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:

Saturday, December 1, 2012

Mathematics from the masses

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 blue links to be taken to the sites mentioned.)

So you think you know how to teach mathematics?

Conrad Wolfram is renown as a “boat rocking” thinker and innovator.  This presentation challenges current educational methods for teaching mathematics.  In an entertaining speech Wolfram asks what is mathematics and then describes it as a four stage process consisting of:
  1. Posing the right question
  2. Putting the question into a mathematical framework or context
  3. Calculating
  4. Converting the answer back into a real world context.
Wolfram states that students spent 80% of their time at stage three - performing calculations, often manually.  He states that this is odd given that computers / calculators can do this much better than any human brain and that we should be concentrating our efforts on the other three stages.  A thought provoking video lasting 26 minutes - watch it over a coffee and think about it long after.
  • Open ended questions.
The maths world is awash with the power of open ended questions rather than a steady diet of closed questions.  But how do we formulate good open ended questions?  This powerpoint, based upon the work of Peter Sullivan, shows how it can be done.
  • Early Counting . Research cited at says that teaching pre-schoolers to count (as opposed to just recite numbers) to 20 is an advantage later in life

Resources that you might find useful.
  • A pinterest site shared by classroom teacher Laura Chandler with lots of resources used by mid-primary teachers.  Not all of it pushes the boundaries of mathematics teaching but much of it would be useful to classroom based teachers at this level.

  • Mathplayground - a good site for classroom teachers with lots of areas to explore - allows effective teaching not just drill and practice.

  • NCES Kid’s Zone.  A collection of web based tools for graphing and probability for primary students.  The applications are accessed via buttons on the top of the screen.
  • A+ click A wide ranging free site from grad 1 to 12 covering wide range of mathematics. Problem and logical thinking questions to suit the needs of most teachers.
  • How many texts are sent every day in your town?  The New York Times provided this bit of maths based on a perhaps surprising statistic - that the number of texts sent last month fell for the first time in history.  However, the linked article provides all the information students would need to extrapolate to your home area. If we assume that the number of texts sent per person is constant across all areas (but we may choose not to accept this assumption - coming up with another figure might be useful in itself) and the population of our area is (???) then how many texts might be sent from our home?  In fact, challenging the statistics provided might be even more fun. What is the average number of text sent each day by members of your class? Would this hold true across all grade levels and classes? How might we find out? Once done, what is our estimate?

  • Not everyone enjoys mathematics.  New research has found that just thinking about doing mathematics can cause headaches in some people.  
  • Still on the brain, Scientific American reports research that suggests that the brain can do mathematics unconsciously. (This might explain the phenomena of students who appear asleep in class but still manage to get some work done.)

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: 

Image from Google images:                                                    ~~~~~~~~~~~~~~~~~~~~~~~~~