Relational Mathematics

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This workshop is one in a series designed to help organisations to improve the creative potential of their teams. Trying to do so in a short 'taster' session is a big challenge, because teamwork is a mixture of psychological, cultural, social and ethical beliefs and processes. Even if we understood all of these tendencies, every team player is an individual with a unique style of commitment, loyalty and team spirit. In seeking to sidestep these complexities, this workshop will explore some very simple mathematics that you can apply in different circumstances to make your teams work more effectively.

Task 1. - complete this QUESTIONNAIRE for testing levels of tolerance to ambiguity. Why? - because we want to raise awareness of several issues:
  1. Perhaps we all think differently (cognitive diversity)
  2. Each approach probably has both advantages and disadvantages
  3. A given approach to reasoning may be more or less useful, depending on the situation/context.

At school, we are expected to believe that addition and subtraction are simple and straightforward, but the logic only works when we can ignore the presence of 'entanglement'. The same mistake applies in economics (the mathematics behind money is non-relational, therefore cannot wisely be applied to living systems).

  • E.g. from Heraclitus to Cratylus
  • My youngest granddaughter tells me she is angry with her teacher, and with the cruel logic that the schools system metes out to children.
    • Her example: it takes four men half an hour to dig a hole, how many holes will they dig in 2 hours? The difficult part of this question is not the number system itself, but how to second guess what the examiner is asking. Is it the same question to ask ' it takes four men half an hour to dig a hole, how many holes will they dig in 22 hours?' Is the workforce hypothetical, or actual? However, the child is seldom, if ever, told the assumptions behind the question. Perhaps the men are able to work, indefinitely, at a constant rate. are self employed, gig economy workers who all work at a standard rate and only contribute to the task only work when they want to rather how can four men maintain a constant does not really exist, or that raises a second-order question that may be obvious to most children -
Task 2. - use the individual scores to create duos (high-scorers partnered with low-scorers). Give each duo the following (3 cup) problem.


Task 3.

Tell them all "We are going to start with a simple mathematical puzzle and we want you to do this together with your partner". (This is called the 3 cups problem). In the beginning position of the problem, one cup is upside-down and the other two are right-side up. The objective is to turn all cups right-side up in no more than six moves. You must turn over exactly two cups at each move.


  1. "Who solved the problem?"
  2. "Did you come to any conclusions about the task itself?"
  3. "Did you know that it is believed to be mathematically impossible?''
  4. "Did anybody find it frustrating to have to work with another person?"
Task 4.
  1. If/where there seems to have been conflicts in partnerships (i.e. between the different cognitive types), add a third person as emulsifier__.
  2. These trios could continue alongside other teams to run the next task.
Task 5.
  1. Create 2 or 3 teams
  2. Tell them all "This is a workshop about fairness."
  3. Give each participant a knife.
  4. Reveal a big block of delicious and attractive cake / cheese
    • (secretly weigh it before the event).
  5. Ask everyone to divide it up equally so each participant gets a fair share.
  6. Ask whether the division was fair, and who got more than whom.
  7. Weigh the cheese to see whether some has disappeared during the process.


  1. Were there any 'winner/s' (or 'losers')?
  2. ''Did the winners set out to get more cheese than anyone else?
    • if so, do they love cheese / or do they just love winning?'
  3. Ask if this was a relevant question (who said it was competitive?)
  4. Ask whether the resource is sustainable.
  5. Ask whether anyone knows about the Law of Diminishing Returns.

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Holding Individuals to Account

When teams underachieve we may be tempted to blame individual team players, rather than looking into the relations that characterise the team. Similarly, when a partnership breaks down people may take sides with one, or other partner. When this happens the 'blamers' might express their judgement as a ratio, percentage, or fraction of a whole number that shows the (assumed) culpability of each partner. For example:

    • 50:50
    • 20:80
    • 35:65 etc.

In this partner-focused approach, each example adds up to 100, because it uses the logic of addition. When adding percentages, the rules of adding up make it impossible to exceed 100%. This logic assumes a world in which 1+1=2. But this does not work the same way with people, ideas or innovations.

