Learning, Literacy and Learner Skill

This year, the New Zealand Ministry of Education displayed a list of terms in a draft instruction sheet for learners of Chemistry.

The proposal is that page 3, labelled for student use, may assist learners to reach the standard and achieve a rudimentary qualification in secondary education (NCEA Level 1).

Learners at this level must have a “comprehensive understanding” of aspects of basic Science. Here’s a sentence from the draft sheet:

A comprehensive understanding means you are able to link ideas to integrate the relevant chemistry through elaborating, justifying, relating, evaluating, comparing and contrasting, or analysing.

The new Bloom’s Taxonomy, a tool for teachers, lists the same and similar terms as in the sentence shown above. It is understood, however, that teachers using Bloom’s taxonomy are either familiar with the meanings of the terms or have the initiative and required education to find out about these for themselves.

Terms like justifying and evaluating are not easy to define clearly, even for some teachers. Most teachers do not draw a clear distinction between the processing skills of contrasting and comparing, say.

Yet these are just a few of the difficult terms that are found in a draft instruction to NCEA Level 1 learners.

As teachers, of course, we must teach/coach/train our learners to be able to recognise the difference between such terms as contrasting and comparing. The learner needs to be introduced to what each of these analysis processes has to offer.

Why do we end up asking kids to get their head round the lingo that teachers may well have difficulty grappling with? I wonder if this intellectuality is really beyond the stage that most NCEA Level 1 learners are at, given that many already have difficulty with literacy at this level.

It seems that responsibility for learning continues to devolve.

It is as if learners are now expected to know the meaning of terms (or at least acknowledge their existence) often before they have the chance to get any real practice in the skills they are the labels for. These are skills that learners may not yet have the developmental ability to permit them to understand.

Where Has Education Gone?

The Skill Mastery Hyperdome - SLENZ Project - Foundation Studies Build, Kowhai

Authentic learning is a solution to some of the problems

that arise in schools, workplaces and in society today.

Isn’t it funny that at a time when training is being heaved out of the workplace by a change of organisational thinking, it must find in-roads to secondary schools where, purportedly, it is desperately needed?

Last week, I attended the Te Aho o Te Kura Pounamu All-School Forum 2010. This was an international occasion for Te Kura.

Three Keynote speakers - Viv White from Australia, Elliot Washor from the United States and Stuart Middleton from New Zealand - gave their perspectives on ‘authentic learning’ in schools.

Stuart could not attend the Forum. He gave a recorded presentation outlining how he saw the history of what has happened globally to education in the past 50 or so years, and how those changes are impacting on what is happening to the youth in society today.

ROSLA

Clearly, the raising of the school leaving age, by several years, has brought about changes in how education is delivered. It has also altered how society looks on school-leavers who go about looking for jobs.

A number of associated changes have accompanied all this.
The origins and reasons for the changes are complex. But the situations for prospective employment of school-leavers are implicit.

Over 40 years ago it was acceptable for kids to leave school without having any formal qualification. There were plenty of jobs for them. They were trained and educated on-the-job, and stories of their successes in life are numerous. Education through ‘the university of life’ was not an uncommon occurrence. As well, night classes became very popular. These provided a useful adjunct to the education of that group of learners.

But the gradual societal changes, brought about through the raising of the school-leaving age and the programs introduced to schools to cope with these, meant that jobs for inexperienced and unqualified youths became less and less plentiful. What’s more, the general calibre of those jobs is now of a lowly nature and night classes are disappearing.

Hands on

Today there is a desperate need for kids who are likely to leave school early to be introduced to vocational possibilities during their remaining school years. It’s being recognised that preparation for the workplace in a hands-on manner and while kids are still attending school, is an effective way to accomplish this.

It so happens that the standards-based qualifications system adopted in New Zealand early this century was adapted from the trades schemes. Argue as you may, there is more training taking place in schools today than was delivered there 40 years ago.

It’s now recognised that the vocational access routes available for learners in schools are still not enough, a situation which is driving ‘authentic learning’ schemes into schools. I agree that more of this is now needed. I just wonder at what society is doing to education.

Knowing what to do

Education is supposed to be preparation for life. It has been said by many educators that “education is knowing what to do when you don’t know what to do”. When schools become geared to providing training for kids so that they can step into a job as soon as they leave school, isn’t there a possibility that ‘preparation for life’ will have to be diminished and/or postponed? When does that start if it has been displaced by the need for more immediate training in schools?

Listen to Elliot Washor speak on Big Picture Learning with Kathryn Ryan.

