Monday, November 4, 2013

The paradox of our age - and our schools


We have bigger houses but smaller families;
more conveniences, but less time;
We have more degrees, but less sense;
more knowledge, but less judgement;
more experts, but more problems;
more medicines, but less healthiness;
We've been all the way to the moon and back,
but have trouble crossing the street to meet the new neighbor.
We build more computers to hold more
information to produce more copies than ever, but have less communication;

We have become long on quantity, but short on quality.  These are times of fast foods but slow digestion;    Tall men but short character; Steep profits but shallow relationships.

It's a time when there is much in the window, but nothing in the room.
The 14th Dalai Lama

The above text from the Dalai Lama reminds us of some of the contradictions of our modern society.  Without wishing in any way to deminish the message of this  inspired writing I’d like to suggest my own additions relating to current educational practice  to his powerful prose.

“We have more instruction but less inspiration,
More trivial answers but fewer significant questions,
More rote and recall, less speculation and imagination,
More “tick the box” but less “think outside the square”,
More “rights”, but fewer “responsibilities”,
More edutainment but less engagement,
More talking but less conversation,
More efficiency but less excellence,
In short, we have more teaching but less learning.

It’s time to put the focus back where it should be - on the child. The needs of “the system” should be replaced by the needs of our students.

I began this post by citing the Dalai Lama.  I’ll end  by citing another known for his cosmological musings - Albert Einstein.   “Many of the things you can count, don't count. Many of the things you can't count really count.”

Education counts. Maybe schooling doesn’t.

Wednesday, March 13, 2013

Teachers - do what you love and do it often.

Burnout is a major problem in schools.  It shouldn’t surprise anyone - teaching is a demanding occupation.  As well as being educators teachers are part social worker, part nurse, part de-facto parent, part careers advisor, policeman, counsellor and even relationship advisors. Balancing the dictates of the curriculum and  the requirements of site administrators with the capabilities of the students is in itself no easy task. Add to that the emotional impact of dealing with students who often come to school with emotional baggage and stressful domestic issues well beyond their control - and often their comprehension.  While addressing these issues teachers also have to keep in mind the special needs of individual children - some of which can be literally life threatening if things go wrong. Into this recipe for stress throw in some seasonal variations such as the lack of hours in the day at peak times like report writing or parent teacher sessions and it is easy to see why burn out is all too common.   Teachers, after all, are only human.

Teachers are people, not robots.  Through observation of effective teachers over three decades I have noticed one important characteristic that effective long-term educators have in common. They all have interests outside of school.  Their careers are important to them, very important in fact, but they have other aspects to their lives.  They are not defined but their job - it is an important part of their lives but only one aspect.They have balance - and they cling to their interests despite the unrealistic pressures put on them by the job.  I think it is aptly summed up by the opening lines of the Holstee Manifesto;

“This is your life. Do what you love and do it often.”

Teachers need to follow their passion for life outside of school in exactly the same way that we encourage our students to do.  Sometimes we have to discover what it is that stops us from doing what we love - and make some changes.   This is probably the only antidote I know for burnout. Outside interests re-energise and re-invigorate people at the personal level. Doing what we love doing makes us healthier people - which in turn helps us to devote real energy to education when we focus on it - and there is plenty of scope and need for that. In the end, our students are the ones who really benefit.  
What greater advice can we give our students than to quote the opening of the Holstee Manifesto to them over and over again? This is your life. Do what you love and do it often.” If that attitude is the only thing our students pick up from us we may well have taught them one of life's most important lessons.

Enjoy this great interpretation of the Holstee Manifesto. 

Holstee manifesto image: Google images.
Video - via youtube.

Wednesday, February 27, 2013

Mathematics from the masses #4

Each month I collect some of the gems from the web that relate to mathematics.  This month there are fewer links than previously - maybe I’m just becoming more selective...

