Dragon Trainer rich mathematical task

I love rich mathematical tasks. Here is one for all levels of schooling. What do you think?

Background to rich tasks

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.

Some important aspects of rich mathematical tasks

Background to Dragonistics data cards

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.

A small sample of Dragonistics data cards

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.

The task

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.

Try it yourself

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.

Possible approaches

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.

Lessons to bring out at different levels

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.

Extension questions

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?

Show them the maths

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.

Transferring

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?

Educating the heart with maths and statistics

What has love got to do with maths?

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.

Teachers are people

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.

Don’t sweeten the broccoli

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.

Teaching mathematics and statistics is an act of social justice

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.

Play and learning mathematics and statistics

The role of play in learning

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.

I suspect that one reason our statistics videos on YouTube are so popular is because they are playful.

helen-has-attitude

Helen has an attitude problem

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.

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

Dragonistics data cards provide light-hearted data which yields worth-while results.

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. Sadly, our Kickstarter campaign was unsuccessful, but we hope to work with an established game manufacturer to bring them to the market by the end of 2017.

We'd love your help.

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?

 

The nature of mathematics and statistics and what it means to learn and teach them

I’ve been thinking lately….

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

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

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.

 

Mathematics teaching Rockstar – Jo Boaler

Moving around the education sector

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.”

Jo Boaler – Click here for official information

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

What does it mean to understand statistics?

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.

And here is a link to a listing of all our videos and how you can get access to them. Statistics Learning Centre Videos

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The importance of being wrong

We don’t like to think we are wrong

One of the key ideas in statistics is that sometimes we will be wrong. When we report a 95% confidence interval, we will be wrong 5% of the time. Or in other words, about 1 in 20 of 95% confidence intervals will not contain the population parameter we are attempting to estimate. That is how they are defined. The thing is, we always think we are part of the 95% rather than the 5%. Mostly we will be correct, but if we do enough statistical analysis, we will almost definitely be wrong at some point. However, human nature is such that we tend to think it will be someone else. There is also a feeling of blame associated with being wrong. The feeling is that if we have somehow missed the true value with our confidence interval, it must be because we have made a mistake. However, this is not true. In fact we MUST be wrong about 5% of the time, or our interval is too big, and not really a 95% confidence interval.

The term “margin of error” appears with increasing regularity as elections approach and polling companies are keen to make money out of sooth-saying. The common meaning of the margin of error is half the width of a 95% confidence interval. So if we say the margin of error is 3%, then about one time in twenty, the true value of the proportion will actually be more than 3% away from the reported sample value.

What doesn’t help is that we seldom do know if we are correct or not. If we knew the real population value we wouldn’t be estimating it. We can contrive situations where we do know the population but pretend we don’t. If we do this in our teaching, we need to be very careful to point out that this doesn’t normally happen, but does in “classroom world” only. (Thanks to MD for this useful term.) General elections can give us an idea of being right or wrong after the event, but even then the problem of non-sampling error is conflated with sampling error. When opinion polls turn out to miss the mark, we tend to think of the cause as being due to poor sampling, or people changing their minds, or all number of imaginative explanations rather than simple, unavoidable sampling error.

So how do we teach this in such a way that it goes beyond school learning and is internalised for future use as efficient citizens?

Teaching suggestions

I have two suggestions. The first is a series of True/False statements that can be used in a number of ways. I have them as part of on-line assessment, so that the students are challenged by them regularly. They could be well used in the classroom as part of a warm-up exercise at the start of a lesson. Students can write their answers down or vote using hands.

Here are some examples of True/False statements (some of which could lead to discussion):

  1. You never know if your confidence interval contains the true population value.
  2. If you make your confidence interval wide enough you can be sure that you contain the true population value.
  3. A confidence interval tells us where we are pretty sure the sample statistic lies.
  4. It is better to have a narrow confidence interval than a wide one, as it gives us more certain information, even though it is more likely to be wrong.
  5. If your study involves twenty confidence intervals, then you know that exactly one of them will be wrong.
  6. If a confidence interval doesn’t contain the true population value, it is because it is one of the 5% that was calculated incorrectly.

You can check your answers at the end of this post.

Experiential exercise

The other teaching suggestion is for an experiential exercise. It requires a little set up time.

Make a set of cards for students with numbers on them that correspond to the point estimate of a proportion, or a score that will lead to that. (Specifications for a set of 35 cards representing the results from a proportion of 0.54 and 25 trials is given below).