Journey

As teacher, I often tramp the journey. Most of the way, I learn more than I teach, which is fortunate as I’m bored easily.

Even the well trodden-paths bear fruit. But I have to be more aware when pacing there; vigilant, else I miss what is to be learnt.

As teacher, most of what I learn on the well-tramped lanes happens as I watch others less familiar with the paths. This learning is the most enlightening, yet so difficult to pass on to others.

I’ve begun to understand why.

As teacher, all learning is a journey. How can a learner explain the destinations to someone who has never been there and seen what they’re like? There is often no measure to compare, no gradation to gauge against, and no foundation to build upon.

And so learning, once accumulated, is not necessarily always useful. At least, not as useful as we might think it should be. And so it is that the adage of teachers ‘filling jugs’ doesn’t really work, no more than their teaching does.

Johnstone’s Information Processing Model, a simplified version which heads this post, suggests that there is a real need to tread the ways often. It implies that learners may not be wholly aware of what’s to be learnt on the way, nor of its significance even if they were.

It also reinforces that perhaps filling jugs doesn’t work so smoothly, that much is spilt in the process – that many approaches may have to be tried before the jugs contain anything useful at all.

The Tale Shouldn't Wag The Dog

I finished writing a print-based learning resource a few weeks ago. I say finished – it was really only a draft before I passed it on to the editors.

I put a copy of it on the Science database and was grateful for feedback received from our now huge Science teaching cohort. It was good to get comment from colleagues who, like me, had been plunged into the sweat of writing learning resources to deadlines.

Storyline an interest

Among many aspects of the draft discussed was its storyline. This attracted interest from my colleagues as it was seen as an effective teaching feature. It also made me think more deeply about strategies I’d used in creating the storyline in the first place. I realised that I hadn’t really planned it.

It became clear with more thought and discussion that a storyline in a learning resource should not be a starting point for writing. My past experience has shown me that building necessary learning around a plot can be difficult. It’s a manufactured process that has the potential to limit severely pathways the writing may follow.

A different approach

Adhering to a storyline can stymie other useful teaching ideas that may otherwise emerge. It can lead to contrived resource components that do not satisfactorily contribute to effective learning.

The storyline that seemed to work so well in this resource was created using an entirely different approach. Here’s how it came together for a NCEA Level 1 generic learning resource on writing a scientific report.

Staid and boring stuff

I started on the second chapter simply because it was the one I had in mind when I put my fingers to the keypad. My plan was for Peter to write a report of something he did during his holidays and hand it in for his Science teacher to assess at the start of term.

I had practically finished that chapter when I'd already decided that it was boring, principally because it involved doing homework for a teacher. Undaunted, I continued with chapter 3, then drafted chapter 1 and was half way through chapter 4 when I really couldn’t pursue my ideas any more. The resource was becoming all too staid and boring.

Change tactics

I went back to chapter 2, stripped it down and rewrote it introducing a friend, Mahi, an intelligent Māori girl who could write good scientific reports.

It was she, not the teacher, who read Peter’s brief report of his visit to a museum exhibition on colossal squid. It was she, not the teacher, who desperately wanted to see the exhibition after reading Peter’s report.

And it was she who tried to follow the deficient instructions in the report. As a result of these shortcomings, Mahi missed out on her planned visit to the exhibition – all good material to use for teaching about informative report writing.

I switched back to rewrite parts of chapter 1, simply introducing the characters Peter and Mahi. I then flicked to the other draft chapters incorporating the growing relationship between Peter and Mahi as friends who supported each other with their interest in Science.

Here’s how chapter 1 begins:


Reading magazines

Peter found a magazine page that had an article about a car that ran on water instead of petrol.

Activity 1A

Read Peter’s magazine page shown below.


1 What is the name of the magazine that this page came from?
__________________________________________________________________

2 Explain two things you see on the magazine page that might suggest that the information about the car is not true.
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________

Peter wanted to find out more about the car, so he went to a car showroom and spoke with a salesman. The salesman laughed at Peter and did not believe that a car like that could be made. Peter still wasn’t sure if the information in the magazine was true.

3 Explain two other things Peter might do to find out if the story in the magazine is true.
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________

Check the answer guide

Proof of the pudding

The rest of the resource fell into place using the theme of collaborative learning between two school chums. Peter followed up Mahi’s library research into giant squids and colossal squids. He then asked her for help when drawing a graph for the report he was writing.

Instead of Peter handing in his report for a teacher to mark, he chose Mahi to help him improve it. This permitted a chummy dialogue between two friends that was not only fitting and appropriate, but had the potential to enhance learner interest.

