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Philip ONeil



Cards - Prompt
Picture being a teacher and putting a class of 25 into groups of 5 for 10 minutes at a time.

Round 1:

Tell the students to form groups with 4 other people. This will happen quickly and naturally. After ten minutes it is time for round 2.

Round 2:

Tell the students to form groups with 4 different people who were not in the round 1 group. This will happen a bit slower but you shouldn't have too many problems. After ten minutes it is time for round 3.

Round 3:

Tell the students to form groups with 4 different people who were not in the round 1 group or round 2 group. This will happen considerably slower and it may seem impossible. If you manage to achieve this, ten minutes later it is time for round 4.

Round 4:

Tell the students to form groups with 4 different people who were not in the round 1 group, round 2 group or round 3 group. For teachers who try to do this, this is usually the time the teacher gives up and decides to let groups form any way.

This process of getting students to form groups with new people is called "Orthogonal Regrouping". The fact that each new group contains nobody from a previous group means that the regrouping process is said to be orthogonal to the previous group. Mathematicians have been developing orthogonal regrouping strategies for over 150 years. In the early days, it was more of a curious problem that mathematicians did because it was interesting and fun but over time they have been finding relevant uses for it. Generally, there is not a simple formula for generating a solution for certain situations and so mathematicians design software for finding solutions to problems for dealing with different total group sizes and sub group sizes to maximise the possible number of rounds are available.

With Mixintools, the complex mathematics behind working the best way to put people into groups has been taken out of the picture. Students have fun cards with insects, fruit, musical instruments, animals and plants and get to interact in 6 groups. Every group they go in is orthogonal to every other group and so they can interact with a large range of other students. The reason that it is near impossible for teachers to execute orthogonal regrouping for 4+ rounds without these cards is that the action that every student makes in every round all affects the probability of a possible orthogonal option in the future. This example of 25 students only works if the 25 students follow highly structured paths, not random selection. For this to happen without planning the paths before hand makes winning the lotto look easy.

It is still early days for discovering the benefits of orthogonal regrouping in larger groups but for generations dance teachers have been teaching the benefits of their students changing groups regularly in different stages of learning. The standard story is the example of the couple who stays together in a dance class for romantic reasons while the rest of the class changes partners and the romantic couple is often the couple who is most likely to form bad habits. The thinking is that regularly regrouping students raises the standard of work that students expect of each other and the cooperativeness of students to help each other.


Do Mixintools allow students to visit every member of the class?

The Mixintools set has been designed to allow every class member to go into 6 different groups. For the situation where the class is between 20-25 students and the group size is 5 people per group, it is possible for every student to share a group with every class member over 6 groups. However, most of the time this is not the case. The aim of the cards is to let the children interact with more than 50% of the class over multiple groups.

What is the point of the software?

The Mixintools software is an Excel based spreadsheet that is designed to allow teachers to assess students in multiple groups and have the average group scores worked out without collecting individual student scores. This means that teachers don't have to worry about calculating the student scores and can focus on assessing groups where the student scores are calculated for the teacher.

What are the different colours on the cards for?

The different colours on the cards serve two purposes. Firstly, the instructions for handing out the cards are to hand out one colour at a time. This is a preventative method for when the number of students is not a perfect multiple of the number of people per group where handing out one colour at a time guarantees that there are no tiny groups at any one time and the range in groups differs only by one. Secondly, the colours offer group stratification benefits. If you want to guarantee that there is different genders in each group, give the boys and girls different coloured cards. If you want to separate certain students, give them the same coloured card. If you want to give a spread of ability in each group, give the higher performing students certain colours and other students other colours.

Why do the sets for 5 groups and 6 groups have the same coloured card for the same plants?

The reason for this is related to the constraints in what the mathematics allows. The sets for 7 or more groups have 6 sets of groups with different colours for every group but the smaller number of groups don't allow for this because it is impossible. Depending on how you wish to use the cards (with or without stratification) this may mean you may want to avoid or use plant groups when using the smaller numbers of sets. It is also worth noting that when using the sets for 6 groups, there is only 5 plant groups and they are groups of 6, not 5. This is also due to mathematical constraints for the cards. The decision was made to keep a uniform design to all cards as despite the constraints that reality brings, it is believed that many teachers will still want to use the plant groups for these cards.

What is orthogonal regrouping?

Orthogonal is a mathematical term that in terms of the use of these cards describes the fact that no two group members share the same group twice. Regrouping is a term to describe forming a new group. Hence combining these two words means forming new groups where no two members share the same group twice. The mathematics behind this term dates back to a famous mathematical problem in 1850 known as the Kirkman's schoolgirl problem where students were put into groups to organise students into different groups of 3 over 7 days to ensure that students got to interact with different friends each day. The mathematics behind this problem has been evolving slowly as depending on each mathematical condition there requires a different arrangement but it has continued to this day where now researchers use computers and programming to learn more about this. To a large extent the educational benefits of orthogonal regrouping are still being realised.

What are the benefits of orthogonal regrouping?

The best analogy of the benefits of orthogonal regrouping is what often happens with "the jealous partner problem" in dance classes. Many dance instructors instruct their students to regularly change partners so that they can learn with different people. Often couples come to learn dancing and refuse to swap partners as they want to dance together. Instructors find that these couples are often the worst performing people in the class as their limited experience leads to them forming bad habits together. It is observed that by regularly changing groups, the class members appear to have higher expectations of each other and generally perform better than class members who stay together all the time. There are times when it is important for people to stay together but often there is more to benefit by allowing students to learn from a broader range of other students.

Why do I need the Mixintools cards for orthogonal regrouping?

Put simply, the mathematics that makes the cards work has already been worked out for you. If you try to get children into groups by asking them to find new students or by some random method, you should find that you will be lucky to get your students interacting in more than 3 groups with new students every time. A lot of work has been done to ensure that the mathematics behind these cards works and so what would be a very complex process is now an easy one because it has been worked out for you.

How long do children go in the groups?

The cards can be used for quick events or slow events. What is often fun for students is to use them for an hour with 10 minutes per group. However they can also be used over months with 2 weeks per group. How you want to use the cards is up to you.