We didn’t always have our mission or aim or vision as well articulated, and if asked we tended to give some vague description like – “we make stuff to help people learn maths and statistics.”

StatsLC identifies as a social enterprise because we are driven by a purpose beyond making profit for shareholders, and our purpose is a social good – in this case education. A social enterprise exists in the continuum between a business which operates for profit, and a charity, which is strictly not-for-profit, but measures its effectiveness in different ways. We wish ultimately to be self-sustaining so that we are not at the mercy of grants or contracts with outside providers.

We, the directors, have spent the last eight months, on and off, working on our purpose, customer focus, financials and operations as part of an Elevate course with Ākina. The course is aimed at social enterprises, and we have been participating with between five and eight other social enterprises based in Christchurch, New Zealand.

At our last session Ākina wanted to know what value we have gained from the course, what it does well and what can be improved. Ākina itself is a social enterprise that helps other social enterprises. Social Enterprise is a popular phenomenon, particularly in our area, where recently Ākina hosted the World Forum.

The first unit of four sessions, one morning per week, addressed our impact. We needed to identify what we are trying to achieve, why and how. We talked about vision, mission and purpose. This would help us later to think about who are our customers and who are our beneficiaries. I still find the delineation between vision, mission and purpose a bit confusing. Our vision has expanded during the course. This is where we are up to now, though it is still a work in progress.

Vision – a world of mathematicians

Purpose: We invent, create and disseminate resources and ideas to enable people to learn and teach mathematics and statistics enjoyably and effectively.

We | invent | resources | to enable people to | learn | mathematics | enjoyably |

create | and | and | and | and | ||

disseminate | ideas | teach | statistics | effectively |

As we considered our impact we realised that we are making an impact. We have over 1000 views of this blog daily. There are over 35,000 subscribers on our Youtube channel. Hundreds of children and teachers have been inspired and enthused by our “Rich Maths” events. You can see more about our impact here: Statistics Learning Centre Impact.

We have not been doing well at specifying exactly what impact we aim to have, and measuring it. Originally our impact was with teachers and learners of secondary and university level statistics. However we are now thinking bigger, and wish to create a world of mathematicians. We truly believe that education is a political act, and knowledge of maths and statistics empowers people, allows greater career choice and enables informed citizenship.

The “customer” or marketing section of the course was the one we felt most in need of, and probably are still most in need of. We learned that we need to ask what problem we are solving and for whom. This has led to serious thought and discussion on our part as we have so many ideas about how we can do good, and for whom. However, the point of social enterprise is that you are not a charity, so need to trade or provide services for money in order to be sustainable. So we need to identify our customers – the people and organisations that will pay money for what we do – either for them or for others.

At the time we were gearing up for a holiday programme, and we used some of the ideas to advertise on Facebook. One outcome of the course is that we have decided we need to employ someone to help with the marketing.

As we already have an accounting package, Xero, and work with an accountant, the need for help here felt less imperative. We have developed different systems in using Xero that will help us analyse our progress. One idea that was valuable was to do with the value of our time. Time and money emphasis did not have to be commensurate in all circumstances. Two sessions on budgets were helpful when thinking about grant applications. We have thought more about cashflow, though a crisis at the end of 2016 had already made us aware of potential problems. We started paying ourselves.

What has become clear throughout the course is that we do not have enough time between the two of us to do all the things we need, as well as maintaining cashflow through contracts. This has helped us to recognise the need to employ someone to cover our areas of weakness, in particular marketing. We also need to develop more passive income streams.

What was extremely valuable in this section was learning about employment contracts and health and safety. We are now formalising our contracts with staff. Being a responsible employer, even for family members, takes a bit of work.

Another useful session concerned governance, management and operations. As a small enterprise, both of us tend to fill all three roles. At this point we need to get some advice at the governance level – even just having someone to ask us questions and to report to periodically. It can be easy to spend too much time chipping away at the coalface, and losing direction. It can also be seductive to spend all our time discussing visionary ideas for future development, rather than getting on and producing. Like most of life, the answer lies in a balance.

A common expression in social enterprise is Mission Drift meaning letting the commercial aspects over-ride the social impact focus or mission.

We tend to suffer from something similar, that I call mission lurch. I’m not sure it is the right term, as it is more that we are adapting our mission in order to align it better with activities that will lead to sustainability. Our problem is that we need to be doing some more activities that bring in revenue to sustain our mission.

One big benefit from participating in the programme has been making contact and building relationships with others in similar circumstances. This builds confidence.

For me the big lessons from this course are

- Articulating our mission
- Confidence to do something big

A year ago I was quite happy to dabble around in the edges of business/social enterprise. We were not really making enough to keep us going, but had hope that something might change. Over the course of 2017 we have had contracts with Unlocking Curious Minds, to take exciting maths events to primary schools. We have also gained contracts writing materials for other organisations. Our success in these endeavours, along with the help from the Elevate course has helped us to think bigger.

Watch this space!

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Our job as teachers at any level is to teach the students we have. I embrace this idea from Dr Kevin Maxwell:

“Our job is to teach the students we have.

Not the ones we would like to have.

Not the ones we used to have.

Those we have right now.

All of them.”

