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The difference between instrumental and relationalunderstanding

published by David Wees on Sun, 06/10/2012 - 16:44

I recently found this article written byRichard Skempthat 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. Youmove 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 itwas 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, 2026, (1976)
http://www.skemp.org.uk/http://www.skemp.org.uk/http://www.skemp.org.uk/http://www.blog.republicofmath.com/archives/654http://www.blog.republicofmath.com/archives/654http://www.blog.republicofmath.com/archives/654http://www.blog.republicofmath.com/archives/654http://www.blog.republicofmath.com/archives/654http://www.blog.republicofmath.com/archives/654http://www.skemp.org.uk/


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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 Wees's blogComments

Structural Learning

PermalinkSubmitted by David Salusbury (not verified) on Mon, 11/26/2012 - 07:00.

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
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Relational and Instrumental Understanding in Math Education

In Richard R. Skemps article, Relational Understanding and Instrumental Understanding we find an excellent expos about the

advantages of relational understanding, one of two different types of teaching and learning in math education. Skemp defines

relational understanding as learning both the what and why of mathematics and instrumental understanding as merely

learning the rules without learning the reason behind those rules. They are similar in that they both teach the student the rules

of math and many times how to apply those rules. Skemp leads his reader to believe that relational understanding includes in

itself most or all of instrumental understanding. Both of these teaching types have been used over the years and each has its

advantages and disadvantages. Skemps opinion is that relational understanding will give to a student a huge advantage in his or

her math education by giving them solid bases on which he or she can develop new ideas. On the other hand, Skemp explains

that in instrumental understanding, the student does not receive sufficient why to continue holding their interest and to h elp

them actually learn and retain usable math, needful throughout their lives. The article explains in great detail this issue which is

of most importance to math educators.


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THEORETICAL FOUNDATIONS OF PROBLEM

SOLVING

A POSITION PAPER

by Richard R. Skemp, University of Warwick, January 1993

1. Our problem as teachers

An important quality of mathematics is its applicability to a wide range of uses in

science, technology, industry and commerce. This is why it is so useful practically,

and so important in today's world of rapidly advancing science and its embodiments in

technology. Many of these uses are routine, but we also encounter new requirementsfor which we do not have a ready-to-hand method. In other words, we have a problem

to solve. This is the essence of a problem, that we want to do something but do not

know how.

So we need two kinds of mental equipment: routine methods for routine situations,

and problem-solving ability for situations we have not met before, or those we have

met but have not yet found how to deal with. The latter may subsequently also

become routine: so we need also to be able to learn new methods, and then routinize

them without detracting from the possibility of future adaptation. This aspect has been

discussed at length inRelational Understanding and InstrumentalUnderstanding(Skemp, 1976, 1989a).

So far, most teachers have shown themselves much more successful at the first than at

the second. Routine methods can be taught as something to be memorized, but in this

form they lack adaptability. Since a new method has to be learnt for each new kind of

task, the memory load in time becomes impossible. Moreover, without relational

understanding students are often in doubt which method to use. Most of us are

familiar with the query "Please, is this an add or a multiply?"

At the NCTM conference at Seattle in 1980 the committee resolved that it was timefor a new approach to the teaching of mathematics, based on problem solving. I

waited with interest to hear how this was to be done, but they never told us. Nor did

the following few years provide any evidence of progress in this direction, to judge by

the title of a paper published in April 1983 by the (American) National Commission

on Excellence in Education.


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This was entitledA Nation at Risk: the Imperative for Educational Reform, and the

remedies proposed include taking more mathematics in high school, increasing the

school day and year, and assigning more homework. In his telling criticism of this

approach, Witney writes: "It should be very clear that we are missing something

fundamental about the schooling process. But we do not seem to be interested in this; .

. . we try to cure symptoms in place of finding the underlying disease, and we focuson the passing of tests instead of on meaningful goals." (Witney, 1985) A decade

later, there is still little or no evidence of progress. In my view this was predictable,

and will persist as long as efforts are based on what is promoted by the powerful

commercial interests of the major publishing companies.

