I love Lego. And I love making up mathematics and statistics activities for people of all levels of attainment. So it makes sense that I would make up maths discussion activities using Lego.

Whenever I have posted my ideas on Twitter (hashtag @Rogonic) and Facebook (Statistics Learning Centre) they have proved popular. So I thought it would be good to put them in a less transient location – this blog.

Here is one to start with:

Which of the models, A to H is most like the one in the middle?

You can ask any question you like. I suggest, “Which of the models, A to H is most like the model in the middle?”

Then listen to what your learners have to say. Feel free to vote here:

I would love to hear what comes of this discussion. Please put your ideas in the comments below. In a follow-up post, I will talk about some of the concepts that might have arisen in discussions. A follow-up activity for your students (or you) is to come up with a new model that is most like the one in the middle, but not exactly the same.

Math/Maths Lego/Legos

Explanation of the placement of the letter ‘s’ with respect to Maths with Lego. The Danish company that makes Lego does not approve of the use of Legos as a word. The plural of one Lego brick is two Lego bricks. In New Zealand we talk about Lego as a collective noun, as in “I am going to play with my Lego” and “Pick up your Lego before I stand on it.” We also follow the UK tradition of talking about the subject of Maths, rather than Math. I am aware that my friends in the US would talk about Math with Legos, but I am not in the US so I reserve the right to talk about Maths with Lego. I am refraining from making any statement about the state of politics in the US at present… With difficulty.

Disclaimer

This website uses Lego (R) bricks to teach mathematical and statistical concepts. There is no official affiliation with the Lego(R) company. Lego is a registered trademark of the Lego company.

Maths is right or wrong – end of discussion – or is it?

In 1984 I was a tutor in Operations Research to second year university students. My own experience of being in tutorials at University had been less than inspiring, with tutors who seemed reserved and keen to give us the answers without too much talking. I wanted to do a good job. My induction included a training session for teaching assistants from throughout the university. Margaret was a very experienced educational developer and was very keen for us to get the students discussing. I tried to explain to her that there really wasn’t a lot to discuss in my subject. You either knew how to solve a set of linear equations using Gauss-Jordan elimination or you didn’t. The answer was either correct or incorrect.

I suspect many people have this view of mathematics and its close relations, statistics and operations research. Our classes have traditionally followed a set pattern. The teacher shows the class how to do something. The class copies down notes and some examples into their books, and then they individually work through exercises in the textbook – generally in silence. The teacher walks around the room and helps students as needed.

Prizes can help motivate students to give answers in unfamiliar settings

So when we talk about discussion in maths classes, this is not something that mathematics and statistics teachers are all familiar with. I recently gave a workshop for about 100 Scholarship students in Statistics in the Waikato. What a wonderful time we had together! The students were from all different schools and needed to be warmed up a little with prizes, but we had some good discussion in groups and as a whole. One of the teachers commented later on the level of discussion in the session. Though she was an experienced maths teacher she found it difficult to lead discussion in the class. I am sure there are many like her.

It is important to talk in maths and stats classes

It is difficult for many students to learn in solitary silence. As we talk about a topic we develop our understanding, practice the language of the discipline and experience what it means to be a mathematician or statistician. Explaining ideas to others helps us to make sense of them ourselves. As we listen to other people’s thinking we can see how it relates to what we think, and can clear up misconceptions. Some people just like to talk, (who me?) and find learning more fun in a cooperative or collaborative environment. This recognition of the need for language and interaction underpins the development of “rich tasks” that are being used in mathematics classrooms throughout the world.

I have previously stated that “Maths learning should be communal and loud and exciting, not solitary, quiet and routine.”

Classroom atmosphere

One thing that was difficult at the Scholarship day was that the students did not know each other, and came from various schools. In a regular classroom the teacher has the opportunity of and responsibility for setting the tone of the class. Students need to feel safe. They need to feel that giving a wrong answer is not going to lead to ridicule. Several sessions at the start of the year may be needed to encourage discussion. Ideally this will become less necessary over time as students become used to interactive, inquiry-based learning in mathematics and statistics through their whole school careers.

“Number talks” are a tool to help students improve their understanding of number, and recognise that there are many ways to see things. For example, the class might be shown a picture of dots and asked to explain how many dots they see, and how they worked it out. Several different ways of thinking will be discussed.

Children are encouraged to think up multiple ways of thinking about numbers and to develop discussion by following prompts, sometimes called “talk moves”. Talk moves include revoicing, where the teacher restates what she thinks the student has said, asking students to restate another students reasoning, asking students to apply their own reasoning to someone else’s reasoning (Do you agree or disagree and why?), prompting for further participation (Would someone like to add on?), and using wait time (teachers should allow students to think for at least 10 seconds before calling on someone to answer. These are explained more fully in The Tools of Classroom talk.

The essence of good discussion is good questions. Question ping pong is not classroom discussion. We have all experienced a teacher working through examples on the board, while asking students the answers to numerical questions. This is a control technique for keeping students attentive, but it can fall to a small group of students who are quick to answer. I remember doing just this in my tutorial on solving matrices, when I didn’t know any better.

Teachers should avoid asking questions that they already know the answers to.

It is not a hard-and-fast rule, but definitely a thing to think about. I like to use True/False quizzes to help uncover misconceptions, and develop use of statistical language. I just about always know the answer to the question, but what I don’t know is how many students know the answer. So I ask the question not to know the answer, but to know if the students do, and to provoke discussion. Perhaps a more interesting question would be, how many students do you think will say “True” to this statement. It would then be interesting to find out their reasoning, so long as it does not get personal!