I believe Maxwell’s focus was on the diverse learning needs we have in our classes. I would like to take another angle on this. If students do not have the needed skills to learn what we are teaching, then we need to teach those skills.

In many subjects, content and the skills are largely uncoupled. For example in history, a skill might be to integrate material from two conflicting sources. You can learn this in multiple contexts, and you do not need to know the history of the world up until 1939 in order to study World War II.

In mathematics, there are clear progressions. It is very difficult to learn about trigonometry if you do not have a good working knowledge of the Pythagorean theory. And learning Pythagoras is built on applying formulas, which is built on basic algebra. I admit, that as I write this I can see other approaches, but the point is that later learning in maths is built on earlier knowledge, understanding and skills. Learning in maths is also built on earlier feelings – a post for another day.

There are two gaps we need to mind. The gap between levels of schooling, and the gap between what the preparation the students need, and what they have. I taught at the University of Canterbury for twenty years, and often heard colleagues complain about the level of preparation in our students. I am ashamed to say that it took me several years to realise that if our students do not have the foundation they need to learn what we are teaching, then we need to do something about it. As a result I created a course that started with making sure students knew how to use < and >, and which is bigger out of 0.04 and 0.2. These are necessary in order to make decisions about rejecting a null hypothesis.

Recently at a workshop I asked a group of about forty teachers how many of them have students starting high school who do not have the necessary knowledge of number skills – basic facts and multiplication tables. Every hand went up. There is a gap. I asked them what they are doing about it. Some suggested working in “Communities of Learning” to help primary schools to prepare the students better. This is fine, but what are they doing now! There was some discussion that if we are teaching lower curriculum levels at high school, they may never cover the materials at higher levels.

For that I have two responses. The first relates to the Maxwell quote I started with. “Our job is to teach the students we have.” Our job is to teach the students we have, the things they need to learn. If our students start high school without a good enough grasp of basic facts, then we need to help them to develop them. And we need to work out good ways to do this. I suspect part of the problem is that secondary maths teachers do not have training or knowledge in teacher beginning maths. Do we believe it is not part of our job?

The second response is that there is no point in moving on to later maths if the students’ foundation is weak. Now I say this with some trepidation as I can picture students being held back until they become fluent in their tables. This is not what I mean. One of the participants in the workshop asked me how I would go about setting up a programme to help such students. Obviously this is not a question I could answer on the spot, but here are some ideas and principles.

Summary:

**Do not**under any circumstances give these students tests with time pressure.- Expect the students to be able to learn what is needed more quickly than they would have done when younger.
- Engage students in deciding what they need to learn and how.
- Integrate the skills into other activities
- Make it fun

Explained:

Read Fluency without fear by Jo Boaler. Read this about Maths trauma. Do not add to the students’ feelings of inadequacy. One possibility if you wish to give a diagnostic test, and want to have some idea of how long they take, tell them they have as long as they need, but after a certain amount of time get them to change to a different coloured pen.

My experience with teaching adults and teens is that once they realise they can learn, they learn quickly. Believe it. I don’t mean that they can answer questions quickly, but that they will be able to progress more quickly as they have better metacognitive skills, literacy, maturity.

This is their learning. Make sure the students know why they need the skills and how they will help them. Talk to students about how they would like to learn them, and let them choose their own reward system if appropriate. Different students will have different areas of weakness, and different ways to improve.

I can’t imagine much worse than an entire maths lesson on basic facts. If we are working on multiplication, this fits well with area calculations. We also need to keep revisiting.

There is a place for well made and used flash cards to improve retrieval. There are multiple posts on using flashcards well. I would recommend them for some students for the last sticky facts, like 6 x 7, 6 x 8, 7 x 8 etc. Those were the ones that got me stumped. However, most knowledge is better gained in context. Create or find rich, open-ended tasks that help develop the skills the students need.

Maths lends itself to games and fun. If you can’t think up a way – find it on line. But if you don’t think it’s fun the students aren’t going to. (Not sure the converse is true, but…)

Our aim at Statistics Learning Centre is a world of mathematicians. My dream is for math trauma to be a thing of the past, and for all citizens to embrace mathematical thinking similarly to literacy. As maths and statistics educators we can work towards this. The most important student you have in your maths class is the one who becomes a primary school maths teacher. Make sure she loves maths!

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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.

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.

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.

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.

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.

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.

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This is written in the week before the 2017 New Zealand General Election and it is an exciting time. Many New Zealanders are finding political polls fascinating right now. We wait with bated breath for each new announcement – is our team winning this time? If it goes the way we want, we accept the result with gratitude and joy. If not, then we conclude that the polling system was at fault.

Many wonder how on earth asking 1000 people can possibly give a reading of the views of all New Zealanders. This is not a silly question. I have only occasionally been polled, so how can I believe the polls reflect my view? As a statistical communicator, I have given some thought to this. If you are a statistician or a teacher of statistics, how would you explain that inference works?

Here is my take on it.

Imagine you have a bowl of seeds – mustard and rocket. All the seeds are about the same size, and have been mixed up. These seeds are TINY, so several million seeds only fill up a large bowl. We will call this bowl the population. Let’s say for now that the bowl contains exactly half and half mustard and rocket, and you suspect that to be the case, but you do not know for sure.