Can we do better ? Yes, if and only if we understand and accept the need for theory.

Common sense alone is not enough, or it would have succeeded long before now.

To begin with a much simpler example, common sense knows that iron sinks. So how

is it that we can make ships of iron which float? To understand this we need a simplebut powerful theory, the principle of Archimedes. Some of us may remember this

from our school days. If we don't, it doesn't matter because we aren't boat builders. It

is this principle which enables us to predict whether a ship made of iron will float or

sink. It takes us far beyond what we can do with common sense. Whoever would

think of making a boat out of reinforced concrete? But these are made, and they do

float. And powerful as it is, the principle of Archimedes doesn't provide enough

theory to make a boat that will be stable. If we were going to set up as boat builders

there is a lot more theory we would need to know.

Likewise, to understand how we can teach for problem-solving ability, common sense

is not enough. We need the power which only theory can provide. And since we are

teachers, it does matter whether or not we know it, because without it we aren't likely

to succeed in helping our children any better in the future than in the past. And the

amount of theory we need is more than will go into the present paper. Nevertheless,

we can make a good start.

Common sense might suggest that the best way to teach students to learn problem

solving in mathematics is to base teachingfrom the starton real-world problems

which require mathematics for their solution. Even a little theory will be enough to

indicate that this is not so.

2. The power of structured knowledge

A while ago I had a real-world problem, that of finding my way home from a school

in an area quite new to me, which I had been visiting with a maths adviser. He gave


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me the following directions for getting back onto my familiar road home. "You need

to go left, left, left, right, half right, and then left onto the A45."

You can imagine what happened. I took no notice of the left turn at the school gate

and was soon lost. It took me a long time to get home. A plan of this kind is closely

tied to action. And if one of the actions is wrong, it's liable to throw out all whichfollow. In this case, if one gets off the path, one doesn't know what to do to get back

on. The cognitive element is low. Traditional teaching, based on habit learning, is of

this kind. It has low adaptability. But would a problem-solving approach, in which I

had to find my own way without even the help I was given, have been any better? I

hardly think so.

In time for my next visit, my friend sent me a street map. Here is part of it. (The

original also has street names.)

Now I had structured knowledge, which is a much more useful and powerful kind. For

a start, I could see where I had gone wrong last time. And even if I did go wrong, I

could use the street names to find where I was, and then work out a new plan for

getting home from there. And in general, this map contains the potential for working

out how to get from any one location to any other location on the same map. Suppose

that I wanted to go home by a route which took in, say, a petrol filling station and a

convenience store to pick up some milk. I did not have a ready-to-hand route, so I had

a problem. Without a map it would have been a difficult one, but with a map it was

easy to work out a way of doing it. To do this by learning all the possible separateroutes would entail memorizing 3080 of these. This shows rather well the problem-

solving power which structured knowledge makes available to us.

This principle is quite general, and the idea of a road map generalizes well into all

kinds of knowledge structures (psychologists call themschemas) with the help of

Tolman's useful metaphor of a cognitive map.


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3. Getting there by a detour

When I was studying for my psychology degree, I read about an experiment which I

have remembered ever since (though unfortunately not the reference). There was a

short wire mesh fence, with food on one side and an animal on the other, thus.

Animal, a dog or a hen.

Fence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Food

The food was clearly visible through the fence, and the hen made straight for it.

Coming up against the fence, it could get no further, and continued indefinitely trying

to get through the fence. The dog, on the other hand, immediately turned aside, ran

round the end of the fence, and got the food. The dog could make a detour to achieveits goal, the hen could not.

It is important that in our efforts to achieve our present goal we act more like the dog

than the hen. And theory, even the small amount described so far, tells us that the best

way to help students to become good at problem solving involves taking them via a

detour, that of building up the necessary knowledge structures. Once they have these,

they will be equipped to solve a number and variety of problems far greater than could

be possible by any direct approach. This is particularly necessary in today's world of

rapidly advancing science and technology. Some of the future problems which our

present students will need mathematical knowledge to solve do not yet even exist.