All this was laid over a framework of teaching and learning. I got a real kick out of how easy it was to incorporate a story line in a learning resource, and from the supportive feedback I received from colleagues.

The proof of the pudding will be when I examine feedback from learners!

What Are Teachers For?

I have heard it said that teachers are assessors and that assessment is part of the teaching and learning process. I often wonder if the meaning of the word ‘assessment’ is sometimes stretched and perhaps misused in contexts to do with learning.

Standard assessment

At the moment, NCEA standards are of two types. There are Unit Standards, awarded to learners who meet all component criteria for a particular standard. There are also Achievement Standards, awarded as Achieved (just a pass), Merit and Excellence.

Re-assessment

A learner, who does not meet the criteria for a unit standard or for an award in an achievement standard, can choose to be re-assessed after a period of re-teaching and further study.

My thinking has never been aligned with the philosophy of teaching to a standard. I don’t believe that’s what ‘education’ is about. However, as a teacher, I have no choice but to accept the assessment system that is now intimately bound with secondary education in New Zealand.

At least re-assessment permits the learner to revisit the learning and allows the teacher to do some more teaching.

Re-submission

NZQA is reviewing, again, the process of assessing learners for NCEA standards in New Zealand. The specific issue that I bring to this post is the matter of what’s called a re-submission. I’ll explain.

When a learner completes and submits a standard assessment test, and it is not clear to the assessor whether the learner has actually met the standard, the test script can be returned to the learner for amendments to be made. These are performed by the learner under test conditions before re-submitting the test script for the assessment to proceed. This process is called re-submission.

Furthermore, during re-submission the learner is not permitted to study on the topic, nor receive any teaching, coaching or advice associated with the standard, before revisiting assessment tasks they performed in the test script. They are only permitted to consider their answers and perhaps amend them.

There have been many debates by teachers about what constitutes a case for re-submission. Performing a re-submission is quite different from entering into a phase of re-teaching followed by re-assessment. Teachers like things to be cut and dry. So many discussions at the moment centre on the criteria for re-submission.

A disservice to learning

As a teacher who is more interested in what learners learn and how they learn, than what they achieve in a standard test, I am often in a bind over the issue of re-submission.

I feel that I’m doing learning a disservice if I do not permit (and cannot permit) learners to revisit the learning. This is precisely what happens, and must happen according to the rules for standard assessment, when re-submissions are permitted for learners who are very close to meeting criteria for a particular standard.

In such instances, I start wondering what my role is as a teacher and educator. I also wonder if I am doing learners a disservice by permitting them to re-submit their test evidence without any further teaching and learning taking place.

I feel that if learners are in need of more teaching and learning, then they should be given the opportunity to receive just that. This opportunity is denied learners who gain a standard on re-submission.

What are teachers for?

Authentic Elearning?

When a learner is in an environment that is relevant to what's being learnt, the likelihood for effective engagement is high. Learning by doing is supposed to be one of the most successful ways to learn.

Both these factors – the relevant environment and the doing – are thought to provide jointly the greatest incentive for a learner to take interest in what is to be learnt. They form the basis for what is referred to as authentic learning.

For as difficult as such situations are to establish and sustain in face-to-face situations, elearning environments can present major barriers to authentic learning that are almost impossible to overcome unless the elearning vehicles are in situ.

Typical examples of these are online tutors for word processing, graphics applications or other computer functions where the learner is involved in using mouse and keyboard to operate a tutorial directly relevant to the application.

I cite the Southern Hemisphere planisphere with a built-in tutorial as one example of an in situ learning vehicle in a junior Science elearning resource.

The examples given above are all very well, but unless a considerable component of what is learnt is transferable to other purposes, the learning acquired by the learner has limited use elsewhere. One of the characteristics of authentic learning is the transferability of the learning to other situations or disciplines.

Two examples where generic and transferable skills can be learnt are online instruction in touch-typing, and the use of a flight simulator as part of training to become an aircraft pilot.

Kallan and Tuxedo presenting a session in building in SL

Recently I was privileged to share in the facilitation of a session sponsored by ISTE, teaching people online to manipulate and assemble prims, the building blocks of Second Life (SL).

As well, part of the duties I perform as an ISTE docent in SL involves assisting and teaching newcomers to that environment by the use of text and voice chat. The learning facilitated in these situations is authentic.

People who come into SL need to acquire new skills. Most who stay to use that environment want to learn skills that can only be acquired online. But other than exercising skills in associated disciplines such as art and design, the skills I teach to newcomers are only useful in Second Life. And here is the conundrum associated with authentic elearning.