Say you take out 10 seeds. The most likely result is that you will get 4,5 or 6 mustard seeds. There is a 65% chance, that that is what will happen. If you got any of those results, you would think that the bowl might be about half and half. You would be surprised if they were all mustard seeds. But it is possible that all ten seeds are the same. The probability of getting all mustard seeds or all rocket seeds from a bowl of half and half is about 0.002 or one chance in five hundred.

Now, if you draw out 1000 seeds, it is quite a different story. If all the 1000 seeds drawn out were mustard, you would justifiably conclude that the bowl is not half and half, and may in fact have no rocket seeds. But where do we draw the line? How likely is it to get 40% or less mustard from our 50/50 bowl? Well it is about one chance in 12,000. It is possible, but extremely unlikely – though not as unlikely as winning Lotto. We can see that the sample of 1000 seeds gives us a general idea of what is in the bowl, but we would never think it was an exact representation. If our sample was 51% mustard, we would not sensibly conclude that the seeds in the bowl were not half and half. In fact, there is only a 47% chance that we will get a sample of seeds that is between 49% and 51%.

Of course we know we are not little seeds, but people. In fact we like to think we are all special snowflakes. (The scene from “Life of Brian” springs to mind. Brian – “You are all individuals”, crowd – “We are all individuals”, single response – “I’m not!”)

But the truth is that as a group we do act in surprisingly consistent ways. Every year as a university lecturer I tried new things to help my students to learn. And every year the pass rate was disappointingly consistent. I later devised a course that anyone could pass if they put the work in. They could keep resitting the tests until they passed. And the pass rate stayed around the same.

People do tend to act in similar ways. So if one person changes their viewpoint, there is a pretty good chance that others will have also. So long as we are aware of the limitations in precision, samples are good indicators of the populations from which they are drawn.

Here is a link to our video about Inference.

I have described why polls generally work. The media tends to dwell on the times that they fail, so let’s look at why that may be.

Sometimes the poll may just be the one that takes an unlikely sample. There is a one in a thousand chance that ten seeds from my bowl will all be mustard – and a one in a thousand chance that all will be rocket. It is not very likely, but it can happen. Similarly there is a teeny chance that we will get a result of less than 45% or more than 55% when we take out 1000 seeds. Not likely, but possible. This is called sampling error, and that is what the margin of error is about. Political polls in NZ generally take a sample of 1000 people, which leads to a margin of error of about 3%. What margin of error means is that we can make an interval of 3% either side of the estimate and be pretty sure that it encloses the real value from the population. So if a poll says 45% following for the Mustard Party, then we can be pretty sure that the actual following back in the population is between 42% and 48%. And what does “pretty sure” mean? It means that about one time in twenty we will get it wrong and the actual following, back in the population is outside that range. The problem is we NEVER know if this is the right one or the wrong one. (Though I personally choose to decide that the polls that I don’t like are the wrong ones. ;))

Non-sampling error and bias

There are other problems – known as non-sampling error. I wrote a short post on it previously.

And this is where the difference between seeds and people becomes important. Some issues are:

When we take a handful of seeds from a well-mixed up bowl, every seed really does have an equal chance of being selected. But getting such a sample from the population of New Zealand is much more difficult. When landlines were in most homes, a phone poll could be a pretty representative sample. However, these days many people have only mobile phones, and which means they are less likely to be called. This would not be a problem if there were no differences politically between landline holders and others. I think most people would see that younger people are less likely to be polled than older, if landlines are used, and younger people quite possibly have different political views. Good polling companies are aware of this and use quota sampling and other methods to try to mitigate this.

The wording of the question and the order of questions can affect what people say. You can usually find out what question has been asked in a particular poll, and it should be reported as part of the report.

Unlike seeds, people do not always show their true colours. If a person is answering a poll within earshot of another family member, they may give a different answer to what they actually tick on election day. Some people are undecided, and may change their mind in the booth. Undecided voters are difficult to account for in statistics, as an undecided voter swinging between two possible coalition partners will have a different impact from a person who has not opinion or may vacillate wildly.

In a volatile political environment like the one we are experiencing in New Zealand, people can change their mind from day to day as new leaders emerge, scandals are uncovered, and even in response to reporting of political polls. The results of a poll can be affected by the day and time that the questions were asked.

On balance, polls are a blunt instrument, that can give a vague idea about who people are likely to vote for. They do work, within their limitations, but the limitations are fairly substantial. We need to be sceptical of polls, and bear in mind that the margin of error only deals with sampling error, not all the other sources of error and bias.

And as they say – the only truly correct poll is the one on Election Day.

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We ask children what mathematicians do, and the answers include, “they do mathematics”, “they get things right”, and “they answer questions.” Hmm.

Recently in guest workshops I asked about 120 pre-service primary/elementary teachers how many see themselves as mathematicians. Each time, there were about 10% who identified as mathematicians. I then asked them, how many would like the children they teach to think of themselves as mathematicians. It was almost 100% to the affirmative. And then I ask, “Do we have a problem?”

