Math Challenge: Do you know what algorithm this is?

David Wees came up with this challenge. Determine what algorithm this code emulates. You will find the answer more informative if you create a table to see the pattern of moves the code makes. The table can then be used to introduce the algorithm formally to your students. Or better yet, get them to build their own tables from the code. Try a range of integers to test the code. What patterns exist?



I will post my answer in a couple of days.

Emulating Evolution by Involving Students

Evolution is one of those topics I really enjoy learning and teaching, particularly with open activities. The War of the Evolutionists Web Scavenger Hunt and Mock Trial, and the Evolution and Adaptation games are just some of my attempts to engage students in ecology and evolution.

Of these The Evolution Game, created by Simon Boswell and Phil Lewis, comes the closest to emulating the process and product of evolution. The rules and gates of the game are the processes of ecology and evolution; it is not a trivia game, a passive one, nor one like charades whose play has little semblance to evolution. As such The Evolution Game exemplifies that rare and powerful quality game that embeds students in their learning by forcing them to deal with and adapt to the “rules” or processes of evolution and ecology. It is however a long-term activity, like Risk, of both strategy and chance.



Just yesterday, I came across another great activity that comes close to emulating both ecology and evolution. The activity, created by Tyler Rhodes and featured by Scientific American, consists of two parts: a student-centered exercise and a technical exercise. The student exercise takes about an hour, or one period, to conduct. The technical one — creating a video — took Tyler, who claims to be expert enough to work efficiently, three months to complete. So feedback in the form of product is delayed, though formative feedback is immediate and embedded in the students’ own activity.

The idea is simple, if not elegant, and follows the same design premise as The Evolution Game and Bernie Dodge’s formula for game design that “Elegance = congruity between the forms of the game and structures within the content“.

Tyler drew a nondescript salamander-like creature and enlisted five independent groups of students (from five schools) to draw copies of this creature.



Once the students compared and discussed the new creatures, Tyler “exposed” the creatures to some ecological stress or change. The students had to vote which creatures perished based on the new ecology and the features of the creatures. According to Tyler, ninety-eight percent of the creatures perished (in a class of 30, where each student drew one creature, one creature survived). Tyler gathered the extinct creatures and repeated the exercise five more times with the survivors, each time with a new ecological event wiping out ninety-eight percent of the creatures. The six generations were kept or labelled apart.

The exercise illustrated branching phylogenetic evolution and coevolution — rather than the defunct linear evolution — as shown in the following drawing, where each “arm” of creatures came from a separate class of students, so giving five arms.


A Wheel of LifeA Wheel of Life © 2012 Tyler Rhodes | more info (via: Tyler Rhodes)
Click on the image to enlarge it.


What is nice about this exercise is that the students actively engage in, and embed themselves in, the process of natural selection by ecological change. They, being the active agents in both the creation and voting off of these creatures, were given the opportunity to experience and learn about the fundamental processes of evolution, much like The Evolution Game.

Tyler designed this process after a “Chinese Whisper” or “Telephone” game, where a message is passed from person to person and changes through mutation as it is delivered. In fact he presented it as a game. The message, however, was visual — the drawing of the creature — and the changing ecology affected the message. Tyler was specifically looking for a way to branch the creature evolution like a phylogenetic tree and his use of the “Chinese Whisper” or “Rumour” game enabled this.

Here is the final video.


The creatures in this video are those from the top-left arm of the Wheel of Life tree illustrated above. Tyler promises four more videos, for each of the remaining four groups of students. And he invites teachers to take his initial nondescript salamander-like creature, repeat his method and e-mail him facsimiles of the creatures created. If teachers take him up on the offer, he can bank, analyze and share some really interesting evolutionary results from the project. His conclusions should be interesting.

For more information on Tyler’s project, visit his blog, Evolution!, documenting his progress.

How can we emulate Tyler’s project for outcomes in our classrooms? Or have you done so already?

Thinking… Please Wait


Hi. It has been a month since I last posted, but I accomplished a lot during that time. And I am very happy I did. I feel like I progressed quite a bit since I last posted.

