[Author’s Warning: This entry involves references to and discussions about the system of Cartesian Coordinates and how it might assist mediators in reality testing and reframing. Readers with a Math Phobia are advised to proceed with cautious abandonment.]

[Author’s postscript: As I finish writing this, it becomes increasingly clear to me that some readers may find the ideas in this entry, if not a bit bizarre, then certainly off the wall. I promise next month, I will write something a little more normal.]

In last month’s entry, I wrote about using the NLP Meta Model to overcome problems of perception. This month’s entry essentially follows on from that and looks at how the mathematical construct of Cartesian Coordinates can assist mediators in reality testing and reframing.

The system of Cartesian Coordinates was created by René Descartes and revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. (I can almost see the eyes of readers glazing over!)

Put simply, if we take 2 lines and intersect them at right angles to one another, 4 quadrants are created. Naming the horizontal line “x” and the vertical line “y”, we label the point of intersection as “0”. To the right of the intersection on the horizontal line “x”, we can make gradations eg. “1”, “2”, “3” etc. To the left, we can gradate negative values eg. “-1”, “-2”, “-3”. Simlarly, above and below the intersection on the vertical line “y”, we can make positive and negative gradations respectively.

This provides us a system by which any point in any of the 4 quadrants created can be notated by a set of coordinates expressed in terms of “x” and “y”. For example, any point in the top right quadrant can be expressed by (x,y). Similarly:

– Any point in the top left quadrant expressed by (-x, y)

– Any point in the bottom right quadrant expressed by (x, -y)

– Any point in the bottom left quadrant expressed by (-x, -y)

“Fine and well”, the reader may say, “and thanks for the math lesson”. What does this have to do with mediation? I’m glad you asked ☺

When a party provides you a sentence e.g. “I can’t agree to this proposal”, the maker of the statement has figuratively (and some would suggest neurologically) created a box within which this thought or idea is kept. Arguing directly with this thought is rarely useful because it is creating a diametrically opposed idea or thought, which is by definition outside this box. They cannot co-exist; kind of like matter and anti-matter.

Mediators and other professional communicators generally sidestep this exercise in futility by asking questions. In this case, the obvious one is “Why not?” A useful question that will hopefully surface hitherto unidentified concerns or interests.

Another way of generating useful questions in this context is to use Cartesian Coordinates. The system of Cartesian Coordinates allows us to take 2 concepts within a sentence and run it through the 4 quadrants. Using the above example “I can’t agree to this proposal”, one could ask, “What would happen if you did?” This sets up a hypothetical frame that causes the maker of the statement to consider both the negative and positive consequences of accepting the proposal. (As an aside, this is a different question from “Why not?” which only surfaces the negative consequences of accepting the proposal.)

If we took the question “What would happen if you did” and translated them to (x, y) coordinates, then x = “What would happen” and y = “If you did”. Expressed formally, (x, y) = (What *would* happen, If you *did*). Seen this way, if should be apparent how 3 different but related questions could be asked.

– (-x, y): What *wouldn’t* happen if you *did*?

– (x, -y): What *would* happen if you *didn’t*?

– (-x, -y): What *wouldn’t* happen if you *didn’t*?

By answering these questions, it causes the maker of the statement to consider possibilities that are outside the box they have created. In fact, there will usually be a question from a particular quadrant where it will be difficult for the maker of the statement to consider and when they eventually think it through, they may find their views changed.

Don’t take my word for it. I invite readers to try this out for themselves. Think of something that you have been hesitating to do or unsure of doing. Then take a piece of paper, draw 4 quadrants on it and ask yourself the questions from each quadrant, noting in each quadrant what the pros and cons of each answer are.

Once you notice the insights this exercise provides, I invite you to try these questions in social situations until you work your way up to using them in a conflict situation.

The question set above is an example of using the Cartesian Coordinates in reality testing. The Cartesian Coordinates can also be used in reframing statements that have 2 concepts connected. Taking an example from my entry last month, “He is a bad father because he comes home late everyday”, this can be expressed formally as (-x, -y) = (bad father, comes home late). The other 3 quadrants can be expressed thus:

– (x, y): “Good father, comes home early” or “Not bad father, not comes home late”

– (-x, y): Bad father, comes home early” or “Bad father, not comes home late”

– (x, -y): “Good father, comes home late” or “Not bad father, comes home late”

It is useful to point out 2 things. First, it is obvious that a subjective judgment call is made as to whether to express “bad father” as “x” or “-x”. It really does not matter at the end of the day as long as one understands that “-x” is the opposite of “x”. Hence, (bad father, comes home late) can also be expressed as (x, y) which the corresponding redefinitions in the other 3 quadrants.

Secondly, expressing the opposite of “x” or “y” can be done either by providing an opposite concept i.e. late vs early or by adding a “not” in front of the concept. Doing the latter is actually more effective albeit ungrammatical.

Going back to our initial formulation above i.e. (-x, -y) = (bad father, comes home late), one could ask:

1. “How does coming home early make him a good father?” (y, x)

2. “How could coming home early make him a bad father?” (y, -x)

3. “How could coming home late make him an even better father?” (-y, x)

4. “Do you know of any bad fathers who come home early?” (-x, y)

5. “Do you know of any good fathers who come home late?” (x, -y)

One can see that in order to formulate the questions, it is permissible to reverse the order of x and y. Of these 5 formulations (and they are by no means exhaustive), the third one is likely to have most effect in getting the maker of the statement to think outside the box.

In closing, I would like to make 4 points about using the Cartesian Coordinates. First, as with using the Meta-Model, do not expect this to be a magic bullet. Some questions work better than others and it may require a string of questions around related topics to shift the maker of the statement.

Secondly, one should be prepared that sometimes, if one asks the right question that truly takes the maker of the statement outside their box, it can seem that they have difficulty considering the question or even hearing you. This is normal and the appropriate response is to repeat the question and ask them to fully consider it.

Thirdly, sometimes, the maker of the statement will provide a trite response where it will be evident that they did not fully consider what you asked. It becomes a judgment call on your part whether to persist. My experience has been that persistence often yields tremendous value.

Finally, sometimes, after having fully considered the question, the answer may make no sense or there may not even be an answer at all. This is ok. Even in these situations, there is a shift in the mindset of the maker of the statement that will be evident in their attitude.

Personally, Cartesian Coordinates is something I have had positive experiences using (in both counseling and the mediation context) and I hope readers will too!

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