I also introduced the idea of maths trauma, that I wrote about in a previous post, and explained that preservice elementary school teachers have been found to have the highest rate of self-identified maths trauma among undergraduate students. The heads were nodding, so I asked who would say they had maths trauma. Nearly one-third of the teachers said they felt traumatised by maths. Some came and talked to me individually after our session, and told me how their fear of maths was restricting the age group they felt they would be comfortable teaching. My message to them, is the very important message I learned from the webinar – “It is not your fault.” They have been taught maths in a way that was not suitable to them, or they many have had one terrible experience that put them off permanently. * It is not their fault*. And we and they need to do something about this.

Now there is quite a gap between being traumatised by maths, and perceiving oneself as a mathematician. I have, before my mathematical renaissance, been known to say that I was not a mathematician, as I saw myself as an operations researcher or statistician, rather than the abstract-focused (I believed) mathematician. I tend to think concretely, and had perceived that that excluded me from the ranks of true mathematicians. I have also written posts outlining the difference between mathematicians and statisticians (and operations researchers).

But these days, I have become a maths activist. Or maybe a maths whisperer? My mission, for the rest of my life, is to help people, and in particular, teachers, overcome their fear or dislike of mathematics and perceive themselves as mathematicians.

Education is a political act, and knowledge of maths and statistics empowers people, allows greater career choice and enables informed citizenship. (Nic Petty)

I have learned a great deal from the MTBOS or Maths Twitter Blogosphere. I hope one day to attend a Twitter Math Camp (#tmc17), but I fear I am destined always for #tmcjealousycamp where all the wannabetheres lurk. One of the best things was to find out about “Becoming the Math Teacher you wish you’d had” by Tracey Zager. The book is organised in chapters focused on what mathematicians do. We found this inspiring and have spent some more time grouping together our own, Zager’s and others’ ideas of what mathematicians do into the following structure:

I put my initial ideas out into the Twitterverse, where they were retweeted and endorsed. I have done some more refining to come up with these.

**Mathematicians work together and alone**. Too often classroom teaching has focussed on individual endeavour, whereas many people prefer to work in groups, where we can bounce ideas around.

**Mathematicians work at different paces.** I recently saw a Tweet quoting Jo Boaler who said “There is a common and damaging misconception in mathematics- the idea that strong math students are fast math students.” The person tweeting added, “ It’s not always about speed.” I replied, “Actually, it is never about speed.”

**Mathematicians work intuitively and methodically. **Sometimes we get a hunch and it turns out to be correct, or useful. Other times we just have to grunt through some ideas and processes to find things out.

**Mathematicians estimate and calculate.** Sometimes we just need an answer near enough. Often we need to have an idea of the near enough answer so we can check our calculations. Sometimes we need to calculate carefully and with precision.

This set of ideas would apply to many subjects, but I have found them really useful to encourage bravery in mathematics.

**Mathematicians rise to a challenge.** When we visit schools with our Rich Maths events, we tell students how mathematicians rise to a challenge. Then when we outline the different activities the can choose from, we tell them that one in particular is very challenging. We have seen many children take great delight in taking on the challenge. You can see it here: Challenging activity

**Mathematicians take risks.** Too often students are so focussed on getting things correct that it seems too risky to try new things and push boundaries.

**Mathematicians persevere. **When we see students struggling with a challenging problem, it is really important as teachers to reinforce the characteristic of persevering, and not to “rescue” them. It can be very difficult to hold back, when you are bursting to help them, but short term help is no help. Encouraging them to keep persevering, and recognising what they have already done is far more beneficial.

**Mathematicians make mistakes and learn.** This is one of the key ideas we emphasise in our visits. It is one of the key principles in the Growth Mindset way of thinking. When we get things right all the time, there is less learning than when we make mistakes. Sometimes really interesting discoveries come from mistakes. I’d like to add a little aside here that maths teachers ALSO make mistakes and learn. If you have never had a lesson fail miserably, you are not taking enough risks!

I will address the remainder of our characteristics and behaviours of mathematicians in a later post.

Here are the five in summary form:

I would love to hear your opinion – is this what your students think mathematicians do? Is it what you think mathematicians do? Do you make enough mistakes?

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A rich task is an open-ended task that students can engage with at multiple levels. I use the following information from the nrich website when I am talking to teachers about rich tasks.

In this task we use our Dragonistics data cards, which are shown here. For a less colourful exercise you could use 24 pieces of card with numbers 1 to 8 on them.

Each dragon has a strength rating of between 1 and 8, shown by the coloured dragon scales on the right-hand side of the card. The distribution of dragon strengths is not uniform, but is clustered around the middle, and depends to a certain extent on other aspects of the dragon, such as their species, gender and behaviour.

The students will already be familiar with the dragon cards, and each group of students has a set of about 24 dragonistics data cards. As there are a total of 288 dragons, each group will have a different set of dragons. Some may or may not have dragons of each strength rating.

**A dragon team trainer says that teams of two dragons chosen at random nearly always have a combined strength of between 7 and 11.
Is this true?
Provide evidence to support your conclusion.**

If you do not have any dragons of your own, make up about 20 pieces of card, with the numbers 1 to 8 on them, so you can explore the problem. Like Tracy Zager, we emphasise the necessity of exploring the maths ourselves before the children.

