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Seminar at Loughborough University

Dr. Christian Bokhove recently gave an invited seminar at Loughborough University:

Using technology to support mathematics education and research

Christian received his PhD in 2011 at Utrecht University and is lecturer at the University of Southampton. In this talk Christian will present a wide spectrum of research initiatives that all involve the use of technology to support mathematics education itself and research into mathematics education. It will cover (i) design principles for algebra software, with an emphasis on automated feedback, (ii) the evolution from fragmented technology to coherent digital books, (iii) the use of technology to measure and develop Mental Rotation Skills, and (iv) the use of computer science techniques to study the development of mathematics education policy.

The talk referenced several articles Dr. Bokhove has authored over the years, for example:

  • Bokhove, C., & Drijvers, P. (2012). Effects of a digital intervention on the development of algebraic expertise. Computers & Education, 58(1), 197-208. doi:10.1016/j.compedu.2011.08.010
  • Bokhove, C., (in press). Using technology for digital maths textbooks: More than the sum of the parts. International Journal for Technology in Mathematics Education.
  • Bokhove, C., & Redhead, E. (2017). Training mental rotation skills to improve spatial ability. Online proceedings of the BSRLM, 36(3)
  • Bokhove, C. (2016). Exploring classroom interaction with dynamic social network analysis. International Journal of Research & Method in Education, doi:10.1080/1743727X.2016.1192116
  • Bokhove, C., &Drijvers, P. (2010). Digital tools for algebra education: criteria and evaluation. International Journal of Computers for Mathematical Learning, 15(1), 45-62. Online first. doi:10.1007/s10758-010-9162-x
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Recent presentation: Mental Rotation Skills

I gave two paper presentations recently at the BSRLM day conference in Brighton. Abstracts and slides are below.

Bokhove, Christian* & Redhead, Ed
University of Southampton
Training mental rotation skills to improve spatial ability
Prior research indicates that spatial skills, for example in the form of Mental Rotation Skills (MRS), are a strong

predictor for mathematics achievement. Nevertheless, findings are mixed whether this is more the case for other

spatial tasks or, as others have stated, numerical and arithmetical performance. In addition, other studies have
shown that MRS can be trained and that they are a good predictor of another spatial skill: route learning and
wayfinding skills. This paper presentation explores these assumptions and reports of an experiment with 43
undergraduate psychology students from a Russell Group university in the south of England. Participants were
randomly assigned to two conditions. Both groups made pre-and post-tests on wayfinding in a maze. In-between
the intervention group trained with an MRS tool the first author designed in the MC-squared platform, which was
based on a standardized MRS task (Ganis & Kievit, 2015). The control group did filler tasks by completing crossword puzzles. Collective ly, the 43 students made 43×48=2064 assessment items for MRS, and 2×43=86 mazes. Although the treatment group showed a decrease in time needed to do the maze task, while the control group saw an increase, these changes were not significant. Limitations are discussed.
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Recent presentation: digital books for creativity

I gave two paper presentations recently at the BSRLM day conference in Brighton. Abstracts and slides are below.

Geraniou, Eirini*, Bokhove, Christian* & Mavrikis, Manolis
UCL Institute of Education, University of Southampton, UCL Knowledge Lab
Designing creative electronic books for mathematical creativity
There is potential and great value in developing digital resources, such as electronic books, and investigating their
impact on mathematical learning. Our focus is on electronic book resources, which we refer to as c-books, an
d are extended electronic books that include dynamic widgets and an authorable data analytics engine. They have been designed and developed as part of the M C Squared project (, which focuses on social
creativity in the design of digital media intended to enhance creativity in mathematical thinking (CMT).
Researchers collaborating with mathematics educators and school teachers form Communities of Interest and
Practice (COI and COP) that work together to creatively think and design c-book resources reflecting current
pedagogy for CMT in schools. We plan to present a number of these books and discuss how they were designed.
We will share our reflections from using one of the c-books for a school study and highlight its impact on students’
learning, but also how c-books could be integrate d in the mathematics classroom.
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As some may know I’ve had an interest in technology for maths for quite some time now. Because of this I am very aware of what the developments are. One of the latest offering from the wonderful online graphing calculator from Desmos, consists of their ‘activities’. Although every maths teacher should stay critical with regard to integrating ‘as is’ activities in their classrooms, I also think they should be aware of this fairly new feature. That is the reason I flagged it up during some ‘maths and technology’ sessions I ran for the maths PGCE. But always as critical consumers.

