Friday, 27 October 2017

Problem... Cake

#Combinations, #Rotation, #Computational Thinking 

The problem was adapted from the 'Ice Cream Cake' problem in Peter Winkler's 'Mathematical Mind Benders' (p111 with solution on pp115-118), and via Stan Wagon's take on it in Gil Kalai's blog.  It's likely origin was as problem in Olympiad 31 (1968) Moscow Mathematical Olympiads (p90).


The values of \(f\left( \theta  \right)\) for \(\left\{ {\theta |\theta  \in \mathbb{N},1 \le \theta  \le 360} \right\}\) can be generated thus:

\[\begin{array}{l}Let\;k = \left\lceil {\frac{{360}}{\theta }} \right\rceil \\\\f\left( \theta  \right) = 2k\;for\;\left\{ {k|k = \frac{{360}}{\theta },1 \le \theta  \le 360} \right\}\\\\f\left( \theta  \right) = 2k\left( {k - 1} \right){\rm{ for }}\left\{ {k|k \ne \frac{{360}}{\theta },1 \le \theta  \le 360} \right\}\end{array}\]
Giving \(f\left( {181^\circ } \right) = 4\) and \(f\left( \theta  \right)\) for all \(\left\{ {\theta |\theta  \in \mathbb{N},1 \le \theta  \le 360} \right\}\):

θ f(θ)
1 720
2 360
3 240
4 180
5 144
6 120
7 5304
8 90
9 80
10 72
11 2112
12 60
13 1512
14 1300
15 48
16 1012
17 924
18 40
19 684
20 36
21 612
22 544
23 480
24 30
25 420
26-27 364
28-29 312
30 24
31-32 264
33-35 220
36 20
37-39 180
40 18
41-44 144
45 16
46-51 112
52-59 84
60 12
61-71 60
72 10
73-89 40
90 8
91-119 24
120 6
121-179 12
180-359 4
360 2

Wednesday, 11 October 2017

On The Inadvertencies of Teaching (Girls & STEM)

The lack of engagement in @TeachFMaths' female-only Twitter #MathOlympiad2017 — in relative comparison, that is, with the engagement generated by the male-only Olympiad (won by Euler, incidentally) — has spiked my interest, particularly after seeing what I think I can uncontroversially describe as a fairly exasperated tweet from @TeachFMaths pop up in my timeline, and especially since today is the international day of the girl, and yesterday was Ada Lovelace Day, an international day celebrating the achievements of women in science, technology, engineering and mathematics.

Yes, of course, @TeachFMaths' (fantastic) math Olympiads are just the kind of innocent, knockabout fun that Dorsey, Glass, Stone, and Williams could almost have made Twitter for (see @richardosman's glorious World Cup of Crisps, for example, or his World Cup of Biscuits), but as it is likely that a large proportion of the audience engaged with the Olympiads are bothered, in some respect, with the business of education, and the fact that this disparity in engagement between the male and female-only Olympiads so patently exists, inadvertently throws up, I think, some pertinent questions that it's worth schools — and particularly we as teachers — ponder a tad.  So, if you will amuse me for a moment or two:

As of today, with the female-only Olympiad still in progress, if we take a quick look at the number of votes cast in each — as a proxy for engagement (ignoring, that is, the secondary tweets, re-tweets and the like generated by the competition, where, incidentally, the disparity is even greater) — we see:

The mean number of votes cast per match up in the male-only Olympiad was 315, or 233 if we remove the two outliers that were the epic Newton v Euclid semi-final, and the final.  The mean number of votes cast per match up after the first ten rounds of matches for the female-only Olympiad is 61.  In other words, the men’s Olympiad attracted around four times the degree of engagement than the women’s Olympiad is currently attracting.  The match — or should that be mismatch — that attracted the least number of votes in the male-only Olympiad, still attracted more votes than the match that has thus far attracted the most number of votes in the female-only Olympiad.