Introduce the exercise as follows:
“I have a computer game, and have set the ratio of wins to losses at a certain value. Each of you has played 25 times, and the number of wins you have obtained will be on your card. It is really important that you don’t look at other people’s cards.”

Hand them out to the students. (If you have fewer than 35 in your class, it might be a good idea to make sure you include the cards with 8 and 19 in the set you use – sometimes it is ok to fudge slightly to teach a point.)
“Without getting information from anyone else, write down your best estimate of the true proportion of wins to losses in the game. Do you think you are correct? How close do you think you are to the true value?”

They will need to divide the number of wins by 25, which should not lead to any computational errors! The point is that they really can’t know how close their estimate is to the true value – and what does “correct” mean?

Then work out the margin of error for a sample of size 25, which in this case is estimated at 20%. Get the students to calculate their 95% confidence intervals, and decide if they have the interval that contains the true population value. Get them to commit one way or the other.

Now they can talk to each other about the values they have.

There are several ways you can go from here. You can tell them what the population proportion was from which the numbers were drawn (0.54). They can then see that most of them had confidence intervals that included the true value, and some didn’t. Or you can leave them wondering, which is a better lesson about real life. Or you can do one exercise where you do tell them and one where you don’t.

This is an area where probability and statistics meet. You could make a nice little binomial distribution problem out of being correct in a number of confidence intervals. There are potential problems with independence, so you need to be a bit careful with the wording. For example: Fifteen  students undertake separate statistical analyses on the topics of their choice, and construct 95% confidence intervals. What is the probability that all the confidence intervals are correct, in that they do contain the estimated population parameter? This is well modelled by a binomial distribution with n =15 and p=0.05. P(X=0)=0.46. And another interesting idea – what is the probability that two or more are incorrect? 0.17 is the answer. So there is a 17% chance that more than one of the confidence intervals does not contain the population parameter of interest.

This is an area that needs careful teaching, and I suspect that some teachers have only a sketchy understanding of the idea of confidence intervals and margins of error. It is so important to know that statistical results are meant to be wrong some of the time.

Answers: T,T,F, debatable, F,F.

Data for the 35 cards:

Number on card

8

9

10

11

12

13

14

15

16

17

18

19

Number of cards

1

1

2

3

5

5

6

5

3

2

1

1

Parts and whole

The whole may be greater than the sum of the parts, but the whole still needs those parts. A reflective teacher will think carefully about when to concentrate on the whole, and when on the parts.

Golf

If you were teaching someone golf, you wouldn’t spend days on a driving range, never going out on a course. Your student would not get the idea of what the game is, or why they need to be able to drive straight and to a desired length. Nor would it be much fun! Similarly if the person only played games of golf it would be difficult for them to develop their game. Practice driving and putting is needed.  A serious student of golf would also read and watch experts at golf.

Music

Learning music is similar. Anyone who is serious about developing as a musician will spend a considerable amount of time developing their technique and their knowledge by practicing scales, chords and drills. But at the same time they need to be playing full pieces of music so that they feel the joy of what they are doing. As they play music, as opposed to drill, they will see how their less-interesting practice has helped them to develop their skills. However, as they practice a whole piece, they may well find a small part that is tripping them up, and focus for a while on that. If they play only the piece as a whole, it is not efficient use of time. A serious student of music will also listen to and watch great musicians, in order to develop their own understanding and knowledge.

Benefits of study of the whole and of the parts

In each of these examples we can see that there are aspects of working with the whole, and aspects of working with the parts. Study of the whole contributes perspective and meaning to study, and helps to tie things together. It helps to see where they have made progress. Study of the parts isolates areas of weakness, develops skills and saves time in practice, thus being more efficient.

It is very important for students to get an idea of the purpose of their study, and where they are going. For this reason I have written earlier about the need to see the end when starting out in a long procedure such as a regression or linear programming model.

It is also important to develop “statistical muscle memory” by repeating small parts of the exercise over and over until it is mastered. Practice helps people to learn what is general and what is specific in the different examples.

Teaching conditional probability

We are currently developing a section on probability as part of our learning materials. A fundamental understanding of probability and uncertainty are essential to a full understanding of inference. When we look at statistical evidence from data, we are holding it up against what we could reasonably expect to happen by chance, which involves a probability model. Probability lies in the more mathematical area of the study of statistics, and has some fun problem-solving aspects to it.