This is great since I experienced debilitating writer’s block with some of the key posts I have been struggling to publish, and this pretty much stalled me. It happens I guess. I had all these things I wanted to do and I wasn’t advancing.

Thank goodness I had another blog and other professional resources to work on.

Last month I took a rest from Digital Substitute so I could catch up on some much neglected projects that I was just raring to work on. In one of my early posts, Math Lab: Revisiting Technology and Imagination, I exclaimed how liberating it was to take a single day off from blogging, tweeting, PD and other professional activities to just play, and I think the post that resulted was one of my favourites to write, and perhaps my second most popular.

I find most of my posts, and certainly my best ones, result from some emotional or playful encounter. So, I consider the sacrifice of one month worthwhile to recharge myself.

Wandering in the Land of Set-aside

So last month I worked on several fronts.

Teaching Resources

I editted my Teaching Resources site, including adding:

  1. notes I wrote, and links to online archives, from several of my recent PD sessions to my Professional Development Index,
  2. resources, and a Slideshare Pak Liam created in response to these pages, to my Green Pea Analogy pages,
  3. focussing questions and points to my Phronesis page, and
  4. a math term etymology document that apparently was very well received given the tweets and requests for links to it on Twitter.

Writing in Play

I also did a lot of writing this month, something that has sadly been long waiting, including the completion of a short story based on a Figment Theme Prompt and working on a chapter in one of my long stories. I participated in Figment Theme Prompts, doing a little writing each day. And I posted to my Stefras’ Bridge blog. Altogether, a great month of accomplishment for my writing.

Stefras’ Bridge

I blogged about another of my hobbies, oil painting, and linked to one of my essays describing the history and folklore behind the earliest form of Mardi Gras. And I posted my first review and interview — with Malyn Mawby.

The Write Group

I also created a new Twitter account for the Write Group, to which I am migrating my English and writing tweeps and adding tweeps specifically for the Write Group. And I continued to work on our wiki, gathering RSS feeds and bookmarks relevant to our group. It doesn’t look like much now, but wait until I get things up and running!

Flickr, Videos and Coding

Finally, I added some photos to my Flickr photostream, I am working on two cartoon videos for Pi Day — there has got to be a better way to do this than drawing all these pictures … but boy does it look neat (for my first true “movie” videos) — and I am refreshing my coding skills with Code Year.


How did I get so busy?

I think I work in there somewhere.

Thinking … please wait.

Our Children’s Gears: Do You Like Dinosaurs?

Do you like dinosaurs?

Or did you when you were a kid?



Dinosaurs are neat. They are big, ferocious and were, quite frankly, very successful. They were also the dominant animals of the Mesozoic, for 180 million years. That is impressive.


I never liked dinosaurs when I was a kid. I found them boring.

I was beguiled by the Palaeozoic and early Cenozoic rather than the Mesozoic. The creatures — plants and animals — that lived then were alien, intriguing and awesome.

I could never put my finger on why trilobites and Paraceratherium interested me more than dinosaurs. But there was a pattern in that interest that cropped up elsewhere in my life.

An underlying ecology

When it came time for me to enter university, I knew exactly what I wanted to be — an ecologist. Not a botanist, not a zoologist, not even a geologist, an ecologist. Don’t get me wrong, I was fascinated in zoology, botany and geology, mostly botany, but I did not want to study one thing.

I was interested in it all. I was interested in how it all fit and worked together. I was interested in how life lived on an erratic Earth. Its individual forms fascinate, but mostly as pieces of the intricate whole.

And that, as I later found out, was why I didn’t like dinosaurs. They ate. They fought. They terrorized the land — not to mention other animals. But, until the last ten to twenty years, for me they never belonged — neither fit nor worked — within a bigger system.

They were boring.

In the last decade or two, that changed, or perhaps I became aware of the “bigger” Mesozoic picture. More Mesozoic palaeoecology has been learned and integrated into other disciplines, as illustrated in Harold Levin‘s The Earth Through Time (I have the 2003 seventh edition published by John Wiley and Sons). And now the dinosaurs belong with, influence and are influenced by a bigger lifescape and ecology. Dinosaurs became more and more interesting as they began to fit and work in the puzzle of life and living in a changing, furious Earth.