Apart from learning that is directly associated with the elearning application or platform, how is authentic learning achieved online?

A Hierarchy of Learning

My work environment has been a bit noisy lately. The school is having the roof renovated. The building’s air-conditioning units have been disconnected during the time of this refurbishment.

It being summer, there’s a need to open windows and to bring in industrial air-conditioning units to maintain a workable atmosphere within the building. At the beginning of this week, contractors wheeled two of these units into the space close to my office area, plugged them into the mains and switched them on.

Other units were similarly introduced at points throughout the building. The Science teachers who worked in the areas, including myself, viewed this activity with amusement. It was evident that the contractors knew nothing about thermodynamics.

It sucks

The heat exchange part of an air-conditioning unit operates in a way similar to a refrigerator. In normal use, the unit sucks warm air from the room through a cooling unit and filter. Fresh air from outside the building is drawn in through windows and other openings to replace the heated air expelled during the process.

The resulting chilled air is blown back into the room while the removed heat is air-pumped to the exterior, usually through a duct in a window.

In the case of the units that we observed, there was no such venting. Instead, the hot air from the action of the cooling unit was being pumped back into the room. It was as if a fridge had been turned on and its door had been left wide open. In such a circumstance, the fridge does nothing more than make a noise and heat the room.

The overall effect of the air-conditioning units being used in this way was not unlike that of using large blow heaters. In no time, people started to complain about the rising temperature.

A little learning

Science is a wonderful thing. Its principles are being utilised in just about every piece of technology that contributes to our lives today. Of course, an understanding of scientific principles isn’t always necessary to use or install the equipment that puts these principles into effect.

There are at least three levels of understanding that can allow one to realise the significance of a scientific idea, such as the thermodynamic principles that were put to use in the construction of the air-conditioning units:

1. It works provided certain conditions are met according to a recipe for installation. For the air-conditioning unit to be effective, it has to be functioning and have its required vents clear, one of which has to be ducted to the exterior.
   
   
2. It works as it follows the thermodynamic idea that heat can be pumped by using a small amount of energy that is eventually released as heat (which is why the fridge with its door open does nothing more than heat up the room).
   
   
3. It works, and its function can be explained by thermodynamic principles:
   
   a) energy can neither be created nor destroyed,
   
   b) heat energy is released when a gas is compressed so that it condenses to a liquid and this same heat is taken in when the liquid is allowed to evaporate – this is what happens in the heat exchange unit of a fridge,
   
   c) some energy will always be wasted when heat energy is pumped using mechanical means – entropy is always increased as a consequence.
   

Understanding at level 3 can be achieved by senior secondary school Physics students.

Level 2 can be understood by able students of Junior Science.

Working recipes that define the factors that are important in level 1 need only be followed when it comes to the correct and appropriate installation of a piece of technology in general circumstances.

The example that I unpack here shows how related learning can apply at various levels to the curriculum. What is significant is that the most elementary levels of learning are still important to the correct use of technologies that involve sophisticated principles in their design and construction.

Sorted

As it happens, the contractors were notified by Science teachers about the correct use of the air-conditioning units which were immediately switched off. Appropriate locations near windows were then found. Necessary ducts to the outside of the building were fitted correctly to the machines within 24 hours.


Through all this, the cicadas continued their sibilant summer chorus.

Using Elearning Resources and All That Gear

The Particle Laboratory, Teal

This week I submitted my plans for three Science learning resources that I will be writing this year. The expectation is that the drafts will have an online component.

I say drafts, for when the resources are built and accessible to the learner, the intention is to amend and refine them, as part of an ongoing process, dependent on analysed feedback from the learner.

Each resource set will provide teaching and learning material for a learner to achieve an associated NCEA Level 1 Science standard.

There will be no formal question-and-answer written tests for these.
All of them include reporting of a sort – a way that learners can show their communication skills – as well as demonstrating their knowledge and understanding of the various aspects of Level 1 Science.

Learning and assessment

I was heartened by Britt Watwood’s response to my last post on elearning and pedagogy.

He kindly included a link to the Virginia Commonwealth University Online Teaching and Learning Resources Guide, which I read. It was a joy to see the inclusion of the terms formative assessment and summative assessment, with appropriate links given so that their use is unequivocal.

Summative assessment will take the form of teacher assessment, driven according to assessment schemes written against the NCEA Level 1 Science standards. But the formative assessment that has to occur before that will consist of an assortment of methods including:

• self-marked booklet study

• computer assessed interactive activities

• teacher feedback

• creating and maintaining learner engagement.

