in public and shout, “It’s OK.  I never learned how to read a book, either.  I was terrible at reading.”  Western culture has a deep respect for, reliance upon, and expectation of a broad and public literacy.  Why, then, do we accept broad proclamations of innumeracy as social badges of honor?  When an adult can’t read, we try to get help.  Why not the same of innumeracy?

I will be the first to admit that much of what happened in most math classrooms in the past (including those when I was a student) may have been suffocatingly dull, unhelpful, and discouraging.  Sadly, most of today’s math classrooms are no better.  Other countries have learned more from American research than have American teachers (one example here).  That said, there are MANY individual teachers and schools doing all they can to make a positive, determined, and deliberate change in how children experience and engage with mathematical ideas.

But in the words of the African proverb, “It takes a village to raise a child.”  Part of this comes from the energetic, determined, and resourceful teachers and schools who can and do make daily differences in the positive mindsets of children.  But it also will take every one of us to change the American acceptance of a culture of innumeracy.  And it starts with enthusiasm.  In the words of Jo Boaler,

When you are working with [any] child on math, be as enthusiastic as possible. This is hard if you have had bad mathematical experiences, but it is very important. Parents, especially mothers of young girls, should never, ever say, “I was hopeless at math!”  Research tells us that this is a very damaging message, especially for young girls. – p. 184, emphasis mine

Boaler’s entire book, What’s Math Got to Do With It? (click image for a link), but especially Chapter 8, is an absolute must-read for all parents, teachers, really any adult who has any interactions with school-age children.

I suspect some (many?  most?) readers of this post will have had an unfortunate number of traumatic mathematical experiences in their lives, especially in school.  But it is never, ever too late to change your own mindset.  While the next excerpt is written toward parents, rephrase its beginning so that it applies to you or anyone else who interacts with young people.

There is no reason for any parent to be negative about the mathematics of early childhood as even the most mathphobic of parents would not have had negative experiences with math before school started.  And the birth of your own children could be the perfect opportunity to start all over again with mathematics, without the people who terrorized you the first time around.  I know a number of people who were traumatized by math in school but when they started learning it again as adults, they found it enjoyable and accessible. Parents of young children could make math an adult project, learning with their children or perhaps one step ahead of them each year. -p. 184

Here’s my simple message.  Be enthusiastic.  Encourage continual growth for all children in all areas (and help yourself grow along the way!).  Revel in patterns.  Make conjectures.  Explore. Discover.  Encourage questions.  Never be afraid of what you don’t know–use it as an opportunity for you and the children you know to grow.

I’ll end this with a couple quotes from Disney’s Meet the Robinsons.

“Math Play” Presentation for Early Childhood teachers

Even though my teaching experiences are all middle and high school, as a PreSchool-12 math chair and father of 3 young children, I’m intensely interested in how math is presented to very young people.

As a result, I’m presenting ideas for teaching math through fun and exploration to about 55 Cleveland area pre-school through kindergarten teachers this morning.  My handout is on Scribd and should show below.  Math is  about Play and Curiosity.  Teach it that way.

http://www.scribd.com/doc/235527694/Learning-Through-Math-Games

 

Which came first: Math Ability or Computational Speed ?

I’ve claimed many times in conversations over the last two weeks that I believe many parents and educators misconstrue the relationship and causality direction between being skilled/fluent at mathematics and being fast at computations.  Read that latter as student accomplishment defined by skill on speed testing as done in many, many schools.  Here is a post from Stanford’s Jo Boaler on math anxiety created by timed testing.

Here’s my thinking:  When we watch someone perform at a very high level in anything, that person appears to perform complex tasks quickly and effortlessly, and indeed, they do.  But . . . they are fast because they are good, and NOT the other way around.  When you learn anything very well and deeply, you get faster.  But if you practice faster and faster, you don’t necessarily get better.