My first instinctual reaction to this disparity was one of almost resignation, that this was not, in other words, unsurprising nor, indeed, something to read anything into; to blow out of proportion.  Of course, there are an inordinate amount of factors that could and would impact on the relative disparity in the degrees of engagement between the male and female-only math Olympiads — and I won't be going into any eigenvalue-ridden factor analysis about it here.  (It is worth pointing out, nonetheless, that @TeachFMaths has more followers now, and thus potential participators for the female-only math Olympiad, than he did when he ran the male-only math Olympiad, and I’m pretty sure we can rule out math Olympiad fatigue, so to speak.)  But I do wonder, from the perspective of a Dad of a daughter, a one-time Engineer turned maths teacher and Headteacher, from the perspective of someone who has seen Bourdieuian socio-cultural reproduction at play first hand day in day out when it comes to the life 'choices' students' make, whether this neutrality of sorts — or this out of sorts neutrality perhaps — to put it quite simply, matters, and matters more that is than we give it credence.

When voting in the female-only Olympiad (put yourself in the position of voting if you haven't, yet), are we voting more in terms of what mathematician we have heard of, rather than what we perceive to be the relative merits of their mathematics, as I would contend would be more the case in the male-only Olympiad?  Are our female-only votes more a function of the relative celebrity status of the mathematicians, as it were, rather than their achievements, in contrast to our male-only votes?  Are our votes simply an act of reinforcement?  (Take Ada Lovelace, for example, relatively unknown ten years ago, she now has a day named after her, and trounced the opposition in the group stage of the Olympiad.)  What are our choices — to vote or not to vote, rather than who to vote for — a function of?

When it comes to our engagement in a fun, onthefaceofit innocuous social-media competition, it does not — or, moreover, should not — matter whether groundbreaking achievements have been made by men, historically, or not.  It is irrelevant to the act of voting, or to the act of choosing to vote or not.  To put it another way, if I am being asked to vote between Isaac Newton and Euclid of Alexandria, and then to vote between say Phoebe Sarah Hertha Ayrton and Linda Goldway Keen, the very act of me voting is — or should be — independent of the makeup of the group from which I am choosing.  The only thing stopping me from voting for the female-only match up would be the fact that — as I am embarrassed to admit — I know very little of either of Ayrton or Keen, and would need to do some research.  Or, perhaps, in choosing not* to vote in the female-only math Olympiad, I am adopting some kind of aesthetic/intellectual distance (cf. Bourdieu, 1984, p34) to distinguish myself from something, to signify something about myself to others.  Maybe my 'detachment, disinterestedness, indifference' (in that order) should really be read as 'disinvestment [my emphasis], detachment, indifference, in other words, the refusal to invest oneself and take things seriously' (ibid.).  (*I am voting, incidentally.) 

And therein lies the issue that perhaps @TeachFMaths' fantastic Olympiads have, in an unintentional, Milgram-esque sense, exposed (whether the issue needed to be or not).  From the perspective of the girls that I teach, whether I am ignorant of, or ambivalent to the achievements of female mathematicians in comparison to their male counterparts, irrespective of the relative degrees of their achievements, the outcome is the dangerous same, namely that I run the risk of reproducing and reinforcing the conditions whereby girls are inadvertently denied access to the positive gender role models they may need (see, for example, Dennehy and Dasgupta, 2017Lockwood, 2006Allen and Eby, 2004).

This is, of course, not, in any sense, to dilute the joyous sense of inspiration that comes from exposure to and appreciation of the amazing achievements and breakthroughs made by the great male mathematicians of history.  We all, boy or girl, need these subject-centred, or intellectual role-models.  And I want my sons and daughter, and the sons and daughters whom I teach, to aspire to be mathematicians, or scientists, or engineers, or the whateveritistheywanttobe's of the future — but we must remember that the daughters we teach may need female role-models as well, in contrast to the sons we teach, by virtue of the simple fact that history is relatively uncluttered by them.  As a teacher and school leader I of course ensure that students are exposed to — and develop an appreciation for — the great achievements of thinkers throughout history, irrespective of gender.  The point that I am labouring to end on is that girls need female role models as well, and that as a teacher it is part of my job to do more than just not deny them this.  I need to think, carefully, differently maybe, about my own, individual, inadvertent behaviours, limitations (of knowledge), and perspectives, that may inconspicuously — but burningly — limit or even curb girls' notions of what they can or cannot do. 

(Thank you to @TeachFMaths for his #MathOlympiad2017 and for making me think again about my own practice, however inadvertent that may have been!)