A popular exam question involves conditional probability. We like to use a table approach to this as it avoids many of the complications of terminology. I still remember my initial confusion over the counter-intuitive expression P(A|B) which means the probability that an object from subset B has the property of A. There are several places where students can come unstuck in Bayesian review, and the problems can take a long time. We can liken solving a conditional probability problem to a round of golf, or a long piece of music. So what we do in teaching is that first we take the students step by step through the whole problem. This includes working out what the words are saying, putting the known values into a table, calculating the unknown values in the table, and the using the table to answer the questions involving conditional probability.

Then we work on the individual steps, isolating them so that students can get sufficient practice to find out what is general and what is specific to different examples. As we do this we endeavour to provide variety such that students do not work out some heuristic based on the wording of the question, that actually stops them from understanding. An example of this is that if we use the same template each time, students will work out that the first number stated will go in a certain place in the table, and the second in another place etc. This is a short-term strategy that we need to protect them from in careful generation of questions.

As it turns out students should already have some of the necessary skills. When we review probability at the start of the unit, we get students to calculate probabilities from tables of values, including conditional probabilities. Then when they meet them again as part of the greater whole, there is a familiar ring.

Once the parts are mastered, the students can move on to a set of full questions, using each of the steps they have learned, and putting them back into the whole. Because they are fluent in the steps, it becomes more intuitive to put the whole back together, and when they meet something unusual they are better able to deal with it.

Starting a course in Operations Research/Management Science

It is interesting to contemplate what “the whole” is, with regard to any subject. In operations research we used to begin our first class, like many first classes, talking about what management science/operations research is. It was a pretty passive sort of class, and I felt it didn’t help as first-year university students had little relevant knowledge to pin the ideas on. So we changed to an approach that put them straight into the action and taught several weeks of techniques first. We started with project management and taught critical path. Then we taught identifying fixed and variable costs and break-even analysis. The next week was discounting and analysis of financial projects. Then for a softer example we looked at multi-criteria decision-making, using MCDM. It tied back to the previous week by taking a different approach to a decision regarding a landfill. Then we introduced OR/MS, and the concept of mathematical modelling. By then we could give real examples of how mathematical models could be used to inform real world problems. It was helpful to go from the concrete to the abstract. This was a much more satisfactory approach.

So the point is not that you should always start with the whole and then do the parts and then go back to the whole. The point is that a teacher needs to think carefully about the relationship between the parts and the whole, and teach in a way that is most helpful.

Difficult concepts in statistics

Recently someone asked: “I don’t suppose you’d like to blog a little on the pedagogical knowledge relevant to statistics teaching, would you? A ‘top five statistics student misconceptions (and what to do about them)’ would be kind of a nice thing to see …”

I wish it were that easy. Here goes:

Things that I have found students find difficult to understand and what I have done about them.

Observations

When I taught second year regression we would get students to collect data and fit their own multiple regressions. The interesting thing was that quite often students would collect unrelated data. The columns of the data would not be of the same observations. These students had made it all the way through first year statistics without really understanding about multivariate data.

So from them on when I taught about regression I would specifically begin by talking about observations (or data points) and explain how they were connected. It doesn’t hurt to be explicit. In the NZ curriculum materials for high school students are exercises using data cards which correspond to individuals from a database. This helps students to see that each card, which corresponds to a line of data, is one person or thing. In my video about Levels of measurement, I take the time to show this.

First suggestion is “Don’t assume”.  This applies to so much!

And this is also why it is vital that instructors do at least some of their own marking (grading). High school teachers are going, “Of course”. College professors – you know you ought to! The only way you find out what the students don’t understand, or misunderstand, or replicate by rote from your own notes, is by reading what they write. This is tedious, painful and sometimes funny in a head-banging sort of way, but necessary. I also check the prevalence of answers to multiple choice questions in my on-line materials. If there is a distracter scoring highly it is worthwhile thinking about either the question or the teaching that is leading to incorrect responses.

Inference

Well duh! Inference is a really, really difficult concept and is the key to inferential statistics. The basic idea, that we use information from a sample to draw conclusions about the population seems straight-forward. But it isn’t. Students need lots and lots of practice at identifying what is the population and what is the sample in any given situation. This needs to be done with different types of observations, such as people, commercial entities, plants or animals, geographical areas, manufactured products, instances of a physical experiment (Barbie bungee jumping), and times.

