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Pizzas and Pies and Fractions – Oh My!

January 31st, 2012

Another Way to Teach Fractions

Fractions have an unenviable reputation of being difficult to teach.  Although most students have extensive practice shading in parts of a rectangle and counting pieces of pie, many students still lack a conceptual knowledge of fractions. In a multiple choice question asking Year 9 students to estimate the nearest correct answer to  11/12 + 7/8, only a minority of students correctly answered 2. The most popular answers were 19 and 20.  It seems that for many students fractions are a confusing nightmare, but why?

Mastering fractions requires a change in thinking about numbers. Moving from the whole number counting system to fractions (also known as rational numbers, i.e. numbers which are not whole numbers) takes students beyond their existing concept of whole numbers which are used to represent the number of things, to a more powerful number concept that can be used to describe not just whole things, but parts of things. 
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By Marg January 31st, 2012
Categories: Mathematics Tags: 0 comments

How does the brain do what it does?

January 25th, 2012

If we could develop artificial intelligence that mimics how our brains seems to work, would we understand the brain better?  Jeff Hawkins is passionate in his conviction that AI will help us understand the brain, and has developed a model for how the brain works which he has called Hierarchical Temporal Memory. Although the name is a little overwhelming, the essentials of his framework seem to work.

While many of us probably instinctively feel that there is an innate unnknowable something that makes us intelligent,   Hawkins considers that it is possible to have a much simpler definition of intelligence. For Hawkins intelligence (for both man and machine) as the ability to recognise patterns, and to make predictions from those patterns (Hawkins, 2004). Most of those predictions happen at an unconscious level, but when a prediction is made and turns out not to be correct then we notice it.  Certainly current research does seem to confirm that the brain is probably continually making predictions, including research on autism and schizophrenia  (Friston, 2011).


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By Marg January 25th, 2012
Categories: CognitiveTheory

Scratch programming fun

June 24th, 2011

While it’s often assumed that programming is difficult to learn,  many  8 to 14 year olds are now learning programming skills using a specially designed programming tool  developed at MIT  called Scratch.  Scratch is a free program that runs on Mac, Windows and Linux. First released in 2007, it now has an extensive following of students around the world. The developers saw Scratch not just as a programming tool, but an opportunity for students to explore and be creative with computers. Although today’s digital natives can ‘read’ computers, until now most wouldn’t have the first idea about ‘writing’ for computers – i.e. programming.  That might change as more students discover Scratch.

Learning to program has so much going for it that it’s probably only a matter of time before it becomes regarded as an essential part of learning for children of all abilities. Not only does it develop logical reasoning and problem solving skills, but programming can creative and challenging at a number of ability levels. Since programming lets students create projects connected to their own particular interests, projects are more likely to be ones that students find relevant and more meaningful,  and so more likely to have the motivation ingredients found in a self-directed learning activity. Perhaps best of all, programming fosters a healthy attitude to mistakes and setbacks. Analysis of results by review and reflection are fundamental  programming skills. Last but not least, programming can also be a lot of fun.
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By Marg June 24th, 2011
Categories: Software Tags: , 0 comments

Why don’t students like school?

May 4th, 2011

The title of this excellent book is perhaps a little misleading – there’s not much in it about why students don’t like school; it’s actually a concise list of nine principles about how the brain learns that can be applied in the classroom by cognitive scientist Daniel Willingham.  The result is a practical and easily readable introduction into research based cognitive psychology, along with practical suggestions for applying the theory to the classroom.

In a nutshell, here are Willingham’s nine key points:

1. People are naturally curious, but not naturally good thinkers: unless the cognitive conditions are right, we will avoid thinking.

We enjoy mental activity and solving problems bring pleasure, but only when the problem is appropriately challenging – not too simple, and not so difficult that it is frustrating.  Appropriate levels of difficulty will engage students provided they have access to enough information to solve the problem.  Cognitive conflict is a great way to stimulate thinking. (If 1/2 plus 1/4 really does = 2/6 (a pretty common assumption amongst those who are struggling with fractions), then why is the answer (2/6) smaller than 1/2  ? ). It’s also good developing a good metacognitive skill – self check that the answer makes sense.
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By Marg May 4th, 2011
Categories: Books Tags: 2 comments

Monster problem for working memory

March 12th, 2011

The following problem1 demonstrates the impact of working memory limitations on processing information. Although there is no difficult conceptual thinking required to solve the problem, it has been reported that most university students require about 30 minutes to find the solution.

Three monsters, one small, one medium and one large, were each holding a globe. The globe came in three sizes only – small, medium and large, and each globe could be expanded or shrunk repeatedly to any one of these sizes but to no other size.

The small monster was holding the medium globe, the medium monster was holding the large globe, and the large monster was holding the small globe. They could change the size of the globes according to the following rules:

  1. Only one globe could be changed at a time.
  2. When two globes are the same size, only the globe held by the larger monster may be changed.
  3. A globe must not be changed to the same size as the globe of a larger monster.

What sequence of changes would allow the monsters to hold globes proportional to their size?

Although each globe changing rule is easy enough to understand, the problem is difficult because it is difficult to hold the rules in working memory. The original creators of the problem (Kotovsky, Hayes and Simon – 1985) found memorizing the rules of the problem to the point where they could be repeated effortlessly made the problem much easier to solve.

While it is easy to assume that a student can’t solve a problem because he/she doesn’t understand what needs to be done, the monster problem indicates that working memory limitations are just as likely to be the cause of the difficulty.

1.  from  “Instructional design for technology” by John Sweller (1999).

see A Deep Understanding of Memory

By Marg March 12th, 2011
Categories: CognitiveTheory Tags: 2 comments

A deep understanding of memory

March 12th, 2011

In his book “Why Don’t Children Like School”,  psychologist Daniel Willingham says that understanding is memory in disguise.  Although this seems the exact opposite of  the widely adopted strategy of making information memorable by making it understandable,   his point is that memory and understanding are a pidgeon pair. Both are necessary for learning – memory improves as understanding improves, and understanding improves as memorizing improves.  Any teaching strategy that neglects the role that memory plays in understanding is likely to be one where the students find conceptual understanding of the topic elusive.

Yup – I’m saying sometimes memorizing is an essential part of understanding,   and sometimes it needs to come before you can understand enough to learn.

Yup – I know that’s not what they teach in teaching college,  but then they don’t teach much cognitive science in teaching college either.  Sure, plenty of Piaget, Vygostkty, Bruner and Gardner, but only a smattering of neurones and synapses.

Here it is – the crash course in cognitive science – a.k.a.  “Your Memory, and Why It’s Important to Know More About It”.

Message understood doesn’t always mean message is remembered.

The brain apparently handles understanding (processing information) and memorizing (storing information) in totally different ways.   Although (fortunately) it doesn’t happen very often,  it is possible to have brain damage which makes it impossible to create new memories. Such an individual is able to reason and understand using any knowledge from memories acquired prior to the injury, but is unable to create new memories. Any newly acquired knowledge obtained from logical reasoning and understanding of already known information will not be remembered for more than a few minutes.

The long and the short of memories

Most theories about how brains think, reason, calculate and memorize involve the concept of two types of memory – working memory (sometimes referred to as short-term memory) and long-term memory. Although not yet fully understood, current theories  about the interaction of these two memory types can help in the creation and design of more effective learning experiences for students, particularly those students with learning difficulties.
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By Marg March 12th, 2011
Categories: CognitiveTheory, Teaching styles Tags: , 0 comments
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