One of the latest offerings is the Marbleslides activities. I first read about it on Dan Meyer’s blog. There are several version of it, with linear functions, parabolas and more. As always the software is slick and there is no doubt the ‘marble effect’ is pretty neat. It reminds me of a combination of ‘Shooting balls‘ (linear functions, Freudenthal Institute, progressive tasks), ‘ Green globs‘ (functions through the globs) and also the gravity aspects of Cinderella. It has already been possible to author series of tasks with the latter widget. I first tried the ‘marbleslide-lines‘. The goals of the activity are:

The activity starts off with some instruction on the use of it. Many questions arise:

  1. Why do the marbles start at (0,7) ?
  2. Are the centers of the starts ‘points’? (this becomes important later on)
  3. Why several marbles? Why not one?
  4. Why do the marbles have gravity?
  5. How much gravity is it? 9.8 m/s^2?

Clicking launch will make the marbles fall and because they fall through the stars ‘success’ is indicated.


I am already thinking: so is the point to get through the points or the stars? And if gravity is at play, does that mean lines do not extend upwards? Any way, I continue to the second page, where I need to fix something. What is noticeable is 1. yes, the marbles again are at (0,7), 2. the line has a restricted domain, 3. the star to the right is ‘off line’. I’m not much more informed about the coordinates of the star, which leads me to assume they don’t really matter: it must be about collecting them. ‘Launching’ shows the marbles only picking up only two of the stars (for movie, see here).

desmos_p2The line has to be extended. The instruction is “change one number in the row below to fix the marble slide”. A couple of things here: what is there to fix? Is something really broken? The formula has a domain restriction. Do we really want to use the terminology of this domain being broken? I removed the domain restriction. This is ‘just’ a normal line. But it doesn’t give all the stars so ‘no success’. Restricted to x<12 no. For x<9 marbles shoot over. x<7 and there is success.

desmos_p2_2This is very much trial and error, partly caused by the gravity aspect.

On page 4 there is a more conventional approach: there is a line with a slope. The prior knowledge indicated in the notes mentions y=mx+b should be known: “Students who know a little something about domain, range, and slope-intercept form for lines (y=mx+b)”. I wonder why this terminology is not used then. Again the formulation of the task is “change one number in the row below to fix the marble slide”.

desmos_p3Because it’s relatively conventional I guess the slope is meant. But am I meant to guesstimate? Or use the coordinates? Does it matter? I first tried 1 (yes, I know that’s incorrect) and I just keep on adjusting.


0.5 seems ok, but 0.45 is ok as well, even 0.43. 0.56 does as well, but 0.57 misses a star because the line runs above it. May I adjust the intercept? I can, so this again promotes trial and error over thinking beforehand, in my opinion. In addition, it does not instill precision.

On page 5 the same thing but now for the intercept.

desmos_p5I’m still curious why the terminology of y=mx+b isn’t used. I guess -2 is expected as nicest fit but I can go as far as -2.7 to get ‘success’, yet -1.4 is ‘no success’. This could be seen by the teacher, of course (well, we can make any confusion into an interesting discussion, of course). It is interesting to see the marbles now start from higher up, by the way. The gravity question becomes more pertinent. How much gravity? And there is a bounce, surely the bounce is more if the gravity or hight is greater? Or not? Apart from the neat animation, what does it add?

Then on page 6 we go to stars that are not on one line (surely too quick?). There again are several answers, which in my opinion keeps on feeding the idea that points (but sure, they are stars) do not uniquely define a line.

desmos_p6From page 7 predictions are asked when numbers are changed. There still is no sign of terminology. It is a nice feature that the complete Desmos tool can then be used to check the answers. This is about functions, unlike the marbles section. Why is the domain still restricted, though? Throughout the tasks it seems as if domain and range are modified to suit the task, rather than a property of functions. Granted, a small attempt to address it is in page 11.