The MathTwitterBlogosphere has many attractive ideas to use in teaching maths.

I rather like “Which one doesn’t belong”, which has echoes of “One of these things is not like the other, one of these things doesn’t belong, can you guess…” from Sesame Street. However, in Sesame Street the answer was usually unambiguous, whereas with WODB there are lots of ways to have alternative answers. There is a website dedicated to sets of four objects, and the discussion is about which one does not belong. In each case all four can “not belong” for some reason, which I find a bit contrived, but it can lead to discussion about which is the strongest case of not belonging.

Whole class and group discussion

Some discussions work well for a whole class, while others are better in small groups or pairs. Matching or ordering paper slips with expressions can lead to great discussion. For example we could have a set of graphs of the same data, and order them according to how effective they are at communicating the aspects of the data. Or there could be statements of possible events and students can place them in order of likelihood. The discussion involved in ordering them helps students to clarify the nature of probability. Desmos has a facility for teachers to set up card matching or grouping exercises, which reduces the work and waste of paper.

Our own Dragonistics data cards are great for discussion. Students can be given a number of dragons (more than two) and decide which one is the best, or which one doesn’t belong, or how to divide the dragons fairly into two or more groups.

It can seem to be wasting time to have discussion. However the evidence from research is that good discussion is an effective way for students to learn mathematics and statistics. I challenge all maths and stats teachers to increase and improve the discussion in their class.

I have been reading further about teaching mathematics and came across this interesting assertion:

Play, understood as something frivolous, opposed to work, off-task behaviour, is not welcomed into most mathematics classrooms. But play is exactly what is needed. It is only play that can entice us to the type of repetition that is needed to learn how to inhabit the mathematical landscape and how to create new mathematics. Friesen(2000) – unpublished thesis, cited in Stordy, Children Count, (2015)

Play and practice

It is an appealing idea that as children play, they have opportunities to engage in repetition that is needed in mastering some mathematical skills. The other morning I decided to do some exploration of prime numbers and factorising even before I got out of bed. (Don’t judge me!). It was fun, and I discovered some interesting properties, and came up with a way of labelling numbers as having two, three and more dimensions. 12 is a three dimensional number, as is 20, whereas 35 and 77 are good examples of two dimensional numbers. As I was thus playing on my own, I was aware that it was practising my tables and honing my ability to think multiplicatively. In this instance the statement from Friesen made sense. I admit I’m not sure what it means to “create new mathematics”. Perhaps that is what I was doing with my 2 and 3 dimensional numbers.

You may be wondering what this has to do with teaching statistics to adults. Bear with…

Traditional vs recent teaching methods for mathematics

Today on Twitter, someone asked what to do when a student says that they like being shown what to do, and then practising on textbook examples. This is the traditional method for teaching mathematics, and is currently not seen as ideal among many maths teachers (particularly those who inhabit the MathTwitterBlogosphere or MTBoS, as it is called). There is strong support for a more investigative, socially constructed approach to learning and teaching mathematics. I realise that as a learner, I was happy enough learning maths by being shown what to do and then practising. I suspect a large proportion of maths teachers also liked doing that. Khan Academy videos are wildly popular with many learners and far too many teachers because they perpetuate this procedural view of mathematics. So is the procedural approach wrong? I think what it comes down to is what we are trying to teach. Were I to teach mathematics again I would not use “show then practise” as my modus operandi. I would like to teach children to become mathematicians rather than mathematical technicians. For this reason, the philosophies and methods of Youcubed, Dan Meyer and other MTBoS bloggers have appeal.

Play and statistics

Now I want to turn my thoughts to statistics. Is there a need for more play in statistics? Can statistics be playful in the way that mathematics can be playful? Operations Research is just one game after another! Simulation, critical path, network analysis, travelling salesperson, knapsack problem? They are all big games. Probability is immensely playful, but what about statistical analysis? Can and should statistics be playful?

My first response is that there is no play in statistics. Statistics is serious and important, and deals with reality, not joyous abstract ideas like prime numbers and the Fibonacci series – and two and three dimensional numbers.

The excitement of a fresh set of data

But there is that frisson of excitement as you finally finish cleaning your database and a freshly minted set of variables and observations beckons to you, with SPSS, SAS or even Excel at your fingertips. A new set of data is a new journey of discovery. Of course a serious researcher has already worked out a methodical route through her hypotheses… maybe. Or do we mostly all fossick about looking for patterns and insights, growing more and more familiar with the feel of the data, as if we were squeezing it through our fingers? So yes – my experience of data exploration is playful. It is an adventure, with wrong turns, forgetting the path, starting again, finding something only to lose it again and finally saying “enough” and taking a break, not because the data has been exhausted, but because I am.

Writing the report is like cleaning up

Writing up statistical analysis is less exciting. It feels like picking up the gardening tools and putting them away after weeding the garden. Or cleaning the paintbrushes after creating a masterpiece. That was not one of my strengths – finishing and tidying up afterwards. The problem was that I felt I had finished when the original task had been completed – when the weeds had been pulled or the painting completed. In my view, cleaning and putting away the tools was an afterthought that dragged on after the completion of the task, and too often got ignored. Happily I have managed to change my behaviour by rethinking the nature of the weeding task. The weeding task is complete when the weeds are pulled and in the compost and the implements are resting clean and safe where they belong. Similarly a statistical analysis is not what comes before the report-writing, but is rather the whole process, ending when the report is complete, and the data is carefully stored away for another day. I wonder if that is the message we give our students – a thought for another post.

