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.

Resources:

• 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.

Hmm....

• 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.

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If you found this post useful you might enjoy my maths page.
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Credits:

All links to original sources.

Image details here.

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 dailybreeze.com 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.

Hmmm....
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 Irishtimes.com 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.

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Credits: All links go to original sources.

Image from Google images:  http://web1.northmead-h.schools.nsw.edu.au/moodle/pluginfile.php/2/course/section/1/maths21-15bpx4o.gif

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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 mathematics” by 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.

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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.

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Credits: All links go to original sources.

Image: via google images: http://www.kingslangley.ps.education.nsw.gov.au/images/Math-Symbols.jpg

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?

• “Stop teaching Calculating, Start Learning Maths” - Conrad Wolfram

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 edweek.org 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?

Hmmm...

• 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.

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Credits: All links go to original sources.
Image from Google images: 
Image from Google images:http://www.ihub.co.ke/blog/wp-content/uploads/2012/10/social-media-banners.jpg                                                    ~~~~~~~~~~~~~~~~~~~~~~~~~  

Vedic maths. Facts + fiction = fantastic

It has been said that truth is stranger than fiction.  That is certainly true when it comes to the story of the origins of “Vedic” maths.  

Vedic mathematics first came to light in a book published in 1965 by the impressively named Bharati Krishna Tirthaji - an Indian mathematician ...and mystic. Tirathji claimed to have rediscovered Vedic mathematics after meditating on some neglected sacred scripts. This is where the story gets a tad bizarre - for there is no evidence in the Sutras of anything vaguely resembling the mathematics that Tirthaji proposed.  He is reported to have spent eight years meditating in a forest in order to discern the hidden mathematics.  This has been likened by Alex Bellos, author of “Alex’s Adventures in Numberland”, as  being akin to a “...a vicar announcing he had found a method for solving quadratic equations in the Bible”. When challenged by other Indian mathematicians who could read Sanskrit to identify the passages upon which he based his mathematics Tirthaji claimed that the messages were in his texts and in no other.  There was also mention of a 16 volume manuscript that Tirathji had written dealing with the subject - but which had somehow been lost prior to publication - indeed prior to him sharing it with a single other person. Such loss apparently did not disturb Tirathji who simply said he would re-write it from memory.  The first, and only book published (posthumously) was but a fraction of the mathematics that he claimed to have re-discovered in the sacred texts.   (A detailed discussion on the story of the “discovery” of Vedic mathematics can be found here.)  

In short, it appears unlikely that the mathematical techniques mentioned were “discovered” in any ancient Vedic text.  Intrigued by the power and simplicity of the mathematics Alex Bellos did some research to see if the techniques had been mentioned elsewhere, and it turns out that one of them, Vertically and Crosswise,  had been published in none other than Fibonnaci’s Liber Abaci (published in 1202 and credited with introducing Hindu-Arabic numerals and calculation techniques to Europe).  Another, All from 9 and the last from 10,  was apparently  a wide spread technique in Europe in the sixteenth-century - so wide spread that it may have been the origin of our current symbol for multiplication.

However, although many would find the story of the “re-discovery” of Vedic mathematics fanciful it is undeniable that the approaches  described in Tirathji’s text actually work.  In some cases they are no more efficient than contemporary approaches but in others they clearly are.  

Two of the techniques describes as “Vedic” maths that can be traced to European sources are All from 9 and the last from 10 and Vertically and Crosswise. Both are worth exploring.

All from 9 and the last from 10 is essentially a quick mental maths trick with limited application - the approach as depicted here can only be used when subtracting numbers from multiples of 10 (in other words 100, 1000, 10 000 etc.) but with minor modification can be used with any number ending in zero.  However it is useful in illustrating the nature of Vedic maths.  With this approach to subtraction you work from left to right and subtract all digits being subtracted from 9 except for the last digit which is subtracted from 10.

1000 - 578 becomes

1000
-  578 (SUBTRACT the 5 and 7 from 9 and the 8 from 10)
= 422

Although limited in application it is easy to see how this might be a useful skill when using money for example.

A technique with wider application is  Vertically and Crosswise which allows rapid calculation by multiplication. Consider 63 x 28. The Vedic method would be;

Step 1. Write the digits being multiplied on top of each other.

6 3

2 8

Step 2. Multiply the numbers in the right hand column (24).  Write the 4 in the units column and carry the 2.

6 3

2 8

                                                                   2  4
Step 3.  Multiply diagonally opposite numbers (cross-wise) and then add the products (multiply 6 X 8 = 48, 2 X 3 = 6, 48 +6=54.  Use the carried 2 from the previous stage = 56. Write the 6 and carry the 5

6 3

2 8

                                                                5  6 4
Step 4. Multiply the left hand column (6 x 2 = 12)  Add the 5 being carried (12 + 5 = 17). Write the 17 to the left of the answer thus:

6 3

2 8

                                                               1 7 6 4


For those who prefer a more visual explanation this video gives a good indication of how the process works.

