The Reflective Educator

Education ∪ Math ∪ Technology

Author: David Wees (page 35 of 97)

June 10, 2012 / 20 Comments

The difference between instrumental and relational understanding

Stanley Park map

 

I recently found this article written by Richard Skemp that Gary Davis (@republicofmath) highlighted on his blog . I recommend reading the whole article. Skemp describes the difference between instrumental and relational understanding, and how the word understanding is used by different people to mean different types of understanding. He also makes the observation that what we call mathematics is in fact taught in two very distinct ways. Skemp uses an analogy to try and explain the difference between relational and instrumental knowledge which I would like to explore. 

Imagine you are navigating a park, and you learn from someone else some specific paths to follow in the park. You move back and forth along the paths, and learn how to get from point A to B in the park, and you may even be able to move quickly from point A to B. Eventually, you add more points to your list of locations to which you know how to navigate. Step off any of your known paths though, and you are quickly completely lost, and you might even develop a fear of accidentally losing your way. You never really develop an overall understanding of what the park looks like, and you may even not know about other connections between the points you know. This is instrumental understanding.

Imagine that instead of navigating the park by specific paths shown to you, you get to wander all over the park. For some parts of the park you may be guided, through other parts of the park, you wander aimlessly. In time, you develop an overall picture of the park. You might discover the shortest paths between two points, and you might not, but you would understand the overall structure of the park, and how each point in the park is related to each other point. If someone showed you a short-cut in the park, you’d probably understood why it worked, and why it was faster than your meandering path. You wouldn’t worry about stepping off the path though, since even if you get lost, you’d be able to use your overall understanding to come to a place you know. This is relational understanding.

Here’s Richard Skemp’s description of the analogy.

“The kind of learning which leads to instrumental mathematics consists of the learning of an increasing number of fixed plans, by which pupils can find their way from particular starting points (the data) to required finishing points (the answers to the questions). The plan tells them what to do at each choice point, as in the concrete example. And as in the concrete example, what has to be done next is determined purely by the local situation. (When you see the post office, turn left. When you have cleared brackets, collect like terms.) There is no awareness of the overall relationship between successive stages, and the final goal. And in both cases, the learner is dependent on outside guidance for learning each new ‘way to get there’.

In contrast, learning relational mathematics consists of building up a conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point. (I say ‘in principle’ because of course some of these paths will be much harder to construct than others.) This kind of learning is different in several ways from instrumental learning.” ~ Richard Skemp, Mathematics Teaching, 77, 20–26, (1976)

Instrumental understanding is really useful when you have to know how to do a specific task quickly, and aren’t too concerned about how this task fits into other similar tasks. Relational understanding is useful when you want to explore ideas further, are unconcerned about your destination, and are more concerned with the process.

Unfortunately, our system tends to favour instrumental understanding too much. While it is useful to be able to get from point A to point B quickly, if one is not aware of one’s surroundings, and doesn’t get to enjoy the scenery, it hardly makes the trip worthwhile.

June 10, 2012 / 2 Comments

Children are not railroad trains

"Timetables! We act as if children were railroad trains running on a schedule. The railroad man figures that if his train is going to get to Chicago at a certain time, then it must arrive on time at every stop along the route. If it is ten minutes late getting into a station, he begins to worry. In the same way, we say that if children are going to know so much when they go to college, then they have to know this much at the end of this grade, and that at the end of that grade. If a child doesn’t arrive at one of these intermediate stations when we think he should, we instantly assume that he is going to be late at the finish. But children are not railroad trains. They don’t learn at an even rate. They learn in spurts, and the more interested they are in what they are learning, the faster these spurts are likely to be." ~ John Holt, How Children Learn (1984), p155

John has certainly identified the problem, the question is, how would we build our system differently?

A lot of people have identified this problem, but I have seen less solutions to it than people expressing their outrage at it. It is certainly true, we do treat children like railroad trains, and expect far too much regularity in how they learn.

Further, our education system has become more like an accelerating railroad train in which each year children are expected to be able to do more sooner. Algebra in 8th grade. Reading in kindergarten. Essays in 5th grade. Why do we feel the need to keep up with the Joneses?