Notes & (Select) Links:

Friday, 6 October 2017

On Ada Lovelace Day

Founded in 2009 by Suw Charman-Anderson, a social technologist, journalist and writer (and, I must say, an enviable talent for naming cats), Ada Lovelace Day (ALD) is more than merely a day of observance, if you like; it is a day when schools should unashamedly revel in a pure celebration of scientific and technological achievement, made, as it happens, by relatively un-celebrated women, whose passion for learning, for understanding and discovery — for education — overcame any socio-culturally formed gender barriers that may have otherwise got in the way (read Project Ada's interview with Suw Charman-Anderson here, discussing why she founded the day.)

As a teacher and school leader, marking ALD with and for our students is absolutely about raising the profile of women in science, technology, engineering and mathematics (STEM) — but, importantly, for our boys, as well as our girls.  The day is also, and indeed, unashamedly about raising awareness of contemporary female STEM role models explicitly for the girls under our care: as Lockwood (2006) suggests, 'women... may derive particular benefit from the example of an outstanding woman who illustrates the possibility of overcoming gender barriers to achieve success... in contrast [to] men'.  But to suggest that we put aside one day in the school year to celebrate the achievements of women in STEM is — or should be — anathema to any institution that bothers itself with the business of education.  Marking ALD is just one of the many ways schools use historical and contemporary achievements to motivate and inspire young minds.  The day is an opportunity to boot for schools to unabashedly reinforce the ethos underpinning our organisations, to reaffirm the values that implicitly or otherwise inform the everyday work we do.  Yes, we are celebrating the historic achievements of women who have excelled in their field, but by also framing their achievements against the contemporary work of today's pioneers, we are cultivating a set of cultural conditions for the future where boys as well as girls — or arguably where boys more than girls — understand that it is not just OK for girls to be STEM nerds too, many quite simply are, so get over it. 

This post is not about me retelling Lovelace's story, or discussing the history or controversy over the extent of her work — others have done this with scope and rigour and you will find or may stumble across links to some excellent sources through this post.  Suffice to say, in the words of Ursula Martin, Professor of Computer Science at the University of Oxford,  'Lovelace saw the potential for the computer to do amazing things', but to really 'understand Lovelace’s work, you need to know the context of the ideas of the time.'  (You can read a short, fascinating interview with Professor Martin about Lovelace's papers here (you may also enjoy perusing Oxford's Bodleian library's blog dedicated to Lovelace).  Through this post I am simply encouraging schools to mark the day, hopefully providing some simple suggestions, or spikes for further exploration, for how this may be done, without any huge organisational implications.

For example, ALD also, coincidentally, marks the start of Global Math Week (October 10-17), and the end of the United Nations' World Space Week (October 4-10).  With this in mind, two potential avenues for exploration with or by students on the day could be around the work of Maryam Mirzakhani, the Fields Medallist who tragically passed away only recently, and perhaps around the story of Valentina Tereshkova, the first woman in space and subject of a current science museum  exhibition (read this from Doug Millard at the Science Museum blog)

You will find a Twitter Moment I created marking Maryam Mirzakhani's death here, with links to a number of sources you may wish to explore and/or share with students, including a brief documentary about her groundbreaking work.  You may also wish to explore with students this fantastic task around topology, Mirzakhani's field of research, devised by Professor Yahya Tabesh, one of Mirzahkani's mentors when she was studying mathematics, and shared by Jo Boaler and the team.  The question is 'can you morph a donut [sic]?'   (You may also want to explore the excellent Women of Mathematics throughout Europe website.)

Whilst, despite the name, ALD is not explicitly about Ada Lovelace, her story, achievements, and visionary insights are more than worth exploring with students.  You will find some other sources of information — and links to other sources — about Lovelace in the Twitter Moment I created here, and the Bodelian library have an excellent short biography of her here.  Rachel Thomas at Plus magazine has written another excellent and accessible article here, and you may want to share this brief video by @YourLifeTeam and narrated by Hannah Fry (@FryRsquared), who also told Lovelace's story in more depth in this superb BBC documentary.  (You may also want to use Hannah's wonderful talk on Benford's Law and 'repeat victimisation', at the Royal Institution for ALD 2014, 'Can Maths Predict the Future?')  If you want to delve a little deeper into the enigma that is Ada Lovelace, consider 'Untangling the Tale of Ada Lovelace', an excellent biographical blog post by Stephen Wolfram.