Can statistics be playful?

For I have not yet answered the question. Can statistics be playful in the way that mathematics can be playful? We want to embed play in order to make our task of repetition be more enjoyable, and learning statistics requires repetition, in order to develop skills and learn to differentiate the universal from the individual. One problem is that statistics can seem so serious. When we use databases about global warming, species extinction, cancer screening, crime detection, income discrepancies and similarly adult topics, it can seem almost blasphemous to be too playful about it.

Helen has a real attitude problem and hurls snarky comments at her brother, Luke. The apples fall in an odd way, and Dr Nic pops up in strange places. This playfulness keeps the audience engaged in a way that serious, grown up themes may not. This is why we invented Ear Pox in our video about Risk and screening, because being playful about cancer is inappropriate.

Ear Pox is imaginary disease for which we are studying the screening risk.

A set of 240 Dragonistics data cards provides light-hearted data which yields satisfying results.

When I began this post I did not intend to bring it around to the videos and the Dragonistics data cards, but I have ended up there anyway. Maybe that is the appeal of the Dragonistics data cards – that they avoid the gravitas of true and real grown-up data, and maintain a playfulness that is more engaging than reality. There is a truthiness about them – the two species – green and red dragons are different enough to present as different animal species, and the rules of danger and breath-type make sense. But students may happily play with the dragon cards without fear of ignorance or even irreverence of a real-life context.

What started me thinking about play with regards to learning maths and statistics is our Cat Maths cards. There are just so many ways to play with them that I can see Cat Maths cards playing an integral part in a junior primary classroom. This is why we created them and want them to make their way into classrooms. You can help by supporting our Kickstarter crowdfunding campaign. Click the picture to pledge and get a box, provide a box for a school, or make a corporate donation.

We’d love your help.

Your thoughts about play and statistics

And maybe we need to be thinking a little more about the role of play in learning statistics – even for adults! What do you think? Can and should statistics be playful? And for what age group? Do you find statistical analysis fun?

Sometimes it pays to stop and think. I have been reading a recent textbook for mathematics teachers, Dianne Siemon et al, Teaching mathematics: foundations to middle years (2011). On page 47 the authors asked me to “Take a few minutes to write down your own views about the nature of mathematics, mathematics learning and mathematics teaching.” And bearing in mind I see statistics as related to, but not enclosed by mathematics, I decided to do the same for statistics as well. So here are my thoughts:

The nature of mathematics

Mathematicians love the elegance of mathematics

Mathematics is a way of modelling and making sense of the world. Mathematics underpins scientific and commercial endeavours as well as everyday life. Mathematics is about patterns and proofs and problem structuring and solution finding. I used to think it was all about the answer, but now I think it is more about the process. I used to think that maths was predominantly an individual endeavour, but now I can see how there is a social or community aspect as well. I fear that too often students are getting a parsimonious view of mathematics, thinking it is only about numbers, and something they have to do on their own. I find my understanding of the nature of mathematics is rapidly changing as I participate in mathematics education at different ages and stages. I have also been influenced by the work of Jo Boaler.

To learn mathematics

My original idea of mathematics learning comes from my own successful experience of copying down notes from the board, listening to the teacher and doing the exercises in the textbook. I was not particularly fluent with my times-tables, but loved problem-solving. If I got something wrong, I was happy to try again until I nutted it out. Sometimes I even did recreational maths, like the time I enumerated all possible dice combinations in Risk to find out who had the advantage – attacker or defender. I always knew that it took practice to be good at mathematics. However I never really thought of mathematics as a social endeavour. I feel I missed out, now. From time to time I do have mathematical discussions with my colleague. It was an adventure inventing Rogo and then working out a solution method. Mathematics can be a social activity.

To teach mathematics

When I became a maths teacher I perpetuated the method that had worked for me, as I had not been challenged to think differently. I did like the ideas of mastery learning and personalised system of instruction. This meant that learners progressed to the next step only when they had mastered the previous one. I was a successful enough teacher and enjoyed my work.

Then as a university lecturer I had to work differently, and experimented. I had a popular personalised system of instruction quantitative methods course, relying totally on students working individually, at their own pace. I am happy that many of my students were successful in an area they had previously thought out of their reach. For some students it was the only subject they passed.

What I would do now

If I were to teach mathematics at school level again, I hope I would do things differently. I love the idea of “Number talks” and rich tasks which get students to think about different ways of doing things. I had often felt sad that there did not seem to be much opportunity to have discussions in maths, as things were either right or wrong. Now I see what fun we could have with open-ended tasks. Maths learning should be communal and loud and exciting, not solitary, quiet and routine. I have been largely constructivist in my teaching philosophy, but now I would like to try out social constructivist thinking.

Statistics

And what about statistics? At school in the 1970s I never learned more than the summary statistics and basic probability. At uni level it was bewildering, but I managed to get an A grade in a first year paper without understanding any of the basic principles. It wasn’t until I was doing my honours year in Operations Research and was working as a tutor in Statistical methods that things stared to come together – but even then I was not at home with statistical ideas and was happy to leave them behind when I graduated.