I fear too many educators and parents are confusing what comes first.  From my point of view, understanding must come first.  Playing with ideas in different contexts eventually leads to recognizing that the work one does in earlier, familiar situations eventually informs your understanding in current, less familiar settings.  And you process more quickly in the new environment precisely because you already understood more deeply.

I think many errantly believe they can help young people become more talented in mathematics by requiring them to emulate the actions of those already accomplished in math via rapid problem solving.  I worry this emphasis is placed in exactly the wrong place.  Asking learners to perform quickly tasks which they don’t fully understand instills unnecessary anxiety (according to Boaler’s research) and confuses the deep thinking, pattern recognition, and problem solving of mathematics with rapid arithmetic and symbolic manipulation.

Jo Boaler’s research above clearly addresses the resulting math anxiety in a broad spectrum of students—both weak and accomplished.  My point is that timed testing–especially timed skill testing–at best confuses young students about the nature of mathematics, and at worst convinces them that they can’t be good at it.  No matter what, it scares them.   And what good does that accomplish?

Enhancing Creativity at Home

I had a great conversation last month with a Westminster colleague, Kay, as school was winding down.  While from completely different academic departments, our daughters are nearly the same age, and so we share lots of parenting ideas and stories.  Here’s some thoughts we developed to use with our kids this summer to enhance creative thinking and enhanced abilities in many areas.

Creativity in Literature:

Our daughters LOVE to read, so our first challenge to our girls would be for them to find a good stopping place 20-40 pages before the end of each book they read. Then we encourage them to describe how they think the story might end and why they think their hypothesis is possible.  If they see multiple ways it could end, they could describe alternates.  They should WRITE their predictions to get their minds to commit and more actively engage.  This need not be anything formal–but it can be if they like.  We just want the girls to think a bit more deeply about what they’re reading and engage.  Use what they know has already happened to draw conclusions about what might have happened.

After their informal (or formal) writing, read the remainder of the book and compare their prediction(s) to the author’s chosen end.  How close did they come to the author’s conclusion?  What information did the author use to end the book that they didn’t use?  Was the author’s conclusion reasonable? Was theirs?

There’s not any particular right or wrong here for us beyond getting our kids to think about what they’re reading and to engage the process.  We also hope they will become more attentive to details in their reading.

Cooking:

Allow the kids to help plan meals; help them understand the daily processes for planning nutritious meals.  Encourage them to participate in cooking, especially for anything you cook from scratch.  What choices do you make and why?

For cooking, talk to them about what the different ingredients do for the resulting dish.  Have them make predictions about what will happen when new ingredients are added or the collected ingredients are prepared.  To the best of your ability, explain what each step does and why it’s important to the final product.

Compare the predictions to the final results.  When some part inevitably turn out differently than predicted, learn why.  Take pictures along the way for you or your child to use when comparing hypotheses and outcomes.  When you come back to a recipe on another occasion, think about how it turned out last time and plan a change or an improvement.  Knowing what your child wants to happen, can she adapt the preparatory steps to accomplish that change?

In the end, this really isn’t all that different from the reading suggestions.  Engage, observe, explore, make predictions, and compare expected and actual results.  All along, use data to explain why you believe your claims are justified.

Conclusion:

This is about where Kay & I ran out of time to chat that day, but we ended with the realization that what we were describing was exactly the scientific process, embedded in arguably “non-science” settings.  As I’ve mused over this for the past few weeks, I’ve realized that this process can be applied almost anywhere:

• athletics (how do you get a better result? Why?),
• mathematics (what kind of answer will I get? Can there be more than one? What will it look like?)
• gardening (how do I get a particular plant to perform a specific way? Is that even possible with that plant?),
• computer programming (getting a computer to do precisely what you ask of it), and so on.

Engage your children or your students. Get them to hypothesize and justify using data.  Teach them (and yourself) to be more alert to patterns and clues about past and future behavior.  Perhaps most importantly, determine if other outcomes are possible and what it would take to get there.  Envision something she or he hasn’t seen or done before and figure out what is needed to make it happen.  Then … make it happen.