This link will take you to a digital transcription — by John Walker (@fourmilab) — of Lovelace's translation of Luigi Federico Menabrean's write up of Babbage's lecture series on the Analytical Engine at the Academy of Sciences in Turin, including Lovelace's famous notes, including Note G, the first published computer program: Instructions for the Analytical Engine to calculate Bernoulli numbers.  (The First Edition of Lovelace's translation is currently valued at $28,000.)

You may also wish to explore and share the videos from the 2016 ALD Live event organised by, and held at the Institution of Engineering and Technology, featuring design engineer Yewande Akinola, planetary physicist Dr Sheila Kanani, science writer Dr Kat Arney, developer Jenny Duckett, mathematician Dr Sara Santos, computational biologist Dr Bissan Al-Lazikani, and climate scientist Dr Anna Jones.  This year's event is taking place at the Royal Institution, and you can find more information about it here.  In addition, have produced an excellent pack of resources for schools, pitched  for use with 11-14 year olds.

Gloucestershire STEM network have produced a worksheet providing an introduction to coding using free online resources that you can access here, and the University of Edinburgh has produced a series of Open Educational Resources to complement their wonderfully hectic ALD schedule.  Adalogical Ænigma produces sets a tricky logical problem every month that are definitely worth a look at; this one published by Alex Bellos in The Guardian is a good place to start.

There are of course an inordinate amount of simple ways that schools and teachers can mark ALD, not least through assemblies, publicising via the school website and/or social media, etc.  Marking ALD most effectively, I believe, however — and as ever — is through our interactions as teachers with students in our classrooms.  Below, for example, is a sheet I affixed to my Y7 class' page in their exercise books on last year's ALD, to invoke their curiosity and questioning — which it certainly did!

Just recently, my little girl succeeded in a making three-strand plait for the first time, alone, after an inordinate amount of time trying, i.e. in her own words, "6 years" (she's 6.)  We will now, of course, be exploring more plait and braid types (see SecureRF's 'Introduction to the mathematics of braids' and NRICH's 'Plaiting and Braiding').  Drenched in the joy at her achievement, with the plait clasped firmly in her hand like a theorem, she ran to me screaming, "Look, I figured it out by myself.  I'm an Ada!"  Marking ALD can make a difference, to someone, even it is just one person.  

Notes & (Select) Links:

Thursday, 5 October 2017

On Thinking Through

I was spiked to write this post by this tweet from @HenkReuling, mathematics teacher at Liemers College in the Netherlands (visit his web page here), where he posed the lovely little problem shown on the left in orange below — inspired by this problem, possibly, shown on the right in white below, one of many such problems created by @sansu_seijin.  We are asked to find the area of rectangle ABCD, given a semi-circle and a quarter-circle.

These geometric problems — and their like — demand the kind of interrogative depth from secondary (11-16) students that is necessary for, and that encourages, their mathematical maturation. They invoke many of the pedagogical conditions I contend we need to concern ourselves with if, indeed, we are concerned about supporting the development of our students' mathematical maturity.  As, indeed, we should be.  But it is important to recognise that such problems will only ever engage this (mathematical) maturation process if we as teachers, as custodians of the knowledge underpinning them, approach and guide our students' interactions with them carefully, sensitively and with intellectual, emotional and pedagogical humility.  In any other such philosophical hands, these problems have the potential to be more than mere missed opportunities, they can undermine the very effects that their design aims — and has the potential — to produce, posing a real and equally as potent threat to a student's mathematical maturation.  

Yes, 'we need to design learning that involves students making meaning,' as David Geurin succinctly puts it, rather than 'just accepting information' (see his blog post here), but here I am also arguing that we need to give as much thought to the design of our teaching to such problems — our interactions with students, how we guide students' own interactions with the problem, what we will say to them and what we will and will not tell them, how we will say what and at what point in the process we will or won't say it — just as much if not more than the thought we give to the design of the problems themselves, or, indeed, the thought we put into what students will do, as if the doing alone equates to learning.