The nature of statistics

Statistics lives in the real world

My views now on the nature of statistics are quite different. I believe statistical thinking is related to mathematical thinking, but with less certainty and more mess. Statistics is about models of reality, based on imperfect and incomplete data. Much of statistics is a “best guess” backed up by probability theory. And statistics is SO important to empowered citizenship. There are wonderful opportunities for discussion in statistics classes. I had a fun experience recently with a bunch of Year 13 Scholarship students in the Waikato. We had collected data from the students, having asked them to interpret a bar chart and a pie chart. There were some outliers in the data and I got them to suggest what we should do about them. There were several good suggestions and I let them discuss for a while then moved on. One asked me what the answer was and I said I really couldn’t say – any one of their suggestions was valid. It was a good teaching and learning moment. Statistics is full of multiple good answers, and often no single, clearly correct, answer.

Learning statistics

My popular Quantitative Methods for Business course was developed on the premise that learning statistics requires repeated exposure to similar analyses of multiple contexts. In the final module, students did many, many hypothesis tests, in the hope that it would gradually fall into place. That is what worked for me, and it did seem to work for many of the students. I think that is not a particularly bad way to learn statistics. But there are possibly better ways.

I do like experiential learning, and statistics is perfect for real life experiences. Perhaps the ideal way to learn statistics is by performing an investigation from start to finish, guided by a knowledgeable tutor. I say perhaps, because I have reservations about whether that is effective use of time. I wrote a blog post previously, suggesting that students need exposure to multiple examples in order to know what in the study is universal and what applies only to that particular context. So perhaps that is why students at school should be doing an investigation each year within a different context.

The nature of understanding

This does beg the question of what it means to learn or to understand anything. I hesitate to claim full understanding. Of anything. Understanding is progressive and multi-faceted and functional. As we use a technique we understand it more, such as hypothesis testing or linear programming. Understanding is progressive. My favourite quote about understanding is from Moore and Cobb, that “Mathematical understanding is not the only understanding.” I do not understand the normal distribution because I can read the Gaussian formula. I understand it from using it, and in a different way from a person who can derive it. In this way my understanding is functional. I have no need to be able to derive the Gaussian function for what I do, and the nature and level of my understanding of the normal distribution, or multiple regression, or bootstrapping is sufficient for me, for now.

Teaching statistics

I believe our StatsLC videos do help students to understand and learn statistics. I have put a lot of work into those explanations, and have received overwhelmingly positive feedback about the videos. However, that is no guarantee, as Khan Academy videos get almost sycophantic praise and I know that there are plenty of examples of poor pedagogy and even error in them. I have recently been reading from “Make it Stick”, which summarises theory based on experimental research on how people learn for recall and retention. I was delighted to find that the method we had happened upon in our little online quizzes was promoted as an effective method of reinforcing learning.

Your thoughts

This has been an enlightening exercise, and I recommend it to anyone teaching in mathematics or statistics. Read the first few chapters of a contemporary text on how to teach mathematics. Dianne Siemon et al, Teaching mathematics: foundations to middle years (2011) did it for me. Then “take a few minutes to write down your own views about the nature of mathematics, mathematics learning and mathematics teaching.” To which I add my own suggestion to think about the nature of statistics or operations research. Who knows what you will find out. Maybe you could put a few of your ideas down in the comments.

My life in education has included being a High School maths teacher, then teaching at university for 20 years. I then made resources and gave professional development workshops for secondary school teachers. It was exciting to see the new statistics curriculum being implemented into the New Zealand schools. And now we are making resources and participating in the primary school sector. It is wonderful to learn from each level of teaching. We would all benefit from more discussion across the levels.

Educational theory and idea-promoters

My father used to say (and the sexism has not escaped me) “Never run after a woman, a bus or an educational theory, as there will be another one along soon.” Education theories have lifespans, and some theories are more useful than others. I am not a fan of “learning styles” and fear they have served many students ill. However, there are some current ideas and idea-promoters in the teaching of mathematics that I find very attractive. I will begin with Jo Boaler, and intend to introduce you over the next few weeks to Dan Meyer, Carol Dweck and the person who wrote “Making it stick.”

My first contact with Jo Boaler was reading “The Elephant in the Classroom.” In this Jo points out how society is complicit in the idea of a “maths brain”. Somehow it is socially acceptable to admit or be almost defensively proud of being “no good at maths”. A major problem with this is that her research suggests that later success in life is connected to attainment in mathematics. In order to address this, Jo explores a less procedural approach to teaching mathematics, including greater communication and collaboration.

Mathematical Mindsets

It is interesting to see the effect Jo Boaler’s recent book, “Mathematical Mindsets “, is having on colleagues in the teaching profession. The maths advisors based in Canterbury NZ are strong proponents of her idea of “rich tasks”. Here are some tweets about the book:

“I am loving Mathematical Mindsets by @joboaler – seriously – everyone needs to read this”

“Even if you don’t teach maths this book will change how you teach for ever.”

“Hands down the most important thing I have ever read in my life”

What I get from Jo Boaler’s work is that we need to rethink how we teach mathematics. The methods that worked for mathematics teachers are not the methods we need to be using for everyone. The defence “The old ways worked for me” is not defensible in terms of inclusion and equity. I will not even try to boil down her approach in this post, but rather suggest readers visit her website and read the book!

At Statistics Learning Centre we are committed to producing materials that fit with sound pedagogical methods. Our Dragonistics data cards are perfect for use in a number of rich tasks. We are constantly thinking of ways to embed mathematics and statistics tasks into the curriculum of other subjects.