So you think you know how to teach mathematics?
  • Mathematics curricula all over the world require teachers to teach with rigor.  But what does it mean?  This brief and insightful piece by NCTM President Linda M. Gojak sheds some light on what it should mean.
  • Every successful teacher knows that motivated students learn more and more easily than less engaged students. But how do you motivate students in the mathematics classroom. This info from nctm has some useful tips, more correctly they are   philosophical positions that generate teacher behaviour that all teachers could benefit from adopting - and not just in mathematics classes.
  • This post at mashable features 8 videos with a mathematical theme showing how much fun can be had with mathematics.
  • Another piece from nctm, this time showing how conservation of materials at school can save money.  Most of the ideas are not fully fleshed out - but a bit of creative thought could spark a series of “How much money could we save by...” questions.

  • ICT Magic is a site dedicated to the use of technology to enhance education. This link goes to a section devoted to some worthwhile maths games etc.

  • I’m really not sure which side of the fence I’m on with this issue - mobile phones in classrooms. Education Week reports on a study that finds up to 39% of middle school students in the USA use their smart phones for help with homework - yet only 6% are allowed to access their devices during class time.  I’ve taught high school classes so I understand the reasons behind the ban during lessons - but I also understand that smart phones are very powerful devices, most have more computing power than the combined computing power in the control room at NASA during the moon landing.  Surely we can use these resources in some way?
  • This piece at Slate tells us that there is now a “new” largest known prime. Discovered by Curtis Cooper of the University of Central Missouri it is  257,885,161-1 - a number so large that it apparently takes 17 425 170 digits to write it.  If you want to get a sense of that click here.  Some interesting discussion and background in this piece - well worth a browse.

All links go to original sources.
Image via google images.

Monday, February 4, 2013

Mathematics from the masses #3

The web is full of great material dealing with mathematics.  Below is a selection of sites, reports and ideas dealing with mathematics in general and the teaching of mathematics more specifically.   The criteria for inclusion is completely idiosyncratic - if it appeals or interests me, or if I think if might be useful to maths educators it gets included.  Enjoy... but be warned - there are some sites listed this month that are seriously distracting and damaging to productivity.

So you think you know how to teach mathematics?

  • You have to be smart to do well at school maths right? Wrong. This report  of a German study finds that motivation and work habits are more important than IQ.  Educators should not be surprised by this finding as IQ as a predictor of success (even if it could be measured properly) has long been deemed of questionable worth as a predictor of anything much.   Attitude is everything –  a “growth mindset” is frequently (always?) found to be more important than innate ability.
  • Maybe the best teachers are students? This study found a program that used older students to tutor younger ones was very effective. Given so many other initiatives fail this might be worth trying in other places...
  • Maybe we simply need to get students to spend more time at school? NPR reports that a number of states in the USA are trialing just that in an effort to curb holiday brain fade and forgetfulness of material “taught” last year. (Maybe making the material worth remembering might be a better approach...)
  • Maybe we should all just up and move to smaller towns and leafy suburbs.  This  study reported in Education Week has found that rural and inner city tend to underperform in mathematics relative to their peers from other areas.  (Again, I’m not sure this is really news to experienced teachers.)
  • I’m not sure if this is new or simply supporting an understanding of teachers all over the globe. A study has found that students who struggle with mathematics use a different part of the brain to those students who are competent at mathematics.  Those that cope well appear to be using a part of the brain that accesses memory for facts - lending support to the old fashioned notion of automatic recall of basic number facts as being important in mathematical performance.   It has to be said that the study involved only a relatively small number of participants (N=43), but given the hi-tech nature of the study this might not be a major indicator.


