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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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Presentation ICME13

This is the presentation I gave at ICME-13:

OPPORTUNITY TO LEARN maths: A curriculum approach with timss 2011 data
Christian Bokhove
University of Southampton

Previous studies have shown that socioeconomic status (SES) and ‘opportunity to learn’ (OTL), which can be typified as ‘curriculum content covered’, are significant predictors of students’ mathematics achievement. Seeing OTL as curriculum variable, this paper explores multilevel models (students in classrooms in countries) and appropriate classroom (teacher) level variables to examine SES and OTL in relation to mathematics achievement in the 2011 Trends in International Mathematics and Science Study (TIMSS 2011), with OTL operationalised in several distinct ways.  Results suggest that the combination of SES and OTL explains a considerable amount of variance at the classroom and country level, but that this is not caused by country level OTL after accounting for SES.

Full paper, slides:

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Slides from researchEd maths and science

Presentation for researchED maths and science on June 11th 2016.

References at the end (might be some extra references from slides that were removed later on, this interesting 🙂

Interested in discussing, contact me at or on Twitter @cbokhove

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Presentatie researchED Amsterdam (Dutch)

Dit is de researchED presentatie die ik gaf op 30 Januari 2016 in Amsterdam. Enkele Engelstalige woorden zijn er in gelaten. Literatuur is aan het einde toegevoegd.

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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.

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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!)