I was asked to provide a nativity scene for the local church. Could I do it in origami… It goes on display, with some other crèches from around the world, on 2nd December, but you get to have a sneak preview!
I was asked to provide a nativity scene for the local church. Could I do it in origami… It goes on display, with some other crèches from around the world, on 2nd December, but you get to have a sneak preview!
So, you’re walking around the Sahara Desert, as one does from time to time, and you come across someone who’s very dehydrated. What do you do?
You would, of course, expect a trained medic to know the best response for that situation, but others may also have well-informed solutions. The point? Water is already an integral part of the human body and without the water this particular body would not have lived in the first place. Although it may be very dehydrated, this human being at one time had enough water to be mobile and do more than barely hang on to life. What is needed is to reintroduce more water to revive them…
Now, let us replace the dehydrated body with the STE of STEM… guess what the water is!
It seems totally ludicrous, to me, to keep seeing articles, blog posts, and so on, which continue to mention “integrating” mathematics with STEM. What does the ‘M’ mean then? Mathematics is already in the other three areas and, truth be told, probably was the underlying foundation on which science, engineering and technology were built. You cannot integrate something which is already there, and we are giving a false impression about the role of mathematics in STEM if we continue to talk about it this way.
Mathematics and language (whether English or otherwise) can stand alone as a pair quite happily by themselves. Even the most primitive societies counted things, solved problems and communicated ideas with each other. The issue STEM faces is not so much about integrating mathematics, as about integrating the mathematics teachers. The isolation of subjects creates certain inter-dependencies, more so with some subjects than others. It is unfair to expect mathematics teachers to teach the mathematics for your subject in their lessons… Similarly, mathematics teachers need to take more responsibility for connecting what they are teaching to as many other things outside mathematics as they can.
Let’s stop talking about integrating mathematics with anything from now on, because it’s already there! Why not talk instead about making a more integrated use of mathematics teachers?
It seems strange to look back on one year of blogging. The posts I thought should be read more, weren’t. The ones I thought would get most visits, didn’t. The things I thought would provoke most comment, haven’t. I’m not sure whether or not I have hit on the formula or balance of contents either. Although this site was my first blog site, I also decided to set up a separate one for my creative writing, and I am also considering a third to curate presentations I have made at various TeachMeets and other educational events. I haven’t written as many posts about music or English language teaching, as I had originally anticipated, neither has my posting been as frequent nor as regular as I might have liked. I also wondered whether or not I should have a blogiversary as some people do and write something celebratory or reflective. That would have been yesterday, though, so you can see I sort of decided against it – always the rebel!
So what’s in store for the second year here? I’m not making any predictions, but expect to see more about music, education reform, and the occasional mathematical titbit – and a lot more paper-based stuff! If you haven’t had a poke about in some of my earlier posts, now might be a good time for a visit!
Jag gjorde en origami session med ett universitet matematik klubb. En av utmaningarna jag ger är att vika en liksidig triangel från en kvadrat. En student använde sin iPhone för att titta på hur man gör det på YouTube. Jag frågade honom varför metoden fungerade. Han hade ingen aning. Han studerade matematik på universitetet och han kunde inte utnyttja den grundläggande geometrin av en 30, 60, 90 triangel.
Eftersom mängden information ökar och tillgången till denna information blir lättare i framtiden, blir det allt viktigare för de studerande att ansluta dessa bitar av information på ett meningsfullt sätt. I matematik, åtminstone, måste vi ansluta den verkliga världen av studenten, som är mestadels online, med till synes virtuella världen av matematiska begrepp. Jag skulle vilja föreslå en möjlig metod.
Först skulle jag vilja att ni se en kort video med bakgrundsmusik och ingen berättarröst. Som du tittar på, vilka frågor eller funderingar har du?
Vid denna punkt skulle jag brukar be dig att dela frågor och tankar med varandra. Tyvärr finns det ingen tid att göra detta nu, dock! En uppenbar fråga jag kan ställa är kan du återskapa rad veck du bara tittade på videon?
Det finns många viktiga matematiska begrepp att utforska i detta enkla exempel. Jag skulle till exempel be dig rita mönstret i vecken exakt med hjälp av ett online-verktyg, såsom GeoGebra.
Vikning rektangeln i mitten och dra 45 graders linjen kan göras med koordinater. Men att göra några veck i en bit papper är matematiskt motsvarar en reflektion, med veck som spegel linjen. Så jag kan också dra halvvägs veck genom att hitta mittpunktsnormalen av två långsidor.
Och så vidare och så vidare och så vidare. Jag slutar där innan matematikerna blir alltför upprörd och resten av er har en panikattack!