  • Wolframalpha is a great site for online calculations.  This blog from the creators shows some really useful automatic displays related to simple calculations - most if not all teachers of mathematics would be able to find a use for this site - so easy to use but with so much potential to explain answers...   I really like the fact that a completed numberline of the calculation is provided almost instantly - which makes it a really handy way of explaining order of operations to primary age children.
  • The PBS Learning media site has lots of useful video / teaching material on a range of subjects -  including mathematics. The parent site can be searched for via subject and grade level. Well worth investigating for some teaching media.
  • In a similar vein, everybody’s favourite site “lluminations” continues to be an almost unbeatable source of sound resources based around active learning principles.
  • On a related theme comes this blog item revealing that use of Interactive White Boards in maths lessons not only improves student attention and participation but leads to improvement in  maths scores.
  • Looking for a really simple site that makes use of your IWB?  This site from harcourtschool allows simple arrow clicking to create visual matchings of equivalent fractions.

  • In the same vein, the old fashioned geoboard gets a virtual makeover at the mathlearningcentre.  It may not be as organic as the “hands on” variety but it is still lots of fun … and has the distinct advantage that the “rubber bands” can’t be flicked all over the classroom.

  • Mathrecap is a site edited by Dan Meyer - presumably the same maths teacher who made a name for himself with his TED talk on teaching mathematics.  The site recaps (which no doubt explains the name) a variety of presentations made at maths conferences around the USA.   A source of some very worthy techniques and good ideas for maths instruction.
  • Normally I’d avoid any article  using the word “cool” in the header - but this time it leads to a collection of maths games suited to primary students.  It’s, … well, ... cool - especially the game called B-Cubed.
  • This NPR piece gives some nice examples of how New Zealand teachers are teaching probability - applying Murphy’s Law (“Anything that can go wrong will go wrong”) to stats. Does butter really always fall butter side down?  Some cartoon graphics provide a good starting point for other like minded engaging explorations.
  • Math Munch is a blog after my own heart - a site dedicated to cherry picking the best of the web’s maths related site and sharing them with people with a similar mind set.  It is updated on a regular basis (usually weekly) and has a host of really useful educational mathematics related sites as well as material that is simply of a mathematical nature.  Highly recommended - you will almost certainly want to put this one in your favourites.


  • Does mathematics make research sounder? Yes - well, apparently we all think so. A recent piece reported on at Freakonomics cites a study where academics were asked to evaluate the strength of research papers - some of which were doctored with meaningless maths out of context.  The maths "enhanced" studies were considered to be the stronger.  The really scary thing here is that the participants all held post graduate degrees.   So what chance does the general public have?
  • A rather depressing report in the Mailonline cites UK authorities requiring teachers to return to “traditional” methods of teaching long multiplication and division and away from more progressive processes based on mental maths and number sense.  Looks like the pendulum of reform didn’t even get to complete a full swing ...  teaching procedural competence replaces conceptual understanding yet again...
  • Is there a relatively simple formula that governs how long we (and just about everything else) will live? This NPR piece tells more … it appears so (well... maybe). Perhaps. Possibly.

If you found this post useful you might enjoy my maths page.
All links to original sources.
Image details here.

Wednesday, January 23, 2013

The emotion of learning

I’m breaking a self imposed rule with this post. I’m writing it while on holiday. The reason is simple,  I’ve achieved a boyhood ambition - decades after the seed was planted.
Today I created fire.
Not with a match.
Not with a lighter.
With a magnifying glass. 

For as long as I can remember I have wanted to do it.  Memories of summer delights as a boy have resurfaced – as a boy / teenager I used to use a magnifying glass to burn my initials into my summer thongs with a magnifying glass.  My public explanation was that it made it easier to find my thongs when “the gang” came over and all our thongs were piled up by the back door.  True enough – but the other truth was that I enjoyed doing it. I enjoyed seeing the snail trail of my initials “magically” appear as a result of nothing more complicated than the sun and some curved glass.  However, given the long, tinder-dry grass in the paddocks that surrounded my boyhood home,  I never attempted to  light a true fire – despite a burning desire to do so (pardon the pun).