4. Forming mathematical concepts, and constructing mathematical knowledge

Another aspect of theory which is of importance in the present context is the process

of abstraction by which we form concepts of any kind. Since mathematical concepts

are a particularly abstract kind of concept, we particularly need the help of theory for

teaching these. My own theory, as presented in a number of publications (see

especially Skemp 1971, 1979, 1989) predicts the following.

The kind of learning situation which best supports the formation of mathematicalconcepts, and the building of mathematical schemas (knowledge structures) is one

which satisfies the following requirements. These predictions have been confirmed

experimentally.


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(i) It should provide embodiments of the concept having the least amount of irrelevant

information which has to be ignored while forming the concept. (The technical term is

'low-noise.')

(ii) There should be a number of examples of the concept, close together in time.

(iii) There should be just one new concept to be learnt at a time, and

(iv) This should be one for which they already have an appropriate schema, so that

they can connect the new concept to it and thereby learn with (relational)

understanding.

The above requirements are difficult or impossible to achieve in 'real-world' problems,

but are specifically catered for inSAIL through Mathematics. Can any reader

suggest real-life problems from which children can learn the mathematical concept of

"addition" which satisfy the foregoing requirements at all, leave alone betterthan Stepping Stones,Crossing,Slippery Slope,Explorers(to name but a few). And

from what real-world problems can students gain understanding of the concepts which

give validity to the method of long multiplication? These concepts, five in all, are

among the main foundations for learning algebra in the future. One could go on and

on with examples of this kind. Real-world problems are also unlikely to provide

enough repetition to consolidate newly-formed concepts, and to establish new skills as

well-established routines.

5. Some categories of mathematical problems

I have already suggested that the essence of a problem is that there is something we

want to do for which we do not have a ready-to-hand method. Solving the problem

means finding one or more suitable methods, and then applying them. For

mathematical problems, we can distinguish the following main categories. A given

problem may be in more than one category at a time, which naturally makes it harder.

For beginners, it would seem desirable to ensure that this is not the case.

Category 1. A verbally stated problem for which a mathematical model has to be

constructed (this is the non-routine part), using mathematical knowledge which we

already have. When this has been done, there is a routine method for (e.g.) doing thecalculation, solving the equation, or whatever. Finally, the result has to be interpreted

in the context of the original problem. These are often called word problems; a more

general term isproblems in applied mathematics.
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Category 2. A mathematical task which is itself of a non-routine nature, but which

can be solved by using mathematical knowledge which we already have. These

areproblems in pure mathematics. For short, we may call them mathematical puzzles.

Category 3. A task in pure or applied mathematics for which we do not yet have the

necessary mathematical knowledge. I think that problems of this kind are necessarilyalso in category 2, but not the other way about. A classical problem for the Greek

mathematicians was to find the length of the diagonal of a unit square, when their

available number systems did not go beyond rational numbers. Historically, this

problem led to the invention of irrational numbers. These we may callproblems

outside our present domain. An important special case of these areproblems in our

frontier zone. Each new concept presented inSAIL through Mathematicsis a

problem of this kind, in a carefully devised and field-tested embodiment which

(unlike real-world problems) satisfies the requirements listed in section 4.

6. Abstracting and re-embodying

We can distinguish three stages of abstraction relating to problem solving (there may

be more).

(i) From real-life situations we abstracta conceptual model, often represented by

words.

(ii) From this conceptual model we abstractthe concepts which matter for our

problem.

(iii) From these concepts we abstracta mathematical model.

We ourselves often do these in rapid succession, from long practice. But abstraction is

often much harder than working with the mathematical model when we have it.

Next, at the abstract level, we manipulate the model in ways which correspond to the

events we are interested in. So we need to be clear about this correspondence, too.

Word problems start at stage (ii). Real-life problems start at stage (i). Finally, we

work in the reverse direction.