Neurovetenskap berättar att en del av lärandet är att kunna relatera ny information till tidigare erfarenheter. Om eleverna bombardera deras sinnen ständigt med ny information, kommer det att bli vårt jobb, som lärare i framtiden se till att de också har tidigare erfarenheter för att hjälpa dem bearbeta den. Om jag får felcitera John F Kennedy:
Fråga inte vad tekniken du bör ta till lektionen, fråga vad som bör teknik tillföra min lektion.
Hur som helst, njut av resten av kvällen!
Du får gärna använda videon med dina elever. Om du gör, låt mig veta hur det fungerar!
Followers of #mathchat and #scichat may be wondering what this #anyqs hashtag is that people seem to be tweeting about. Dan Meyer’s at it again! For anyone who hasn’t heard of Dan (), here’s his much-viewed TEDxNYED talk. He started off WCYDWT and the #WCYDWT hashtag, which he vigorously tries to defend as a brand. That is an abbreviation of “What can you do with this?” – the prompting of “How much, how big, how long, how many…?” types of questions using a photograph or a video.
The #anyqs hashtag, which Dan launched on 5th May 2011, seems to me, at least, to be a refinement of the #WCYDWT idea. Dan’s follow-up post Dissents Of The Day: Danielson, Pickford, Scammell says:
The point of the #anyqs challenge is to evoke a perplexing situation so skillfully that the majority of your students will wonder the same question (whatever that is) and the rest of the class won’t find that question unnatural or uninteresting, even if it wasn’t the first question that struck them.
The ‘rules’ are much tighter than #WCYDWT – one picture or one video with a duration under 1 minute, viewers should respond with the first (mathematical) question that springs to mind. A few others have ‘jumped on the bandwagon’, I have set up my own pages to collect questions for myself, but this raises another question: “What do I do with all of the questions once I have collected them?”
It seems to me there are two main approaches you could take, once you have first tested the visual prompt outside your class:
The first approach is probably a return to the confusion or flapping about, the “excessive cognitive load resulting from a unfocused problem space” Dan refers to in his response to a question about the difference between #WCYDWT and #anyqs (comment 14). I suspect, judging by the comments, advice and friendly prods from Dan, that his intent is to adopt the second approach.
If (when!) I get approach two working myself, it could/should lead to the situation where my own students are making #anyqs challenges for each other, with me doing the friendly prodding and giving advice. If students can get caught up in designing these types of prompt, perhaps they’ll increase their awareness of the mathematics around them, and forget that maths lessons aren’t supposed to be fun… And if they don’t get to the stage where they can create their own challenges, maybe they’ll have fun doing mine… (ooh, did I just say fun again?)
I am assuming you are here because someone knows you teach mathematics and has got you onto Twitter as part of your professional development. Or maybe you are a student or parent who is interested in issues related to mathematics and someone told you about #mathchat.
I have still to reach my one-year anniversary on Twitter, but probably a large proportion of those following me know me through #mathchat. So what is #mathchat? Basically, #mathchat is one of the many hashtags used by educators to hold discussions on topics of immediate relevance to them. There is a whole raft of educational chats and hashtags on Twitter, but #mathchat concerns itself mostly with issues surrounding the teaching and learning of mathematics from beginning to end, K-death (for the North Americans!).
I am usually credited with founding #mathchat, but it’s probably fairer to say that I revived it in it’s current form. The revival started because of a comment made by a middle school teacher who said he wished there was something for mathematics teachers which was like #edchat. I got the ball rolling, we had a TwtPoll to decide on day and time, and I decided that the same topic should be discussed on a different day at a different time to allow more input or opportunity for those on both sides of the Atlantic to take part. I’d like to see a third discussion in the Pacific region too – any takers in Aus/NZ?
What is the purpose of #mathchat? This excerpt from the #mathchat wiki will give you an idea:
The aim is to provide a forum for anyone involved with Mathematics – whether as an educator, a student or an interested party – to discuss and share ideas about issues affecting them at this particular time.
Like most of the other educational hashtags on Twitter, the people who contribute to #mathchat are passionate about teaching mathematics and helping others come to terms with what is involved in learning mathematics. What is really special for me, though, is that not long after its revival, #mathchat expanded beyond the ‘topic of the week’ to become a place where teachers and students can ask questions at almost any time of the day or night and get helpful responses, usually within 5 or 10 minutes… If you want to know what #mathchat means, just send out a tweet “What does #mathchat mean to you?” and wait for the answers! And if you have any further questions or need help, feel free to tweet me. I’ll look forward to tweeting you some time!
