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.

## David Salusbury says:

Agree with you 100%, David!

I did my Master’s thesis on structural learning, basing a lot on Skemp’s dual learning types.

Problem is, we teach many subjects, math in particular, using the instrumental approach.

A great little example:

Why do we “flip and multiply” when we divide fractions? Answer: Just do it…it works!!

Students who just learn instrumentally, eventually come unstuck; those who learn relationally take longer, but go further!

See a brilliant study called “Benny” by Stanley Erlwanger (my advisor at Concordia) back in the 90s. Benny was a 100% relational learner, who tried playing the game to explain everything…with surprising results!

I left Math teaching 12 years ago to work in government, but still love to do work on the side…

David

November 26, 2012 — 10:00 am

## David Wees says:

The frustrating thing is that there are so many relatively simple to understand systems which help prevent teaching to instrumental understanding. The struggle has been getting teachers to USE them, or even recognize that this is an issue.

Thank you for the reminder of the Benny research. I read about it on Raymond Johnson’s blog a while back, and it is useful as a refresher.

November 26, 2012 — 2:01 pm

## aashumi says:

DAvid this is a fantastic explanation have been struggling for days to get my head around Skemp Wow finally makes sense now

thank you

January 10, 2014 — 3:31 pm

## Bob stott says:

Great article David and reflects the global interest in complexity science and complex adaptive systems – away from linear to nonlinear thinking.

Best,

Bob Stott

August 23, 2015 — 9:19 am

## Christ Jos says:

thanks a lot for sharing your understandings, they generated a better understanding between instrumental and rational understanding after reading this.

September 17, 2015 — 8:16 pm

## Liz says:

some teachers don’t have the conceptual knowledge. They only know the procedural knowledge because that was how they were taught. So a teacher who only has an instrumental understanding will only transfer that part of understanding to his/her students. It is an error that has gone on for…forever. How do u solve that?

November 11, 2015 — 9:37 pm

## David Wees says:

This is a good point. There are three ways that I’m personally working on this problem.

1. Writing curriculum that emphasizes students making connections between different mathematical ideas. In reading and using this curriculum, I hope that teachers will develop more of those connections themselves.

2. Focusing on supporting dialogic teaching where connections across different content areas and mathematical ideas are intentionally the focus of the discussions. This means that on a regular basis teachers will be prompted to support students in making connections, and as those students make those connections, their teacher will too.

3. Modeling teaching approaches with teachers that focus on relational understanding and then supporting teachers in rehearsing and practicing these approaches before being expected to use them with their students.

However, I work quite locally and many people in my position also have an instrumental understanding of mathematics, so this idea that the system propagates itself is definitely an issue, and will be for a long time.

November 12, 2015 — 2:33 pm

## Liz says:

Thanks

November 16, 2015 — 6:43 pm

## Turtle Toms says:

David,

Thanks so much for this analogy. I’ve been searching for easy to understand ways to explain why teachers are being asked to teach differently. This is a beautiful one, as so often I hear folks ask, “Why don’t you just teach them the simplest way to get from point a to point b?”

Now, thanks to you, I have the perfect response for that particular context!

Best

Turtle

December 21, 2015 — 3:06 pm

## Ryan says:

Love this analogy! Matches Principles to Actions p25 – “The depth of understanding is related to the strength of connections among mathematical representations that students have internalized.” Pair these with the Lesh Model and we have a triple-decker sandwich of awesomeness!

January 11, 2016 — 11:34 pm

## David Miles says:

Thanks, David. This analogy outlines the issues very nicely. I will share this with my maths department.

February 14, 2016 — 10:37 am

## Mark Dunn says:

I am a newcomer to maths tutoring. I have been fascinated by these issues for years but, not being a teacher, did not worry about this much.

Now I am thinking much more about pedagogy and it’s relation to supporting students who have significant difficulty with maths.

I like the analogy, but for me it omits an important feature of the park. In much of the park, the image is great, but in some parks, there are rivers, some are fast flowing and deep. Anyone helping people explore that park needs to advise explorers of the dangers of the river and how to recognise when they are near one and need to take care.

If the river divides interesting parts of the park, there may be limited ways to cross. There may be just a few slippery stepping stones to use.

Some explorers may be blind or have other impairments that make it hard for them to recognise they are in danger.

Now, I know this is only an analogy and it is easy to take an analogy too far.

My point is I know I can enjoy getting lost in these interesting, fun, but not very practical ways of working. I can lose sight of the practical needs that drive the development of mathematical techniques and the very different needs students may have for learning fairly mundane maths, e.g. for exams or practical tools for a particular task they need to tackle.

So, I suggest the relational approach is valuable, but it has dangers, there could even be monsters lurking in those rivers for some unsuspecting explorers!

I’d be interested in reactions to this image. For me I hope this could lead to a richer understanding of the students ( explorers) and their needs. I really don’t want to retraumatise students! Sadly, this often can be minimised as a little maths anxiety. For some, there is real trauma and they hang onto their useful, if very limited, well worn paths. Giving them confidence to explore is important, but real care is needed next to a waterfall please!

October 7, 2021 — 5:14 am