Predicted result in the physical world or word problem we re-embody Result of these

manipulations

Putting our plan into action we re-embody Making a mental plan
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7. The two approaches compared

The three diagrams which follow offer an overview of the differences which have

been discussed, between the direct approach, which suggests (reasonably enough at a

common-sense level) that the best way for students to learn real life problem-solving

with the help of mathematics is to start by giving them problems of this kind; and thetheory-based approach, which suggests that students will get there more successfully

by a detour. First, help them to build up the structured mathematical knowledge which

is essential for solving for solving real-life problems. They will then have the mental

equipment they need for successful real-life problem solving.


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References

Skemp, R. R. (1976). "Relational and Instrumental Understanding in the

Learning of

Mathematics," Mathematics Teaching, no. 77. [Also the Prologue in

Mathematics in the Primary School.]

Skemp, R. R. (1971, 2nd edn 1986).The Psychology of Learning Mathematics.

Harmondsworth: Penguin Books.

Skemp, R. R. (1979). Intelligence, Learning, and Action: A Foundation for

Theory and Practice in Education. Chichester: Wiley.

Skemp, R. R. (1989).Mathematics in the Primary School. London: Routledge.

The American Commission on Excellence in Education. (1983).A Nation at Risk:

The Imperatives for Educational Reform.

Witney, H. (1985). "Taking Responsibility in School Mathematics Education."

Proceedings of the Ninth International Conference for the Psychology of

Mathematics Education, vol. 2, The State University of Utrecht.

Skemp Activities Home Page|SAIL Different?|SAIL Components|Concept Maps|Sample Activities

NCTM Standards|Problem Solving|Availability
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Instrumental vs. RelationalRichard Skemp (photo ICME)

There's an article I assign nearly every teacher prep class I teach: Richard Skemp's Relational Understanding and Instrumental

Understanding, Mathematics Teaching in the Middle School, September 2006 (A reprint from a 1976 art icle for the British

Mathematics Teaching.) It's a bit of a tough slog, but touches on such a fundamental teaching issue, that I always get new things

out of rereading it.

(Skemp'sobituary;Republic of Math's takeon this article.)

Recently I had a conflict with a class and an inservice, so I designed an online class for the students to try. Mixed results. On a 0 to

5 rating for being worthwhile, students gave it just better than a 2.5. On the other hand, if it could have replaced our usual class

time, it may have indicated trouble in that direction.

But as a part of the class, they responded to questions on a Google doc, and their responses make for interesting reading in

themselves, so I thought I would share selections.

1) What is the point of starting off with the Faux Amis story?

(Afaux amisare two words in different languages that sound similar but mean differently. Sopa (soup) and soap (jabn) are my

favorite from Spanish. Skemp says that the ways we use "understanding" are as different as if they were faux amis.)

* I think that sometimes we may think we are saying one thing, but really we are saying something entirely different. This is , but

should not be the case, when we are talking about how students understand mathematics.

* The Point of the Faux Amis story seems to get the reader interested in a scenario outside of mathematics. Its interesting that what

sounds like library really means bookstore and so on. Once this story is presented, it is easier to grasp what Skemp is trying to say

about different meanings in the word understanding and mathematics.

*The Faux Amis story provides a generic overview of the topic covered by the article. It explains that though we may mean to say

one thing, our students may understand what we are saying in a completely different context. As teachers, we must brainstorm the

different ways our lessons could be interpreted by our students in order to ensure they grasp the content we teach.

2) What is your favorite example of rule without reason? Why?

* Right now my favorite rule without reason is geometric mean. While tutoring last night we were talking about similar righ t

triangles. Turns out if you find the altitude from the right angle of a triangle, the length of the altitude is the geometric mean of thetwo parts of the hypotenuse. So if the altitude cuts a hypotenuse into 9 and 16, then the geometric mean is (9/x)=(x/16), so x^2=144,

and x=12. You can also use the geometric mean to find the lengths of the sides. Why? Beats me.

* I would say my favorite rule without reason is the fact that x^0 = 1. This is always true but honestly I dont even know why. It seems

that many rules in math are like this, easy to remember and implement but they dont make any sense at all.

