A Rate of Learning

Acknowledgments

The inspiration for this post comes from this tweet by Heather Julian (Twitter @mathrules13) and re-tweeted by Jill B (Twitter @jillgrafton):mathrules13

Background

I will take it as a given the readers are familiar at least one set of skill/attainment level descriptors going from “can recognise” to “can exploit”, and agree that different people have different levels of recall for different things. I will also take it as a given that readers are familiar with Maslow’s hierarchy of needs.

This post does not refer to any publications, scholarly or otherwise. It is merely one person’s thoughts, and it is intended merely to provoke discussion.

The Teaching-Learning Equation

There are a number of phrases associated with the acquisition of skills and knowledge from a source outside the learner, each phrase having its own level of acceptability in different contexts. Examples include “teaching”, and “delivery of learning”. My own sense is that such phrases can obscure the dynamics of the teacher-pupil situation. Given the context implied by Heather’s tweet, I propose a scenario where the teacher:

  • has full command of the subject
  • is totally self-confident
  • has excellent communications skills
  • has good empathy
  • has a good grasp of the pupil’s current knowledge base in the subject

(I can hear some of you thinking back to your own school days and thinking “I wish!”.) Also, to do justice to Heather’s question, I think that we need to focus on what is going on in the pupil’s mind (and here I am forced to rely on my own memories of school, as augmented by how some of my own students have demonstrated learning).

A Model of Learning

My own model of learning is constructivism. In the context of numeracy, a grasp of whole numbers is necessary before the idea of fractions can have any meaning, for example, and the rest of this post is written accordingly.

I find the words “truly remember” and “meet our standards” to be thought-provoking, as it raises questions about what level of memory is being envisaged, and whose standards are being applied during assessment. Using the “quadratic formula” as an example, memory may range from recalling “I know that phrase means something” to being able to write it down upon the instant (as well knowing what it is used for), and its consequent relationship to skill/attainment level descriptors.

Extending the words “truly remember” and “meet our standards” to what the learning experience means for the pupil can also be revealing. Again from personal experience at high school, I was in one cohort where everybody was expected to learn lines of poetry and recite them in class. I learned what I needed to in the preceding 48 hours, recited, and promptly forgot them one hour later. In terms of the teacher’s expectation that we should all enjoy poetry, it was for me a disaster: his standards were irrelevant to me at a personal level.

At the opposite extreme, and at the same school and at about the same time, I had a number of experiences that I found to be highly relevant. I have room here to describe only one of them. The subject was mathematics. The class had been working though material for a few days, and I was not quite sure of how all the bits and pieces fitted together. Towards the end, the teacher spoke a single sentence and everything suddenly fell into place. That learning has stayed with me for over 40 years, this in stark contrast to the poetry experience.

Discussion

Teachers may have aspirations for their pupils. Sometimes those aspirations are selfish. At other times they are focused on the pupil. I have memories of both from my school days. It is for this reason that I posited the scenario above.

We are now into the complex area of the teacher’s motives, skills, knowledge and abilities and the interaction of these with the pupil’s motives, skills, knowledge and abilities. I argue that it is this complexity that rules out any simple answer or answers to Heather’s question, and that perhaps we should instead be asking ourselves “What can I do to help this pupil towards self-actualisation in as quick and as efficient way as possible?”.

Conclusion

It’s not an easy job!

A Philosophy of Instructional Design

Acknowledgment

The inspiration for this post comes from Temitope Ogunsakin and his comment on this post where he asks “What would you say is your Instructional Design philosophy?”. This post is by way of answering his question.

Objective

The objective of this philosophy is to make the learning experience as easy and enjoyable as possible for the learner. This has two outcomes. Firstly, it maximises the effective of the learning experience for the learner. It also fulfills part of the social contract between educator and learner, that of mutual respect.

This philosophy uses a number of guidelines as a way of meeting that objective. Discerning readers will notice that I have avoided using the word “rules”, as rules tend to be prescriptive and thereby interfere with reaching the objective.