What is great about this exercise is that you can explore it experimentally or theoretically. It has a low entry point, as encouraged on Youcubed. This is sometimes called “low floor, high ceiling”.For younger children, it is a good start to take pairs of dragons, add their strengths, and write down the answer. Then they need to work out a recording method, possibly a tally table. You can have discussions about what it means for the dragons to be chosen randomly. You can also discuss what “nearly always” means.

Recently I used this task with a group of ten-year-olds. After they had made an attempt at solving it, I asked what they thought would be the most common team strength, and one said 9 or 10 because it is in the middle. I should have explored this idea further. What I did do, was start working out how many different combinations were possible. It is not possible to have a team of strength 1, and there is only one way to get a team of strength 2. How many ways to get strength 3? By the time we got to strength 6, they could see a pattern, that the number of combinations is one less than the total strength. So then I jumped to the other end of the distribution, asking “What is the strongest team we could possibly get?” As it happened, they did have two dragons of strength 8 in their set of dragons, so they correctly estimated the answer to be 16. So then I asked how many different ways they could get 16, and using their previous rule, they suggested 15 ways. Then when I asked them to tell me what they were, they realised that there was only one way. From there we started working down the numbers. Unfortunately this was during a holiday programme, so I didn’t have time to pursue this further. However we will be using this exercise in our rural rich maths events.

There are three main ways to approach this problem. The first is to experiment by randomly taking pairs of dragons, and recording their total strengths. A simple theoretical model involves thinking about all the possible outcomes and seeing what proportion of the outcomes lies between the chosen values. Then a more refined model would take into account the distributions of strengths for the given dragons. The learners may well come up with some interesting other ways to go about this.

A teacher can encourage further thinking with questions such as:

Would this answer be the same for every group of dragons? Is it possible to find a set of dragons so that the only team strengths are between 5 and 11? What would happen if you had teams of three dragons. Does it make a difference if you select one team at a time, and shuffle, or divide into lots of teams and record, before shuffling? How many different team possibilities are possible? What if you only had green dragons – would this make a difference?

It is important to point out the mathematical skills they are exercising as they tackle rich tasks. This task improves number skills, encourages persistence and risk-taking, develops communication skills as they are required to justify their conclusion. At higher levels it is helping to develop understanding of probability distributions, and you could also introduce or reinforce the idea of a random variable – in this case the team strength.

It would also be interesting to look at the spread for single dragons, two dragon teams and three dragon teams. With enough repetitions (and at this point a spreadsheet could be handy) the central limit theorem will start to be apparent. As you can see, there is great potential to expand this.

We need to look at ways this is also applicable in daily life, and not just for dragon trainers. The same sort of problem would occur if you had people buying different numbers of items, or different weights of suitcases. You might like to think of the combined strengths as similar to total scores in sports events. At higher levels you might discuss the concept of independence.

So rich – so many possibilities! Thoughts?

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Too often mathematics is seen as pure and apolitical. Maths teachers may keep away from concepts that seem messy and without right and wrong answers. However, teachers of mathematics and statistics have much to offer to increase democratic power in the upcoming NZ general elections (and all future elections really). The bizarre outcomes for elections around the world recently (2016/2017 Brexit, Trump) are evidence that we need a compassionate, rational, informed populace, who is engaged in the political process, to choose who will lead our country. Knowledge is power, and when people do not understand the political process, they are less likely to vote. We need to make sure that students understand how voting, the electoral system, and political polls work. Some of our students in Year 13 will be voting this election, and students’ parents can be influenced to vote.

There are some lessons provided on the Electoral Commission site. Sadly all the teaching resources are positioned in the social studies learning area – with none in statistics and mathematics. Similarly in the Senior Secondary guides, all the results from a search on elections were in the social studies subject area.

In New Zealand, our MMP system throws up some very interesting mathematical processes for higher level explorations. Political polls will be constantly in the news, and provide up-to-date material for discussions about polls, sample sizes, sampling methods, sampling error etc.

It would be great to hear from anyone who uses these ideas. If you have developed them further, so much the better. Do share with us in the comments.

These suggestions for lessons are listed more or less in increasing levels of complexity. However I have been amazed at what Year 1 children can do. It seems to me that they are more willing to tackle difficult tasks than many older children. These lessons also embrace other curriculum areas such as technology, English and social studies.

Make a ballot box, make a voting paper. Talk about randomising the names on the paper. How big does the box need to be? How many ballot boxes are being made for the upcoming election? How much cardboard is needed?

Make a time series graph of poll results. Each time there is a new result, plot it on the graph over the date, and note the sample size. At higher levels you might like to put confidence intervals on either side of the plotted value. A rule of thumb is 1/square root of the sample size. For example if the sample size is 700, the margin of error is 3.7%. So if the poll reported a party gaining 34% of the vote, the confidence interval would be from 33.3% to 37.7%.

You can get a good summary of political polls on Wikipedia.

Figure it Out, Number sense Book 2 Level 4 – has an exercise about finding fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.

This is a guide to running an analysis on the level of representation of different geopgraphic areas in the news. The same lesson could be used for representation of different parties or different issues.