It is their place in ecology that fascinates me, not their ferociousness nor their reputation.

The point? Even as a child, I was geared toward ecology.

An overarching Universe



My enjoyment of astronomy also stems from the same root. I am fascinated by the Earth’s place and development in the Solar System, and of the Solar System’s place and development in the Universe.

I look at a star as I do a handful of sand and I wonder about its past, about its surroundings, its environment, its present and its future. I wonder about what it interacted — or will interact — with, what it influences or what it is influenced by. I similarly wonder (to the same depth) about the Universe that the star represents and the Earth and rock that the sand typifies.

I remember encountering an ant crawling on a moss and seeing its ecology and the ecology of the ecosystem where it lived. I had no words for these concepts, but I distinctly remember seeing the ant interacting with its environment. I barely noticed the ant outside of this frame. I was in grade two. And I still see ants and stars and handfuls of sand this way.



A far-sweeping magic

Story exists in this way too. With story we build our cultures, societies, histories, skills and technologies. But we also build our spirit and curiosity.

Story exists in a bigger context, constructed of reality and imagination and wonder.

Arthur C. Clarke coined, “Any sufficiently advanced technology is indistinguishable from magic”. I prefer to replace “advanced” with “exotic”, meaning unfamiliar or novel or not (currently) understood.

Ants and sand and stars, ecology and math and story and language are magic. There is always more of them to explore.

Story is a form of species-changing magic. And writing transmits this magic into the minds of generations and far-flung peoples.

In writing fantasy (which I mostly do), one creates the rules of a given world and studies how a story fits and works within that world. It is intriguing to witness story unfold even as one writes it. I am always surprised by what story reveals, about what it says about the world it explores, influences, interacts with and is influenced by.



Story is a key part of my life and has been for as long as I can remember. I am geared toward it like I am ecology and astronomy.

The gears of our children

In his essay forward, The Gears of My Childhood, to his 1980 book Mindstorms: Children, Computers, and Powerful Ideas, Seymour Papert eloquently describes how gears shaped how he perceived the world and approached learning when he was a child.

We all have our gears where what we learn ceases to be flat and static and becomes multidimensional and living. Papert describes vividly how gears of different sizes fit together to produce meshing products in a multiplication table. I tried to describe here how interactions are part of my guiding gears. I have students who are ranchers. Others are athletes, artists, scientists, writers. And of course they are each interested in more than one thing and are geared toward truly fundamental world views.



Imagine viewing multiplication as representation of meshing gears. What most affects us, influences our world view and shapes how we perceive and interpret what we later encounter has a great effect and affect on what we learn and how we do it.

Teaching toward our children’s gears might help them understand and learn what we are teaching. It also might allow them to more easily own what they learn, extend it beyond our teaching and keep it for a lifetime. Teaching the student more than the students and the lesson content facilitates her engagement with and conceptualization of the outcomes we wish him to learn.

Papert recounts his discovery that others do not share his world model of the gears, but have different models instead. We have to teach students we know in multiple ways to help them learn what we want them to learn. We have to know and value our students to help them realize their influence, their potential and their dreams.

We might have a class under our charge, but that class consists of unique individuals geared by unique world models. The function and art of teaching is to change behaviour not people. Our gears are as precious as our names. Sometimes all we own are these two things. We need to be careful to nurture and engage our children’s gears so that they might serve our children well in our multi-layered societies.



This post was inspired by David Wees’ draft of a keynote he was invited to present at the 2012 University of Alberta Faculty of Education Technology Fair.

Math: My Solutions to mathhombre’s Problems

Part of the job of a teacher is to model her or his Subject-specific thinking to help students understand the subject techniques and the Subject processes, patterns and metacontent. To this end, I dedicate this post to modelling how I tackled the math problems, created by John Golden as part of a math final for preservice teachers, that I mentioned in my last post.

mathhombre's Twitter avatar I enjoy these types of problems. They are chuck full of math — and Math — from geometry and trigonometry to algebra and arithmetic. But even more so, they require mathematical thought, imagination, creativity, logic and perseverance to solve them. Had John expanded the questions to ask for metacognitive reflection and recording, at least the trigonometry question could easily pass as a Math lab.