* The most prominent rule without reason for me right now is the fact that the product of two negative numbers is a positive

number. I never learned why this is, and honestly I still couldnt sufficiently explain why. Even worse, I cant think of any context to

put it in.

* I agree with this statement. When we were trying to express and explain the mult iplication of negative numbers in class, I had

absolutely no idea how to explain why this is nor create a context for multiplying two negative numbers.

3) Does the authors idea of looking for your own examples and his three reasons for it make sense? Why?

* Understanding is best shown with someones own words. There is no way to copy down and memorize an original thought unless i tis ones own. Using ones own words to describe a mathematical concept is the best indication of fluency. Examples made in this

fashion are easy ways to build relational understanding, the more diff icult of the types of understanding.

4) Explain Skemps two kinds of mismatches (in the classroom) in your own words.

* There are two primary areas of mismatch that happen due to the Faux Amis of understanding mathematics. The first is that the

teacher is striving for students to have relational understanding, but the student is satisfied with instrumental understanding. In other

words, the teacher wants the student to understand why the problem works, but the student just wants to get the right answer. This
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is more common and probably more evident in the classroom. The second mismatch is that the student wants to understand the

problem relationally, but the teacher is only teaching instrumentally. In other words, the teacher wants the students to repeat the

same steps, or the same procedure, or just follow the rule for each problem. The student wants to understand why, or maybe wants

to make more connections to other problems, but the teacher is limiting that.

5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?

*I think that most people would say a student trying to understand instrumentally while being taught by a teacher who wants them to

understand relationally, is the more common mismatches. However from my personal experience it is the other way around. I

experienced instrumental teaching throughout high school and including some college courses. I also think that a student trying to

understand instrumentally while being taught by a teacher who wants them to understand relationally, is more of a problem for the

teacher because the teacher is trying to teach them for a deeper understanding but many students just want the surface knowledge

needed to get something done as soon as possible.

6) What are Skemps faux amis in mathematics teaching? Is either one an issue in your math major classes here in GVSU?

No one tackled this! Skemp describes that mathematics itself is a faux amis, with two entirely different kinds of mathematics being

taught: instrumental mathematics and relational mathematics. Instrumental math is a collection of procedures, where the goal is to

recognize the correct procedure and apply it efficiently. Relational math is knowing what to do, why to do it, and why it works.

7) Would you add any advantages to his list for instrumental mathematics?

* I think if I could add one advantage to the instrumental mathematics it would be it is more time convenient, because overall I feel

like teaching for instrumental understanding is just trying to get the students to be able to solve the problem by plugging and

chugging. If that is all you are doing as a teacher and that is what the students are learning then the time it would take to cover the

lesson would be much less, giving the teacher a lot more time overall to cover all the stuff that needs to be covered in the

mathematics curriculum.

* I would add that instrumental understanding is advantageous for relational understanding. Being able to instrumentally understand

a particular topic would benefit the learner when trying to reach the goal of relational understanding. A mathematics teacher could

use the mold of procedures, and with the proper questioning of students about those procedures, to encourage students to think

deeply about why? This will bring students closer to understanding important concepts, so they can ultimately reach relational

understanding.

8) Would you add any advantages to the list for relational mathematics?

* Relational mathematics gives a meaning behind the instrumental aspects. It is said that math is so highly disliked by students

because they see no purpose, and are constantly told that it will make more sense later on. This is not acceptable! Drawing upon

the music example, we wouldnt expect a student in middle school to learn music instrumentally and then be able to identify and

draw upon that knowledge in their everyday life. I would definitely say that the relational understanding is going to be key in getting

students to appreciate math again.

* Going along with the previous statement, relational mathematics gives meaning behind instrumental mathematics. Giving meaning

to instrumental techniques can assist students in the leap from a mathematical problem to a real-life, meaningful situation for

students.

9) Do you agree with the advantages that he lists for the two types?