As I said in the comments under the photo, I chose a quote from my own post:
In thinking back to my own school days, not all of the memories I have are about the exciting times. Some of them are quite bad and, in many ways, I’d like to forget them. What I do not remember so clearly are the periods in-between. There must have been (long) periods when I was just learning (or being educated, if you prefer) because, let’s face it, I wouldn’t be where I am now without it.
The change in focus in the campaign, from people pouring out thoughts inside a 500- word limit towards an image and a reflection on a prior post, is an important one. As a composer, I find my creativity comes in spurts. I can ‘knock out’ an arrangement in a couple of hours, but making something new, usually from scratch, needs more time and greater contrasts in pacing. I am trying to write a poem a day during April, when the US celebrates National Poetry Month. I am managing to keep up so far and, if successful, I may try it again in October for the UK version.
It requires a lot of self-discipline to publish something which I feel is worthy of my own standards… let alone what other poets may think. Churning something out on a daily basis is part of polishing one’s art, though, and honing that self-discipline. Children nowadays are bombarded with fast-moving images, sounds and experiences. It would be easy to say that teaching methods should match this type of MTV-style – juicy sound-bites and factoids that will be needed for the exam, and so on. What is going to be more challenging, though, is to ensure that there are periods of quiet reflection built into that. Excitement and stimulation are vital for engagement, but is life one big adventure or a series of little ones each preparing for the next?
Yes, I have some bad memories from school. Yes, I have some memories of exciting adventures, too. Maybe it was the times between when the learning happened though…
OK, before everyone starts screaming about sexism, I used to go to my father for help with my maths homework, and this post is a reflection on the problem (grand)parents of my generation and/or younger are facing, or going to face, as the use of digital technology increases. Never mind about Gen-X or Gen-Y, the current cohort of students is definitely going to (have to) be e-Gen.
I have been ‘pushed’ towards this post because a number of different articles, talks and reflections have crystallized into a form which I feel satisfactorily explains or, at least, exemplifies what I think has always been a fundamental approach towards my own methodology in teaching mathematics, or which have helped me to clarify my own thoughts on it. It took another change in technology to bring it to the fore (or should that be four!). Be prepared for a bit of a ramble, but I feel I have to write this down because… I just do that’s why!
First was an article in the BBC News magazine, September 2010, by Rob Eastaway (@robeastaway on Twitter, if you aren’t following him, and are interested in maths education, why not?) called : “Why parents can’t do maths today.” Basically, Rob explained that most of the problems you are having now, started with the National Numeracy Strategy (1999) which was aimed at primary education (in England). Of course, in my case, it was all about the new maths back in the ’70s…
Second was an article/broadcast by Keith Devlin (@nprmathguy now he could at least have used the ‘s’, being British and all…), last week (as of writing) on NPR in the US, called: “The Way You Learned Math Is So Old School.” Now, I’m not going to make any comments about the United States being a bit behind here – oops, just did! – but the content is essentially the same as Rob Eastaway was discussing in his item for the BBC.
Sir Ken Robinson (@SirKenRobinson) gave a talk to the RSA last year (2010), which I had a little rant about, and then I watched him again in a live presentation, streamed by Learning without Frontiers (edited highlights here), where he was again talking about creativity. I forgive him his harping on about maths, because he didn’t/doesn’t get it and admits as much, but in the interim I also watched John Cleese in a presentation he gave at the Creativity World Forum in Flanders in 2008. The main thing I took from John Cleese was that creativity can be taught, and I have long believed this myself. Interestingly, he also said “To know how good you are at something requires the same skills as to be good at that thing,” and “Most people who have no idea what they’re doing have absolutely no idea that they have no idea what they’re doing.” This takes some thinking about! Essentially, this means if you’re hopeless at maths, you lack exactly the skills you need in maths to know that you’re hopeless at it… However, he goes on to say “Teachers, who may not realize that they are not themselves creative, may not value creativity, even if they can recognize it.” (Yes! Yes! Yes!)