The newspapers and online will be full of graphs and other graphical representations. Keep a collection and evaluate them for clarity and attractiveness.

This inquiry uses a mixture of internet search, mathematical modelling, estimation and calculation.

- How many electorates are there?
- How many polling booths per electorate?
- How many people per booth?
- How long are they employed for?

- Is the location of polling booths fair?
- What is the furthest distance a person might need to travel to a voting booth?
- What do people in other countries do?

This link provides a thorough explanation of the system. A project could be for students to work out what it is saying and make a powerpoint presentation or short video explaining it more simply.

Overhang occurs when a party gets more electoral MPS elected than their proportion allows. Here is a fact sheet about overhang and findings of the electoral review. Students could create scenarios to evaluate the effect of overhang and find out what is the biggest overhang possible.

How might the previous two election results have been different if there were not the 5% and coat-tailing rules?

Different ways of assigning areas to electorates get different results. The Wikipedia article on Gerrymandering has some great examples and diagrams on how it all happens, and the history behind the name.

Statistics should be analysed in response to a problem, rather than just for the sake of it.

Suggested Scenario: A new political party is planning to appeal to young voters, under 30 years of age. They wish to find out which five electorates are the best to target. You may also wish to include turn-out statistics in your analysis.

Resource: Enrolment statistics by electorate – some graphs supplied, percentages for each electorate.

In the interests of better democracy, we wish to have a better voter turnout. Find out the five electorates with the best voter turnout and the worst, and come up with some ideas about why they are the best and the worst. Test out your theory/model by trying to predict the next five best and worst. Use what you find out to suggest how might we improve voter turnout.

Resource: Turn out statistics – by electorate or download the entire file

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Trauma is a deeply distressing or disturbing experience. Many people in my home town of Christchurch still suffer from post traumatic stress disorder (PTSD) as a result of our earthquakes five or so years ago. I know I will never be the same again. The trauma began with the original terrifying experience of having the ground move in a way you never thought was even possible. It was perpetuated by over eighteen months of never knowing when the next earthquake (deceptively called aftershock) would hit. And the trauma still continues for many as they struggle to sort out their homes, and jobs, and their families. (Even now the thought of earthquakes can bring me to tears, and heavy machinery undertaking drainage work happening in my street is not helping.)

People might question if the impact of bad maths experiences can really be likened to the trauma people experience as the result of a series of earthquakes. I listened recently to a webinar about maths trauma, hosted by Global Math Department, and presented by Dr Kasi Allen. Math Trauma: Healing Our Classrooms, Our Students, and Our Discipline The webinar occurred in April 2016, but thanks to the amazing global maths community, it is still available and has had over 1000 views. Dr Allen calls herself a “math activist who studies math trauma and promotes teaching mathematics for social justice”. I see myself and the work we do at Statistics Learning Centre in that vein also.

I have reproduced a few of the ideas in the webinar, but would recommend visiting it yourself to get the full value.

Dr Allen’s proposition is that what is commonly called math anxiety is probably better described as math trauma. She teaches preservice elementary school teachers. A watershed experience has been seeing people bolt from the room in tears, simply looking at the syllabus at the start of a maths course.

I am frequently told by people that they do not have a maths brain, could never learn maths, that they are not a maths person. I have had middle-aged women tell me of formative experiences that happened over fifty years previously that have shaped their relationship with mathematics. Recently I asked my Facebook friends both mathematically inclined and not so mathematically inclined about how they picture numbers. Time and again their responses included the statement that they are not good at maths.

The term “math anxiety” dates back to the 1950s and is still used today. There are decades of research into how math anxiety disproportionately affects students who are female, low income and non-white. What Dr Allen (and I) found disturbing was that among college students, undergraduate education majors are the most maths anxious, both in terms of number and severity. These are the people who are entrusted with teaching mathematics to the next generation. Primary school teachers too often have an unhealthy relationship with maths – that is NOT their fault. They were taught in a way that did not work for them and they carry the burden with them.

Dr Allen suggests that maths trauma is a more fitting description than maths anxiety. Jo Boaler talks about people as having been maths traumatised. The negative experiences people have with mathematics, are described as painful and damaging. Traumatic events can be grounded in everyday life, and do not need to come from one catastrophic event. It is the subjective response that matters. Dr Allen gives the following definition:

“Math trauma stems from an event, a series of events, or a set of circumstances experienced by an individual as harmful or threatened such that there are lasting adverse effects on the individual’s functioning and well-being in the perceived presence of mathematics.”

Dr Allen has suggestions to help heal maths trauma. One suggestion is to acknowledge past negative experiences and their effects. We can listen and express sympathy and even apologise for the harm people have felt. We can provide opportunities for students to tell their maths stories. We can help them nurture their mathematics identities.

We also need to work on prevention of maths trauma. Classroom culture is important. Students need to feel safe and brave and they need to move. And we need to end traumatizing traditions. I have reproduced a screen shot of the slide about ending traumatizing traditions. Timed tests in mathematics have to stop. Now. Forever.