Today, decades after my teenage years faded into the fog of time, I finally tried to light a fire using nothing  more than the material I wanted to burn and a magnifying glass.  In theory there was no doubt that it would work and I knew that it would take marginally more effort than using a match. But I wanted to do it – simply because I wanted more than intellectual knowledge that it would work. I wanted to actually do it.   So I did.

It occurred to me while I was holding the magnifying glass that there were broader lessons in the action.  I needed to maintain focus – my own and that of the magnifying glass. The pin point of concentrated solar energy had to be held over the same spot and I had to concentrate to ensure that this happened. It is probably dignifying the action a little to say that I needed to employ a technique but in a sense it was true.  If I didn’t hold the glass in the correct way nothing would happen.    I also needed to clarify my own definition of success.  Was singeing the paper the same as creating fire?  I decided “no” - but it was probably acceptable as an encouraging sign.  Was creating smoke (after mere seconds) creating fire?  Again I decided “no” – but it was probably proof of concept.  What did I mean by fire? I decided that it was flame that consumed the material that I wanted burned and that only that could be considered success.  After passing through the first stages towards success I needed to adjust my approach by  getting some smaller twigs from the garden – which, in academic terms could be considered refining my practice in the face of observable results.

So I was learning.  But the feeling of success and achievement when the flame burst into life was out of all proportion to the lack of effort required to achieve it.  It turns out the lighting a fire with a magnifying glass is as easy as it is reported to be.  But the key aspect for me is that now I know it, I’ve done it myself.  Second hand knowledge has been replaced by experience. 

The other aspect is that this was a fiercely personal objective. I doubt any of my family or friends share this desire to “achieve” this goal.  In world terms it is insignificant. But in personal terms it brought a glow to my face that was not simply a reflection of the fire. 

It occurs to me that this is true of all personal learning.  Is this why musicians try to reproduce the sound of their favourite guitarist when the mp3 player can do it effortlessly for them?  Is this why painters use pigment to capture the landscape when a camera can do a more accurate job in an instant?  Is this the feeling of truly significant personal learning?  If it is, then surely we as teachers owe it to our students to let them experience this success in Iearning or achieving something of personal significance?   I wonder how many discipline problems would fade away and how the motivation levels of our students would increase if this feeling was a regular part of our classrooms? 
Easily the most popular post on this site is one I wrote dealing with PBL.  If I may say so my-self it provides a good summary of the concept and links to a host of  very useful resources.    My experiences today re-affirm my reasons for writing that post; we need to meet the personal interests of our students in our programs.  Adding the emotion of learning to our classrooms may be a key ingredient in a truly motivated learning environment.

Truly personal learning is significant; it produces emotion in the part of the learner – not just intellectual advancement.  If it doesn’t then maybe it isn’t as significant to the learner as it is to the teacher.
Image via google images:

Thursday, December 20, 2012

Mathematics from the masses #2

The web is awash with wonderful resources for teaching mathematics - plus some sites pedalling pedagogy best avoided.  What follows is a purely personal selection of some of the more interesting and worthwhile sites that beamed out from my screen this month. (Follow the underlined blue links to be taken to the sites mentioned.)

So you think you know how to teach mathematics?

