Rich maths with Dragons

Thanks to the Unlocking Curious Minds fund, StatsLC have been enabled to visit thirty rural schools in Canterbury and the West Coast and provide a two-hour maths event to help the children to see themselves as mathematicians. The groups include up to 60 children, ranging from 7 to 12 years old – all mixed in together. You can see a list of the schools we have visited on our Rich Maths webpage. And here is a link to another story about us from Unlocking Curious Minds.

What do mathematicians do?

We begin by talking about what mathematicians do, drawing on the approach Tracy Zager uses in “Becoming the Math teacher you wish you had”. (I talk more about this in my post on What Mathematicians do.)

• Mathematicians like a challenge.
• Mathematicians notice things and wonder
• Mathematicians make mistakes and learn
• Mathematicians work together and alone.
• Mathematicians have fun.

You can see a video of one of our earlier visits here.

Each child (and teacher) is given a dragon card on a lanyard and we do some “noticing and wondering” about the symbols on the cards. We find that by looking at other people’s dragons as well as our own, we can learn more. As each of the symbols is explained, there follows an excited buzz as children discuss whose dragon is stronger or older, or has more dangerous breath.  We wonder if green dragons are more friendly than red dragons and work together, making a human data table, and using proportional thinking to draw some conclusions.

A small sample of Dragonistics data cards

Mixed group work

Next, in randomly chosen, mixed level groups of three, the children perform their own statistical investigations. They have randomly assigned roles, as dragon minder (looking after the cards), people minder (making sure everyone is participating) or record minder (making sure something gets written down). They take their roles seriously, and only occasionally does a group fail to work well. The teachers are free to observe or join in, while Shane and I go from group to group observing and providing guidance and feedback. All learners can take part at their own level.

As we visit a variety of schools we can see the children who are more accustomed to open-ended activities. In some schools, and with the older children, they can quickly start their own investigations. Other children may need more prompting to know where to begin. Sometimes they begin by dividing up the 24 cards among the three children, but this is not effective when the aim is to study what they can find from a group of dragons.

Levels of analysis

It is interesting to observe the levels of sophistication in their analysis. Some groups start by writing out the details of each individual card. I find it difficult to refrain from moving them on to something else, but have come to realise that it is an important stage for some children, to really get to understand the multivariate nature of the data before they begin looking at properties of the group. Others write summaries of each of the individual characteristics. And some engage in bivariate or multivariate investigations. In a sequence of lessons, a teacher would have more time to let the learners struggle over what to do next and to explore, but in our short timeframe we are keen for them to find success in discovering something. After about fifteen minutes we get their attention, and get them to make their way around the room and look at what the other groups are finding out. “Mathematicians learn from other mathematicians”, we tell them.

Claims

Sometimes groups think they have discovered everything there is to know about their set of dragons, so we have a range of “claims” for them to explore. These include statements such as:

• Is this true? “There are more green dragons than red dragons.”
• Is this true? “Changeable dragons are less common than friendly or dangerous dragons.”
• Is this true? “There are more dragons younger than 200 than older than 200.”
• Is this true? “Fire breathing dragons are mainly female.”
• Is this true? “There are no fire breathing, dangerous green dragons.”
• Is this true? “Strong dragons are more dangerous.”

Some of the claims are more easily answered than others, and all hint at the idea of sample and population in an intuitive rather than explicit way. Many of them require decisions from the learners, such as what does “mainly” mean, and how you would define a “strong” dragon?

The children love to report back their findings.  Depending on the group and the venue, we also play big running around games where they have to form pairs and groups, such as 2 metres different in height, one of each behaviour, or nothing at all the same. That has proved one of the favourite activities, and encourages communication, mathematical language – and fun! Then we let them choose their own groups and choose from a range of mathematical activities involving the Dragonistics data cards.

The children work on one or more of the activities in groups of their own choice, or on their own. Then in the last fifteen minutes we gather them together to revisit the five things that mathematicians do, and liken it to what they have been doing. We get the children to ask questions, and we leave a set of Dragonistics data cards with the school so they can continue to use them in their learning. It is a blast! We have had children tell us it feels like the first time they have ever enjoyed mathematics. Every school is different, and we have learned from each one.

A wise intervention

The aim is for our event to help children to change the way they feel about maths in a way that empowers them to learn in the future. There has been research done on “wise interventions”, which have impact greater than their initial effect, due to ongoing ripples of influence. We believe that helping students to think about struggle, mistakes and challenge in mathematics in a positive light, and to think of themselves as mathematicians can reframe future events in maths. When they find things difficult, they may see that as being a mathematician, rather than as failing.

Lessons for us

This is a wonderful opportunity for us to repeat a similar activity with multiple groups, and our practice and theory are being informed by this. Here is an interesting example.

At the beginning of the open-choice section, we outline the different activities that the children can choose from. One is called “Activity Sheets”, which has varying degrees of challenge. It seems the more we talk up the level of challenge in one of the activities, the keener the children are to try it. Here is a picture of the activity:

The activity involves placing nine dragons cards in position to make all of the statements true. Originally the packs included just 20 dragons, and by swapping in and out, it is challenging. However, when you have just nine dragons to place, it can be very difficult. Now for the first few visits, when children rushed to show us how they had completed their sheet, we would check it for correctness. However, through reading, thinking and discussion we have changed out behaviour. We wish to put the emphasis on the learning, and on the strategy. Peter Johnston in his book, “Choice words: how our language affects children’s learning” states,

“The language we choose in our interactions with children influences the ways they frame these events and the ways the events influence their developing sense of agency.”