Multiplication Practice Plus Creativity

I hope this post is particularly helpful for parents and teachers of elementary school children.  Through my Twitter network last week I found via @Maths_Master‘s Great Maths Teaching Ideas ‘blog a 2010 post summarizing Dan Finkel’s Damult dice game. Recognizing that “practicing times tables can be unmotivated and boring for kids,” Damult is an attempt to make learning elementary multiplication facts more entertaining. I offer some game variations and strategies following a description of the game.

Here’s Dan’s game:

image via Wikipedia

Each player takes turns rolling 3 dice. First to break 200 (or 500, etc.) wins. On your turn, you get to choose two dice to add together, then you multiply the sum by the final die. That’s your score for that turn.

Simple; no bells, no whistles. For example, I roll a 3, a 4, and a 6 on my turn. I could either do (3+4) times 6 for 42 points, OR (3+6) times 4 for 36 points, OR (4+6) times 3 for 30 points. I’ll take the 42 points.

I spent some time playing this with kids the other day and I saw that (1) it was genuinely fun, and (2) it gives you almost all the multiplication practice you could ask for. In fact, it gives even more, because the choice of which dice to add and which to multiply reveals some interesting structure of numbers. Seriously, get a kid hooked on this game, and it’s the equivalent of dozens or hundreds of times table practice sheets.

It’s a fun activity idea by itself.  Damult combines a bit of luck and memory, and rewards the ability to recall multiplication facts.  As an added bonus, it requires players to be able to manipulate objects in their heads–how many different ways can the three given dice be manipulated in summation stage to create unique products? How can a player ensure that she has found the biggest product for her score?  Try the game!

CONNECTIONS AND EXTENSIONS:

This is a great opportunity for parents to engage with their  children as they learn multiplication facts.  Parents and teachers could play along, or the learner might be the only player, talking out loud so that the teacher or parent can “hear the thinking.”

I love that the game completely randomizes the multiplication tables.  This significantly enhances recall as memory is not tied to particular patterns or positions on fact pages. Players must adapt to each random roll.

In any variation, there obviously should be a discussion among all players about what products were found to confirm the results. Make the game more formative or more competitive, depending on the experience level of the players.  In more competitive variations with experienced learners, if a product was miscalculated and claimed, you might decide that no score should be recorded for that round.

If you’re guiding someone on this it is critical that you DO NOT give answers.  Students need to explore, hypothesize, discover errors, learn how to communicate their conclusions in clear and concise language, and to learn how to defend their findings while also learning how to admit flaws in their reasoning when faced with contradicting data.  Experimenting and discovery is always deeper, richer,and more long-lasting than just being told.  Remember the Chinese Proverb: “I hear and I forget. I see and I remember. I do and I understand.” Always seek understanding.

The first comments on Dan’s post noted that while one player was summing and multiplying, the other player(s) were largely disengaged. Also, the game could drag on as unconfident players tried to make sure  they had explored every possibility.  To address that and several other possibilities, I offer the following Damult variations.  Some more complex variations are toward the end. Read on!

Finally, if you’ve read my ‘blog much, you know that I’m a huge fan of leveraging technology for math learning, but this is one of those situations where I think you should 100% unplug. To learn multiplication facts is to learn some of the basic grammar and vocabulary that makes the language of mathematics work.  You simply can’t communicate mathematically with an underlying awareness of how the structure of the language works.

GAME VARIATIONS:

Variation 1: Adding a timer to the game could cure the slow-down issue. Depending on the age of the child and his/her familiarity with multiplication, the timer can be longer or shorter.  If the skill levels of the players are unequal, make the timer unequal.  (I love the adage, “Fair is seldom equal, and equal is seldom fair.”)