Accounting for Teams

As Paul Romer, the economist, said, “...possibilities do not add up. They multiply.” (Romer, 1991).

Many everyday things (e.g. cooking/gardening/raising a family) show us that when we combine complex things we produce side-effects that change the quality, or character of the ingredients themselves. Part of the trick is knowing when to focus on the parts and when to focus on the relationships among the parts. Teams only work when they come together as team members. For this reason we will focus on team relations, rather than the actions of individuals.

  1. Create 2 or 3 teams.
  2. Tell them "This is NOT a workshop about fairness".
  3. Ask each team member to write down as many funny/useful/wise anecdotes as possible (silently, in 5 minutes).
  4. Ask them to choose 1 (the funniest/most useful/wisest), stand up and tell it to the rest of the team.
  5. Ask each team to compile a list and to count the number of jokes/stories etc..
  6. Give a prize to the team with the MOST jokes.
  7. Give a prize to the FUNNIEST team.
  8. Give a prize to the MOST MEMORABLE/USEFUL team.
  9. (perhaps) give a prize to the team that:
    • a) Didn't enjoy themselves...
    • a) Found it painful...
    • c) Didn't know any jokes...
    • d) Couldn't see any point in the exercise...


  1. In the last exercise we finished by discussing the mathematical logic of 1+1=2.
  2. We asked everyone to donate ONE story/joke but we ended up with everyone knowing ('X' number of) jokes.
  3. The numbers are clear but the outcomes were not of equal value (?)
  4. Ask what the teams gained from the exercise:
    • e.g. Fun <?>
    • e.g. Experience of working together <?>
    • e.g. Insights into their team members <?>
    • e.g. Bonding on a social level <?>
    • e.g. Witnessing different levels of thinking <?>
  5. Ask whether anyone knows about the Returns to Scale.

What is Synergy?

  • Combining things can produces synergy
    • does this mean that 1 + 1 = >2?
    • (We normally assume that the two things are 'positives')
  • But what would happen if we combine 2 negatives?
    • Is it true that (-1) + (-1) = (+)1?
  1. Create 2 or 3 teams.
  2. Tell them "This is NOT a workshop about problem-solving....and it's more difficult".
  3. Ask the teams to work in PAIRS.
  4. Each participant writes down ONE particular PROBLEM or LACK.
  5. Each pair reveals their problem/lack to the other.
  6. The task is to work out what would happen when the 2 problems are combined.


  • Did anyone find learn anything, or find anything useful?
    • e.g. did it help to solve anyone's 'problem', or reduce anyone's 'lack'?
    • e.g. Did the combination produce:
      • A: 2 problems?
      • B: 1 BIG problem?
      • C: Some new opportunities?
      • D: Some synergies?
  • Synergy is a free bonus that is usually a surprise.
    • synergy “the behaviour of whole systems unpredicted by the behaviour of their parts taken separately” (Fuller, 1971).

Painful Synergies

Putting together a bunch of very high-achievers to work as a team frequently produces surprisingly poor results. (See The Apollo Syndrome).

Relationships Don't Add Up

This suggests that, when we set up a relationship, we might expect to get more out than what we put in. In percentage terms we should not be surprised to get more than 100% of what we started with. To many, this may seem counterintuitive and wrong. However, when we analyse the mathematical ratios between the number and team members and the number of relationships among them, this anomaly begins to make sense. Hence,

  • In a team of 8, each team member is responsible for 25% of all relations
  • In a team of 4, each team member is responsible for 50% of all relations
  • In a team of 3, each team member is responsible for 66.6% of all relations
  • In a team of 2, each team member is responsible for 100% of all relations

Harnessing the Surplus

  • (When) can 2 + 2 = 6?
  • In the last workshop we worked with pairs.
  • In this experiment, one/all/none of the pairs found that the combining of 2 problems created an additional outcome.
    • If so, this is like a duet...you can hear both voices/instruments...but you also hear the COMBINATION.
  • We can see this as a synergy.


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