The Wolfram brothers. Yes, there are two. Let’s not confuse Stephen (who designed Mathematica, the Wolfram Alpha search engine, and wrote A New Kind of Science) with his (much) younger brother Conrad. [Oh, they’re both British by the way, just thought I’d mention that to the morons in the UK who are not investing in ICT research or education in computing technology… that would be the government, wouldn’t it?]. Now, Stephen clearly has delusions of grandeur, based on his idea that everything can be computed, but I’ll let him off, because he’s clearly a physicist – even though he mentioned Leibniz and not Newton! I still remain to be convinced that a computer/application/whatever can give me an answer to a somewhat vague question, though. “Single biggest idea of the Century,” hmm, I’m definitely reserving judgment on that one! By the way, I love what Mathematica and Wolfram Alpha provide…
I’m not sure that Stephen’s TED talk got blogged about, or retweeted, as much as his brother’s more recent one – Teaching kids real math with computers. Now, Conrad, we call it maths in the UK, and you should know that, particularly because you were giving your talk in Oxford… (Argh! It’s maths… maths… write it out 1,000 times… ) I don’t agree that mathematics has changed so much just or only because of computers, although I would agree that computers have speeded up the process. Learning Ancient Greek and Latin gave me excellent foundations in expanding my vocabulary in English, as well as helping with my other language acquisition, and I don’t remember ever being taught Ancient Greek in my maths class. Not using paper in mathematics, sorry, totally disagree… I am switching off now. Where is the connection between real world (origami) and the algebra/geometry to represent it on the computer? How are computers helping with visualization of 3D or understanding geometry and topology better? Yes, I agree with the programming, and almost always recommend the Project Euler site for that, but that is really only the algebraic part of mathematics. Teachers have to be able to understand (all or) most of the concepts before they can start throwing computers at students, and hoping they will ‘get it’. Personally, I got very little that was new from this talk because it’s something I’ve being doing (trying to do) already… Don Cohen (aka the Mathman) has also been doing things like this for many years, from which he drew up his calculus map.
Key points, as I see or expand on them, from the Wolfram brothers’ talks:
I posted a little about my programming background when I introduced my first online manipulative. This post is essentially a follow-up to that one. I don’t think that anyone would disagree that the introduction of new technologies, applications and, in particular, computational power are increasing exponentially. It is ridiculous to ask human beings to perform tasks which can be carried out more efficiently and effectively by machines. We are currently entering a transitional phase between what I see as a tech-available and a tech-reliant society. Just considering what has happened during my own lifetime in mathematics, the impact is going to be very much greater than being able to use a calculator, and then being required to use one, in examinations.
When I was taught mathematics, algorithmic methods were taught and used because they helped to make the calculations more efficient. There were books of tables of sines, cosines, logarithms and so on. Part of my O-level required the use of a slide rule. By the time I came to take my A-levels, calculator use was allowed, but only certain types – none of your programmable calculators if you please! Move on to 1995, when I did my PGCE, and there’s a debate brewing about whether or not graphing calculators should be allowed, let alone required. The introduction of this type of technology allows the user to go much farther in the areas of mathematics which they can explore: wherein lies the problem…
Allowing or encouraging the use of new technology is great, and I am all in favour of it. However, the curricula, examinations, qualifications, systems of training and so on which are in place have had a good deal of time and money invested in them, and they are not going to be changed overnight. I think it is reasonably accurate to say that the effects of a new approach to content or methodology are going to take at least five, probably more like ten, years to start to make themselves known.
So that is why, when you were at school ten or more years ago, the mathematics (and let’s be fair many other subjects) you were taught are not and will not be the same as your children are being or going to be taught. The introduction of powerful calculating machines means that mathematics can now be taught or explored at deeper levels at earlier ages, and in ways that children understand. This also means there needs to be a paradigmatic shift in thinking to accommodate the freedom from the slavery of tedious calculation. This is not a new idea, Wilhelm Schickard wrote to Johannes Kepler in 1623 about his progress on developing a machine to completely automate the tedious calculations required to do astronomy at that time, despite the fact that John Napier had only recently discovered logarithms, which speeded up the process of multiplying and dividing a lot!
The e-Gen, as I called them earlier, are going to be bombarded with huge amounts of information which is easily accessible. More time will need to be invested on improvement of critical thinking skills, assessing and filtering the information, making connections and models in the virtual world which help us to understand the real world better… For mathematics, this means being able to understand number systems, thinking in a more algebraic way for programming, thinking more geometrically for design, thinking more probabilistically to handle data, and so on. That is why you probably won’t be able to help your child or grandchild or nieces and nephews with their maths, and it’s probably just as well you can’t! However, you can help by asking them to try and explain what they’re doing and why it works – if you have the patience to do so – since the latest changes to the curriculum are asking students to communicate more and show their understanding of the subject. Now, if you don’t mind, I’m going back to read my book of tables and play with my slide-rule!
Not long after starting to use Twitter and starting blogging, I was asked to summarize a discussion on #edchat. Later, I reflected on a similar issue after three months of running #mathchat. Last night (22nd February 2011), #edchat was again rehashing the same old ground about educational reform. My views have not changed that much since either of my earlier posts.