Variation 2: Why must only one player be active? The players could take turns rolling the dice while both record scores based on what they find.  If a particular combination was not noticed by one player, that player doesn’t get to consider it for his/her score.

Variation 3 – As an aside, notice that Dan implicitly claims there are only 3 possible sums from a 3-dice roll. Will that always be the case?  Can you convince someone why your solution is correct?  

(For 3 dice the maximum number of possible sums is 3. When and why would there be fewer products?)

Variation 4 – How many multiplication facts are possible using only 3 dice?

This would be a great number sense exploration.  Some may try it by gathering lots of data, others may have more sophisticated reasoning.  I suggest that you or your students hypothesize an answer first along with some reason why you think your hypothesis is correct.  Different answers are OK, and you can always revise your hypotheses if you get evidence leaning in another direction. No matter what, have fun exploring and learning. 

(Middle School extension: Damult creates products of axb where a can be any integer 1 – 6 and b can be any integer 2 – 12.  That gives 66 different products if you count different arrangements (3×4 and 4×3) as different products. Can you or your student see why? How many outcomes are possible if you look only at the product result and not at the factors which created it?)

Variation 5 – After discovering or just using the answer to the last variation, you could use a table of multiplication facts and see how quickly different facts and be “discovered” from rolls of the dice.  After rolling 3 dice, mark off all multiplication facts you can using the sum-then-multiply combination rules posed at the beginning.  This might be a fun way for early learners to familiarize themselves with multiplication patterns.

NOTE: If you play variations 4 or 5 as a game, you’ll likely want (or need) to stop before all possibilities are found.  Some (eg, 6×12 and 1×2) will be pretty uncommon from dice rolls.

Variation 6 – You could make a Bingo-like or a 4 or 5-in-a-row game.  The first person to mark off a certain number of facts or the first to get a certain number in a row would be a winner.

Variation 6 – If you try the last few variations, you’ll see that some products occur much more frequently from the dice rolls than others.  This could be used to introduce probability. Which products are more likely and why?

As an example, I suspect 3×7 could happen six times more often 1×2.  Can you convince yourself why 3×7 is so much more likely?  Can you see why 3×7 is exactly six times more likely than 1×2?

Variation 7 – Why restrict yourself to 3 dice? When just starting out, using more than 3 dice would definitely be a frustration factor, but once you’ve got a good grip on the game, consider rolling 4 dice and allow players to multiply the sum of any 2 or 3 of the dice by the sum of the remaining dice.

By my computation, using 4 dice means there are up to 7 possible combinations in a given roll.  Can you prove that? Being able to consistently find them all is likely to be a very difficult challenge, but it is a phenomenal and early opportunity to stretch a young person’s mind into considering multiple outcomes and reliable ways to guarantee that you’ve considered all possibilities.

Variation 8 – Why go for maximum products and being the first to get to 200 or 500 points?  Why not try for a low score (like golf), seeking minimum products  and being the last to exceed 100 or 200?

Variation 9 – Stealthy Calculus:  OK, my analysis on this one goes way deeper than is necessary to play the game, but sometimes knowing more than is necessary can give insights and can help you lead others toward developing “math sense”–a truly invaluable skill.

LOW LEVEL – After you’ve played this a few times, ask the player(s) if there is some strategy that could be used to guarantee the biggest (or smallest) possible product for any roll.  This could be a great mathematical experiment for which the solutions are not at all intuitive, I think.  Some might figure it out quickly and others might need to gather lots of data, comparing products from lots of rolls before distilling the relationship.

If you’re guiding someone on this it is critical that you DO NOT give the answer.  Students need to explore, hypothesize, learn how to communicate their conclusions in clear and concise language, and to learn how to defend their findings while also learning how to admit flaws in their reasoning when faced with contradicting data.  If you don’t know the answer, stop reading now and figure it out for yourself. I provide an answer in the next paragraphs, but experimenting and discovery is always deeper, richer,and more long-lasting than just being told.  Remember the Chinese Proverb: “I hear and I forget. I see and I remember. I do and I understand.” Always seek understanding.