Once again I invite you to visit his post and take a crack at his problems before reading on.

The remainder of this post contains my solutions to these problems. (I went overboard with the trigonometric problem. I blame it on play and slightly improved health.)

Trig Problem 2

The first problem John offers is a circle geometry / trigonometry problem. He presents a diagram and simply asks the test taker to “Figure out some of the missing information in the diagram”. The only information the student is sure of from this task is that some information — more than one piece — is missing. Interpretation, logic and problem solving are all the tools the student has to continue with the question.

As I mentioned in my last post, what I really like about this question — and the next one — is its openness. The student isn’t asked for all possible missing information. He can choose what she wants to explore and discover.

I began my activity to this task by stating two assumptions on which the rest of my work rests. I assumed that point C is the center of the big circle. I also assumed that point A is the center of the small one. You might think this obvious given the diagram, but I like stating these assumptions right away. That way everyone knows where I stand and the rest of my work can not be faulted based on ambiguity on these points.

I then stated the explicit givens, that angle ADE is 30°, angles AED and ACD are 90°, and segment AC is 3 units long. (All of this and the rest of the original problem diagram is in light blue in the answer diagram below.)

Here is a diagram of my answer.



Click to enlarge or visit the GeoGebra construct from which it derives.

Joy, Pythagoras

Based on the assumptions that C is the center of the big circle and A is the center of the small one, I calculated and labelled the radii for the big and small circles. So CD, CI, CL and CM are each 6 units long and AC, AG, AH, AI, AJ and AK are each 3 units long.

This made apparent that triangle ADC is a right triangle with legs CD of 6 units and AC of 3 units, allowing (through Pythagoras’s Theorem) hypotenuse AD to be calculated as square root 45 or 3√5, which is also the hypotenuse of triangle AED.

Making angles

Equally obvious to the use of Pythagoras’s Theorem to calculate the common hypotenuse of triangles ADC and AED, is the use of the 180° (or π) “rule” for use in triangle AED.

This concept is so well known, it has almost become pneumonic. Yet, think on how profound the notion that the internal angles of all triangles always add up to a common constant, and that this constant, 180°, is also found in: the angles that accumulate to make a line, the conversion from degrees into radians and the command that sends us back the way we came.

In triangle AED, angle DAE is 60° because angle AED is 90° and angle EDA is 30° and these add up to 180°.

Attempting a similar algorithm in triangle ADC requires some labelling. I set the measure of angle ADC to unknown x, which makes the measure of angle CAD to 90-x by the 180° rule. But the fun does not stop there. Since the angles touching line EF on one side at common point A also add up to 180°, angle FAC is 30+x (60+(90-x)+(30+x)=180).

One could circle point A applying this rule to label all angles surrounding A, but another way is to apply the Vertically Opposite Angle Rule (yet another pneumonic one can ignore since the 180° rule produces it) to determine that angle JAK is 60°, IAJ is 90-x and HAI is 30+x.

Similar rules can be used to determine that all the angle measurements about points C and E are 90°, but also to calculate that angles CFA, LFN, ABE and VBW are 60-x in measure. (Oh, by the way, I added points B, L, N, M, P, Q, R, V, W, S, T and U to the diagram. The last five are hidden off the edge, but are used solely to allow labelling of angles VBW, UDT, TDS and LFN vertically opposite of and equal in measure to angles ABE, EDA, ADC and CFA respectively.)

The value of x, and consequently of the other angles, is calculated applying the Law of Cosines in triangle ADC.

AC² = AD² + CD² – 2(AD)(CD)cos(x)
3² = (3√5)² + 6² – 2(3√5)(6)cos(x)
9 = 45 + 36 – (36√5)cos(x)
cos(x) = [(-72)/(-36)](1/√5)
cos(x) = 2/√5
x = 26.57°

From this, 90-x = 63.43°, 30+x = 56.57° and 60-x = 33.43°. Using the 180° rule to back check these values confirms them.

