Abstraction and domain mapping overcomes common knowledge conflict
In a problem that looks simple but somehow wrong and violates the commonly known concepts, power of abstraction may often overcome the apparently unknown and uncertain state and helps to reach the elegant solutions. On the way to the solution, usually a higher level rich concept is created the precise form of which was hidden just around the corner.
We will try to give body and shape to these vague words through the process of solving two similar simple looking problems with a twist. These two problems reportedly taken from 10 year old kids’ problem sheets appeared in a leading social network stream and caused quite a sensation, as majority of the adults found the problems quite difficult and unsolvable.
We will also go through the possible reasons behind the difficulties.
The problems
Find the perimeter of the two figures below.
Try to solve the problems before going ahead. We will first analyze and explore the first problem.
The importance of known and unknown quantities (or variables) in solving a mathematical problem
The first shape has six edges where lengths of only two edges are given. We are accustomed to dealing with perimeter, area and other properties of a rectangle with four edges and two given length values. As the lengths of opposite sides are equal for a rectangle, the number of unknowns reduces to zero.
In this apparently simple problem, unfortunately we are at a loss about how to deal with the extra number of unknowns. As the bounded shape has similarity with a rectangular shape we can consider it closest to a rectangle and classify it as a deformed rectangular shape. It has six edges with two given length values and so we are left with an uncomfortable number of 4 extra unknowns.
Basic concept is the commonly known concept
The basic concept of perimeter is,
$\text{Perimeter of a bounded shape }=\text{ Summation of lengths of bounding edges}$.
The number of bounding edges may be 1, as in the case of a circle or an ellipse where the bounding edge must be a curved line.
When the bounding edges are straight lines, more than two and minimum of three straight lines are required to form a bounded region. The familiar bounded shape with three straight line edges is a triangle.
Deductive reasoning helps to transform a hypothesis to a fact of reasonable certainty
Our problem shape is radically different from that of a triangle because apparently all angles between two adjoining edges seem to be right angles.
This is not stated and is an assumption or hypothesis. But this becomes a fact when we consider the reasoning that,
With only two given values for lengths of six edges, if the angles between the adjoining edges were not all right angles, the problem would have been unsolvable.
This is deductive reasoning.
Thus we find our problem shape closer to a rectangular shapee than any other bounded shape. This is so, as only rectangle shares this important characteristic of all perpendicular adjoining edges with our problem shape.
To be truthful, at the first go, our mind attempted to find the perimeter of a rectangle only, but stopped short at the deformed edge.
So we are partially happy to be left with our first major conclusion,
The given shape is closest to a rectangular shape in comparison with any other bounded shape.
The satisfaction is partial as we still have to deal with four unknown length of edges. But at this point of time we have a gut feeling that the solution must come through the basic commonly known concept of perimeter of a rectangle,
$\text{Perimeter }=\text{ twice the sum of length of two adjoining sides}$.
Problem transformation
The problem is now transformed to,
How to convert the four unknown lengths of four edges to the two known lengths of two edges.
If we can do that we get our solution using rectangle perimeter concept. This seems to be the only way to go.
But the concept gap looked squarely at our face. How to bridge the concept gap?
In reality, though fairly experienced in solving problems in general for more than half a decade, we stopped short for close to a minute and truthfully felt a bit embarrassed about it. After all, this problem that we encountered in a G+ stream sometime ago, was reportedly meant for testing 10 year old children.
Past experience based domain mapping came to our rescue, formation of rich concept of shortest walk along perpendicular paths
At this point of the problem solving process, the experience of walking along a two axis system of roads from one point to another came to our rescue.
The following is such a road system.
On such walks from point A to point B we formed the concept that,
However many turns we take, we will always have to cover the same distance walking from point A to point B, if we take care to move always towards B, and not away from it.
In more specific terms the ‘if’ condition translates to, "Always moving South or East direction, if we define B to be situated directly South-East of A."
Abstraction helps us to ignore even the N-E-S-W direction system
In fact when we start our walk from A to B we do not think about North, South, East or West. We think about the only characteristic of the road system,
All roads that we have to walk along are perpendicular to each other at all crossings. It is two sets of parallel roads cutting perpendicularly across each other.
In this case we need to be aware of only four mutually perpendicular directions. The knowledge of the reference directional system of N-E-S-W is not necessary.
This perpendicularity of two parallel sets of roads is the main concept based on which we move unerringly from A to B ensuring always the shortest distance coverage if we never backtrack even for once.
This is the rich concept based on which not only do we traverse the shortest path from point A to point B in a two axis road system, but also would form the basis of solving our perimeter problem.
To state the rich concept of shortest walk along perpendicular roads,
All paths from point A to point B in a mutually perpendicular set of parallel roads will have least walk length if at no point of the walk the direction of movement is reversed.
The situation is depicted in the following figure with two shortest distance paths from point A to point B in colors red and blue for ease of identification.