For those who are unfamiliar with the term, formative assessment is a means used by a learner to reflect on what’s been learnt and understood. A course of action may be followed to do further learning if required.

Formative assessment can involve a teacher who provides feedback to the learner. It takes the form of automatic computer feedback in interactive elearning resources. Or it can be a checklist of answers or explained processes to supplementary examples given in a printed resource book.

A rudimentary example of formative assessment is a list of answers to clues in a crossword puzzle.

Down to Earth

One of the standards involves investigating an astronomical or Earth science event. When writing my draft resource for this standard, I will be pulling on all appropriate techniques in elearning that are available to me and my cohort of learners:

• keeping a balanced approach to what is e-offered to the student

• keeping in mind the fundamental principles of creating and maintaining elearning engagement

• elearning pedagogy (Oops! Did I really use that description?)

• use of existing resources appropriate to the teaching, such as the learning resource on drawing scientific graphs – this resource provides computer created formative feedback

• use of existing internet resources such as the wealth of specific webcam portals – these resources can be incorporated as a series of possible internet links

• use of possible internet resources filtered through the use of specific, teacher created criterion focused internet searches.

I will keep in mind the usefulness of games-based learning and will try to remember all that’s considered to be elearning myth.

Elearning Patterns and Predictions

The Band Shell - ISTE Island

In 1972 I took my first permanent teaching job in an elite Edinburgh secondary school. Just the year before, it had been a private school. James Gillespie’s High School (JGHS) always prided itself on its high standard of education.

A maximum class size was among many standards that JGHS maintained. It strove to have no more than 25 learners per class. The belief was that a finer relationship between teacher and learner could be attained. Through this practice, significantly better educational successes were achieved.

The number of scholars who left JGHS to take up business careers, or went on to higher education, was proof enough of this accomplishment.

It was a fantastic start for me as a teacher to be with classes of this size. But it was certainly no preparation for the learning environments that confronted me in teaching positions I took up after that time.

In 1974, I had a form class of 36 learners in a school that maintained an average class size of 35.

An elearning counterpart

Almost 30 years later, research into educational achievement through elearning methods suggested that there was an optimum size for elearning groups led by a teacher.

In any group of elearners, there will always be those who can be considered ‘active’ – others who learn despite their apparent inactivity in engagement with a teacher or facilitator – and those who are neither active nor engaged significantly in learning. Time has to be distributed fairly in attending to the needs and wants of each these groups.

An e-teacher who has more than 15 to 20 ‘active’ learners in a group is always extremely busy. When the number of active learners increases much above 20, strategies have to be developed and practiced to cope with the constant learner-teacher activity that inevitably occurs in those environments.

Symptoms of stress and exhaustion are inevitable in any teacher who succeeds in engaging enough learners so that the active group comprises much above 25. My own experiences of this have been confirmed by those of my colleagues teaching in e-environments similar to my own.

A recent experience

The Band Shell - ISTE Island

Recently, I’ve been actively engaging in discourse with groups of people in Second Life (SL). In the months that I’ve assiduously studied in this networking environment, the number of ‘friends’ in my Friends List has slowly increased. The size of my list is now well over 60.

Within that group is a sub-group of 15 to 20 friends who actively use instant messaging (IM) to contact me whenever I am online. It is a great experience to network with people in this way. Not all of this is done locally – that is to say, my avatar is not necessarily appearing on the same screen as the avatars of those who are IMing me.

Of late, I have found it difficult to be online while following any intended single pursuit. Two weeks ago, I became an ISTE docent, as avatar Kallan. In adhering to the expected commitment that comes with this office, I discovered that my group of ‘active’ friends caused me significant concern when attending to docent duties on campus.

At first, I was torn between being seen to be ignoring my friends, while attending to a duty which I enjoyed. Even sending messages of apology to IMing friends was an activity that I found distracting while attending to docent duties.

I’ve since learnt from other colleagues in similar SL situations that they often simply ignore the incoming IMs during the time that they are occupied with more immediate activities. Obviously they too have difficulty reading and responding to incoming IMs when engaged in other cerebral activity.

Elearning environment

In translating this to the elearning environment, I can see clearly how a teacher can become stressed and overworked. There is the part that is played by commitment. There is also the aspect of prioritisation.

Who gets priority from the teacher in a learning environment?

When a teacher is actively assisting several learners together, how does she cope when a learner puts forward, out of the blue, a desperate plea for help?