MUCH HIGHER LEVEL – As a calculus teacher, the very first fact that struck me was Damult’s implied goal: Getting the largest possible product from any roll of three dice.  That’s an optimization problem, and I knew from calculus that the greatest possible product of two numbers whose sum was constant happens when the two numbers are as close as possible to being equal.  Likewise, the smallest possible product happens when the two factors are as far apart as possible.  (If you recall some calculus of derivatives, I encourage you to prove these for yourself.  If anyone asks, I could write a future post with the proof.)

In Dan’s initial example above in which 3, 4, and 6 were rolled, I stopped reading after the first sentence of paragraph 2 (pausing to think and draw your own conclusions is a great habit of the mind) for a few moments as I thought, “I know 3+4 and 6 are as close to equivalent as I can get, so 7*6=42 is the greatest possible product.”  I didn’t even look at the other possibilities, I knew they were less. This fact was established (unnecessarily for me) in the end of the paragraph.

Without calculus, I propose students try making tables of their data.  They’ll have up to three unique products (Variation 3) and will need to explore the data before hopefully discovering the relationship. If a young person doesn’t discover the relationship, Don’t tell him/her! it is far better to leave a question as unanswered to think on and answer another day than to have a relationship given unearned.  Value comes from effort and discovery. Don’t cheat young learners out of that experience or lesson.

Conclusion: Don’t just play a game. Be creative! Strategize! Encourage young ones not just to play, but to play well. Children are quite creative in free play as they continually make new and adapt old “rules”.  Why should intellectual play be any different?  I’d love to see what variations others discover or have to offer.

Gender and Expectations Lessons from Research

A couple reports from NPR yesterday have me thinking about some of the articles I’ve compiling in my Diigo library about what I’ve learned about gender and expectations differences in parenting and teaching.  I don’t have anything particular to tie together here, but I thought these four resources were more than I could comfortably tie together in a coherent Tweet thread, so I thought I’d gather them into an impromptu ‘blog post.

Girls May Get More ‘Teaching Time’ From Parents Than Boys Do via @NPR.  Excerpts:
… ” ‘How often do you read with your child?’ or ‘Do you teach them the alphabet or numbers?’ … Systematically parents spent more time doing these activities with girls.”
… “Since parents say they spend the same amount of time overall with boys and girls, Baker’s analysis suggests that if parents are spending more time with girls on cognitive activities, they must be spending more time with boys on other kinds of activities.”
… “The costs of investing in cognitive activities is different when it comes to boys and girls. As an economist, he isn’t referring to cost in the sense of cash; he means cost in the sense of effort.”

Gender Gap Disappears in School Math Competitions via sciencedaily.com .  Excerpt:
… “Most school math contests are one-shot events where girls underperform relative to their male classmates. But a new study by a Brigham Young University economist presents a different picture.  Twenty-four local elementary schools changed the format to go across five different rounds. Once the first round was over, girls performed as well or better than boys for the rest of the contest.”
… “It’s really encouraging that seemingly large gaps disappear just by keeping [girls] in the game longer.”

A broader look at school expectations leading to enhanced math performance:  What Distinguishes a Superschool From the Rest via ideas.time.com .
… “The difference seems to lie in whether a school focuses on basic competence or encourages exceptional achievement. While almost all the schools saw it as their responsibility to cover the math knowledge necessary to do well on the SATs, the authors noted that “there is much less uniformity in whether schools encourage gifted students to develop more advanced problem solving skills and reach the higher level of mastery of high school mathematics.”
… “The fact that the highest achieving girls in the U.S. are concentrated in a very small set of schools, the authors write, indicates ‘that almost all girls with the ability to reach high math achievement levels are not doing so.’ ”

Girls, Boys And Toys: Rethinking Stereotypes In What Kids Play With via @NPR.  Excerpts:
… Some toy companies are re-thinking gender-specific marketing and branding.
… “I think what they were worried about was causing gender identification needlessly — to turn off passive learning, passive expression down the road, even passive economic opportunity for girls or boys if they felt they couldn’t do something because of societal norms,”
… “It’ll be interesting to see how this changes the attitudes of parents and of kids over time or whether or not it does. There may be some hard-wired differences,”

 

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 and .