On page 13 the stars are back again. The first attempt with whole numbers is exactly right y=2x+4{x<5}. Some marbles fell to the right of the line though. Nevertheless, there was ‘success’. But there also was success on page 14 like this:

desmos_p14_2From page 15 there are challenges. The instruction says to “Try to complete them with ONLY linear equations and as FEW linear equations as possible.” These are a lot of fun, but I struggle to see the link to slope-intercept form y=mx+b. It is not mentioned explicitly. There is no real attempt to link to the terminology. I fear it will remain a fun activity with a lot of creative trial and error.

desmos_p17I’ve also looked at the parabolas activity. The same features are apparent here: functions are collections of points (rather: stars) and functions have to be found that go through them. The assertion is that transformations of graphs are somewhat addressed concurrently, but the trial and error aspect makes me doubt this. It also deters from general properties of graphs like roots, symmetry, minimums, maximums. I can see a role for playful estimates but in my opinion they must be anchored in proper terminology, precision and properties of graphs. Furthermore, I was inclined to sometimes just use lines. There was no feedback as how this was not permitted. One could even say a line also is polynomial, so why wouldn’t I. The trial-and-error nature might further incentivise these creative solutions. Great, of course, if you know transformations already but not if the activities are meant to strengthen skills and understanding (did I ever say they go hand in hand? 🙂

Some of these aspects might be mitigated by the editing feature that will be released soon, but surely not all answers to fundamental but friendly critique will be “do it yourself”? Another nice feature of course, also in other software, is that you can see student work. Yet I feel with some of these fundamental issues not properly addressed, misconceptions might arise. I think that the marble animation is at risk of obfuscating what the tasks should be about. It might lead to more engagement (fun!) but if it does not lead to learning or even might lead to misconceptions, is that helpful? Firstly, I think the scaffolding of tasks should be more extensive with a clear link to maths content. Secondly, I would reconsider the confusion between ‘points on a line’ and ‘stars to collect’. I hope Desmos can iron out some of these issues, because one thing is sure: the falling marble effect remains a joy to behold. However, pedagogically, I think as it stands it needs to be developed further.

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Games in maths education

This is a translation of a review that appeared a while back in Dutch in the journal of the Mathematical Society (KWG) in the Netherlands. I wasn’t able to always check the original English wording in the book.

Computer games for Maths

Christian Bokhove, University of Southampton, United Kingdom

51iyzu1DTlL._SX326_BO1,204,203,200_Recently, Keith Devlin (Stanford University), known of his newsletter Devlin’s Angle and popularisation of maths, released a computer game (app for the iPad) with his company Innertubegames called Wuzzit Trouble ( The game purports to, without actually calling them that, address linear Diophantine equations and build on principles from Devlin’s book on computer games and mathematics (Devlin, 2011) in which Devlin explains why computer games are an ‘ideal’ medium for teaching maths in secondary education. In twelve chapters the book discusses topics like street maths in Brasil, mathematical thinking, computer games, how these could contribute to the learning of maths, and concludes with some recommendations for successful educational computer games. The book has two aims: 1. To start a discussion in the world of maths education about the potential for games in education. 2. To convince the reader that well designed games will play an important role in our future maths education, especially in secondary education. In my opinion, Devlin succeeds in the first aim simply by writing a book about the topic. The second aim is less successful.

Firstly, Devlin uses a somewhat unclear definition of ‘mathematical thinking’.: at first it’s ‘simplifying’, then ‘what a mathematician does’, and then something else yet again. Devlin remains quite tentative in his claims and undermines some of his initial statements later on in the book. Although this is appropriate it doesweaken some of the arguments. The book subsequently feels like a set of disjointed claims that mainly serve to support the main claim of the book: computer games matter. A second point I noted is that the book seems very much aimed the US. The book describes many challenges in US education that, in my view, might be less relevant for Europe. The US emphasis also might explain the extensive use of superlatives like an ‘ideal medium’. With these one would expect a good support of claims with evidence. This is not always the case, for example when Devlin claims that “to young players who have grown up in era of multimedia multitasking, this is no problem at all” (p. 141) or  “In fact, technology has now rendered obsolete much of what teachers used to do” (p. 181). Devlin’s experiences with World of Warcraft are interesting but anecdotical and one-sided, as there are many more types of games. It also shows that the world of games changes quickly, a disadvantage of a paper book from 2011.

Devlin has written an original, but not very evidenced, book on a topic that will become more and more relevant over time. As avid gamer myself I can see how computer games have conquered the world. It would be great if mathematics could tap into a fraction of the motivation, resources and concentration it might offer. It’s clear to me this can only happen with careful and rigorous research.