Second suggestion is “Practice”. And given the choice between one big practical project and a whole lot of small applied exercises, I would go with the exercises. A big real-life project is great for getting an idea of the big picture, and helping students to learn about the process of statistical analysis. But the problem with one big project is that it is difficult to separate the specific from the general. Context is at the core of any analysis in statistics, and makes every analysis different. Learning occurs through experiencing many different contexts and from them extracting what is general to all analysis, what is common to many analyses and what is specific to that example. The more different examples a student is exposed to, the better opportunity they have for constructing that learning. An earlier post extols the virtues of practice, even drill!

Connections

One of the most difficult things is for students to make connections between parts of the curriculum. A traditional statistics course can seem like a toolbox of unrelated but confusingly different techniques. It takes a high level of understanding to link the probability, data and evidence aspects together in a meaningful way. It is good to have exercises that hep students to make these connections. I wrote about this with regard to Operations Research and Statistics. But students need also to be making connections before they get to the end of the course.

The third suggestion is “get students to write”

Get students to write down what is the same and what is different between chi-sq analysis and correlation. Get them to write down how a poisson distribution is similar to and different from a binomial distribution. Get them to write down how bar charts and histograms are similar and different. The reason students must write is that it is in the writing that they become aware of what they know or don’t know. We even teach ourselves things as we write.

Graphs and data

Another type of connection that students have trouble with is that between the data and the graph, and in particular identifying variation and distribution in a histogram or similar. There are many different graphs, that can look quite similar, and students have problems identifying what is going on. The “value graph” which is produced so easily in Excel does nothing to help with these problems. I wrote a full post on the problems of interpreting graphs.

The fourth suggestion is “think hard”. (or borrow)

Teaching statistics is not for wusses. We need to think really hard about what students are finding difficult, and come up with solutions. We need to experiment with different ways of explaining and teaching. One thing that has helped my teaching is the production of my videos. I wish to use both visual and text (verbal) inputs as best as possible to make use of the medium. I have to think of ways of representing concepts visually, that will help both understanding and memory. This is NOT easy, but is extremely rewarding. And if you are not good at thinking up new ideas, borrow other people’s ideas. A good idea collector can be as good as or better than a good creator of ideas.

To think of a fifth suggestion I turned to my favourite book , “The Challenge of Developing Statistical Literacy, Reasoning and Thinking”, edited by Dani Ben-Zvi and Joan Garfield. I feel somewhat inadequate in the suggestions given above. The book abounds with studies that have shown areas of challenge or students and teachers. It is exciting that so many people are taking seriously the development of pedagogical content knowledge regarding the discipline of statistics. Some statisticians would prefer that the general population leave statistics to the experts, but they seem to be in the minority. And of course it depends on what you define “doing statistics” to mean.

But the ship of statistical protectionism has sailed, and it is up to statisticians and statistical educators to do our best to teach statistics in such a way that each student can understand and apply their knowledge confidently, correctly and appropriately.

Pedagogical Content Knowledge

Pedagogical content knowledge for Statistics

Pedagogical content knowledge means knowing how to teach a specific subject, discipline or context. There is a school of thought that the skill of teaching is transferable between subjects, so long as the teacher knows the content. However others argue that teaching strategies differ sufficiently across disciplines to create individual but overlapping bodies of knowledge, called pedagogical content knowledge. To me it is clear that different skills and approaches are needed in the teaching of different disciplines. The methods for teaching a foreign language differ largely from those for teaching history or science or cake decorating or jazz piano. There are also commonalities in all teaching, such as the need to build a relationship between the teacher and student, and building on students’ previous knowledge.

I first learned about the concept of “pedagogical content knowledge” in one of my favourite books – How People Learn: Brain, Mind, Experience and School. This book brings together research into how the brain works, and how people learn, in such a way that teachers can gain from it in their practice. Regarding pedagogical content knowledge, it states “Expert teachers know the kinds of difficulties that students are likely to face; they know how to tap into students’ existing knowledge in order to make new information meaningful; and they know how to assess their students’ progress.”

I fear that one of the reasons that the subject of statistics is not as popular as it deserves to be, is because almost all the teachers at all levels lack pedagogical content knowledge with respect to teaching statistics. I am not saying that the teachers are bad teachers, or ill-meaning, or unintelligent. I am saying that most teachers of statistics do not really know how to teach statistics.