Assumptions

This philosophy makes a number of assumptions about the learner. At the risk of perhaps stating the obvious, it may be worthwhile making those assumptions explicit, and it provides readers with an opportunity to challenge those assumptions.

All Learning is Built on Previous Knowledge

While the claim “All learning is built on previous knowledge” may seem bold, and it sidesteps the question of how babies start acquiring knowledge, I would argue that it is a useful starting point for discussing later learning. By way of example, the teaching of grammar relies on learners being able to recognise sentences, which in turn relies of the recognition of words. In a similar fashion, the ability to use money relies on (among other things) the ability to recognise and understand the meaning of digits.

Different Learners Learn at Different Speeds

It is perhaps a common mantra that different learners learn at different speeds, so it is worthwhile checking a few sources to see how widespread this view is. A quick search on Google found this from the Board of Studies Teaching & Educational Standards NSW, this from the Australian Curriculum, Assessment and Reporting Authority, and this from the New England Complex Systems Institute, all of which support this view.

Adults and Children Learn Differently

Adults bring a number of skills and personal attributes into the learning situation that can be harnessed to good effect. These include, but are not limited to, life experiences across a wide range of subjects, independent research skills, and a thorough grasp of what they want to achieve from their learning.

All Learning Requires the Sharing of an Idea

With the claim “All learning requires the sharing of an idea” I can see readers saying “But what about psycho-motor learning? And affective learning?”. While subjects such as pure mathematics and philosophy are entirely cognitive, and the claim is unlikely to be challenged in those arenas, I would also argue that “the idea comes first” in both other areas. In the case of psycho-motor skills, the idea is often shared by way of demonstration, and followed by learner practice Examples include how to use a hammer, how to drive a car, and how to use a paint brush. With affective learning, the idea is often first shared by asking a question: “How would you feel if …?”, followed by a discussion. Agreeing the ground rules with a new cohort is another example.

Tasks Analysis

I have designed and used task-analysed teaching material from nearly 20 years, and I have found it to be one of the most effective tools in my educational toolkit. I have found that the ideal length of a learning task is between five and 15 steps.

The Guidelines

  • Audience identification: Unless I have a very good idea about what the learners already know, and what it is that they wish to learn, I have nothing to work with: I have no foundations upon which to present new ideas, and I have no direction as to which ideas I should choose. By way of example, I delivered learning in Microsoft project a group of older people who were experienced project planners using manual techniques and who wanted to learn how to use Project instead.
     
  • Matching the learning material to the learners’ existing knowledge base: Using the same scenario as above, I learned that my audience were in the business of installing water distribution infrastructure, so examples, exercises and questions about pipe laying, construction of pumping houses and supply of electrical power were obvious choices.
     
  • Bite sized chunks of learning: My comment on the ideal size of task-analysed activities translates well into other learning activities when it comes to planning the amount of material to be shared between signposts. Even in an extended whole group discussion, the learners themselves can raise such signposts, and the learning goes on to new ideas.
     
  • Structure: I regard structure as crucial. This is a direct consequence of the idea that learners build their new understanding on the foundation of previous knowledge.
     
  • Humour: Humour can be used tool, though not all learners seem to recognise it at the time. One cohort only came to understand the significance of a Mr Thread as the CEO of the Brazil Nut and Bolt Company in their coursework once they had finished their exams. Other humour can be injected ad lib while speaking.
     
  • Personal relevance: Without personal relevance, the learning material is likely to be less effective. The mathematics of the motor car was so interesting to one youth-at-risk that it turned his life around and he became a productive member of society.
     
  • Relevant technologies: The idea of of using technology as a ploy to engage students strikes me as at best silly. Having said that, I will make heavy use of technology where it is relevant. With the Microsoft Project course, it was appropriate to make the following design decisions:
    • All the learning materials, apart from Microsoft Project itself, were entirely web-based. There were no paper handouts.
    • All learners had individual computers, each with a copy of Microsoft Project already loaded.
    • All learners were invited to bring one of their current projects along with them using whatever technology suited them.
    • All learners were expected to work with both Microsoft Project and the web-based learning materials open on their computer at the same time.
    • All learners were expected to leave the course with a fully functional copy of one of their projects in Microsoft project.