What strategies can she employ to ensure that this learner, who may well be one who has never before communicated directly with her, gets the necessary immediate support?

What provisions must administration provide in a school to ensure that the number of active learners in a teacher’s group is kept to a manageable level?

How can administration ensure that teachers are not forced to adopt strategies to disengage themselves from needy learners in order to protect their own stress levels?

On Creativity

I took my daughters out for a coffee the other day. Just before we left, Hannah who is age 19 and the older of the two, stepped out of her bedroom looking like a Vogue model. I was charmed that she took time and effort to change before going out for coffee with her old dad.

“I like your outfit, especially the belt”, I said as we got into the car.

“Two dollars in a second-hand shop”, she quipped with a cheeky smile.

That’s one way a second year Art student at AUT demonstrates her creativity to the world – by choosing appropriate clothes to dress smartly on a minimal budget.

Did she have to study Art to express her imagination this way? Certainly not. Anyone can do it if they have a mind to.

But Hannah worked very hard to secure her place in her course at AUT. When she studied Art at High School, she created many different things. Not everything she did was as successful as she’d like it to have been. But she did it all the same. It all contributed to her portfolio – time after time.

To me, the difference between the artist and the non-artist
is that the artist is the one who does it.
– Helen Garner

If a thing’s worth doing . . .

When my son, Jack, was to be married a few years back, I wrote a waltz in honour of his lovely Irish bride, Ailish. She had told me that she loved dancing to waltzes. As it happened, she liked the tune I wrote for her. I commissioned a local Irish dance band to play at the wedding.

A musician friend of mine who was a member of the band obviously didn’t like my waltz. When the band was rehearsing it under my supervision, he asked me why I composed stuff like this.

I explained that the alternative was that I did nothing at all. Then the bride would not have her own waltz for her wedding day. He quietly picked up his violin and played the music – beautifully.

Creativity has to be nurtured

Creativity is so delicate a flower that praise tends to make it bloom, while discouragement often nips it in the bud. Any of us will put out more and better ideas if our efforts are appreciated.
– Alex F Osborn

The connection between Art and creativity is so intimate, it is almost impossible to define a difference between the two. One thing is clear, and that is that creativity has first to exist before Art comes into being. Creativity tends to be the easier to recognise, while identifying Art can often be an elusive and subjective assignment.

Schools have a big part to play in encouraging creativity in learners.
But they are not the only important influences, and this has been demonstrated by the work of some of the world’s greatest artists, many of whom were influenced by factors well beyond the precincts of the school grounds.

The importance of what is within us

The mind inhabits a complex organ. The self has to be nourished from within. Two undoubtedly brilliant artists in two discrete disciplines, who lived their lives in different countries and in separate centuries, had very similar views on what nurtures and brings out creativity from within. Check them out.

Go cherish your soul;
expel companions;
set your habits to a life of solitude;
then will the faculties rise
fair and full within.
– Ralph Waldo Emerson

When I am . . . completely myself, entirely alone . . . or during the night when I cannot sleep, it is on such occasions that my ideas flow best and most abundantly.
– Wolfgang Amadeus Mozart

On the Hypotenuse

The area of the square on the hypotenuse of any right-angle triangle
is equal to the sum of the areas of the squares on the other sides.
Pythagoras' Theorem

I was hopeless at Mathematics when I began High School. Teachers despaired at my ineptitude. But I had a natural interest in Science.

It was mainly due to good Scottish teaching, and a genuine willingness on my part to engage in learning, that I studied Mathematics to a level that let me reach the highest academic levels in Science.

It was also due in part to the curriculum I followed. I believe that it taught me how to think. It wasn’t that it taught me to think. I could do that. It taught me how to think, and to be able to think in specific ways.

In the 1970s, I taught High School Mathematics to senior level.

Recently, I have become familiar with fundamental changes in the way mathematical thinking is taught in High School in New Zealand.
I suspect that these are similar to changes in the Mathematics curricula in other western countries.

Get it right

The Euclidean proof of the useful Theorem of Pythagoras was one of the things I had to learn for my early qualifying examinations at High School. My mother learnt the same proof when she attended High School.

Euclid’s proof was taught in New Zealand schools until changes were made in the curriculum towards the end of last century (20^th). It is now supplanted by what I’d call a non-proof. What young minds are required to study about the Pythagorean Theorem in New Zealand today is certainly not a proof of it.

Euclid’s proof is based on the properties of rectangles and of triangles that are each of the same shape and size – as well as associated geometry ideas involving these shapes drawn between parallel lines. There are several hundred proofs of this fundamental theorem. Some are outlined in Wikipedia. A few I learnt at High School.