As for mathematical operations, anything goes!  The basic +, -, *, / along with exponents, roots, decimals (4.4 or .4), concatenation (44), percentages, repeating decimals (), 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.

View this document on Scribd

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 .

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.

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, of a senior’s level 3.0 .  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 ( ), and realized she could just bury the extra 4 there ( ).  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 and argued that from his perspective, it was simpler than the level 1.0 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.

Arrangements Connections for Young Students

Mathematics is not arithmetic.

The latter is a set of symbol manipulation rules that dominates most of what we teach in school.  Mathematics, on the other hand, is a science of patterns.  It is a way of logical thinking, making sense of forms and arrangements–sometimes applied and sometimes purely imagined.  It involves looking at the implications of what we know and pushing that knowledge as far as we can to see what else can be learned based solely upon connections we can make from our assumptions.

Within the last few weeks, I’ve discovered a great daily ‘blog run by @Five_Triangles “for (but not limited to) school years 6-8.”  I’d argue that those posts are great for a broader range of ages. I gave my 3rd grade daughter one of the puzzles during breakfast.  We had some great conversations then and on the way to school.  I share those below.  Another offering extends that thinking in a way that may not be immediately obvious to young people.

Here’s the part of the post I used at breakfast.

For my daughter, I saw this problem presenting two different possibilities–the obvious arithmetic problem and a mathematics extension.  The arithmetic requires very basic subtraction facts and wee bit of trial-and-error (a GREAT mathematics skill!) to tease out a solution.  Part of the mathematics here, in my opinion, involves asking a “What if?” question.

I posed this problem to my 3rd grade daughter and after randomly dropping in some numbers at first and seeing some frustration, I said to her, “I wonder what sorts of numbers subtract to give 3.”  Her frustration evaporated as she started making a list of several possibilities for such digits. She noted that there were far more possibilities for these difference than space in the problem allowed.  I encouraged her to keep trying.  We never explicitly discussed the problem’s set up with a four-digit number subtracted from a five-digit number, but I saw her try a couple different first digits before realizing that the first character of the five-digit number clearly had to be “1”.  A little more experimentation and she had an answer.

She thought the puzzle was over–after all, school has trained her to think that once she had “an” answer, she must have found “the” answer.

That’s when I prompted some mathematics.  I asked if she could find another answer.  A few other prompts and she had found 6 different solutions.  I asked her how she found them.  “Easy,” she replied.  “You just put the number pairs in different orders.”  She found through trial-and-error that the five-digit number always started “12…” and therefore the four-digit number started “9…”.  Checking her list of differences leading to 3 left no other possibilities.  Everything else was flexible, thus her six different answers.

• Can you explain why the five-digit number must start “12…”?
• Once I had the “12…” and “9…”, I knew there were at least 6 solutions  before I had found even the first one.  My daughter wasn’t ready for this thought, but can you explain why this is true?
• Can you find all 6 answers?
• Better: Can you explain why there cannot be any more?

The second part of the problem (with the same rules and a different result) is definitely tougher.

You can quickly conclude that the first digit of the five-digit number must be 4 or 3, but it’s definitely more challenging to tease out the rest.  Rather than dealing with the entire problem at once, I suggest another great mathematics strategy:  Simplify the problem.  Using only the digits 1 to 9, can you find all possibilities that would result in the beginning of the problem?