* I do agree with the advantages and the disadvantages that he talks about. I feel as thought that each one does have its place in

teaching and that there are many advantages to instrumental mathematics even though I feel like relational mathematics is better in

the long run. I feel as though there are many situations where instrumental teaching does have its place and where trying to teach

relational mathematics would just cause problems and would make things more confusing to the students. Many times such as

learning that a negative multiplied by a negative is a positive is better when it is just given to the students instrumentally as

compared with trying to explain why which they will learn in time.


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10) Whats an example of relational understanding in your non-math life?

* In my non-math life, Ive gained a great deal of relational understanding throughout my participation in cheerleading. In cheer, I

was told to learn the position of flyer when stunting. To perform a stunt, you must first know what it is. The difference between an

elevator and an extension scorpion would have little relevance to anyone who did not have some type of instrumental

understanding of the sport. However, to successfully execute a stunt, you must understand exactly what you need to do with your

body. A flyer needs to keep straight legs, pull up her own weight, smile ext, but these are all instrumental things too. To gain a

relational understanding in cheer, I had to practice the stunts, and through this I learned body position, working with my stunt group,

how/when to alter my positions ext. These are all things that someone who had not tried flying would not necessarily understand.

Instrumental understanding in cheer is something that any person could gain through studying the sport, however, for relational

understanding participation is vital.

* During the summer I work for a motorcycle safety course as a range assistant and when I started the job, I had no previous

knowledge of how to ride a motorcycle. As I did all the miscellaneous jobs on the range, I heard the same safety t ips over and over

and saw hundreds of students practice. I learned where the controls were on the bikes and the best techniques for going around a

curve safely or stopping quickly in an emergency. I heard the information over and over and could recite it, but I had no idea what i t

was like to actually drive a motorcycle. Last summer one of the coaches offered to teach me. It wasnt until I actually started riding

myself that I could begin to really understand what it was all about. I knew the little details but I didnt have the whole picture, without

practice and first-hand experience I couldnt put it all together and actually control the bike. Until I reached relational understanding,

there was no way I could succeed at riding.

* I started coaching a 3rd and 4th grade basketball team this year through a program called Upward, which aims to provide children

with a faith based, competitive sports league. The league strives for equality amongst its players, and to accomplish this, the

coaches make a roster of their players in order of ability. Come game time, there is a specific line-up procedure where the top 5

players start the first game, then the 2nd best-6th best start the second game, and so on. This is done so that all girls get equal

playing time, and so that equally talented players are playing with each other. This prevents a teams top 5 players competing

against another teams bottom 5 players.

11) Whats an example of relational mathematics understanding for you? How do you know?

* For me an example of relational understanding in mathematics would be my visual understanding of geometry and the knowledge

that the use of a ruler and a compass will allow me to draw objects in euclidean geometry. This is a relational understanding

because it is fundamentally what the original philosophers thought of when they constructed the system. The ancient Greek people

did not condone the use of abstract thought as it exists today.

12) So, what about your classroom? Will you teach for one, or the other, or both? Why?

* Ideally I would teach for relational understanding because even though its much harder to teach and for students to learn, it sticks

a lot better and it aids them in making connections to other mathematics. I think that this isnt necessarily an achievable goal,

however. With all of the standards I need to cover I wont always be able to dedicate the time needed to reach a relational

understanding. I already run into this in my tutoring. Just last night the student I tutor, Al, didnt know how to find the similarity ratio

of similar triangles. I was able to very quickly show him how to do it and he can now do i t, but I dont think he truly understands why.

I could not have, however, spent my entire hour with him trying to teach him this because he has a test today and there were other

things he didnt understand either. Im guessing I will have similar experiences in a classroom.

* I agree with this answer above, I think every math teacher should be striving to teach relational understanding but sometimes it is

just impossible due to the time constraints we are presented with as educators. I think the best teachers out there can find the most

important topics that are either continued on later in mathematics or are most pertinent to that specific class and teach thoseunits/areas for relational understanding, this was the students will be exactly where they need to be later on in their math career.

And as for the other units that dont take such an important role in the curriculum I think the teacher just needs to cut their losses

and teach it instrumentally so they can better cover the more important material. That is why I think teaching both is important, but

teaching all relational understanding is ideal.

