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Web Papers

Over decades Douglas Williams has written papers and articles and given addresses on issues and ideas in mathematics education. Some are listed here in alphabetic order of title. The list will be added to from time to time.

Feel free to quote from these sources or print and distribute to stimulate educational debate. We ask only that you acknowledge the source.

Other teachers have also written for our site. You will find many of these pieces in Research & Stories.

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Aha! Teaching Maths is Simple An Ocean of Possibilities Assessment, Recording and Reporting in Mathematics
Curriculum 2000 How Can Solving the World's Hardest Problem Inform Mathematics Teaching? Learning to Work Like a Mathematician
Mathematics Education is not an Enigma Maths Not At The Movies Menu Maths & Other Models for Making Mathematicians
  School Mathematics is Learning to Work Like a Mathematician  

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Red Square  Aha! Teaching Maths is Simple (PDF file)
  • Spring 2010
  • First published in Primary Mathematics, Mathematical Association, UK, Spring 2010, Volume 14 Issue 1.
  • The article is as presented in Primary Mathematics except that the References section has been updated.
  • Beginning with the proposition that 'Teaching maths is simple ... but subtle.', the article moves through examples leading to the statement 'Let the kids do it. The subtlety is in being the teacher who can facilitate this.'; and draws near to conclusion with 'This is the simple but subtle description ...(of the Working Mathematically Process)... that has brought success to hundreds of classrooms from K to 12.'. The examples are drawn from Calculating Changes and as the article unfolds the reader develops a deeper understanding of the philosophy and practice of this Mathematics Centre network.

Red Square  An Ocean of Possibilities (Video & PDF file stored on Cube Tube)

  • March 2010
  • First published in Mathematics Teaching, Association of Teachers of Mathematics, UK, March 2010, Issue 217.
  • The Working Mathematically Process is exposed through Task 216, Square Pairs, a problem that has been researched by two modern mathematicians. It becomes clear that the reason for mathematics education is simply to develop and expand that process. Elements of best practice teaching craft likely to captivate, fascinate and absorb students in this process are also explored. It is proposed that there is a broad ocean of possibilities similar to Square Pairs available when the decision is made to approach learning this way.
Red Square  Assessment, Recording and Reporting in Mathematics (PDF file)
  • April 1997
  • Explore the concept of planning assessment when planning a lesson or unit of work rather than it being added on as an after thought. Alternative assessment techniques in mathematics education are explored.
  • This paper has contributed to the PD from MC workshop day of the same name and is provided as a reference in that session.

Red Square  Curriculum 2000

  • October 1999
  • What would a curriculum for the 21st century look like? A paper to stimulate debate.

Red Square  How Can Solving the World's Hardest Problem Inform Mathematics Teaching? (PDF file)

  • November 30th 2009
  • Text of an address to the Secondary Convention, Mathematical Association of Western Australia.
  • Revision of Learning to Work Like a Mathematician (2002) enhanced by DVD and slide show and additional photo show. Slide show and photo show display in Acrobat Reader - use arrow keys to move through the slides.
  • Video of the BBC Horizon Program Fermat's Last Theorem can be found on the web. Search Fermat's Last Theorem BBC. For the purpose of this address only the first seven minutes is necessary. Finish at Andrew's words "Mathematician's just love a problem..."

Red Square  Learning to Work Like a Mathematician

  • December 5th 2002
  • Text of an address to the Annual Conference, Mathematical Association of Victoria.
  • A development of the philosophy of Curriculum 2000 that builds the mathematics curriculum around the ways of knowing of a mathematician.
Red Square  Mathematics Education is not an Enigma (PDF file)
  • Edited version of the keynote address given at the Association of Teachers of Mathematics (ATM) UK Easter Conference 2012 at Swansea University, Wales
  • First published in two parts in Mathematics Teaching, Association of Teachers of Mathematics, UK, Issue 230, September 2012 and Issue 231, November 2012.
  • In both issues the article was introduced through these words from the editor:
    Doug Williams presented the opening address at Conference 2012 on the theme of 'Enigmas'. This is storytelling about the learning and teaching of mathematics at its best. The descriptions are of real classrooms with real teachers, and real learners. As with many good stories this is the narrative of a journey. A journey that seeks to change approaches to the 'craft' of teaching, to consider what is of primary importance in the classroom, and to develop a curriculum based on the notion of 'working mathematically'. But the curriculum is not just about what is learned. Far more important is 'how' the learning should be enabled, and this is the 'craft' of teaching. Where does this journey start for learners of mathematics? It starts with an interesting problem. - Margaret Jones

Red Square  Maths Not At The Movies (PDF file)

  • June 2006
  • First published in Vinculum, Mathematical Association of Victoria, Volume 43, Number 2, June 2006
  • As working like a mathematician becomes the focus of your teaching, even textbook problems are approached differently. In this article a classic 'stupid' textbook problem is examined in the cold grey light of dawn. Elements of the Working Mathematically Process become critical to its solution.

Red Square  Menu Maths & Other Models for Making Mathematicians (PDF file)

  • November 2013
  • First published in Mathematics Teaching, Association of Teachers of Mathematics, UK, November 2013, Issue 237.
  • School mathematics is about learning to work like a mathematician in best practice classrooms. In this article the concept of giving learners a choice - because mathematicians have a choice about the problems they will investigate - is explored through various menu-based models. In the same sense as a menu offers a diner a choice of what to savour, so Menu Maths offers students a choice of how to 'get stuck into' their mathematics. But this is not a free-for-all. The teachers control the menu - in fact work particularly hard to provide the best menu items they can find. Usually there is a common content thread within the menu which allows teachers to enrich the learning with related whole class investigations - perhaps even with text-based material!

Red Square  School Mathematics is Learning to Work Like a Mathematician (PDF file)
  • March 2023
  • Temporarily suspend your current view of school mathematics and explore an approach that can successfully change your classroom framework without losing any of the maths you value. Perhaps school mathematics really is learning to work like a mathematician rather than just learning maths.
  • Making deliberate teaching craft decisions to encourage students to learn to work like a mathematician is highlighted and supported throughout the article.
  • Garden Beds is the example problem used with a suite of several lessons described. The lessons work anywhere from Year 4 to Year 8. However the article is appropriate to every Year level and it contains a reference to a photo story illustrating very clearly that Year 2 children are equally willing to plunge into the problem.
  • The article is 8 pages and with its embedded practical, hands-on experiences is a perfect stimulus for a DIY professional development session from one hour to half a day.
  • A discussion focus for reviewing the session might be: Compare and contrast the way time is used in the example lesson suite, with the way we currently use time.

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