This minimum distance between point A and point B, the two opposite corners of the rectangular road grid system, is, L1+L2, the sum of two sides of the rectangular grid. This is the sum of path lengths of all the other stepped shortest distance paths, as also the sum of lengths of two sides of the rectangle APBQ.
It is now just one step more to the perimeter of the rectangle APBQ as, 2(L1+L2)=2(12+10)=44 cm, which is the same for all the other stepped paths forming two deformed sides of the rectangle.
The rich concept is devoid of mathematics and is of experiential abstract nature
The rich concept was derived from the experiences of walking along two axis roads, not in terms of mathematics but in terms of a sense of shortest path coverage rule. Whatever underlying mathematics was there was assimilated and absorbed in the experience based abstract concept so much so that even a person not knowing any mathematics can very well form the rich concept of shortest walk along perpendicular roads.
As the concept is more abstract than the precise mathematical formula of perimeter, and also not commonly taught or publicly known, we do not classify the concept as a basic publicly known concept. One needs to form the concept from experience of walking along perpendicular roads getting rid of even the North-South reference direction system, and knowing and sensing only four mutually perpendicular directions. Here the concept becomes more abstract than the specific perimeter concept.
The concept is more basic, abstract and instinctive in nature and thus not only solves the problem of perimeter of one irregular shape, but also provides the solution for all such shortest distance paths with no backtracking.
Such concepts have the potential to provide solution to a wide range of problems because of its abstraction and more fundamental nature.
Domain mapping in action
Finding no other way to the solution of the simple looking problem, our mind automatically searched the problem solving resource base in our mind and pulled out the rich concept from a different domain of walking along perpendicular roads and handed us the solution in a few moments.
Pattern of perpendicularity linked the two structures in two different domains
We have always found domain mapping to be an extremely powerful technique towards reaching efficient and elegant problem solution. But without fail the two domains involved have a common abstract pattern using which our mind linked the two domains and mapped the problem from one to the other. Thus, the abstraction and pattern identification skills form the stepping stones to domain mapping.
Coming back to our problem, the concept from the domain of walking is mapped to the domain of mathematics.
One of the more powerful skills in the area of efficient real life and academic problem solving is the ability to map concepts from one domain to another.
Defining the problem solving layer above any other activity or knowledge layer makes this domain mapping possible.
What happens in case of backtracking – Problem 2
The following is the figure in problem 2.
This problem of finding the perimeter of the bounded area is exactly same as the in problem 1 in nature except that we have backtracked in the reverse direction for 2 cm. And as we have gone away from our desired direction towards B by 2 cm, for reaching B along the shortest path from this point on, we must walk an extra distance of 2 cm to compensate for the deviation.
That makes the perimeter 2x2 cm=4 cm extra to the perimeter of the rectangle 2(9+11)=40 cm.
Knowing the concept of shortest path walk and also how to compensate for deviation of direction of walk provided us the solution easily as 40+4=44 cm.
Key concepts used:
- Deductive reasoning: Outcome: To resolve extra number of unknowns the shape of a rectangle should provide the solution.
- Perimeter concept: Naturally, to find the perimeter we have to be aware of the concept of perimeter.
- Problem transformation: The problem has been transformed to the task of reducing the number of 4 unknown lengths to zero.
- Domain mapping: earlier experience based rich concept of shortest walk along perpendicular roads was mapped to the perimeter problem domain and transformed the length of the jagged portion of the path to the two sides of the rectangle.
- Abstraction and identification of domain linking commmon pattern formed the basis for domain mapping to take place.
- Rich concept of shortest walk along perpendicular roads was an abstract concept devoid of even the reference North South direction system and was formed out of experience.
- Exception handling in the form of dealing with backtracking enriched the power and applicability of the concept further.
Note: Even though most of us are aware of the rich concept of shortest walk, usually in our mind the concept is,
- Not precise in nature, and
- Tied to the domain in which it is created.
These two barriers together prevent smooth and automatic domain mapping. These barriers to domain mapping along with the huge conflict of extra number of unknowns with the common concept of finding perimeter resulted in the general difficulty in solving the problem, especially by significant percentage of adults attempting to solve the problem. Our take for the contrast is, young minds are intuitively more flexible in forming and applying new rules in problem solving than adults.
Aside: When dealing with perpendicular directions of N-S-E-W, we don’t remember the exact relations between adjacent directions. It always confuses us (meaning specific us, not you). Instead we remember the truth:
Sun rises in the East,
And the rule,
If you face East your stretched right hand will point to South.
The third rule, of North-South and East-West opposition can never be forgotten. This is one other application of less facts more procedures approach. The problem it solves is, How to remember North-South-East West directions correctly. One fact and two rules.
Procedures or rules having more contextual and action oriented information, are easier to remember and faster to apply to the isolated facts or concepts.
This results in,
Fast and smooth formation of large number of new concepts from a few factual concepts and a few procedures or rules.
High degree of skill in forming and using less facts and more procedure based concept models plays a very important role in modern life by reducing monumental usually unconnected factual memory load thereby increasing energy, efficiency and effectiveness of thinking mind.
We will discuss this fundamental concept in a separate session.