Let us look at some different groups of teachers, starting with the most influential and consequently worst paid.

Primary (elementary) school teachers)

My experience of primary school teachers is that they generally are less comfortable teaching mathematics than reading and writing. Their knowledge and understanding of statistics ranges between trivial and incorrect. Their pedagogical content knowledge for statistics is pretty low. These teachers often teach incorrect graphing methods, and may well perpetuate the idea that probability relates to dice, coins and counters. It is not really their fault. There is such a broad curriculum at that level, that it must be challenging to cover all possibilities in their training. Having said that, a well-funded initiative in professional development could address this issue.

High school teachers

Mostly statistics at high school level  is taught by mathematics teachers, from a mathematical background rather than a statistical one. I have already written about the problems when mathematicians fail to treat statistics as an allied but separate discipline from mathematics. I was greatly heartened last week to meet with forty committed teachers of high school statistics who are embracing the new approach of the New Zealand curriculum toward statistics. They have seen how interesting the subject is and are helping students to make real progress in their learning. This is testament to the dedication and collaboration of the teachers themselves, and the efforts of bodies such as Census @ School and my own Statistics Learning Centre, which are helping to support these teachers. The support from the official channels appears criminally lacking, unco-ordinated, and at times even conflicting.

These teachers were at my workshop on teaching statistical report-writing, because they were aware of their own inadequacies in this area. (Though some were doing a fantastic job already). It is hardly surprising that they feel unprepared for teaching this material when their expertise has been in teaching trigonometry, algebra, measurement and calculus. The pedagogical content knowledge for teaching statistics is very different from teaching mathematics. Statistics is, compared with mathematics, an inexact science, where context is vitally important, and where different correct approaches will produce different numbers as answers to a problem. In statistics the words used are critical, and one word can change the meaning of the sentence completely.

Fortunately there is research undertaken on how better to teach statistics, and the body of pedagogical content knowledge is increasing. Another of my favourite books is “The Challenge of Developing Statistical Literacy, Reasoning and Thinking”, edited by Dani Ben-Zvi and Joan Garfield. This brings together the results of thinking and experimentation to improve the efficacy of statistics teaching. One problem identified by Garfield some years ago was that even students who received A passes in statistics often had a very poor understanding of even the most basic concepts of the subject. It is exciting to read the progress that is being made in developing strategies for teaching statistics in a way that promotes deep understanding that transfers to other problems and disciplines. It is also exciting to live in New Zealand where the findings of the research have been applied to the development of a national curriculum in statistics.

Khan academy

I’d just like to pop in a reference to Khan Academy because, sadly, it has a great influence. I believe that many of the mathematics Khan Academy videos are fairly well taught, in a “boy-next-door” sort of way. However the statistics videos perpetuate the mathematical view of statistics, as they are a product of an archaic curriculum. Khan has NO pedagogical content knowledge of statistics. This is abundantly clear in the approach and errors. I have covered this in earlier posts.

AP Statistics

Advanced Placement Statistics is an American invention of which I have only a tenuous understanding. It appears to be a subject taken at high school level, examined nationally and can count for credit at a tertiary institution. Consequently, though the level is of first year college level, it is taught by high school teachers, which may or may not be to the advantage of the students. I suspect the level of pedagogical content knowledge among the teachers is highly skewed with  a very large bulge at the low end and very thin tail to the high end. (To me the word skewed goes the wrong way, so I prefer to describe the outcome).

Higher education

Statistics at universities is taught by a wide range of people. Teaching assistants have the advantage of recent experience learning the material and may thus be better able to see the challenges of learning the discipline. There will be truly great teachers of statistics among them. Some instructors specialise in the teaching of statistics and help to advance the corporate body of pedagogical content knowledge. Some academics really don’t care about teaching, and just present the material as painlessly (to them) as possible before they head back to their research.

Developing pedagogical content knowledge

I fear I have stated a problem, with very little in the way of solution. Sometimes it is a good start to identify that the problem exists. Part of my aim in my workshop is to validate the efforts of teachers in what is an unfamiliar environment, and explain why they are feeling out of their depth. This diagnosis helps to remove the blame from the teacher, who are then smart enough, with a few suggestions, to work to develop a solution.

It is my intention that this blog is part of a solution. The aim is that through my musings and the comments of others we are able to encourage progress in the teaching of statistics in such a way that will thrill and excite the masses! Failing that, at least not put them off statistics altogether.