To elaborate, simply doling problems like these out to the dareIsayit more able young mathematicians under our care, as I have seen done, in the name of 'extension', in the mistaken yet often held belief that we are feeding our students' intellectual appetite and that this alone is enough, is, in effect, differentiation's equivalent of a sneaky bag of crisps; the mars bar from the back of the cupboard.  Whilst exposure to and engagement with such multi-layered and thus relatively complex activities is unarguably crucial for mathematical maturation, we cannot expect all of our students to become mathematically mature (or, to put it more accurately, mathematically matur-er, matur-ing) simply from exposure to such problems alone.  We may not be able — or maybe, moreover, it would be silly and more than unproductive of us to try — to teach students how to be mathematically mature, directly, but we must teach our students the tenets of mathematical maturity, or, rather, we must model for them what it is to be mathematically mature, to behave in a mathematically mature manner when we are befuddled by the beautiful sixes and sevens of uncertainty.  

Such problems may on the face of it sate a student's appetite for intellectual challenge, initially, but the burgeoning maturation of the budding young mathematician will soon and at best stall — intellectually in nubibus  if s/he is left lingering, frozen in the fathoms of unfamiliarity, dizzied by the bends of conceptual deprivation.  And it is not to boot OK to simply show them how it's done afterwards we need to problematise with and alongside our students, to expose them to the habits of thought, emotional responses and ways of approaching an unfamiliar problem that in essence define what it is to be mathematically mature.

It is often through problems such as these that our younger, more able secondary students experience struggling with something, for the first time, gently, at the limits of their knowledge.  They are often some of the first developmental experiences young secondary students have of realising, and specifying moreover, that there is a gap in their knowledge and/or understanding that they need to fill — Vygotsky's 'Zone of Proximal development' for real, perhaps — an information-gap 'feeling' that fuels the developmentally important (to students' mathematical maturity) tenet of curiosity (cf. Lombrozo, 2017; Golman and Loewenstein, 2016; Noordewier and van Dijk, 2015; Loewenstein, 1994.)

   (See Seth Chaiklin's paper and David Didau's great post problematising Vygotsky's ZPD)

Such problems are also often the problems that spike some of the first emotional reactions young secondary students have to an appreciation that no matter how 'clever' they are, how 'clever' they feel they are, or are — or have been — told they are, they still need the support and guidance of someone who knows a bit more.  And in the hands of a skilled teacher, these problems can also be one of the first experiences young secondary students have where the surety that you just absolutely know something, that you just know it's right but can't explain it, is — to put it bluntly — no longer good enough.  (As Mortimer Adler put it, 'The person who says he knows what he thinks but cannot express it usually does not know what he thinks.')  In short, in the right hands these problems are, for the budding young mathematician, potently formative ones, but in the hands of the unthinking, such problems have the potential to do damage, more so perhaps than we may appreciate.

Our teaching in mathematics is thus not — or should not be — just about transmitting knowledge, developing skills and deepening understanding, it is also about modelling the fundamentally emotional responses to what Andrew Wiles describes as the 'state of being stuck' (in Ben Orlin's typically wonderful post) — and thus the approaches to problems that may faze us, that may knock us for six, that dare to tell us we're not as smart as we thought we were — problems, in short, that remind us who's boss.  (Listen to — and be inspired by — Richard Feynman talk about persistence and problem solving here.)

It is in this respect, as I hopefully more thoughtfully proselytised in a previous post, that I contend that as teachers of mathematics it is incontrovertibly our job to encourage, explicitly with thoughtfulness and planning, our students' mathematical maturation.  It is our job to help our students discover for themselves the joys and, indeed, frustrations of mathematical thinking.  And we do this by actively promoting, through our curriculum (our teaching) — or by exposing, in other words, our students to — a rich variety of problems and situations, such as the examples given, that demand:

  • An intellectually honest and creative approach to problem solving.
  • Perseverance and persistence in the face of difficulty.
  • Resilience and retrospection in the face of error, frustration and disappointment.
  • Fearlessness, insight and intuition in the face of complexity and unfamiliarity.