If this is part of an answer, the six digits not used in those three boxes must have an arrangement that subtracts to 333.  Unfortunately, none of these actually pan out.  Convince yourself why this must be true.  Students need to learn that not finding an answer is OK.  Knowing that there’s not a solution is actually a solution–you’ve learned something.

Extending the beginning of the problem to

eventually shows that the five-digit number could start “412..” with the four-digit number starting “79..”.  That means the remaining four digits must have exactly two arrangements for precisely the same reasons that the first problem had six solutions.  Can you find the two arrangements that satisfy the 33333 problem?  In case you want to check, I list the answers at the end of this post.

The next week provided another puzzle using the arrangements idea.

The problem doesn’t yield a straightforward solution that can be solved.  Instead, laying out all possible finishing arrangements and testing the veracity of the claims leads to a solution.  Again, there are three entries, so this problem is (mathematically) just like the 3333 subtraction problem above–both have six possible arrangements.  Helping a young person see this connection would be a great thought achievement.

Start by listing the six possible 1st, 2nd, and 3rd place arrangements of the letters A, B, and C:  A-B-C, A-C-B, B-A-C, B-C-A, C-A-B, & C-B-A.   As an example, if the boys finished A-B-C, all three boys would have told the truth, so that finish doesn’t satisfy the problem requirement of one false statement.  Comparing each arrangement to the boys’ statements eventually shows that only one of these arrangements satisfies the problem’s requirement that exactly one of the three boys made a false statement.

A good mathematical extension would be to see if there are any other questions that could be asked from the boys’ statements.  Is it possible that all three told the truth?  Is it possible that only one was truthful?  Are there any other possible outcomes?  Do any of these have unique outcomes given the boys’ statements, or do some have multiple possibilities?

CONCLUSION:  I fear that too often school and students stop at a single answer and don’t explore other possibilities.  Asking “What if” is a critical question in all of science and mathematics.  It inspires creativity, wonder, and exploration.  It doesn’t always yield results, so it also helps motivate stamina.  Convincing yourself that there are no (more) solutions is itself an intellectual accomplishment.

We need more of this.

SOLUTIONS:

• 3333 solutions: 12678-9345, 12687-9354, 12768-9435, 12786-9453, 12867-9534, & 12876-9543.
• 33333 solutions: 41268-7935 & 41286-7953.
• Competition solution:  A-C-B

Multiplication Puzzle for the Very Young

I just read a recent post on NRICH Mathematics that asked readers or students to list four consecutive whole numbers and compare the products of the outer pair of numbers in the list to the product of the inner pair.  For example, if you used the list {4, 5, 6, 7}, you would have and .  Nothing particularly exciting seems to be here, but try another list of four consecutive whole numbers.  Grab a calculator if you want to be particularly daring or obnoxious with the members in your list.  Do you notice anything now?

I argue the beauty of mathematics as the “science of patterns” kicks in after you find these products for a few different lists.

LEVEL 1:  For the very young who are just learning to multiply, I think this is a GRAND problem.  No proof required.  It’s just crazy cool that those two products always have the same relationship.  Allowing calculators to permit young explorers to try lists beyond their ability to hand or mentally compute enhances the mystery, in my opinion.  

I just played this with my eldest daughter.  She first wrote {19, 20, 21, 22} when I asked her for a list of consecutive numbers.  When I then asked her for the products, she asked if she could use a smaller list.  She opted for {3, 4, 5, 6} and {1, 2, 3, 4} without seeing the pattern.  When I offered a calculator for her original list, she got 418 & 420.  Surprised that they were so close, she said, “Wow, they’re only 2 apart!” I asked if that happened other times.  She looked at her simpler two lists and exclaimed, “Cool!”  I asked if that always happened.  She said, “No.  It couldn’t.”  When I asked for a list where it wouldn’t, she suggested {401, 402, 403, 404}.  The outer product was 162004.  You should have seen her face after she pressed enter on the inner product to get 162006.  “Maybe it does always work!”  Then she asked if she could move on to clean her desk.  Game over … for now.