    For the record, the course was extremely successful.

  • Immediate feedback: Some subjects, such a computer programming, Excel and Microsoft Project, have immediate feedback as an inherent quality of the product being learned. Immediate feedback is now normally built into web pages where this is relevant. When I write such pages, I build in such feedback at the design stage.
     

A Final Comment

I would never present all of the above as a single serving to anybody who was learning about instructional design. Having said that, it would not surprise me if experienced instructional designers read it in less than a minute, and then wanted to add their own ideas to the material presented here.

A Foray Into Learner Experience Design

Acknowledgment

The inspiration for this post came from Joyce Seitzinger (web site: www.lxdesign.co, twitter @catspyjamasnz) and her leadership in the arena of learner experience design.

Background

I have developed a range of different materials for different subjects since the mid 1990s. In terms of generating a positive emotional response, the most successful of these was a paper handout where learners had to puzzle out for themselves how to assemble fragments to computer code to achieve a required outcome. The responses ranged from quiet, but still audible, expressions of satisfaction to loud exclamations of success. But for all the materials that I developed the concept of learner experience design was something of an assumption: I was too focused on producing material that was concise, complete and accurate. The time has come for me to turn this around and produce something focused primarily on learner experience design, and treat the subject matter as a given. (The result of this exercise can be found here.)

Discussion

I wanted to design and build something completely from scratch. This immediately eliminated all existing learning management systems and other software aimed at creating interactive learning experiences. This had a downside: learners would not have a record of their progress.

There then came the issues of the target learners, and the subject matter. My own work with adult learners immediately suggested this group, and my own familiarity with mathematics suggested the topic of fractions.

The criteria for success came next. I chose the following:

  1. How well did the result reflect the expectations and background of mature learners?
  2. Did the result strike an effective balance between visual monotony and visual overload?
  3. How well did the material draw on experiences that mature learners are likely to have?
  4. How easily would learners be able to navigate their way around the material?
  5. Was the material chunked logically?
  6. Was each chunk of an appropriate size for the concept or concepts that it contained?
  7. Did all the material make a coherent whole?
  8. Were there opportunities for self-assessment?

Experience

My original intention was to provide a complete guide on working with fractions (addition, subtraction, multiplication, division, and simplification) and designed the front page accordingly. When I came to writing material for each of the chunks, two things became very apparent. The chunks were in the wrong order, and there was too much material for all the navigation points to be shown on a single display. This failed two of the success criteria: navigation, and logical chunking. Being as I was both a content creator and a subject expert, I took the decision to omit multiplication and division from the result. I also discovered that I needed an extra chunk to precede the multiplication and division chunks that I had not thought about when I first chunked the material. While the original order might have made sense from the viewpoint of keeping related ideas together, it would have been a disaster from a pedagogical viewpoint.

The choice of what examples to use also arose. My own experience of materials about fractions left me feeling somewhat jaded: just how often are pizzas divided into equal segments in these materials? I used examples and photographs of items in my own house: coins, a lemon, a box of eggs, and an empty avocado tray. I also produced a graphic of a fuel gauge.

I had a very particular idea of how I wanted fractions to be displayed. All the usual packages failed to match my requirement, so it was a case of write my own application to do that, and use GIMP to process the images into something suitable for display on a web page.

Criticisms of the Result

While the result could be described as “adequate” when it comes to the success criteria, the following observations could be made:

  1. The result will not work on mobile devices: the required display size is too large, and the result has no mobile equivalent.
  2. The learner requires an HTML5 web browser.
  3. The use of an avocado tray in the context of an avocado farm probably lies outside the direct experience of most people.
  4. Some learners may find the style too terse.
  5. The chunk on simplifying fractions properly belongs to the (non-existent) pages concerned with multiplying and dividing fractions.
  6. The result has not been trialed.
  7. The result is not compliant with standards such as SCORM.

Lessons Learned

The next time that I prepare such material, I will be able to do so with a more informed perspective.