The question is, how do we (Statistics Learning Centre) help teachers to recover from maths trauma, so they can feel the fun and excitement that can be had in maths? Teachers matter for themselves, as well as for the good they can do their students. Maths educators need to be part of the solution and part of the prevention – to be maths activists. People are not born with maths trauma and it does not exist in all cultures. We need to do better.

So here is my question. Do you or someone you know suffer from maths trauma? Let me tell you now – it is not your fault. It is not their fault.

What needs to happen for you to feel better about maths? What needs to happen so that maths trauma can be eliminated from our schooling?

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I really like graphs. I like the way graphs turn numbers into pictures. A good graph is elegant. It uses a few well-placed lines to communicate what would take a paragraph of text. And like a good piece of literature or art, a good graph continues to give, beyond the first reading. I love looking at my YouTube and WordPress graphs. These graphs tell me stories. The WordPress analytics tell me that when I put up a new post, I get more hits, but that everyday more than 1000 people read one of my posts. The YouTube analytics tell me stories about when people want to know about different aspects of statistics. It is currently the end of the North American school year, and the demand is for my video on Choosing which statistical test to use. Earlier in the year, the video about levels of measurement is the most popular. And not many people view videos about statistics on the 25^{th} of December. I’m happy to report that the YouTube and WordPress graphs are good graphs.

Spreadsheets have made it possible for anyone and everyone to create graphs. I like that graphs are easier to make. Drawing graphs by hand is a laborious task and fraught with error. But sometimes my heart aches when I see a graph used badly. I suspect that this is when a graphic artist has taken control, and the search for beauty has over-ridden the need for truth.

Three graphs spurred me to write this post.

The first was on a website to find out about property values. I must have clicked onto something to find out about the property values in my area, and was taken to the qv website. And this is the graph that disturbed me.

Sure it is pretty – uses pretty colours and shading, and you can find out what it is saying by looking at the key – with the numbers beside it. But a pie or donut chart should not be used for data which has inherent order. The result here is that the segments are not in order. Or rather they are ordered from most frequent to least frequent, which is not intuitive. Ordinal data is best represented in a bar or column chart. To be honest, most data is best represented in a bar or column chart. My significant other suggested that bar charts aren’t as attractive as pie charts. Circles are prettier than rectangles. Circles are curvy and seem friendlier than straight lines and rectangles. So prettiness has triumphed over truth.

It may be a little strong to call bad communication lack of truth. Let’s look at another example. In a way it is cheating to cite a pictogram in a post like this. Pictograms are the lowest form of graph and are so often incorrect, that finding a bad one is easier than finding a good one. In the graph below of fatalities it is difficult to work out what one little person represents.

A quick glance, ignoring the numbers, suggests that the road toll in 2014 is just over half what it was in 2012. However, the truth, calculated from the numbers, is that the relative size is 80%. 2012 has 12 people icons, representing 280 fatalities. One icon is removed for 2013, representing a drop of 9 fatalities. 2011 has one icon fewer again, representing a drop of 2 fatalities. There is so much wrong in the reporting of road fatalities, that I will stop here. Perhaps another day…

And here is the other graph that perplexed me for some time. It came in the Saturday morning magazine from our newspaper, as part of an article about inequality in New Zealand. Anyone who reads my blog will be aware that my politics place me well left of centre, and I find inequality one of the great ills of the modern day. So I was keen to see what this graph would tell me. And the answer is…

I have no idea. Now, I have expertise in the promulgation of statistics, and this graph stumped me for some time. Take a good look now, before I carry on.

I did work out in the end, what was going on in the graph, but it took far longer than it should. This article is aimed at an educated but not particularly statistically literate audience, and I suspect there will be very few readers who spent long enough working out what was going on here. This graph is probably numerically correct. I had a quick flick back to the source of the data (who, by the way, are not to be blamed for the graph, as the data was presented in a table) and the graph seems to be an accurate depiction of the data. However, the graph is so confusing as to be worse than useless. Please post critiques in the comments. This graph commits several crimes. It is difficult to understand. It poses a question and then fails to help the reader find the answer. And it does not provide insights that an educated reader could not get from a table. In fact, I believe it has obscured the data.

Graphs are the main way that statistical analysts communicate with the outside world. Graphs like these ones do us no favours, even if they are not our fault. We need to do better, and make sure that all students learn about graphs.

Here is a suggestion for teachers at all levels. Have a “graph a day” display – maybe for a month? Students can contribute graphs from the news media. Each day discuss what the graph is saying, and critique the way the graph is communicating. I have a helpful structure for reading graphs in my post: There’s more to reading graphs than meets the eye;

Here is a summary of what I’ve said and what else I could say on the topic.

- The choice of graph depends on the purpose
- The text should state the purpose of the graph
- There is not a graph for everything you wish to communicate
- Sometimes a table communicates better than a graph
- Graphs are part of the analysis as well as part of the reporting. But some graphs are better to stay hidden.
- If it takes more than a few seconds to work out what a graph is communicating it should either be dumped or have an explanation in the text
- Truth (or communication) is more important than beauty
- There is beauty in simplicity
- Be aware than many people are colour-blind, or cannot easily differentiate between different shades.

Late last year I posted four graphs of the same data and asked for people’s opinions. You can link back to the post here and see the responses: Which Graph to Use.