Challenges of implementation

I am aware that many of you readers are not primary or secondary teachers. There are so many barriers to getting mathematics taught in a more exciting, integrated and effective way. Primary teachers are not mathematics specialists, and may well feel less confident in their maths ability. Secondary mathematics teachers may feel constrained by the curriculum and the constant assessment in the last three years of schooling in New Zealand. And tertiary teachers have little incentive to improve their teaching, as it takes time from the more valued work of research.

Though it would be exciting if Jo Boaler’s ideas and methods were espoused in their entirety at all levels of mathematics teaching, I am aware that this is unlikely – as in a probability of zero. However, I believe that all teachers at all levels can all improve, even a little at a time. We at Statistics Learning Centre are committed to this vision. Through our blog, our resources, our games, our videos, our lessons and our professional development we aim to empower all teacher to teach statistics – better! We espouse the theories and teachings explained in Mathematical Mindsets, and hope that you also will learn about them, and endeavour to put them into place, whatever level you teach at.

Do tell us if Jo Boalers work has had an impact on what you do. How can the ideas apply at all levels of teaching? Do teachers need to have a growth mindset about their own ability to improve their teaching?

Here are some quotes to leave you with:

Mathematical Mindsets Quotes

“Many parents have asked me: What is the point of my child explaining their work if they can get the answer right? My answer is always the same: Explaining your work is what, in mathematics, we call reasoning, and reasoning is central to the discipline of mathematics.”
“Numerous research studies (Silver, 1994) have shown that when students are given opportunities to pose mathematics problems, to consider a situation and think of a mathematics question to ask of it—which is the essence of real mathematics—they become more deeply engaged and perform at higher levels.”
“The researchers found that when students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problems, they became curious, and their brains were primed to learn new methods, so that when teachers taught the methods, students paid greater attention to them and were more motivated to learn them. The researchers published their results with the title “A Time for Telling,” and they argued that the question is not “Should we tell or explain methods?” but “When is the best time do this?”
“five suggestions that can work to open mathematics tasks and increase their potential for learning: Open up the task so that there are multiple methods, pathways, and representations. Include inquiry opportunities. Ask the problem before teaching the method. Add a visual component and ask students how they see the mathematics. Extend the task to make it lower floor and higher ceiling. Ask students to convince and reason; be skeptical.”

All quotes from

Jo Boaler, Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching

It is interesting to provide a trade stand at a teachers’ conference. Some teachers are keen to find out about new things, and come to see how we can help them. Others studiously avoid eye-contact in the fear that we might try to sell them something. Trade stand holders regularly put sweets and chocolate out as “bait” so that teachers will approach close enough to engage. Maybe it gives the teachers an excuse to come closer? Either way it is representative of the uneasy relationship that “trade” has with salaried educators.

Money and education

Money and education have an uneasy relationship. For schools to function, they need considerable funding – always more than what they get. In New Zealand, and in many countries, education is predominantly funded by the state. Schools are built and equipped, teachers are paid and resources are purchased with money provided by the taxpayer. Extras are raised through donations from parents and fund-raising efforts. However, because it is not apparent that money is changing hands, schools are perceived as virtuous establishments, existing only because of the goodness of the teachers. This contrasts with the attitude to resource providers, who are sometimes treated as parasitic with their motives being all about the money. It is possible that some resource providers are in it just for the money, but it seems to me that there are richer seams to mine in health, sport, retail etc.

Statistics Learning Centre is a social enterprise

Statistics Learning Centre is a social enterprise. We fit in the fuzzy area between “not-for-profit” and commercial enterprise. We measure our success by the impact we are having in empowering teachers to teach statistics and all people to understand statistics. We need money in order to continue to make an impact. Statistics Learning Centre has made considerable contributions to the teaching and learning of statistics in New Zealand and beyond for several years. This post lists just some of the impact we have had. We believe in what we are doing, and work hard so that our social enterprise is on a solid financial footing.

StatsLC empowers teachers

Soon after the change to the NCEA Statistics standards, there was a shortage of good quality practice external exams. Even the ones provided as official exemplars did not really fit the curriculum. Teachers approached us, requesting that we create practice exams that they could trust were correct and aligned to the curriculum. We did so in 2015 and 2016, at considerable personal effort and only marginal financial recompense. We see that as helping statistics to be better understood in schools and the wider community.

We, at Statistics Learning Centre, grasp at opportunities to teach teachers how to teach statistics better, to empower all teachers to teach statistics. Our workshops are well received, and we have regular attenders who know they will get value for their time. We use an inclusive, engaging approach, and participants have a good time. I believe in our resources – the videos, the quizzes, the data cards, the activities, the professional development. I believe that they are among the best you can get. So when I give workshops, I do talk about the resources. It would seem counter-productive for all concerned, not to mention contrived, to do otherwise. They are part of a full professional development session. Many mathematical associations have no trouble with this, and I love to go to conferences, and contribute.

I am aware that there are some commercial enterprises who wish to give commercial presentations at conferences. If their materials are not of a high standard, this can put the organisers in a difficult position. Consequently some organisations have a blanket ban on any presentations that reference any paid product. I feel this is a little unfortunate, as teachers miss out on worthwhile contributions. But I understand the problem.