An Essay on Instructional Design

Acknowledgments

I thank both Jo Hart (Twitter @JoHart, blog johart1.edublogs.org/) and Michael Graffin (Twitter @mgraffin, web site www.mgraffin.com/) for inspiring me to write this post.

On Learning About Instructional Design

When Jo mentioned instructional design in conversation, I realised that while I was acquainted with the term and that it was to do with designing and building educational experiences for learners, I was quite ignorant of exactly what it is that constitutes instructional design, and I regarded this as a quite unsatisfactory state of affairs. A little bit of research soon threw up the following resources:

As I worked my way through these articles, I realised that I had been here before 40 years ago, though in a different context. I was looking at how systems are first analysed, and then designed.

On Analysing Systems

In the business context, the phrase “systems analysis” means producing a model of how part or all of the business works, usually with a view to improving the way that the business operates. This is then followed by “systems design” with the outcome that existing systems are modified to meet the current business objectives, usually increased profits in the case of commercial business.

Translating this into the educational arena, this means producing a model of how education works at the classroom level, and then designing educational experiences to meet the current educational objective, to wit better educated people.

Further Comparison

There are many formal systems for undertaking systems analysis in the business context. This matches the plethora of models of how people learn.

There are many systems design methodologies in the business context. Again the same is true in the educational context: one has only to look at the differing viewpoints in the five links above.

As one trainer in systems analysis and design once said to a class of which I was a part, it is about having a toolbox of methods, and choosing an appropriate method for the situation that you are currently looking at. I think the same holds in the educational context.

The Author’s Toolbox

The concept of instructional design is relevant to all three domains of learning: affective, psycho-motor, and cognitive. However, due to my limited experience in two of them, I address only the cognitive domain.

The model of learning that is presented below results from using the following tools:

  • Observing how learners succeed, and how learners fail, when going through a learning activity.
  • Observing learners’ emotional responses to a range of learning environments and learning activities.
  • Observing the differences in knowledge between different learners in the same cohort.
  • Observing the different rates at which different learners learn.
  • Observing the different different types of question asked by learners of different ages.
  • Observing how students react to different styles of presentation.
  • Observing group dynamics
  • Observing how learner motivation changes over time, and considering what might be causing those changes.
  • Observing how understanding of high school mathematics is dependent on understanding previous concepts.
  • Observing similarities and differences between academic endeavour and commercial endeavour.
  • Observing change of subject matter over time.
  • Asking questions.

A Model of Learning

I claim to have some learning in the areas of mathematics and of information technology, and I use both contexts in what I am about to say.

It seems to me that it is impossible to have the concept of a fraction until you have grasped the concept of a whole: there is a whole cake, take a piece out of it, and you have a fraction of a cake. The same applies to information technology: until you are acquainted with spreadsheets, the term “cell address” is meaningless. In both cases, understanding the second idea is crucially dependent on having a good grasp of the first idea. This is a constructivist approach.

There is also the question of “Why bother to learn?”, this addressing the issue of student motivation. This plays a crucial part when it comes to designing delivery activities and materials.

Relevance of information also plays a part. Applied mathematics changes very slowly, while practice in information technology changes very rapidly. Information that was relevant 10 years ago may be entirely irrelevant today: the floppy disk serves as a paradigm for this. This has implications not just for the professional development of educators, but also for the materials that they use.

The way that learners prefer to study also differ. At one extreme, there are learners who much prefer to work through a learning activity on their own. At other extreme, some learners need a lot of support from their teachers and peers.

Devices for Developing Learning Activities and Materials

The ideas offered below are neither complete nor prescriptive. They again come from my toolbox. Readers must decide for themselves which of those ideas are relevant to their own context.

  • Identifying goals, sub-goal, and sub-sub-goals until you end up with something small enough to be a single learning activity. Verifying that the sequence in which the material is going to be present is logical. This is an iterative process that continues until the summative assessment, if any.
  • Project management:
    • establishing existing levels of knowledge in the learners
    • identifying which learning goals will be achieved by when (aka scheduling)
    • identifying the human resource implications (crucial if you happen to be team teaching)
    • identifying costings where appropriate.
  • Identifying the resources needed for any learning activity, and either locating same or preparing your own.
  • For every planned learning activity, checking the following:
    • relevance
    • completeness
    • correctness, particular for task-analysed activities
    • unambiguousness
    • accessibility
  • Incorporating feedback from the learners into existing and planned learning activities.