Designing a new system will be tremendously difficult. We have an enormous amount of cultural inertia in our current system. It is a difficult problem! How can we take a system wherein we fund students to attend school at a ratio of one teacher for every 20 children (on average) and find ways for each of these children to learn everything we feel is important in order for them to become adults?

Here are some suggestions, which are by no means exhaustive.

  1. Trim the list to that which is really important.
  2. Cultivate a desire to learn more, and the ability to learn for oneself.

 

June 9, 2012 / 0 Comments

What it feels like to learn to read

Where is My Bat? – Marathi

 

I’m in the middle of reading John Holt’s "How Children Learn" and almost hidden in one section of the book is a fabulous professional development activity for teachers.

"One day I took a sheet of printing in some Indian language, and tried to find the words that occurred most often on the page. It was amazingly difficult. At first the page looked like nothing but a jumble of strange shapes. Even when I was concentrating on one short, common word, it took a long time before I could recognize that word at sight and pick it out of the others. Often I would go right by it without noticing it." ~ John Holt, How Children Learn (1982), p136

I think that every teacher (or parent) who teaches children how to read should have this experience. In fact, if I could find a way to translate this experience into all of the other subject areas, I certainly would. The experience of feeling what it is like to be a complete novice learning something is incredibly valuable. We too often think that because we said something, our students will remember it, or will have learned it.

"We are so used to the feeling of knowing what we know, or think we know, that we forget what it is like to learn something new and strange. We tend to divide up the world of facts and ideas into two classes, things we know and things we don’t know, and assume that any particular fact moves instantly from "unknown" to "known." ~ John Holt, How Children Learn (1982), p136

I’ve embedded a children’s book (licenced under a Creative Commons license from a very interesting organization). I recommend taking the time to try and figure out what the story is, and what some of the words might mean. You can try this activity out alone, with a friend, or as a professional development activity with your staff. If you find a way to create a similar activity for science, math, or any other subject area, please let me know.

Try reading this book yourself so you get the experience of what it is to learn something challenging and new. I’ll try it as well, and let’s compare notes.

June 9, 2012 / 0 Comments

Why people often do not accept the research

Via the @BCAMT email list-serve:
 

"[T]here is an interesting (and disturbing) literature on situations in which information does not change prior biases or decisions. The word I have seen is ‘motivated reasoning’.

…

Interestingly, I ran into a problem of ‘motivated reasoning’ with a class of future teachers. The question is: when would research about the teaching and learning of mathematics change their classroom practices. A common response to articles, given some practice in critiquing research, was:\

– if I agree with the conclusion, the article was reliable;
– if I disagree with the conclusion, then here are x reasons why the article was not reliable and I should not change my practices!" 

Dr. Walter Whiteley


Dr. Whiteley works with pre-service teachers, and would like me to point out that they are still in the middle of articulating their own personal theories of how learning and education work, thus they lack experience in schools from the other side of the desk. It is therefore possible that this is an issue isolated to pre-service teachers.

On the other hand, I have seen people vehemently defending a position that has no merit simply because they are unwilling (or unable) to see that the evidence is mounted against them. I have also noticed many times that months later, this person has changed their perspective, sometimes claiming that the opposite to what they had previously believed was their belief the whole time, so maybe that argument influences their thinking later, and they are more willing to change on their own.

It takes enormous strength of will to remind ourselves of our cognitive biases, and act against our instinct to defend our mistakes. I can’t say I’ve succeeded at this all that much. Does anyone?

 

June 8, 2012 / 0 Comments

Imagine something different

See this piece of paper?

Piece of lined paper
(Image credit: D Sharon Pruitt)

 

Throw it away.

Imagine the limitations of the piece of paper shown above do not influence how you share the record of learning your students have done, with their parents, and the wider community.

Now remember the history of grading, which started with one William Farish (in Western culture – Chinese culture has been apparently giving grades to students for many centuries for the purpose of sorting their children into social classes.). William Farish (re)invented grades as a way to increase the number of students he could "teach’ for the purposes of lining his pockets (at the time, more students meant more money).