These calculations take care of all the obvious angles that the problem suggests are missing.

Measuring sides

Observation of the diagram to this point reveals that quadrilateral AEDC is not a kite since AC ≠ AE (AH = AC and AE = AH + EH), angle DAE (at 60°) ≠ CAD (at 63.43°) and angle EDA (at 30°) ≠ ADC (at 26.57°).

The lengths of AE and DE need to be calculated.

I used the Law of Sines to calculate the lengths of AE and DE. This is fairly easy since the sines of 30°, 60° and 90° produce easy ratios, 1/2, √3/2 and 1 respectively. The calculation looks like this.

sin(30°)/AE = sin(60°)/DE = sin(90°)/(3√5)
1/(2AE) = √3/(2DE) = 1/(3√5)
AE = 3√5/2, DE = 3√15/2

I repeated this process to calculate the lengths of AF, CF, BA and BE.

sin(33.43°)/3 = sin(90°)/(AF) = sin(56.57°)/(CF)
AF = 3sin(90°)/sin(33.43°) = 5.45 units
CF = 3sin(56.57°)/sin(33.34°) = 4.54 units

sin(33.43°)/(3√5/2) = sin(90°)/(BA) = sin(56.57°)/(BE)
BA = 3√5sin(90°)/2sin(33.43°) = 6.09 units
BE = 3√5sin(56.57°)/2sin(33.43°) = 5.08 units

It did not escape me that triangles ABE and ACF, with equivalent interior angles, are mathematically similar.

Having calculated the length of CF as 4.54 units and the length of CL as 6 units, it holds that FL = CL – CF = 1.46 units.

Unfortunately, the same can not be said for calculating the lengths of EQ, EP and FN, which remain undetermined. Perhaps there is a trick to calculating them. I was considering the power of a point using secants BD (through Q) and BM (through I) to calculate EQ, but the lengths of DQ, BQ, MI and BI are unknown. Unfortunately, the length of “chord” RQ, subtended by angle EDA and creating triangle QDR, also can not be determined because the length of DQ is again unknown. With point A and angle DAE mathematically set, the length of chord PN might be calculable, paving the way to calculating the lengths of EP and FN. I have not yet figured out how to do this. This leaves EQ, EP and FN unmeasured.

Measuring JR

The length of JR is 1.02 units. But I cheated to determine this!

Whereas all the other determined values described in this post are calculable from the initial information provided in this post and John’s original problem, calculating the length of JR required mechanical measurement of another length, mainly the length of DO. GeoGebra did the measuring, and had it not done so, the length of JR would be undetermined like those of EQ, EP and FN.

Wait! So why didn’t I measure the lengths of EQ, EP and FN? For the lengths of these segments, I have no theory to back my measurements with. For calculating the length of JR, I do.

The length of JR can be calculated using the Side-Splitter Theorem. (Funny, I used to call this the Parallel Projection Theorem. The Side-Splitter Theorem sounds like a comedy act.)

Segment AC splits triangle DOR parallel to leg OR of that triangle. The proportions of triangle DCA are already known to be CD:AC:AD = 6:3:3√5. So knowing the length of DO or OR can provide the length of DR in triangle DOR.

Here is where the mechanical measurement comes in. I used GeoGebra to measure (not calculate) the length of DO at 9.60 units. From this, using proportional scaling of similar triangles DCA and DOR, I calculate the lengths of OR and DR as 4.80 units and 10.73 units respectively.

The length of JR = DR – DA – AJ = 10.73 – 3√5 – 3 = 1.02 units.

Still if this were a real exam and I did not have access to GeoGebra, I would not be able to determine JR’s length.

Why stop there?

That essentially covers the basic values of John’s original problem that any preservice teacher would reasonably be expected to determine on a test. But why stop there?

I am under no time pressure and I saw right from the start that I can calculate values associated with those circles. I am sure John did not expect such a step, but here I am noticing patterns and recognizing opportunities. So, here John are my bonus marks.