In light of the increased use of information technology in learning activities, it is perhaps worthwhile going into some detail about making online learning activities more accessible. At the risk of stating the obvious, merely converting an existing printed document into an online form does nothing to increase the accessibility of the material being learned. The advantages of an online learning environment include, but are not limited to:

  • Immediate feedback on assessment tasks
  • Access to live data
  • Increasing or reducing the challenge presented to the learner on the basis of the learner’s answers (adaptive assessment)
  • Opportunities for independent research
  • Choice of route towards a learning objective
  • Audio and video material
  • Being updated for new information – current news stories are relevant here
  • Being updated for correctness

The choice of route towards a learning objective takes on an even bigger role in online learning. While printed material tends to very linear, online learning lends itself to having multiple pathways, so signposting becomes very important.

An Invitation

Please add your observations and additions in the Comment box below. Thank you.

Silos and Connectivism

Acknowledgment

My thanks go to Shelly Terrell (Twitter: @ShellTerrell, blog http://www.shellyterrell.com/about.html) for prompting me to write this post.

Background

The story starts with an experience towards the end of my high school career when I was studying physics and chemistry (among other subjects). During one particular week the topics being taught were so closely related that I imagined that the two teachers were talking to each other about what they were teaching. I asked one the teachers about this, and he said that they both worked quite independently. This left me with a puzzle for over 45 years. With current discussion in the educational community on the matter of silos and connectivism, now seems like a good time to give that puzzle an airing.

Of Subject Interdependency

Subjects at most high schools are taught quite independently of each other, and yet the links between them are obvious. The study of literature is crucially dependent on a thorough understanding of the language in which the literature is written, often English in English speaking countries. Similarly the study of science is dependent on a reasonable grasp of mathematics.

Speaking from my own background in the sciences, I was learning the mathematics that I was to need in science classes typically one or two years beforehand, a subject that I have always enjoyed. As a consequence, I found the mathematical content of science classes to be trivially easy.

Development – Part 1

There is a theme that I have encountered regularly for decades which is that school leavers avoid reading physics or chemistry at university because in their minds they contain too much mathematics, and instead opt for one of the “softer” sciences such as environmental science or psychology.

There is an area of mathematics known as “statistics”. Understanding statistics depends on first understanding some other “simpler” areas of mathematics. (If you must know what those areas are, they are algebra and calculus.)

At the time of writing Swinburne University of Technology offers a Bachelor Degree in Psychology with a compulsory unit “Foundations of Statistics“. Also at the time of writing Curtin University includes what I would regard as high school mathematics, but without any statistics, in its physical science courses. These two examples come from a few minutes research on the Internet, and reinforce the idea that softer sciences need more powerful mathematical techniques than the physical sciences to obtain meaningful results at the undergraduate level.

We now have all the ingredients necessary to describe a potential problem. Anybody leaving year 12 and opting to study a “soft” science on the basis of their weakness in mathematics is making a big mistake. The cause is perhaps obvious: high school biology, for example, is less about mathematics and more about form and function than either physics or chemistry, leading students to think mistakenly that this will carry over into their university studies.

Development – Part 2

The preceding section identifies two issues. The first issue is the apparent compartmentalisation of knowledge, and is the burden of this post. The second issue of people not understanding what is needed to study soft sciences is beyond the scope of this post, and may be the subject of a future post.

The teaching of different subjects by different teachers is a paradigm based on silos of knowledge. It is up to the student to develop an understanding of how those subjects are related. The idea of a holistic approach to teaching in high schools seems to be regarded as being revolutionary. Finland may be the first country in the world that has addressed the issue of helping students understand the links between traditional subjects. As the Independent newspaper puts it “Subjects scrapped and replaced with ‘topics’“.