At first glance using two digits like this the process might easily be dismissed as a novelty and only marginally more efficient than  the “traditional” approach.  But when three or four digit numbers are being calculated (or indeed any  large number) the same method is employed and needs only one line of working.  It is much quicker than the standard approach to this problem - much more efficient.

Those wishing to explore some of the techniques might like to visit this site which provides opportunities to practice some of the techniques and this link to a free manual for teachers. This latter resource contains much that is not purely related to “vedic mathematics” and offers some sound techniques for teaching mental calculation techniques.There are more techniques than presented here to examine and explore - far more than could be presented in a post such as this.

I believe it would be unwise to teach this approach to young children. However, to students who have been introduced to and understand the principles and concepts of conventional mathematics this approach might add interest  and intrigue - and would promote mathematics as something worthy of exploration.    

There is much of worth in Vedic mathematics.  The murky story of its “re-discovery” should allow the story teller that is  in every effective teacher to create the setting for some effective exploration of mathematics.  At worst it still has to be better than providing yet another worksheet...

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Credits;

• Image of Tirthaji: http://www.vedicmaths.org/Introduction/History/Tirthaji.jpg
• Links provided in text to original sources.
• Alex Bellos, 2010, Alex’s Adventures in Numberland”

If you enjoyed this post you may enjoy my other maths related posts available via the maths page or by clicking here.

Ancient maths = modern learning

There are many issues of concern associated with mathematics teaching today.  One is the emphasis on procedural competence regardless of conceptual understanding. Another is the almost total lack of historical significance of mathematics. This is a pity as one can inform the other.

In our busy classrooms we tend to rush from one mathematical concept to another - often without any explanation given to the students. It is my experience that if we take the time to develop student interest in a topic then sound performance often follows  -  the “diversions” demonstrate that mathematics is more than learning algorithms  and makes the broad topic of mathematics more interesting - which engages the students - and engagement leads to understanding which in turn leads to higher performance.

My views on this are shaped not only by my experience as a teacher - but as a student. My primary years have been largely shrouded in the mist of time but I can still remember being stunned and amazed when I was briefly shown alternate historical methods of calculation. It was a revelation. Maths could be done more than one way? People once did maths differently - and accurately - and it worked … every time?  Ancient peoples were smart enough to do mathematics?  They discovered this stuff?

I’ve covered the same material myself in classrooms and  recognise the same lightbulb moment of understanding in students as they make the same discovery.  It is often the beginning of lasting interest in mathematics.

Today, thanks to web-based video we can show students “historical” mathematics as a way to scaffold and extend their mathematical knowledge.

One of the favourite “ancient” forms of mathematics taught in schools is often known as “Egyptian multiplication”.  It turns out that “Egyptian” part of the label may not be overly accurate - but the technique surely is.  This clip provides a good explanation of the process - together with a brief explanation as to why it works.

Ethiopian multiplication (also known as Egyptian)

From Egypt / Ethiopia / Elsewhere  we can make the short historical journey to the lattice method explained here.

Lattice Multiplication

The use of the diagonal columns will have parallels with the use of vertical columns in our current systems of calculations which will not escape the children and provides a useful opportunity to discuss the use of columns in modern calculations.
From there it is a short step to Napier’s Bones.  
Napier's Bones

                                        

 (This is a good clip despite it’s simplicity as it allows the students to study the structure of the “bones” and perhaps identify the pattern in the rods for themselves.) The clip below shows how the bones or rods were used.

The sense of amazement and wonder is enhanced if students actually complete some calculations themselves - in which case the pro forma here will be useful.  Once the students are capable of using the “bones” it is a useful extension to get them to use them to multiply two two digit numbers - and let them discover the technique themselves if they can.


The discussion around why devices such as Napier’s Bones were useful and necessary is a worthy way to spend some time.  Our students find it difficult to understand that it was once the norm for humans to perform all calculations, not silicon chips.  It is also worth raising awareness the universal education is a relatively new social phenomena - and discussing the impact of that.

Another avenue that can be useful is to look at numbers without using numerals at all (well, at first at least). This clip looks at the ancient Greek interest in square numbers and demonstrates the concept effectively using nothing more sophisticated than stones - which is all the Greeks needed to discover the concept.

Counting like an Ancient Greek

I’m not suggesting that the techniques presented here be “taught” in isolation - more that they can be explored and then, once students are familiar with the techniques, compare and contrast them to current methods.  It is often at this stage that students really develop an understanding of the strength of the place value system that we use today - after which their fluency with formal algorithms often improves significantly.  
If nothing else students will have enjoyed interacting with mathematics - and without a worksheet in sight.

If you enjoyed this post you may enjoy my other maths related posts available via the maths page or by clicking here.