* I feel teaching both is important, and would do so in my classroom. Relational understanding is the goal, but would be near ly

impossible to reach without instrumental understanding. I feel the idea is to use ones understanding of mathematical procedures to

develop understanding of underlying concepts. After reaching a conceptual understanding, students will be able to make

connections with new material and be able to develop new procedures to reach a relational(conceptual) understanding. Its the

relational understanding that will allow students to think deeply and independently, helping them become adequate problem solvers


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and ultimately successful learners.

* In my future classroom, I think that I would incorporate both types of understanding into my curriculum. I think that instrumental

understanding is an easy way for students to find solutions to problems and see the process. Instrumental understanding by itself,

however, is not enough. Relational understanding gives meaning to the processes being completed and helps students relate to

concepts and context.

* In my class, I will try to include more rational understanding into the lessons I will teach. As this can be incredibly dif ficult while

teaching some subject matter, I understand that I will need to also teach instrumental understanding for some units. Once students

have explored the subject matter further, I hope to have students build upon their instrumental understanding and that it will change

into a relational understanding. Many of the standardized assessments students take only require an instrumental understanding of

concepts, but in order for students to become better thinkers, they need to have mastered a level of rational understanding. Though

it may seem pointless to teach anything but instrumental understanding, in the long run, students will gain more problem solving

abilities and learning skills from mastering a relational understanding.

* My objective is to teach to students in a way that would enable them to gain relational understanding but I also think that there will

be some instructional understanding. I feel as if the type of understanding used will be based on the school system. Further gaining

relational understanding,in my opinion takes longer to grasp then instrumental understanding. So for this reason instrumental

understanding may be used so that there is less t ime taken for each lesson. But in regards to this my ultimate is just to use

relational.

* I feel like that when I teach I will use a lot of both types. I feel like if I had the time I would teach relational understanding and I feel

as though for really important topics I will teach this way, however I also feel like with all of the requirements there is no way you can

teach like this all of the time and be effective at it. Because of this I feel like I will end up teaching instrumentally most of the time just

simply because it will allow me to get through the material faster and help the students understand best in the little amount of time

given to cover each topic. However if individual students are confused or there is extra time available I will t ry to teach using

relational understanding.

Skemps and the baseball coach

I love discussing the Richard Skemp article, Relational Understanding and Instrumental Understanding, (Mathematics Teaching inthe Middle School, September 2006) with preservice secondary teachers. Not first thing in a course, but after some experiences withdeep mathematics learning, and some groundwork in thinking about teaching, it is amazing. I've recorded two blogposts about it

already withmy reading guideand withvideo of the novice teachers' summaries.

The article discusses how there are two views of understanding that makes discussing learning difficult. Instrumental understandingis associated with rote earning and mastering, and relational understanding is connected, transferable learning. Skemp puts i tbetter, thank goodness he wrote about it.

This past semester, one student in particular made a strong connection that seemed to also make a lot of sense to the otherteachers. He was willing to share it on video, so,here it is! Thanks to Ryan for his work in making sense of the article and hiswillingness to share.
http://mathhombre.blogspot.com/2011/02/instrumental-vs-relational.htmlhttp://mathhombre.blogspot.com/2011/02/instrumental-vs-relational.htmlhttp://mathhombre.blogspot.com/2011/02/instrumental-vs-relational.htmlhttp://mathhombre.blogspot.com/2011/10/skemp-discussed.htmlhttp://mathhombre.blogspot.com/2011/10/skemp-discussed.htmlhttp://mathhombre.blogspot.com/2011/10/skemp-discussed.htmlhttp://www.youtube.com/watch?v=1u3AZ7n5dj4http://www.youtube.com/watch?v=1u3AZ7n5dj4http://www.youtube.com/watch?v=1u3AZ7n5dj4http://www.youtube.com/watch?v=1u3AZ7n5dj4http://mathhombre.blogspot.com/2011/10/skemp-discussed.htmlhttp://mathhombre.blogspot.com/2011/02/instrumental-vs-relational.html

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