What would you do differently to share your student’s evidence of learning, if the limitations of the paper above did not exist, and if your purpose was neither to sort students into social classes or line your pockets by being able to teach more students? 

June 7, 2012 / 3 Comments

Looking for feedback on this puzzle game

I’m working on a block puzzle game. The objective is to cover the entire puzzle area with blocks of various sizes. So far I’ve got the basic structure up (it will only run in web browsers that support the Canvas HTML element, so Safari, Firefox, Google Chrome, and maybe Opera). Scoring for the game depends on what types of blocks are used (you’ll notice those little 1 by 1 squares are worth no points).

I’m looking for feedback on how to improve the puzzles.

https://davidwees.com/javascript/blockgame/

Some ideas I’ve had are:

  • Restrict what playing pieces the players can use.
  • Randomize the playing pieces to which the players have access.
  • Allow more access to different kinds of shapes, such as triangles, pentiminoes, heximinoes, etc…

 

Update:

Here is some feedback I’ve received as well from other sources.

  • Change the images that turn into the blocks into pictures of the blocks. @joshgiesbrecht
  • Change from scoring blocks to a par system (like golf) where players get scored on the number of blocks used. @joshgiesbrecht
  • Make the point that one gets a higher score from using larger pieces more obvious. @joshgiesbrecht
June 6, 2012 / 0 Comments

An Unfamiliar Revolution in Learning

This video, shared via the Good blog is a must watch. Find six and a half minutes to watch this video, and ask yourself what changes would be necessary in your school to make it more like this one.

 

The work that this school does on teaching empathy, and understanding what it feels like to be another person, is an incredibly valuable life-skill. The abstract reasoning that one gains as one learns empathy has to have side-benefits for academic reasoning as well. If I know what it feels like to be you, and what you likely feel like, I may be able to better make predictions about other types of objects in the world as well.

I particularly like the five habits of mind the school has used for their conceptual framework:

  • Evidence – How do you know?
  • Conjecture – What if things were different?
  • Connections – What does it remind you of?
  • Relevance – Is it important? Does it matter?
  • Viewpoint – What would someone else say? How would someone else feel?

 

June 2, 2012 / 1 Comment

Nobody remembers names

Almost everyone I meet tells me when I first introduce myself that they are horrible at remembering names. I am patient with them and am happy to repeat my name for this person several times.

Why should we expect someone to remember our name the first time? It’s essentially a random piece of information which has no relationship to who we are as people. We learn names by immersion (other people around use the name), by repetition, in context, and by using the name ourselves.

So why would we expect our students to remember disconnected facts without immersion, repetition, context, or use?

May 31, 2012 / 0 Comments

Annie Fetter on the development of math teachers

 

Annie gives a very short talk that highlights some of the issues in math education, and which I can tie to work various people have done on learning.

Everyone who is trained to become an educator has some fairly strong intuitive sense of what it means to be an educator. They have seen educators work, and they know how to copy the behaviours of the teachers they have seen. Unfortunately, often we want to change teachers behaviours, and so we must address the misconceptions that teachers have about learning head-on.

If you do not address the misconceptions that people have, chances are very good that they will incorporate the new information you present (in almost anyway that you present it) into their existing misconceptions and as a result, not change their behaviours at all. This is a problem that numerous educators have discovered (it seems independently of each other) and one which definitely has implications for teacher education.

Annie’s observation that her teaching college in 1988 was already talking about inquiry based learning, and some pretty serious reforms in mathematics education, and then her description of her beginning practices which were so different, gets at the heart of this issue. She was "taught" that inquiry based mathematics is an effective pedagogy, but she didn’t hear it. She probably did hear it, but she thought that her notion of what inquiry based education meant was the same as what she was doing. She was unable as a beginning teacher to see how different her techniques were than what she was being taught to do.

So if we want to change teacher education, we definitely need to assume that the student teachers coming in have an understanding of what it is to teach, and that much of what they understand is misguided and just plain wrong, and we need to incorporate the wrongness of this approach into our instruction of teachers.

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