  • The 2011 TIMSS study was out this month - and produced much hand wringing all over the globe.  This link takes you to a site where you can access the report - and commissioners comments etc.  It seems that maths education is in trouble just about everywhere.
  • This is probably not news to most mathematics teachers - an article from the broader press that highlights the changes away from textbook focussed instruction and the improvements that can come with it.  US site reports on the success of teaching for meaning rather than memory in a local high school.
  • A collection of maths instructional videos at Mathcasts. Suitable for  high school students.
  • From the Detroit News comes an article of schools installing analog clocks in order to help children tell the time.  It goes on to mention their experiences that children can use digital clocks more easily - but that this doesn’t necessarily translate into genuine understanding of the concepts involved. A welcome example of ensuring conceptual understanding over procedural competence.
  • From England’s Guardian newspaper a sensible discussion about reform of the mathematics curriculum in England - with the observation that you can’t necessarily “cherry pick” techniques from one country and transport them to another, but recognising that the approach of some countries does seem to produce better results than others.
  • Also from England is this piece on the BBC site discussing the fact that rushing through the curriculum does not allow for the development of deep understanding. It is an issue for teachers all around the globe.   This is an issue more worthy of discussion than the hand wringing that has been seen around the world as the latest round of international testing results have come out.  Surely getting the curriculum right, and creating space for genuine understanding of the key developmental concepts is more important than the current league table results? Fixing the former will address the second.
  • Vedic mathematics is a fascinating - and neglected area. This post / video at Firstpost provides an introduction to those who are not familiar with it - and might prompt further exploration. A personal blog about the fascinating story behind the “rediscovery” of Vedic mathematics might also interest.
  • Why is the sky blue? This simple question takes some serious explanation - and is not without mathematics to make sense of the science.  Would suit upper high school students.
  • The “12 days of Christmas” gets a fair run in many mathematics classrooms at this time of year.  This self produced video on youtube uses it to explore triangular numbers and goes on to the classic “handshake at a (New Years Eve) party for good measure. Fun and has potential if pitched at the right grade level.  (Apologies to the anonymous creator - can’t attribute due to lack of details in the clip.)

Just for fun

  • This prediction trick has been around in more traditional forms for a while - but it has been given a retweet for the youtube generation by Richard Wiseman. Lots of fun.  
  • This video is great - The Amazing Anamorphic Illusions. If you like the illusions there are links to the images so you can “perform” the same trick yourself.
A collection of TED talks about mathematics.
  • 8 math talks to blow your mind -  hard to elaborate on the title really. My favourite was / is Benoit Mandelbrot’s talk on fractals - but that’s just a personal preference. They are all worth watching over a cup of coffee.
  • Is Zero an even number” asks the BBC. Brief piece re the public perception of zero.

More from before
  • In Maths from the Masses #1 I linked to some research suggesting that some people literally feel pain at the thought of “having” to “do” mathematics.  This is piece from the Newstatesman follows a similar vein  - but interestingly, people’s fear of mathematics seems to disappear when they actually start “doing” it (as opposed to merely completing endless calculations).   This is a widespread issue and impacts on  parents. How do parents who suffer from maths anxiety support their children at home?  This piece from the addresses this issue via the lens of confidence rather than calculations and contains some sensible advice.

Math page
If you enjoyed this post you might enjoy exploring my maths page which features other posts of a similar nature - some with video worth using with students, and some recreational maths developed to share a love of mathematics.
Credits: All links go to original sources.

Image from Google images:

Thursday, December 13, 2012

How to be a great maths teacher #2 - what the research continues to say

There is no shortage of people telling teachers how to improve education.  Sometimes it seems that all the educational experts are either cutting hair or driving taxis - or perhaps in public office.  So it is useful to find clearly written advice based upon educational research and free from economic motivations. The International Academy of Education (IAE) publishes a series of pamphlets that distills educational research into useful summaries of current teaching techniques that have been found to promote student learning.  ”Improving student achievement in mathematicsby Douglas Grouws and Kristin Cebulla provides a succinct  summary of how effective educators  can approach their teaching.