Devlin, Keith. (2011). Mathematics Education for a New Era: Video Games as a Medium for Learning.

ICT Math Education Tools

Wuzzit trouble for Android

Since begin November there is an Android version of Wuzzit trouble in the Play store. I assume it’s the same as the iOS one. A blog post about the game is here.


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Graphical calculator vs Online

Inspired by this blogpost: The CASIO graphical calculator FX-9860G SD emulator, still in use is some classrooms in the Netherlands, on the left for y=sin(1/x), an online tool on the right. Both resized to a width of 263 px, ratios kept the same.



(Of course TI would argue that you therefore need the TI-nspire CX full color with a whopping 320 by 240 pixels, and other features comparable to an old Nokia phone. But hey, that’s just me, it’s all about the pedagogy!)


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Maths iOS app: Wuzzit Trouble

wuzzit1NPR Math Guy Keith Devlin, from Stanford, recently followed up his promise from his 2011 book “Mathematics Education for a New Era”  and released a free maths game Wuzzit Trouble with his company Innertubegames. Of course, let me start by applauding the fact that someone follows up his theoretical ideas by practicing what he preaches, and releasing something as practical as a game. Naturally, I would have hoped it was released multi-platform, or even better, as a web application, but that could easily be construed as nitpicking. Apparently, there will be an Android version as well. I hope so, because I would be very weary promoting a game that only works on one brand.  After having read some positive reviews I gave the game a shot on the iPad3.

...consist of levels...
…consist of levels…

The game follows the structure we know from many apps, like Angry birds: you have levels (75 in total) that require you to solve a puzzle. The better you solve the puzzle, in this case the minimum number of moves, the more stars you earn. The story is about Wuzzits, cute characters that have to be saved. This ‘story’  is not really a story, of course, certainly not the immersive ones we know from blockbuster games like World of Warcraft or Bioshock Infinite. Maybe this isn’t really a fair comparison as these games are on a different level altogether, but that’s what you get as you mention World of Warcraft as a good practice (in the 2011 book).  In this respect I don’t think one of the statements from the 2011 book has been met: it shouldn’t really feel as if you’re doing maths. It is pretty clear that it’s about maths. No problem, I think, because maths and puzzles could be fun, anyway. The levels themselves consist of target numbers that need to be constructed by turning a cog, in the case below with 5. The cog can be turned to the left and to the right. The keys have to be collected by making the numbers by turning the cog. The stars are bonus numbers.

wuzzit9The interface is quite intuitive, and one really only needs a one page help page to get going. This is great, but also the case because the app has a limited scope: integer partitions, if there’s not a lot you can do then you don’t need a lot of instruction. In some cases the lack of maths notation has been applauded. I’m not sure about this; if players see this a maths game -and I think they will- why not introduce or use maths notation as well? I’m thinking of Dragonbox, which uses symbols but later connects them explicitly to maths syntax as well. Based on the number of moves someone needed to wuzzit5collect all the keys, the game awards you with keys, which means -in the narrative- you have rescued some Wuzzits. I didn’t really see how a lot of children would be motivated to find ‘the partition with least moves’  to win three stars. I think they’d just rather progress. I also wonder how this is different, or more interesting, than just timing exercises.

Frankly, after the positive reviews and an ambitious book, I had expected a bit more from the app; especially the scope could have been wider. At the NCETM conference I tried out Beluga learning (iOS only unfortunately) and found it much more enticing, and also Dragonbox (multiplatform, yay!).Sure, the rationale behind this game seems more grounded in a combination of procedural fluency and conceptual understanding, but at the moment just is too limited. I certainly wouldn’t yet say “Video Games Are The Perfect Way To Teach Math, Says Stanford Mathematician”. Innertubegames describes the game/app as a ‘framework’  that could easily consist of more activities, which will probably be called labs. This shows a lot of potential. I hope the makers can do their ambitions justice. Making good games is a time- and money-intensive exercise. You can have the framework but if there isn’t any money or you need a lot of time, then it would defeat the purpose of framework. As if you’ve made a search engine for five items. This will especially be a challenge if you want to put extra (mathematical) thought in the apps/games. I hope they will succeed in making some more and wish them good luck.

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Software for geometric proofs in secondary schools

A week ago I attended a seminar at the School of Education with visitors from Japan. One of the visitors was Professor Mikio Miyazaki. He showcased some of his work on a flowchart tool for (geometric) proofs at Schoolmath. I loved it and would love to see this integrated as widgets in the Digital Mathematical Environment, for example. I will provide an overview in some screenshots.