The Open Market model – supply and demand

I believe that there is value in a market model for resources. People have suggested that we should get the Government to fund access to Statistics Learning Centre resources for all schools. That would be delightful, and give us the freedom and time to create even better resources. But that would make it almost impossible for any other new provider, who may have an even better product, to get a look in. When such a monopoly occurs, it reduces the incentives for providers to keep improving.

Saving work for the teachers, and building on a product

Teachers want the best for their students, and have limited budgets. They may spend considerable amounts of time printing, cutting and laminating in order to provide teaching resources at a low cost. This was one of the drivers for producing our Dragonistics data cards – to provide at a reasonable cost, some ready-made, robust resources, so that teachers did not have to make their own. As it turned out we were able to provide interesting data with clear relationships, and engaging graphics so that we provide something more than just data turned into datacards.

Free resources

There are free resources available on the internet. Other resources are provided by teachers who are sharing what they have done while teaching their own students. Resources provided for free can be of a high pedagogical standard. Having a high production standard, however, can be prohibitively expensive for individual producers who are working in their spare time. It can also be tricky for another teacher to know what is suitable, and a lot of time can be spent trying to find high quality, reliable resources.

Teachers and resource providers – a symbiotic relationship

Teachers need good resource providers. It makes sense for experts to create high quality resources, drawing on current thinking with regard to content specific pedagogy. These can support teachers, particularly in areas in which they are less confident, such as statistics. And they do need to be paid for their work.

It helps when people recognise that our materials are sound and innovative, when they give us opportunities to contribute and when they include us at the decision-making table. Let us know how we can help you, and in partnership we can become better bed-fellows.

In many school systems in the world, subjects are taught separately. In primary school, children learn reading and writing, maths and social studies at different times of the day. But more than that, many topics within subjects are also taught separately. In mathematics we often teach computational skills, geometry, measurement and statistics in separate topics throughout the school year. Textbooks tend to encourage this segmentation of the curriculum. This causes problems as students compartmentalise their learning. They think that something learned in mathematics can’t possibly be used in Physics. They complain in mathematics if they are asked to write a sentence or a report, saying that it belongs in English.

I participated in an interesting discussion on Twitter recently about Stretch and Challenge. (Thanks #mathschat) My interpretation of “Stretch and challenge” is ways of getting students to extend their thinking beyond the original task so that they are learning more and feeling challenged. This reminds me a lot of the idea of “Low floor High Ceiling” that Jo Boaler talks about. We need tasks that are easy for students to get started on, but that do not limit students, particularly ones who have really caught onto the task and wish to keep going.

Fractions

As a statistics educator, I see applications of statistics and probability everywhere. At a workshop on proportional thinking we were each asked to represent three-quarters, having been told that our A5 piece of paper was “one”. When I saw the different representations used by the participants, I could see a graph as a great way to represent it. You could make a quick set of axes on a whiteboard, and get people to put crosses on which representation they used. The task of categorising all the representations reinforces the idea that there are many ways to show the same thing. It also gets students more aware of the different representations. Then the barchart/dotplot provides a reminder of the outcome of the task. Students who are excited about this idea could make up a little questionnaire to take home and get other family members to draw different fractions, and look at the representations, adding them to the graph back at school.

Measurement

Measurement is an area of the mathematics curriculum that is just begging to be combined with statistics. Just physically measuring an object leads to a variation in responses, which can be graphed. Getting each child to measure each object three times and take the middle value, should lead to a distribution of values with less spread. And then there is estimation. I love the example Dan Meyer uses in his Ted talk in 2010 of filling a tank with water. Students could be asked their estimate of the filling time, simply by guessing, and then use mathematical modelling to refine their estimate. Both values can be graphed and compared.

Area and Probability

Area calculations can be used nicely with probability. Children can invent games that involve tossing a coin onto a shape or shapes. The score depends on whether the coin lands within the shape, outside the shape or on a line. They can estimate what the score will be from 10 throws, simply by looking at the shape, then try it out with one lot of ten throws. Now do some area calculations. Students may have different ways of dealing with the overlap issue. Use the area calculations to improve their theoretical estimates of the probability of each outcome, and from there work out the expected value. Then do multiple trials of ten throws and see how you need to modify the model. So much learning in one task!

Statistics obviously fits well in much topic work as well. The Olympics are looming, with all the interest and the flood of statistics they provide. Students can be given the fascinating question of which country does the best? There are so many ways to measure and to account for population. Drawing graphs gives an idea of spread and distribution.

There is so much you can do with statistics and other strands and other curriculum areas! Statistics requires a context, and it is economical use of time if the context is something else you are teaching.

Can you tell me some ways you have incorporated statistics into other strands of mathematics or other subject areas?

Data cards are a wonderful way for students to get a feel for data. As a University lecturer in the 1990s, I found that students often didn’t understand about the multivariate nature of data. This may well be an artifact of the kind of statistics they studied at school, which was univariate (finding the confidence interval for the mean of a set of numbers) or bivariate at best. And back then, when statistical analysis was done by hand calculation, this was all you could expect. How times have changed!

At the NZAMT (NZ Association of Mathematics Teachers) conference in 2015, both Dick de Veaux and Rob Gould suggested in their keynote addresses that students need to be exposed to multivariate data. Rob endorsed the use of data cards to enable this. Data cards are a wonderful tool for all levels of learning. In the New Zealand “Figure it out” series, there are several lessons that use data cards, generally made by the students themselves. We were inspired by this and have developed a set of 240 data cards with information about dragons, to help teachers and students learn and be successful in their statistical endeavours. In an earlier post I discuss the pros and cons of fictional data.