All of the proofs I learnt permitted me to understand, not only how the Pythagorean Theorem can be proved, but also what it is to know the significance of a universal proof. It is a way of thinking that permits a learner to appreciate that a proof needs to be watertight.

What does it prove?

Euclid’s proof can be applied to ANY right-angle triangle, not just a few special triangles. Therein is the difference between what used to be taught and what is now taught.

In High School today, a learner in Mathematics is instructed on how to show that a given right-angle triangle has the Pythagorean property. The instruction is not about any right-angle triangle, but applies only to a particular right-angle triangle that the learner draws.

Essentially, learners follow a recipe that shows that the Theorem may work for their triangle. It provides no real understanding of proof.

Here’s what they are instructed to do:



Learners in Mathematics are shown how to draw a right-angle triangle on a grid, using simple mathematical drawing equipment. They are also shown how to draw squares on each of the three sides of the triangle. By counting the number of grid squares that make up each of the squares, they can show that the Theorem is followed approximately.

Higher thinking

While this recipe permits the learner to practice skills in using drawing equipment, it provides no understanding of a mathematical proof.
It does not even show conclusively that the Theorem works. However carefully it is done, not every attempt at adding the squares will show that the Theorem is precise. Check out the squares shown above.

One can argue over the need for the knowledge of how to prove the Pythagorean Theorem. But the significant learning is nothing to do with that knowledge. The thinking skills learnt that permit the learner to understand what a watertight proof is all about are really what are far more useful and relevant to higher thinking skills.

This is the whole point of teaching the proof. It has the potential to permit the learner to realise the significance of a theorem that can be applied universally, and why it has this property. It is a way of thinking analytically that is not being taught today.

Elearning Prediction / Hope 2010 – Next 10 Years

My prediction and hope for 2010 are tempered by what I learnt and gathered in 2009 and before that. However, my hope does not match my prediction.

I suspect that what I hope happens will not come to pass.

I’ll not dwell on the particular. That is too narrow for the future. Instead, I’ll attempt to look to the bigger picture of where a Nirvana in elearning might lie.

Surmise

Commercialism and consumerism will rise with renewed vigour, despite the recent and global economic collapse that many experts say was brought about through, among other things, flagrant practices of commercialism and consumerism.

Rather than learn from past errors and misguided pathways, society will resume its hazardous journey and continue to career along a wavering and obstacle-strewn path to uncertain success.

Why do I feel this way? There are many reasons. I will cite only one elearning example here.

At the beginning of last decade (2000 – 2003) I watched the rise and fall of what might have been a brilliant concept in learning resource development – that of the learning object.

I may be wrong here – I don’t think I am. But my feeling is that financially pushy commercial factors, far larger than the budgets for learning itself, launched into the sky and eventually nosedived to destruction what could have been a worthwhile elearning concept, in the form of the learning object.

Pandora relic

I have one hope for this year and for the rest of time.

Sheryl McCoy’s recent post, Another Balkanized Technology Rip-Off, puts into words exactly how I feel about the way commercialism and consumerism have continued to hinder the betterment of society.

The drive to sell, through a strategy of planned obsolescence despite genuine need, continues to come in the way of establishing real expertise in the use of technology. It stymies creativity. It comes in the way of progressing to better things, while purporting to advance and progress towards improvement. It wastes time, resources and money.

I have worked through a decade of watching incompatibles, non-connectables, lack of connectivity and even incompatibility between different versions of the same commercial devices/applications/appliances.

I’ve wracked my senses, grappling with upgrades to versions of machines, computers and their applications. All in an attempt to continue to use these for the purposes that I had worked to acquire consummate ease in and reasonable expertise in.

What I discovered was that I was grounded, once again, when the latest version (of whatever) was released. It made me consider seriously and review any further dubious opportunities for ‘upgrading’.

Valuable opportunity

I hope that we can learn from the mistakes of the last decade, and of last century. Let’s not push mistakes into the past as history to be forgotten. Let’s not claim we are assured success by simply ‘moving right along’.

Let’s start putting to good use what can be salvaged in learning from our past mistakes and successes, and move to a richer and prosperous future.

Thinking and Learning

A well-meaning friend once lent me a book and said, “You should read this. You’ll learn how to become very rich.” He also told me that the secret clue to becoming wealthy was found on almost every page, and that it also occurred several times on the first page.