When we simply checked their work, we retained our position as “expert”. Now we ask them how they know it is correct, and what strategies they used. We might ask if they would find it easier to do it a second time, or which parts are the trickiest. By discussing the task, rather than the result, we are encouraging their enjoyment of the process rather than the finished product.

We hope to be able to take these and other activities to many more schools either in person or through other means, and thus spread further the ripples of mathematical and statistical enjoyment.

The role of context in statistical analysis

The wonderful advantage of teaching statistics is the real-life context within which any applicaton must exist. This can also be one of the difficulties. Statistics without context is merely the mathematics of statistics, and is sterile and theoretical.  The teaching of statistics requires real data. And real data often comes with a fairly solid back-story.

One of the interesting aspects for practicing statisticians, is that they can find out about a wide range of applications, by working in partnership with specialists. In my statistical and operations research advising I have learned about a range of subjects, including the treatment of hand injuries, children’s developmental understanding of probability, the bed occupancy in public hospitals, the educational needs of blind students, growth rates of vegetables, texted comments on service at supermarkets, killing methods of chickens, rogaine route choice, co-ordinating scientific expeditions to Antarctica and the cost of care for neonatals in intensive care. I found most of these really interesting and was keen to work with the experts on these projects. Statisticians tend to work in teams with specialists in related disciplines.

Learning a context can take time

When one is part of a long-term project, time spent learning the intricacies of the context is well spent. Without that, the meaning from the data can be lost. However, it is difficult to replicate this in the teaching of statistics, particularly in a general high school or service course. The amount of time required to become familiar with the context takes away from the time spent learning statistics. Too much time spent on one specific project or area of interest can mean that the students are unable to generalise. You need several different examples in order to know what is specific to the context and what is general to all or most contexts.

One approach is to try to have contexts with which students are already familiar. This can be enabled by collecting the data from the students themselves. The Census at School project provides international data for students to use in just this way. This is ideal, in that the context is familiar, and yet the data is “dirty” enough to provide challenges and judgment calls.

Some teachers find that this is too low-level and would prefer to use biological data, or dietary or sports data from other sources. I have some reservations about this. In New Zealand the new statistics curriculum is in its final year of introduction, and understandably there are some bedding-in issues. One I perceive is the relative importance of the context in the students’ reports. As these reports have high-stakes grades attached to them, this is an issue. I will use as an example the time series “standard”. The assessment specification states, among other things, “Using the statistical enquiry cycle to investigate time series data involves: using existing data sets, selecting a variable to investigate, selecting and using appropriate display(s), identifying features in the data and relating this to the context, finding an appropriate model, using the model to make a forecast, communicating findings in a conclusion.”

The full “standard” is given here: Investigate Time Series Data This would involve about five weeks of teaching and assessment, in parallel with four other subjects.(The final 3 years of schooling in NZ are assessed through the National Certificate of Educational Achievement (NCEA). Each year students usually take five subject areas, each of which consists of about six “achievement standards” worth between 3 and 6 credits. There is a mixture of internally and externally assessed standards.)

In this specification I see that there is a requirement for the model to be related to the context. This is a great opportunity for teachers to show how models are useful, and their limitations. I would be happy with a few sentences indicating that the student could identify a seasonal pattern and make some suggestions as to why this might relate to the context, followed by a similar analysis of the shape of the trend. However there are some teachers who are requiring students to do independent literature exploration into the area, and requiring references, while forbidding the referencing of Wikipedia.

This concerns me, and I call for robust discussion.

Statistics is not research methods any more than statistics is mathematics. Research methods and standards of evidence vary between disciplines. Clearly the evidence required in medical research will differ from that of marketing research. I do not think it is the place of the statistics teacher to be covering this. Mathematics teachers are already being stretched to teach the unfamiliar material of statistics, and I think asking them and the students to become expert in research methods is going too far.

It is also taking out all the fun.

Keep the fun

Statistics should be fun for the teacher and the students. The context needs to be accessible or you are just putting in another opportunity for antipathy and confusion. If you aren’t having fun, you aren’t doing it right. Or, more to the point, if your students aren’t having fun, you aren’t doing it right.

Some suggestions about the role of context in teaching statistics and operations research

• Use real data.
• If the context is difficult to understand, you are losing the point.
• The results should not be obvious. It is not interesting that year 12 boys weigh more than year 9 boys.
• Null results are still results. (We aren’t trying for academic publications!)
• It is okay to clean up data so you don’t confuse students before they are ready for it.
• Sometimes you should use dirty data – a bit of confusion is beneficial.
• Various contexts are better than one long project.
• Avoid the plodding parts of research methods.
• Avoid boring data. Who gives a flying fish about the relative sizes of dolphin jaws?
• Wikipedia is a great place to find out the context for most high school statistics analysis. That is where I look. It’s a great starting place for anyone.