This then raises the question of what we should be asking our high school teachers to be teaching. While many teachers might feel threatened by this, I would have expected each of my physics, chemistry and mathematics high school teachers to be comfortable teaching across all three subject areas. A similar case could be argued for history and geography. You could probably suggest your own combination of subjects.

The Future

The reaction from teachers in being asked to engage in cross-disciplinary teaching is perhaps predictable. It came as no surprise to me that the target of the above link includes the words “the reforms have met objections from teachers and heads“. But as Marjo Kyllonen, Helsinki’s education manager, has said “There are schools that are teaching in the old fashioned way which was of benefit in the beginnings of the 1900s – but the needs are not the same and we need something fit for the 21st century.“. I suspect it may be many years before this approach becomes the norm in Westernised countries. I think that it will take the efforts of educators with this vision that will help to shorten the timescale. I see social media, physical conferences and online conferences as being essential communications tools for those educators to talk to each other and the wider community.

I will confess to my own impatience for the advent of holistic teaching.

Maths Teaching – a Rant

The Teaching of Mathematics – a Rant

 
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My Experience of Maths as a Child

When I was a child at school, I learned my maths from teachers with a wide range of understanding of the subject. I was fortunate in that one of my teachers in kindergarten taught it extremely effectively.

As my schooling went on, my understanding of maths continued to develop. I still use this as a basis for much of my own work today.

There was a theme that ran through all of my teachers’ delivery: “All the answers in mathematics are known.”, and as a student I was expected to learn them.

Other Sciences in the News

Ever since I can remember, there have been reports in the popular media about progress in the sciences, but that reports of progress in mathematics were distinctly lacking. Recent examples in the sciences include developments in astronomy and information technology. As an impressionable child, the message that I learned was that while the physical sciences was an area of active research (in all its different fields), nothing was happening in mathematics. This, of course, was a completely false impression.

My Own Experiences as a Teacher of Maths

It was my pleasure to work with a youth-at-risk student who had both the desire and capacity to learn mathematics. This was up to Year 10 level. At the end of the course I asked him if he thought that all the answers to all the problems in mathematics were solved, and he answered with an emphatic yes. When I told him that there were myriad as yet unsolved problems in mathematics, he was incredulous.

At the same time I also tutored a year 9 student with her mathematics. From the nature of the feedback from her Mathematics teacher, and also looking retrospectively at my own feedback to her, I expect that she also would come to the notion that all the problems in mathematics were solved. She was not a strong enough student to cope with the alternative notion, so I forebore to mention it.

And So to University

Well, I went to university to read mathematics, and the result was a disaster. Okay, I may not have been bright enough to make the grade, but I am convinced that the culture shock from “all the answers are known” to “here we address unsolved problems” did not help, and I strongly suspect that I am not alone.

A More General Malaise

Mathematics is an enabling tool with which to do science. No year goes by but I hear of university science lecturers bemoaning the lack of mathematical understanding in the year’s undergraduate intake. By way of example, young people go into environmental science courses without appreciating that they need a very good grasp of statistics, and then find themselves in a remedial maths course just to catch up.

Just to make matters worse, the Training Packages used in Australia no longer identify mathematics as an explicit employability skill. The message is clear: numeracy is not seen as being important. I am no longer surprised by shop assisants who are totally thrown when I hand over $20.30 for an item costing $15.30: I am now just saddened. Mind you, this dropping of mathematics from the list of employability skills comes as a result of research done in Australia into the needs of Australia’s employers. I wonder how employers will cope with an even less numerate workforce in the medium-term future.

What Needs to be Done?

The message needs to be put out to the whole of the Australian primary and secondary educational establishment that mathematics is important. We are handicapping ourselves economically as a nation by remaining as innumerate as we are. We are also failing to properly prepare those who go onto university degree courses in whatever science subject: this does nothing for the future development of the country.

And on a personal note, I think that it is high time that Year 12 teachers in mathematics should better prepare those students who a going on to study mathematics at university: there are lots of challenges for the students to take on!