Part of the power and beauty of mathematics lies in showing that patterns are universal and aren’t limited to numbers we can manipulate quickly in our heads.  I think calculators added to my daughter’s wonder.  I’d love to see my daughter going up to one of her teachers, posing the problem, and predicting the answer without ever knowing the numbers the teacher (or anyone else) had picked.  I think I’d smile even bigger if she had a calculator at hand to offer the adult some “help” if needed!  Math is magical.  Play it up!

LEVEL 2a:  Extend to all integers.  NRICH suggests that the lists need to be whole numbers.  That just isn’t true.  You can start with any integer.  My eldest has been playing with adding negative numbers lately, so I may see if she’s interested in multiplication of negatives.  I’ll think about how to make that idea make sense to her.  At some point in the future, I’ll bring this problem up again and she’ll get an even bigger kick out of seeing that it doesn’t just apply to ordinary and ridiculously large numbers, but negatives, too.

LEVEL 2b:  Proof for the very young.  The NRICH site offers two solutions from “students”.  Whether she’s real or fictional, the approach “Alison” uses is one that I think some sophisticated young learners could grasp long before they learn what a variable is.  Granted, the geometric understanding of multiplication technically works only for specific (not generic) products, but if you set up a few of these, your young one might start to see how the areas grow as the list numbers grow, but the differences in the areas remain constant.

NOTE:  LEVELS 2a and 2b, in my mind, are pretty interchangeable, depending on the readiness and interest of your young learners.  As with all things for young people, throw out the line.  If the interest isn’t there, save the idea for another day.  If you get a nibble, prepare to play!!!

LEVEL 3:  Extend to any arithmetic sequence.  The suggestions NRICH makes for extending the problem all dance around the idea that this property works for any list of four consecutive elements of any arithmetic sequence.  The difference between the two products depends solely on the common difference of the sequence and is completely independent of the initial term in the sequence.  Try {1.1, 1.2, 1.3, 1.4}.  The difference in the outer and inner pair products will be the same as for {98.8, 98.9, 99.0, 99.1} simply because both lists increase by 0.1.

LEVEL 4:  Algebra.  Those who remember their algebra classes may have jumped right to an algebraic justification.  That’s what I did, and that’s the solution “Charlie” gives on the original NRICH post.  In a way, I think I cheated myself out of seeking the pattern as my daughter discovered it.  Whenever your young ones are ready to deal with the magic and power of variables, try out proving this for integers.  When they’re ready for more, prove it for all arithmetic sequences with any initial term.  You’ll know they’re strong when they can argue on their own why the initial term is irrelevant.

LEVEL 5:  More Algebra.  This “trick” extends to to any arithmetic sequence of any length.  With algebra, one can determine a formula for the difference between the products of the last terms and the next-to-last terms.  I think a talented middle school student or young high school student who knows how to handle very generic cases could find that formula.

And it all starts with playing with some little numbers.

Creative Subtraction for the Very Young

OK, this is far from a revolutionary idea, but I was impressed with my eldest daughter’s approach to a subtraction problem. What I present may already be used by others, but it was new to me.

She was interested a few months ago in learning what negative numbers were and what caused them.  Of course, I dove right in and explained.  From time-to-time she’s asked for more, but mostly I thought it was an idea that had gone fallow.  Then she wrote the following last night.

I had learned column subtraction from right to left with “borrowing.”  It was an algorithm that I understood; many classmates struggled.  My daughter worked left to right by column/place value.  So, she computed 70-40 and wrote 30.  Then she used her knowledge of negative numbers and asked me, “So 0 minus 2 is -2, right?”.  She then wrote -2.  The rest was easy.  What impressed me was her adaptation of an algorithm she had been using for partial sums. If a student understands negative numbers, I wonder if this might be a cleaner approach.  The thinking is also similar to what one might use when calculating using an abacus.  Cool.