The real advantage of using data cards to teach sampling is that it is difficult, and approaching prohibitive, to record and analyse all the information. When you have a spreadsheet of data on a computer, to take a sample is contrived and can confuse students. They wonder why you would not simply analyse all the data for the population.Physically collecting data can take more time than is practical. With the data cards, we know we cannot easily process the data from all 240 or 480 dragons (depending on how many boxes you use.) Sampling then becomes a sensible solution. Different groups of students take different samples, and perform their own analysis, leading to similar, but not identical results. This shows the concept of variation due to sampling in a concrete and memorable way.

Some decades ago I developed a set of counters of four different colours, with data with different means and standard deviations. I used these to teach about the concept of sampling, and the students did ANOVA analysis on them to see if the means of the four groups were the same. This was a good way to teach this principle. However there were two limitations. The first limitation is that the data is not multivariate. There are just two

The old technology – two variables, and no embedded context

variables, colour and the number. And the second limitation is that there was no context. I made up a context to go with it, something around sales I think, as this was for an MBA class, which partly overcame that problem.

I’d like to think that I have learned from all the reading, research, experience, seminars etc on how to teach statistics that I have participated in. Consequently, were I to teach an MBA Quantitative methods class again, I think I would use the Dragon data cards. We have recently produced this lesson plan, that teaches about the concept of sampling and variation due to sampling. Dragon data cards could also be used for teaching about the mechanics of sampling, such as stratification and systematic sampling. There needs to be a story behind the analysis or there is no point to the conclusion. In the lesson previously alluded to, the scenario is that we are building separate shelters for male and female dragons, and it would be useful to have an idea of the relative strengths of male and female dragons.

Evidence and Distribution

Using data cards gives a wonderful opportunity to explore the concepts of evidence and of distribution. The students lay out their cards in a nice bar chart arrangement, and say, “See – there is a difference.” Teachers should then ask for evidence. Students need to be able to articulate what evidence there is for the effect they have observed, and place it in context. We have found this to be a useful process when teaching students of all levels.

With regard to distribution, if we work only with numbers, and find the medians of the two groups and observe that the median is higher for one group than the other, this is rather limited information. By observing the distribution of the dragon cards between the two sexes, we can see that there is overlap. It is not a clearcut difference. Additionally we may observe other effects, such as due to colour, which we might like to explore further in another journey around the Statistical Enquiry Cycle.

Data cards are a win

It is fascinating that the concept of data cards is so new. It seems like an obvious idea, and makes concrete some very tricky abstract ideas. Data cards are useful at almost any level of understanding. As the need for understanding of statistics grows, there has been an emphasis on finding out better ways to teach for understanding. Clearly data cards are a win!

There is a push for teachers and students to use real data in learning statistics. In this post I am going to address the benefits and drawbacks of different sources of real data, and make a case for the use of good fictional data as part of a statistical programme.

There are two main types of real data. There is the real data that students themselves collect and there is real data in a dataset, collected by someone else, and available in its entirety. There are also two main types of unreal data. The first is trivial and lacking in context and useful only for teaching mathematical manipulation. The second is what I call fictional data, which is usually based on real-life data, but with some extra advantages, so long as it is skilfully generated. Poorly generated fictional data, as often found in case studies, is very bad for teaching.

Focus

When deciding what data to use for teaching statistics, it matters what it is that you are trying to teach. If you are simply teaching how to add up 8 numbers and divide the result by 8, then you are not actually doing statistics, and trivial fake data will suffice. Statistics only exists when there is a context. If you want to teach about the statistical enquiry process, then having the students genuinely involved at each stage of the process is a good idea. If you are particularly wanting to teach about fitting a regression line, you generally want to have multiple examples for students to use. And it would be helpful for there to be at least one linear relationship.

I read a very interesting article in “Teaching Children Mathematics” entitled, “Practıcal Problems: Using Literature to Teach Statistics”. The authors, Hourigan and Leavy, used a children’s book to generate the data on the number of times different characters appeared. But what I liked most, was that they addressed the need for a “driving question”. In this case the question was provided by a pre-school teacher who could only afford to buy one puppet for the book, and wanted to know which character appears the most in the story. The children practised collecting data as the story is read aloud. They collected their own data to analyse.

Let’s have a look at the different pros and cons of student-collected data, provided real data, and high-quality fictional data.

Collecting data

When we want students to experience the process of collecting real data, they need to collect real data. However real time data collection is time consuming, and probably not necessary every year. Student data collection can be simulated by a program such as The Islands, which I wrote about previously. Data students collect themselves is much more likely to have errors in it, or be “dirty” (which is a good thing). When students are only given clean datasets, such as those usually provided with textbooks, they do not learn the skills of deciding what to do with an errant data point. Fictional databases can also have dirty data, generated into it. The fictional inhabitants of The Islands sometimes lie, and often refuse to give consent for data collection on them.

Motivation

One of the species of dragons included in our database

I have heard that after a few years of school, graphs about cereal preference, number of siblings and type of pet get a little old. These topics, relating to the students, are motivating at first, but often there is no purpose to the investigation other than to get data for a graph. Students need to move beyond their own experience and are keen to try something new. Data provided in a database can be motivating, if carefully chosen. There are opportunities to use databases that encourage awareness of social justice, the environment and politics. Fictional data must be motivating or there is no point! We chose dragons as a topic for our first set of fictional data, as dragons are interesting to boys and girls of most ages.