This is a reflection on a slightly different aspect of education, as part of the purpos/ed #500words series.
The question posed: “What is the purpose of education?” could as easily be rephrased: “What is education?” Are we discussing the system of schooling by government, ways of raising children as ‘useful’ members of society, deciding what should be taught, or something completely different? Behind this, a voice keeps asking: “What is the purpose of school?” That is the question I’m responding to here, since it is easy to forget schools in this whole debate. Some of the time I spent in school I remember as an exciting adventure. Some memories are less positive, and not all the well-meant advice was helpful, based as it was on examination results.
First numbers, based on England, since education systems vary so much, and I am, at heart, a statistician!
Child enters school at 4 and leaves at 16.
School year is 195 days.
Child spends 6 hours per day in school.
There have been four leap years.
Hours alive = 365 * 16 * 24 + 4 * 24 = 140256
Hours in school = 195 * 12 * 6 = 14040
By 16, 10% of a student’s life has been spent in school. The impact whilst there, though, is very much greater and, in many cases, can be damaging to their future because of perceived failure to learn (based almost solely on exam results).
Malcolm Gladwell suggested that 10,000 hours of practice was the key to success in any field. Daniel Coyle suggests that the 10,000 hours needs to be directed towards the formation of myelin in three specific ways: training, motivation and coaching.
Would the current 14,000 hours spent in English schools not be better directed to nurturing and helping children to find something they are good at, and have a passion for, so that when they leave school at the age of 16 they feel successful?
We’d have to abandon fixed curricula and subjects, of course, with students working in dynamic groups, teaching and helping each other, guided by the teachers in the school. Problems with maths? Go to Mr Y. History? Ms A. French? Mme J. Want to learn an instrument, play a sport, start a school farm…? If we must have accredited qualifications at age 16, and being pragmatic we need to, we could take the final two years (only around 4,000 hours after all) to help students focus on their strengths and build up a portfolio of evidence of actual learning and progress: evidence which allows them to leave school saying “I can…” and show concrete examples of their work.
Not every child is going to be a genius, but after 10,000+ hours in school they ought to be leaving as successful learners of something…
Hej alla. Om du inte vet om Twitter, jag ska ge dig en kort inledning till några av de funktioner och hur den kan hjälpa dig.
Först måste du registrera dig för ett Twitter-konto, om du inte redan har ett.
När du registrerar dig för Twitter, är de språk du kan använda för menyerna begränsat till italienska, spanska, engelska, franska, tyska och japanska. Men kan du skicka en tweet på alla språk som du kan skriva.
Det finns två hemsidor på svenska som du ska ha nytta av. Den första är ett blogginlägg som skrevs för snart två år sedan. Den andra är en hemsida som ger tips och nyheter om Twitter. Om du är osäker om hur du använder Twitter, bör dessa webbsidor bidra till att klargöra några av fördelarna.
Det är viktigt att sätta några uppgifter i Bio avsnitt och att sätta en bild på din profil. Du behöver inte sätta ett fotografi av dig själv, men du bör lägga något som kommer att hjälpa andra att känna igen dina tweets. Du bör också ge en URL där andra kan hitta mer om dig, till exempel din hemsida eller blogg. Jag har en speciell sida på min blogg för Twitter-användare. Jag använder också ett South Park tecknad som min bild …
När du har skapat ett Twitter-konto, rekommenderar jag att du använder ett program för att organisera dina tweets. Jag brukar använda TweetDeck och ibland TweetGrid. Detta hjälper dig att följa hashtags och lägga märke till när folk har skickat tweets till dig eller om dig.
För de flesta lärare kommer de mest användbara funktion i Twitter förmodligen utbildning hashtags, till exempel #edchat ofta används för allmän utbildning, lärare i Storbritannien använder ofta #ukedchat, måttlig jag en diskussion två gånger i veckan för lärare i matematik med #mathchat och så vidare. Du kan hitta en lista av de flesta utbildning hashtags på en sida som inrättats av Jerry Blumengarten och det finns en annan sida som listar det vanliga utbildning diskussioner eller chattar. Den vanliga chattar vanligtvis har en egen hemsida med arkiv och detaljer om hur man deltar. Till exempel kan det #mathchat hemsida finns på Wikispaces. Den # mathchat sidan har en widget som översätter till svenska, men det är inte 100 procent korrekt!
Om du är intresserad av för att mildra eller starta en diskussion med hjälp av Twitter, har jag skrivit ett inlägg (på engelska) om detta. Jag hoppas att du njuta av resten av din dag och jag ser fram emot att se dig på Twitter någon gång!