The circumference (2π*radius) of the small circle is 6π since its radius is 3 units. That of the big circle is 12π from its radius of 6 units.

The central angles of these circles produce isosceles triangles with radial legs of either 3 units (small circle) or 6 units (big circle) and chords subtended by these angles.

In particular, the small circle’s central angles GAH and JAK at 60° suggest equilateral triangles AHG and AKJ. They also suggest that arcs GH and JK have arclengths of π. Since these triangles are equilateral, GH and JK are 3 units long just like the radius of the circle. This implies an arclength of one radian or π. Independently, 60° = (60°/180°)π = π/3, while the arclength of an arc of a circle is calculated as the measure of its central angle times the radius of the circle (angle*radius). In this case angle*radius = (π/3)*(3) = π.

Since the other central angles of the small circle are not 60°, their angles times three are used to calculate the arclengths of their arcs and the Law of Cosines is used to calculate the lengths of their chords.

The length of CG, for instance, is 3.15 units as calculated below.

(CG)² = (AC)² + (AG)² – 2(AC)(AG)cos(angle CAG)
(CG)² = 3² + 3² – 2(3)(3)cos(63.43°)
CG = 3.15 units

Arc CG, corresponding to this chord and similarly subtended by central angle CAG, has arclength 1.06π units ((63.43°/180°)*3 = 1.06).

For the big circle, there are four isosceles triangles with 6 unit-long radial legs and 6√2 unit-long chords (calculated using Pythagoras’s Theorem since the central angles around C are 90°). The corresponding arcs are all 3π in arclength.

I used the 180° rule for triangles, used in the Making Angles section, to calculate the other angles of each of these isosceles triangles.

For triangle CAG, angles AGC and GCA are each equal to 180° – angle CAG = 180° – 64.43° = 58.29° in measure.

And that exhausts the values I calculated.

What I missed

I noticed upon reviewing my answer diagram that I forgot to calculate the “exterior angles” around B, D and F, that is angles WBI, EBV, NFC, KFL, CDU and SDE. However, I did calculate these in the GeoGebra construct using the 180° rule for lines.

Angle WBI = angle EBV = 180° – 33.43° = 146.57°
Angle NFC = angle KFL = 180° – 33.43° = 146.57°

Angle SDE = angle CDU
180° – angle TDS – angle EDA
180° – angle ADC – angle UDT
180° – 26.57° – 30°
Angle SDE = angle CDU = 123.43°

Geometry Problem 1

Relative to the circle geometry / trigonometry problem above, John’s line geometry problem is trivial. There is not much to do with it except fill in the missing angles following the given information.

I started out by assuming that lines JK and LM were parallel to each other. This was a pretty good assumption, given that John was testing student-teacher knowledge of parallel lines, perpendicular lines, transversal lines and associated angles.

Here is my answer. I will leave the explanation to you.



Click to enlarge or visit the GeoGebra construct from which it derives.

Benefitting Our Students

Solving an unfamiliar problem in front of, or better yet, with our students can greatly benefit them. They get to see how an unfamiliar problem is solved. They get to see how we approach and think on a problem. They get clearer — or messier — explanations as we think aloud. They even get to see us make mistakes and how we recognize and resolve them.

Sometimes modelling mathematical activity when faced when an unfamiliar task or challenge can flop on us and backfire on our kids. But how we deal with these flops is also informative.

Most of the time, our flops are minor — and often silly (yes, 1 + 2 = 3, not 4) — and our modelling, after correcting the error, leads to a correct answer.

But what our students gain is our thinking, reasoning and problem solving expertise. They learn the difference between solving and solution, between proving and proof. They learn to be — and how to be — creative, logical, persistent and progressive in our tackling of a problem. They learn to touch Math, instead of just math.

And that is where we want our students to be.

Gone are the days where the answer to a particular problem is satisfactory. Now thinking is where it’s at. We have to model our thinking and discuss solving with our students. We have to teach them to think, so they can excel beyond our meager abilities.

Our students deserve better and modelling is one key way (another would be rich math problems and labs) to ensure they get the meta out of their math.