1. This is the entry screen. The flowchart tool is part of a larger environment that stores student information.


2. The materials are presented in a nice overview with levels. The stars do NOT denote difficulty but in how many ways you can actually proof the theorem that is presented.


3. I will choose the section on congruency. Students are presented with a geometry task and are asked to prove the theorem presented (I did not yet manage to find out what the difference between elementary mode and advanced mode is). In this particular example there are four stars, so four possible ways to prove it with the help of congruency. Students have to fill in the flowchart by choosing a strategy/action and providing angles and sides. I love the fact that I can just drag and drop angles and sides to the answer boxes and they will appear there.



4. Having filled in the flowchart the answer can be checked. One of the four stars is coloured yellow.


5. Wrong answers are provided with feedback and an indication where the mistake is:


6. Another final example:


It was interesting to hear that this project faces a challenge that many educational tools face: converting flash and java tools to HTML5 format. I’m still quite disappointed that the Apples, Adobes, Googles and Oracles of the world did not manage to provide a transition period.

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More on Openbadges

I seem to get involved into many #openbadges discussions on Twitter lately. A while back I wrote on my blog about this topic. I think it was quite well-balanced, acknowledging the positive points but also having soOpen badges website

Open badges website

me questions. I sent the mail to one of the leads in openbadges as well and got a useful reply, albeit referring to reactions to earlier critical posts here and here. Both sources raise similar points, which is comforting but doesn’t get me closer to a possible answer. The rest could have been for the Google Group. Well, I didn’t go there, as I had just written an extensive post. The final line in the reply (see below the first post) was : “intrinsic/extrinsic is *itself* merely a construct, and the recognition of which badges are valuable is an emergent property of the ecosystem.” Later on I had to mail. Well, I did that, so I think that ended with ‘We agree to disagree’ (well, I agreed ;-)).

The discussion came to the front again when I was included in this tweet:

This point was one of the points raised on Twitter and also in the aforementioned blogpost. I never really got an answer. Retracing the discussion on twitter it seemed to have started with a a link to a post called “Let’s ban the sticker, stamp and star” and then a comment that OpenBadges were much different because they were ‘intrinsic’ and stickers ‘extrinsic’. I don’t agree, both have both sides, if we can even see it that black and white. Badges are issued ( Stickers are issued. Badges are earned, stickers are earned.  My point is that I don’t agree with the fact they are presented as a lot different. Of course the scale differs. And it’s online in the cloud, so those are all positive points. But different with regard to motivation, I don’t think so.Badges can be another tool in the vocabulary of teachers and students, but like any tool they can be used in good and bad ways. Potential? Sure. But stickers had potential too! 😉ob

The point on having 1000s of them and ‘control’ over them came up as well; it actually was the topic of the tweet ‘that started it all’. The answer would be ‘metadata’. Well, I wasn’t talking about how you are going to find the badge(s) you want, I was talking about the way the value of badges is determined.

(Note: it was pointed out that metadata is more than just information on location, but also a pointer to criteria and evidence:

Fair enough. But that wasn’t the point, the point was that metadata -in my opinion- will not ‘solve’ the institutional issue. How can we evaluate these criteria and evidence between badges? What if there are 1001 Algebra 101 badges from different institutions? Or someone makes his/her own badge? It’s nice that an individual has an overview of his/her badges, but how can this be useful in the workplace? I worry that it will be just as hard and difficult as before with CV’s, but looking slightly different. Suggesting that OpenBadges will change this is wishful thinking.)

It also has been suggested that that too is  “the recognition of which badges are valuable is an emergent property of the ecosystem.”. To me, that sounds like market thinking, but worded differently. Just like ‘the market’ it will depend on the user how much he/she values the badge. Just like the fact that this is pretty hard to do when it comes to cars, houses or insurances, this -in my opinion- will be even harder for educational goals. Does this mean I won’t have anything to do with them? No. I’ve added a Justin Bieber badge to my developer Blog, worked in Moodle with them (in combination with SCORM) and even added them as an experiment to a forthcoming European project (that I will hopefully get, not sure yet). I will keep on thinking about this, hopefully encountering more valid viewpoints than “do your homework” and “shakes head”.