A meaningful question

Here I refer again to that excellent article that talks about a driving question. There needs to be a reason for analysing the data. Maybe there is concern about food provided at the tuck shop, with healthy alternatives. Or can the question be tied into another area of the curriculum, such as which type of bean plant grows faster? Or can we increase the germination rate of seeds. The Census@school data has the potential for driving questions, but they probably need to be helped along. For existing datasets the driving question used by students might not be the same as the one (if any) driving the original collection of data. Sometimes that is because the original purpose is not ‘motivating’ for the students or not at an appropriate level. If you can’t find or make up a motivating meaningful question, the database is not appropriate. For our fictional dragon data, we have developed two scenarios – vaccinating for Pacific Draconian flu, and building shelters to make up for the deforestation of the island. With the vaccination scenario, we need to know about behaviour and size. For the shelter scenario we need to make decisions based on size, strength, behaviour and breath type. There is potential for a number of other scenarios that will also create driving questions.

Getting enough data

It can be difficult to get enough data for effects to show up. When students are limited to their class or family, this limits the number of observations. Only some databases have enough observations in them. There is no such problem with fictional databases, as you can just generate as much data as you need! There are special issues with regard to teaching about sampling, where you would want a large database with constrained access, like the Islands data, or the use of cards.

Variables

A problem with the data students collect is that it tends to be categorical, which limits the types of analysis that can be used. In databases, it can also be difficult to find measurement level data. In our fictional dragon database, we have height, strength and age, which all take numerical values. There are also four categorical variables. The Islands database has a large number of variables, both categorical and numerical.

Interesting Effects

Though it is good for students to understand that quite often there is no interesting effect, we would like students to have the satisfaction of finding interesting effects in the data, especially at the start. Interesting effects can be particularly exciting if the data is real, and they can apply their findings to the real world context. Student-collected-data is risky in terms of finding any noticeable relationships. It can be disappointing to do a long and involved study and find no effects. Databases from known studies can provide good effects, but unfortunately the variables with no effect tend to be left out of the databases, giving a false sense that there will always be effects. When we generate our fictional data, we make sure that there are the relationships we would like there, with enough interaction and noise. This is a highly skilled process, honed by decades of making up data for student assessment at university. (Guilty admission)

Ethics

There are ethical issues to be addressed in the collection of real data from people the students know. Informed consent should be granted, and there needs to be thorough vetting. Young students (and not so young) can be damagingly direct in their questions. You may need to explain that it can be upsetting for people to be asked if they have been beaten or bullied. When using fictional data, that may appear real, such as the Islands data, it is important for students to be aware that the data is not real, even though it is based on real effects. This was one of the reasons we chose to build our first database on dragons, as we hope that will remove any concerns about whether the data is real or not!

The following table summarises the post.

Real data collected by the students

Real existing database

Fictional data (The Islands, Kiwi Kapers, Dragons, Desserts)

It is possible to get a passing grade in a statistics paper by putting numbers into formulas and words into memorised phrases. In fact I suspect that this is a popular way for students to make their way through a required and often unwanted subject.

Most teachers of statistics would say that they would like students to understand what they are doing. This was a common sentiment expressed by participants in the excellent MOOC, Teaching statistics through data investigations (which is currently running again in January to May 2016.)

Understanding

This makes me wonder what it means for students to understand statistics. There are many levels to understanding things. The concept of understanding has many nuances. If a person understands English, it means that they can use English with proficiency. If they are native speakers they may have little understanding of how grammar works, but they can still speak with correct grammar. We talk about understanding how a car works. I have no idea how a car works, apart from some idea that it requires petrol and the pistons go really, really fast. I can name parts of a car engine, such as distributor and drive shaft. But that doesn’t stop me from driving a car.

Understanding statistics

I propose that when we talk about teaching students to understand statistics, we want our students to know why they are doing something, and have an idea of how it works. Students also need to be fluent in the language of statistics. I would not expect any student of an introductory or high school statistics class to be able to explain how least squares regression works in terms of matrix algebra, but I would expect them to have an idea that the fitted line in a bivariate plot is a model that minimises the squared error terms. I’m not sure anyone needs to know why “degrees of freedom” are called that – or even really what degrees of freedom do. These days computer packages look after degrees of freedom for us. We DO need to understand what a p-value is, and what it is telling us. For many people it is not necessary to know how a p-value is calculated.

Ways to teach statistics

There are several approaches to teaching statistics. The approach needs to be tailored to the students and the context of the course. I prefer a hands-on, conceptual approach rather than a mathematical one. In current literature and practice there is a push for learning through investigations, often based around the statistical inquiry cycle. The problem with one long project is that students don’t get opportunities to apply principles in different situations, in such a way that will help in transfer of learning to other situations. There are some people who still teach statistics through the mathematical formulas, but I fear they are missing out on the opportunity to help students really enjoy statistics.

I do not propose to have all the answers, but we did discover one way to help students learn, alongside other methods. This approach is to use a short video, followed by a ten question true/false quiz. The quiz serves to reinforce and elaborate on concepts taught in the video, challenge students’ misconceptions, and help students be more familiar with the vocabulary and terminology of statistics. The quizzes we develop have multiple questions that randomise to give students the opportunity to try multiple times which seems to help understanding.

This short and entertaining video gives an illustration of how you can use videos and quizzes to help students learn difficult concepts.