I read the book. It is well written – a study of human nature – an interesting compendium of anecdotes and tips, directed mainly at sales-people, but not exclusively. I found the clue to becoming wealthy several times before I’d even finished reading page one.

But it didn’t help me become wealthy.

The book was Think And Grow Rich by Napoleon Hill. The clue centred on the word ‘desire’. To become rich, first you must have the desire to do so.

Of course, the main reason I could not use the book for its intended purpose was that I had no burning desire to become wealthy. I’d like to be, but the necessary burning desire isn’t there.

The book did make me rich, however, but not in a financial way. It made me think about how some people can be so engaged in learning, they suck up skills and knowledge as if they were vacuum cleaners. It gave me clues as to how some people seem to learn, almost as if by instinct, and never stop learning.

Teaching and learning

When I taught Mathematics at Rongotai College, Wellington, I always asked for an Upper Fifth Mathematics class when classes were allocated to teachers at the beginning of the year. These classes were of students who had failed to qualify in Mathematics the year before.

Every year, I had a lot of fun with the learners in the Upper Fifth. I really enjoyed teaching them. I used to hold classes after school for those learners who felt they needed some extra help and tuition. It was rewarding. The after-school class was always full, though not all who attended were from my own Mathematics class.

There was one attribute common to all who attended.

Hooks for learning

Teachers speak of engagement, and practice strategies to improve student engagement in learning. Some put a lot of effort into stimulating interest in their learners and this is admirable. They look for and find the hooks that catch some learners and get them engaged.

Learning, and achieving through learning, is a bit like growing wealthy. There are many contributing factors, not all of which are related necessarily to innate ability. The learner who has a burning desire to learn will learn, despite apparent handicaps, whether it is learning to become a musician, learning to play chess or learning to read.

But to do this, they must also be able to think to apply their desires effectively. It is in teaching learners how to think that permits those who have the desire to learn to reach their goal. It ignites the fire of learning within them.

Few people think more than two or three times a year. I have made an international reputation for myself by thinking once or twice a week. – George Bernard Shaw

Compassion - What I Learnt About Fishing

As a child, I was never aware of any real ability I had, other than to get into mischief or catch cold. I discovered these rich talents early on. Even as a ten year old, I bumbled along with no appreciation of any real capabilities I might have had as a human being.

My grandfather used to take me fishing. I got a buzz from the wonderfully natural places we visited. My head got filled with the summer sounds and scenery of these spots. But I was hopeless at catching a fish, having too much compassion for the poor hapless creature to get any enjoyment from the event.

I recall stabbing my finger with a fishhook, being more astonished at how easily the needle-sharp device entered the tissue than the searing pain it caused. It stuck firmly, deep in my fingertip.

My grandfather was annoyed and looked at me sternly for a moment. He fumbled in the pocket of his fishing jacket and took out a small pair of pliers, holding them tightly in his hand.

With his other hand, he grabbed mine and lifted the injured finger so high that my shoulder hurt. I watched to see what he would do.
I thought he might pull out the hook with his pliers but I was too dumb and curious to close my eyes and brace myself.

He deftly pinched the hook in the jaws of the pliers and gave it a powerful and sudden twist. What I saw made my eyes pop.

Contrary to what I’d hoped, he didn’t pull out the hook. Instead, the business end of it reappeared through the tip of my finger – a tiny fluted barb, tinged with the blood that dripped from the newly pierced hole.

Grandfather carefully snipped off the barb with his pliers and swiftly pulled out the remains of the hook. He explained that the barb would have torn my finger apart if he'd remove it the same way it went in.

As I held my sore finger, wrapped tightly in a piece of bandage,
I reflected on what it might be like for a poor fish who unwittingly takes the bait.

I stood beside a brooklet, that sparkled on its way,
and there beneath the wavelets, a tiny trout at play,
as swiftly as an arrow, he darted to and fro,
the gayest of the fishes among the reeds below,
the gayest of the fishes among the reeds below.

Angler there was standing, with rod and line in hand,
Intent upon the fishes, a sportive fearless band,
“`tis vain” said I “good neighbour, to fish a brooklet clear”
The fish will surely see you upon the bank so near.
The fish will surely see you upon the bank so near.

But skillful was the angler, and artful too,
The crystal brooklets depths defying, he hid the fish from view, and then he skill renewing,
the fishes unheeding took the bait,
and I was left lamenting, my tiny troutlet’s fate,
and I was left lamenting, my tiny troutlet’s fate.

Video - The Trout
 
Daniel Barenboim, Itzhak Perlman, Pinchas Zukerman, Jacqueline du Pré and Zubin Mehta