The interesting thing is not which graph was selected as the most popular, but rather that each graph had a considerable number of votes. My response is that it depends. It depends on the question you are answering or the message you are sending. But yes – I agree with the crowd that Graph A is the one that best communicates the various pieces of information. I think it would be improved by ordering the categories differently. It is not very pretty, but it communicates.

I recently posted a new video on YouTube about graphs. It is a quick once-over of important types of graphs, and can help to clarify what they are about. There are examples of good graphs in there.

I have written about graphs previously and you can find them here on the Collected Works page.

I’m interested in your thoughts. And I’d love to see some beautiful and truthful graphs in the comments.

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This morning at the Twitter chat for teachers, (#bfc630nz) the discussion question was, How and what will you teach your students about life this year? As I lurked I was impressed at the ideas and ideals expressed by a mixed bunch of teachers from throughout New Zealand. I tweeted: “I wonder how often maths teachers think about educating the heart. Yet maths affects how people feel so much.”

My teaching philosophy is summed up as “head, heart and hands”. I find the philosophy of constructivism appealing, that people create their own understanding and knowledge through experiences and reflection. I believe that learning is a social activity, and I am discovering that mathematics is a social endeavour. But underpinning it all I am convinced that people need to feel safe. That is where the heart comes in. “People do not care how much you know until they know how much you care.” Relationships are vital. I wrote previously about the nature of teaching statistics and mathematics.

In the culture of NZ Maori, when someone begins to address a group of people, they give a mihi, which is an introductory speech following a given structure. The mihi has the role of placing the person with respect to their mountain, their river, their ancestors. It enables the listeners to know who the person is before they begin to speak about anything else. I am not fluent in te reo, so do not give a mihi in Maori (yet), but I do introduce myself so that listeners know who I am. Learners need to know why I am teaching, and how I feel about the subject and about them. It can feel self-indulgent, thinking surely it is about the subject, not about me. But for many learners the teacher is the subject. Just look at subject choices in high school students and that becomes apparent.

Recently I began studying art at an evening class. I am never a passive learner (and for that reason do feel sympathy for anyone teaching me). Anytime I have the privilege of being a learner, I find myself stepping back and evaluating my responses and thinking of what the teacher has done to evoke these responses. Last week, in the first lesson, the teacher gave no introduction other than her name, and I felt the loss. Art, like maths, is emotionally embedded, and I would have liked to have developed more of a relationship with my teacher, before exposing my vulnerability in my drawing attempts. She did a fine job of reassuring us that all of our attempts were beautiful, but I still would like to know who she is.

I suspect that some people believe that maths is a dry, sterile subject, where things are right or wrong. Many worksheets give that impression, with columns of similar problems in black and white, with similarly black and white answers. Some attempt to sweeten the broccoli by adding cartoon characters and using bright colours, but the task remains devoid of adventure and creativity. Now, as a child, I actually liked worksheets, but that is probably because they were easy for me, and I always got them right. I liked the column of little red ticks, and the 100% at the end. They did not challenge me intellectually, but I did not know any better. For many students such worksheets are offputting at best. Worksheets also give a limited view of the nature of mathematics.

I am currently discovering how narrow my perception of mathematics was. We are currently developing mathematical activities for young learners, and I have been reading books about mathematical discoveries. Mathematics is full of creativity and fun and adventure, opinion, multiple approaches, discussion and joy. The mathematics I loved was a poor two-dimensional faded version of the mathematics I am currently discovering.I fear most primary school teachers (and possibly many secondary school maths teachers) have little idea of the full potential of mathematics.

Some high school maths teachers struggle with the New Zealand school statistics curriculum. It is embedded in real-life data and investigations. It is not about calculating a mean or standard deviation, or some horrible algebraic manipulation of formulae. Statistics is about observing and wondering, about asking questions, collecting data, using graphs and summary statistics to make meaning out of the data and reflecting the results back to the original question before heading off on another question. Communication and critical thinking are vital. There are moral, ethical and political aspects to statistics.

I cannot express strongly enough that the teaching of mathematics and statistics is a political act. It is a question of social justice. In my PhD thesis work, I found that social deprivation correlated with opportunities to learn mathematics. My thoughts are that there are families where people struggle with literacy, but mostly parents from all walks of life can help their children with reading. However, there are many parents who have negative experiences around mathematics, who feel unable to engage their children in mathematical discussions, let alone help them with mathematics homework. And sadly they often entrench mathematical fatalism. “I was no good at maths, so it isn’t surprising that you are no good at maths.”

Our students need to know that we love them. When you have a class of 800 first year university students it is clearly not possible to build a personal relationship with each student in 24 contact hours. However the key to the ninety and nine is the one. If we show love and respect in our dealings with individuals in the class, if we treat each person as valued, if we take the time to listen and answer questions, the other students will see who we are. They will know that they can ask and be treated well, and they will know that we care. When we put time into working out good ways to explain things, when we experiment with different ways of teaching and assessing, when we smile and look happy to be there – all these things help students to know who we are, and that we care.

As teachers of mathematics and statistics we have daunting influence over the futures of our students. We need to make sure we are empowering out students, and having them feel safe is a good start.

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