Their research suggests that teachers can improve mathematical learning via;
1.Opportunity to learn
The extent of the students’ opportunity to learn mathematics content bears directly and decisively on student mathematics achievement.  “As might be expected, there is also a positive relationship between total time allocated to mathematics and general mathematics achievement.”  No surprise there I suspect - but this aspect is worth contemplating; “Short class periods in mathematics, instituted for whatever practical or philosophical reason, should be seriously questioned. Of special concern are the 30-35 minute class periods for mathematics being implemented in some middle schools.”
2. Focus on meaning
Focusing instruction on the meaningful development of important mathematical ideas increases the level of student learning.  Teachers should stress “...the mathematical meanings of ideas, including how the idea, concept or skill is connected in multiple ways to other mathematical ideas in a logically consistent and sensible manner.”
3. Learning new concepts and skills while solving problems
Students can learn both concepts and skills by solving problems.  This clearly addresses the “chicken and egg” issue of some teachers - it is NOT necessary to teach specific computation techniques BEFORE addressing real life applications. “There is evidence that students can learn new skills and concepts while they are working out solutions to problems.”  
4. Opportunities for both invention and practice
Giving students both an opportunity to discover and invent new knowledge and an opportunity to practise what they have learned improves student achievement. The research finds that in the USA over 90% of class time is spent on practicing routine procedures.  In Japan about 45% of instructional time is spent practising routine procedures, 15% applying procedures in new situations and 45% inventing new procedures or analysing new situations. Like to predict which system is ranked higher in international comparisons?
5. Openness to student solution methods and student interaction.
Teaching that incorporates students’ intuitive solution methods can increase student learning, especially when combined with opportunities for student interaction and discussion.  Student interaction - sharing their solutions and the how they approached maths tasks - makes for enhanced student learning.  The notion of a good classroom is a quiet classroom with children working in isolation is simply not supported by the research. Students learn better when they interact - which, if the social-constructionist theory of learning is applied,  is what we would expect.
6. Small group learning
Using small groups of students to work on activities, problems and assignments can increase student mathematics achievement.  Again, co-operative methods of teaching featuring both group goals and individual accountability are associated with enhanced student learning. Teachers would be advised to select mathematical tasks that lend themselves to group exploration rather than simply getting students to “work together” on standard tasks.
7. Whole class discussion
Whole class discussion following individual and group work improves student achievement.  The adult in the room need not be the only teacher in the class.
8. Number sense
Teaching mathematics with a focus on number sense encourages students to become problem solvers in a wide variety of situations and to view mathematics as a discipline in which thinking is important.  Number sense - that feeling accomplished people get when they get an answer that “doesn’t look right” - is an important part of developing mathematical skills...and it requires that students are actually thinking about what they are doing, why they are doing it,  and estimating / predicting internally what sort of result would be reasonable.
9. Concrete materials
Long-term use of concrete materials is positively related to increases in student  mathematics achievement and improved attitudes towards mathematics.  So, a warning sign of a less than effective teacher may be the pile of worksheets students are expected to complete. Combine this with a lack of manipulatives or concrete support materials and students have a problem - and it isn’t the mathematics.
10. Students’ use of calculators
Using calculators in the learning of mathematics can result in increased achievement and improved student attitudes.   Study after study support this notion. The use of calculators enhance mathematics learning.  Why? It lets the students think about the mathematics, not the calculation.

Grouws and Cebulla add a caveat to their list of behaviours - the quality of the implementation of the teaching practices listed above greatly impact upon student learning; for example, it is not only whether manipulatives are used but how they are used that determines effectiveness.  

Much of this list will not be new to those with an interest in mathematics education. However, it seems to my casual eye that mathematics classrooms are often still the domain of worksheets with a focus on procedural competence rather than conceptual understanding, places of compliance rather than engagement.  Reading and discussing research findings such as this may help us improve the quality of our mathematics teaching. Reading the original pamphlet would be worthwhile for all teachers with an interest in the area.

An earlier post - “How to be a great mathematics teacher - what the research says” was mined from another pamphlet in the IAE’s “Educational Practice Series”.  The post summarises the content of that pamphlet and provides links to the original source material - which is well worth reading.

Those with an interest in improving mathematical pedagogy might like to read a related post dealing with the work of Alistair McIntosh - Improving numeracy with the 7Cs.

Those with a general interest in mathematics might enjoy the maths page on this site which collects a range of posts dealing with mathematics.

Credits: All links go to original sources.
Image: via google images: