Archive for CognitiveTheory

Grappling with a concept

Concepts are slippery creatures.  Students seem to understand a particular topic one day, and then a week later the same student has apparently forgotten all the deep understanding recently acquired. Especially, it seems, when it comes to mathematics.  The videos of Cena and Jonathon are dramatic example of students who seems to understand place value, but on closer examination at a later date it is evident that this concept is still very vague for both students.

The initial knee jerk reaction here is to assume that this is evidence that the initial teaching was in some way flawed and the students probably never really had adequate conceptual understanding.

For me the real question is why children have so much difficulty developing their own understanding a concept that has become effortless for us.  If as teachers we have a better appreciation of why some concepts can be so elusive then we are more likely to provide appropriate learning experiences.  Current thinking in both neuroscience and artificial intelligence strongly suggests that concepts are the ability to identify patterns and make inferences from those patterns.

Here’s my attempt at explaining why conceptual understanding is fragile, and what can be done to help students acquire conceptual knowledge.

1) Provide many opportunities to manage the concept in various scenarios. The more often a pattern is identified, the less mental effort is required to recall the memory, and the greater the likelihood that the knowledge can be transferred to novel scenarios.

Each student seemed able to explain place value with numbers on the board, but subsequent questioning reveals significant conceptual gaps. This might indicate that either the previous learning was incomplete or partly forgotten, or it could be that the knowledge didn’t transfer to a different environment. Counting activities at the table are probably very familiar to the students, so it’s not so surprising that their previously well-established one-to-one correspondence type counting concept overpowers the newer concept of grouping things into tens and then counting those.

Brains learn by recognising patterns, and patterns don’t exist unless there is sufficient repetition for the pattern to become apparent. The flip side of this is that established learning – the already recognised patterns – is hard to change, which is handy for retaining existing learning but becomes a problem if the initial concept is incomplete or incorrect . When you think about it, it’s only logical that old learning should be hard to change. If it was easy to change, then most learning would only be transitory. Place value requires students to modify their previous learning of one to one correspondence and regard groups of tens and hundreds as special cases of one.

Our own familiarity with the decimal system of counting may make it difficult to appreciate the complexity of the place value concept. I thought I’d try myself out with a few simple addition, subtraction and basic multiplication operations in hexadecimal, and found myself struggling very quickly. In theory counting in hexadecimal or octal is conceptually no more difficult than counting using decimals, but in practice – well it’s something we need to practice before it can be done without considerable effort.

2) Conceptual knowledge usually take time to acquire and consolidate over a period of days or even weeks. Even those lightbulb learning moments can easily fade away from memory if they are not regularly reviewed. Learning seems to require sleep followed by reactivation of the memory of the pattern before it becomes readily available for use with minimal effort. Our experience of remembering jokes is an all too familiar example for most of us of the elusive quality of conceptual knowledge. A good joke requires conceptual understanding of the unexpected inference, innuendo or play on words in the punchline, yet even with this conceptual knowledge most of us struggle to successfully retell the well understood joke a few months later without an occasional review beforehand.

3) Manipulatives and concrete aids provide opportunities to learn in a specific context, while conceptual knowledge requires the ability to generalize from specific contexts. Hands on manipulatives are a great aid to understanding, but not a panacea. The activities and materials need to be carefully managed to provide opportunities to be sure that the focus is on the desired learning experience. Both Cena and Jonathon count accurately and confidently, but their previous experience of counting without grouping into ten seems to overwhelm their developing concept of place value. They both quickly forget about using groups of ten and revert to their already well established of counting sequentially one at a time starting with one.  While we easily generalize the concept of place value to any scenario requiring numbers more than ten, to these children it is quite likely counting stars on the board and counters at the table are almost two different concepts that have yet to consolidate into a larger general principle. It would be interesting to know how the students would have answered the questions with the counters if they had had more practice in both counting the counters by groups of ten and creating larger numbers from multiples of ten at the table immediately prior to the session.

4) Once short term memory is overloaded processing information becomes overwhelmingly difficult. It’s virtually impossible to form a concept from connected dots if you can’t keep track of all the required dots. At the board students were accurately answering fairly specific questions about the tens numbers, but the next day when the presenter’s references to the multiples of ten were less direct both students seemed to struggle to keep track of the information. This was particularly evident when students were asked to take away a number that required the subtask of breaking a ten group into ten ones.  Both students found it difficult to keep track of the tens and the units at the same time, focusing on one to the exclusion of the other.  By switching to using the less mentally taxing counting procedures that they have already mastered they were able to reduce the mental effort.

The take away message from these two videos is that if we to want to develop students’ conceptual understanding, then we need to understand what processes need to occur for that conceptual understanding to develop. The videos also highlight that conceptual understanding is more likely to be gradually developed than acquired through a sudden insight. As Marilyn Burns, the interviewer in these videos, observes,  “confusion and partial understanding are natural to the learning process” . (Marilyn Burns is passionate about improving how mathematics is taught in school. She discusses her vision in the second half of this video clip).

How does the brain do what it does?

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 Artificial Intelligence will help us understand the brain, and has developed a model for how the brain works which he has called Hierarchical Temporal Memory (HTM). 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. He defines 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 (the ball will bounce when it hits the ground) and is wrong then we notice it (the ball didn’t bounce – what’s happening). Certainly current research does seem to confirm that the brain is probably continually making predictions, including research on autism and schizophrenia  (Friston, 2011).

Read more

Monster problem for working memory

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

A deep understanding of memory

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. Read more

Learning styles- think visual

It’s common knowledge by now that we all have different learning styles. Some of us are visual learners, some auditory, some kinaesthetic and so on. What’s not so commonly known is that cognitive scientists and neuroscientists seem to unanimously agree that the multiple intelligence learning style theory doesn’t stand up to scrutiny. Apparently research consistently shows that the learning effect of matching learning styles to presentation methods is insignificant. Even people who consider themselves to be auditory learners recall information that is presented visually than information that they hear. For optimum learning, a combination of hearing and listening works best of all.

The old maxim of a picture being worth a thousand words rings true for two reasons – it’s not just that we are better at processing visual information, but our ability to recall visual information is over five times better than our ability to recall textual and auditory information. Not only do we recall visual information better, but we are able to recall it for longer. A combination of visual and auditory information is even better than either used in isolation. No wonder blackboards were such a big hit when they were first introduced to the classroom – it was a big improvement on just talk.

Cognitive scientist Daniel Willingham covers the topic in depth in the article Do Visual, Auditory, and Kinesthetic Learners Need Visual, Auditory, and Kinesthetic Instruction   He also has a somewhat dry but logical presentation on the same topic (for the visual/auditory learners).


Brenda Keogh and Stuart Taylor at Manchester University found that  teaching science  by using cartoons  was a very effective method of engaging students.(pdf article).  The cartoons presenting a scenario and cartoon characters with varying viewpoints about the scenario, encouraging students to examine the validity of each viewpoint. They found that students found the system engaging,  made it was easy to provide differentiated learning, and it was also easy for teachers to assess the understanding of the students.

Great but not all of us can come up with suitable cartoons. Dan Meyer’s suggestion is to always have a camera with you, so that you can take photos whenever you see something that might have teaching value in the classroom. He finds that it makes it easier to develop lessons that students can relate to.  Check his 3 minute video on why he finds it so effective.

dy/av : 002 : the next-gen lecturer from Dan Meyer on Vimeo.

I highly recommend Daniel Willingham’s book, Why Don’t students like School.  Donna Bills has neatly summarized nine principles found in this 9 chapter book .   More in depth articles about learning are at Daniel Willingham’s website.