Such problems may (or may not) be 'easy' for us, the teachers posing them, but this is — or maybe should be — a pedagogical irrelevance.  We must remember that there is a lot going on in such problems for young minds and our job is to help students learn how to deal with this messiness.  Indeed, this is fundamental to the mathematical maturation process and should be our default position, or mindset, if you will.  If we are going to encourage depth in understanding, if we are bothered about supporting the development of our students' mathematical maturity, as the curators and custodians of their intellectual capital we have to have the mindset that students will find such problems messy, difficult.  When thinking about our teaching in the frame of our students mathematical maturation, therefore, when planning the use of such problems with our students we need, in a nutshell, to adopt the 'Feynman Technique', as described with typical clarity by the ever inspiring Farnam Street Blog, and as I have adapted here:

Firstly, we imagine we are teaching a young child, focusing on ideas (rather than algorithms or definitions), and using simple language*, because this young child will not have access to the complicated subject-specific vocabulary that we would otherwise — in all likelihood, and perhaps to the detriment of deep real learning — use.  We then review what we have imagined, identifying the gaps we encountered in our own knowledge (pedagogical as well as subject), where we get 'stuck', or where we can feel that our students will get stuck, and which therefore allows us to go back to the source of our knowledge — the first principles — where profound learning occurs and mathematical maturation develops.  Lastly, we organise our thoughts, simplifying things where necessary and formulating them into a naturally flowing narrative, a simple 'story that flows'.  (*Do read this also; yet another illuminating post from the Farnam Street Blog, exploring extracts from James Gleick's biography of Richard Feynman, focusing on Feynman's ideas for the teaching mathematics to children.)

Our planning, thus, is (or should be) about crafting how our students will interact with the problem, and how we will best be able to guide them towards creating their (full) solution, with mathematical — if you like, first principle — integrity.  We can do this, the Feynman Imagine stage, by simply visualising talking to a student about / through the problem, akin to visualising moves in a chess game: the moves we plan to make are a function of how we anticipate our opponent(s) will respond.  Or, perhaps more pertinently — as we do not expect a problem to respond to our line of attack, as it were, in anything other than a mathematically consistent way — it is perhaps more akin to visualising the moves we would make on a messed up Rubik's cube to get to the most efficient solution.

The diagram below, my scribbles on the commute home, where I am 'Thinking Through' — in Feynman style — the problem from the perspective of a Y8 class, is key to my personal process of planning.  It allows me to anticipate where my students will get 'stuck', appreciate why, and decide  how to go about what it is I can do about it.  Note that the 'key conceptual step' scribbled down identifies the point in the process where I believe my students will be most 'stuck', will be most tempted to ignore their uncertainty and keep going without evidence, or proof.  This is key* because again it is the point in the lesson where the deepest, most profound learning will take place, and where mathematical maturation will thus be most influenced.  (*Perhaps hinge is a better word.)

   Thinking Through the problem from my students' perspective (scribbles on the commute home)

Whilst I am not advocating the scripting of lessons, I am suggesting that a key part of the planning process when addressing and sharing such developmentally potent problems is this cognitive visualisation, scribbled down or not — playing through of the lesson in terms of how students will interact with the problem and how we should want them to.  At the same time, I imagine myself talking through the problem with my students (rarely noted down as I have below for the purposes of this post), walking the classroom whilst doing so, noting 'stuff' down, modelling ‘movement’ on the diagram, and anticipating students' responses (or lack of).  Questions with 'can', 'what', 'would', and 'is' stems are invariably followed up, where necessary (i.e. if students don't explain themselves fully), with 'how', 'why', 'why not', 'are you sure' and 'what if' stems.

Thinking Through the possible question stream.

We work through the problem ourselves, careful to cognitively empathise with our students' likely experiences and feelings, even if we devised the problem, and establish a pedagogical narrative.  We identify the point in the problem that learning hinges on, and where other segués in learning are most likely to most naturally occur.  We then interrogate / test these segués for possible gaps in students' knowledge and/or conceptual 'leaps' that may hinder or undermine learning, and think about what we may need to do to support children through them, whether breaking away from the problem is necessary to develop the knowledge and/or understanding that is as yet insecure or missing, but that is needed if students are to fully engage with the problem.  In short, we guide our students' interactions with the problem carefully, sensitively and with intellectual, emotional and pedagogical humility, modelling moreover that it is not just OK to be bedazzled by the shock of complexity and unfamiliarity, that it is not just OK to be stuck — it's actually quite exciting.

Notes & (Select) Links: