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.

Advertisements

Papamoa College statistics excursion to Hamilton Zoo

Pizza in the park

Pizza in the park

Last week I had a lovely experience. I visited the Hamilton Observatory and Zoo as part of a Statistics excursion with the Year 13 statistics class of Papamoa College.

The trip was organised to help students learn about where data comes from. I went along because I really love teachers and students, and it was an opportunity to experience innovation by a team of wonderful teachers.  The students travelled from Papamoa to Hamilton, stopping for pizza in Cambridge. When we got to the Hamilton Observatory, Dave welcomed us and gave an excellent talk about the stars and data. I found it fascinating to think how much data there is, and also the level of (in)accuracy of their measurements.  I then gave a short talk on the importance of statistics in terms of citizenship, and how the students can be successful in learning statistics. I talked about analysis of the Disney Princess movies and the Zika virus.

Turtle

My favourite animal of the day

The next morning we went over to the Hamilton Zoo for breakfast followed by a talk by Ken on the use of data in the Zoo. That too was fascinating, and got my brain whirring. Zoos these days are all about education and helping endangered species to survive. They have records of weights of all the animals over time, making for some very interesting data. Weights are used as an indication of health in the animals. Ken shared pictures of animals being weighed – including tricky keas and fantastically large rhinos. The Zoo also collects a wide range of other data, such as the visitor numbers, satisfaction surveys, quantity of waste and food consumption. We visited the food preparation area and heard how the diets are carefully worked out, and the food fed in such a way as to give the animals something to think about.

Papamoa stats class

Dr Nic and the teachers and students of Papamoa College give statistics two thumbs up!

Though most of my work these days is in the field of statistics education, a part of my heart still belongs to Operations Research. I saw so many ways in which OR could help with things such as diets, logistics etc. I’m not saying that they are doing anything wrong, but there is always room for improvement. Were I still teaching OR to graduate students I would be looking for a project with a zoo.

I am sure the students benefited from the experience of seeing first-hand the use of data in multiple contexts. I was glad to be able to meet with the students
and talk to many about the assignments they will be doing throughout the year. Each student has the opportunity to choose an application area for the multiple assessments. I was impressed with their level of motivation, which will lead to better learning outcomes.

Well done team at Papamoa!

 

A Statistics-centric curriculum

Calculus is the wrong summit of the pyramid.

“The mathematics curriculum that we have is based on a foundation of arithmetic and algebra. And everything we learn after that is building up towards one subject. And at top of that pyramid, it’s calculus. And I’m here to say that I think that that is the wrong summit of the pyramid … that the correct summit — that all of our students, every high school graduate should know — should be statistics: probability and statistics.”

Ted talk by Arthur Benjamin in February 2009. Watch it – it’s only 3 minutes long.

He’s right, you know.

And New Zealand would be the place to start. In New Zealand, the subject of statistics is the second most popular subject in our final year of schooling, with a cohort of 12,606. By comparison, the cohort for  English is 16,445, and calculus has a final year cohort of 8392, similar in size to Biology (9038), Chemistry (8183) and Physics (7533).

Some might argue that statistics is already the summit of our curriculum pyramid, but I would see it more as an overly large branch that threatens to unbalance the mathematics tree. I suspect many maths teachers would see it more as a parasite that threatens to suck the life out of their beloved calculus tree. The pyramid needs some reconstruction if we are really to have a statistics-centric curriculum. (Or the tree needs pruning and reshaping – I think I have too many metaphors!)

Statistics-centric curriculum

So, to use a popular phrase, what would a statistics-centric curriculum look like? And what would be the advantages and disadvantages of such a curriculum? I will deal with implementation issues later.

To start with, the base of the pyramid would look little different from the calculus-pinnacled pyramid. In the early years of schooling the emphasis would be on number skills (arithmetic), measurement and other practical and concrete aspects. There would also be a small but increased emphasis on data collection and uncertainty. This is in fact present in the NZ curriculum. Algebra would be introduced, but as a part of the curriculum, rather than the central idea. There would be much more data collection, and probability-based experimentation. Uncertainty would be embraced, rather than ignored.

In the early years of high school, probability and statistics would take a more central place in the curriculum, so that students develop important skills ready for their pinnacle course in the final two years. They would know about the statistical enquiry cycle, how to plan and collect data and write questionnaires.  They would perform their own experiments, preferably in tandem with other curriculum areas such as biology, food-tech or economics. They would understand randomness and modelling. They would be able to make critical comments about reports in the media . They would use computers to create graphs and perform analyses.

As they approach the summit, most students would focus on statistics, while those who were planning to pursue a career in engineering would also take calculus. In the final two years students would be ready to build their own probabilistic models to simulate real-world situations and solve problems. They would analyse real data and write coherent reports. They would truly understand the concept of inference, and why confidence intervals are needed, rather than calculating them by hand or deriving formulas.

There is always a trade-off. Here is my take on the skills developed in each of the curricula.

Calculus-centric curriculum

Statistics-centric curriculum

Logical thinking Communication
Abstract thinking Dealing with uncertainty and ambiguity
Problem-solving Probabilistic models
Modelling (mainly deterministic) Argumentation, deduction
Proof, induction Critical thinking
Plotting deterministic graphs from formulas Reading and creating tables and graphs from data

I actually think you also learn many of the calc-centric skills in the stats-centric curriculum, but I wanted to look even-handed.

Implementation issues

Benjamin suggests, with charming optimism, that the new focus would be “easy to implement and inexpensive.”  I have been a very interested observer in the implementation of the new statistics curriculum in New Zealand. It has not happened easily, being inexpensive has been costly, and there has been fallout. Teachers from other countries (of which there are many in mathematics teaching in NZ) have expressed amazement at how much the NZ teachers accept with only murmurs of complaint. We are a nation with a “can do” attitude, who, by virtue of small population and a one-tier government, can be very flexible. So long as we refrain from following the follies of our big siblings, the UK, US and Australia, NZ has managed to have a world-class education system. And when a new curriculum is implemented, though there is unrest and stress, there is seldom outright rebellion.

In my business, I get the joy of visiting many schools and talking with teachers of mathematics and statistics. I am fascinated by the difference between schools, which is very much a function of the head of mathematics and principal. Some have embraced the changes in focus, and are proactively developing pathways to help all students and teachers to succeed. Others are struggling to accept that statistics has a place in the mathematics curriculum, and put the teachers of statistics into a ghetto where they are punished with excessive marking demands.

The problem is that the curriculum change has been done “on the cheap”. As well as being small and nimble, NZ is not exactly rich. The curriculum change needed more advisors, more release time for teachers to develop and more computer power. These all cost. And then you have the problem of “me too” from other subjects who have had what they feel are similar changes.

And this is not really embracing a full stats-centric curriculum. Primary school teachers need training in probability and statistics if we are really to implement Benjamin’s idea fully. The cost here is much greater as there are so many more primary school teachers. It may well take a generation of students to go through the curriculum and enter back as teachers with an improved understanding.

Computers make it possible

Without computers the only statistical analysis that was possible in the classroom was trivial. Statistics was reduced to mechanistic and boring hand calculation of light-weight statistics and time-filling graph construction. With computers, graphs and analysis can be performed at the click of a mouse, making graphs a tool, rather than an endpoint. With computing power available real data can be used, and real problems can be addressed. High level thinking is needed to make sense and judgements and to avoid wrong conclusions.

Conversely, the computer has made much of calculus superfluous. With programs that can bash their way happily through millions of iterations of a heuristic algorithm, the need for analytic methods is seriously reduced. When even simple apps on an iPad can solve an algebraic equation, and Excel can use “What if” to find solutions, the need for algebra is also questionable.

Efficient citizens

In H.G. Wells’ popular but misquoted words, efficient citizenry calls for the ability to make sense of data. As the science fiction-writer that he was, he foresaw the masses of data that would be collected and available to the great unwashed. The levelling nature of the web has made everyone a potential statistician.

According to the engaging new site from the ASA, “This is statistics”, statisticians make a difference, have fun, satisfy curiosity and make money. And these days they don’t all need to be good at calculus.

Let’s start redesigning our pyramid.

Introducing Probability

I have a guilty secret. I really love probability problems. I am so happy to be making videos about probability just now, and conditional probability and distributions and all that fun stuff. I am a little disappointed that we won’t be doing decision trees with Bayesian review, calculating EVPI. That is such fun, but I gave up teaching that some years ago.

The reason probability is fun is because it is really mathematics, and puzzles and logic. I love permutations and combinations too – there is something cool about working out how many ways something can happen.

So why should I feel guilty? Well, in all honesty I have to admit that there is very little need for most of that in a course about statistics at high-school or entry level university. When I taught statistical methods for management, we did some probability, but only from an applied viewpoint, and we never touched intersection and union signs or anything like that. We applied some distributions, but without much theoretical underpinning.

The GAISE (Guidelines for Assessment and Instruction in Statistics Education) Report says, “Teachers and students must understand that statistics and probability are not the same. Statistics uses probability, much as physics uses calculus.”

The question is, why do we teach probability – apart from the fact that it’s fun and makes a nice change from writing reports on time series and bivariate analysis, inference and experiments. The GAISE report also says, “Probability is an important part of any mathematical education. It is a part of mathematics that enriches the subject as a whole by its interactions with other uses of mathematics. Probability is an essential tool in applied mathematics and mathematical modeling. It is also an essential tool in statistics.”

The concept of probability is as important as it is misunderstood. It is vital to have an understanding of the nature of chance and variation in life, in order to be a well-informed, (or “efficient”) citizen. One area in which this is extremely important is in understanding risk and relative risk. When a person is told that their chances of dying of some rare disease have just doubled, it is important that they know that it may be because they have gone from one chance in a million to two chances in a million. Sure it has doubled, but it still is pretty trivial. An understanding of probability is also important in terms of gambling and resistance to the allures of games of chance. And more socially acceptable gambling, such as stockmarket trading, also requires an understanding of chance and variation.

The concept of probability is important, and a few rules of probability may help with understanding, but I suspect the mathematicians get carried away and create problems that are unlikely (probability close to zero) to ever occur in reality. Anything requiring a three-way Venn Diagram has moved from applied problem to logic puzzle.This is in stark contrast to the very applied data-driven approach used in teaching statistics in New Zealand.

Teaching Probability

The traditional approach to teaching probability is to start with the coin and the dice and the balls in the urns. As well as being mind-bogglingly boring and pointless, this also projects an artificial certainty about the probabilities, which is confusing when we start discussing models. If you look at the Khan Academy videos (but don’t) you will find trivial examples about coloured balls or sweets or strangely complex problems involving hitting a circular target. The traditional approach is also to teach probability as truth. “The probability of getting a boy is one-half”. What does that even mean?

I am currently reading the new Springer volume, Probabilistic Thinking, and intend to write a review and post it on this blog, if I can get through enough before my review copy expires. It is inspiring and surprisingly gripping (but I don’t think that is enough of a review to earn me a hard copy to keep.). There are many great ideas for teaching in it, that I hope to pass on in due time.

The New Zealand approach to teaching probability comes from a modelling perspective, right from the start. At level 1, the first two years of schooling, children are exploring chance situations, playing games with a chance element and describing possible outcomes. By years 5 and 6 they are assigning numeric values to the likelihood of an occurrence. They (in the curriculum) are being introduced to model estimates and experimental estimates of probability. Bearing in mind how difficult high school maths teachers are finding the new approach, I don’t have a lot of confidence that the primary teachers are equipped yet to make the philosophical changes, let alone enact them in the classroom.

Deterministic and Probabilistic models and thinking

The way we understand and make sense of variation in the world affects decisions we make.

Part of understanding variation is understanding the difference between deterministic and probabilistic (stochastic) models. The NZ curriculum specifies the following learning outcome: “Selects and uses appropriate methods to investigate probability situations including experiments, simulations, and theoretical probability, distinguishing between deterministic and probabilistic models.” This is at level 8 of the curriculum, the highest level of secondary schooling. Deterministic and probabilistic models are not familiar to all teachers of mathematics and statistics, so I’m writing about it today.

Model

The term, model, is itself challenging. There are many ways to use the word, two of which are particularly relevant for this discussion. The first meaning is “mathematical model, as a decision-making tool”. This is the one I am familiar with from years of teaching Operations Research. The second way is “way of thinking or representing an idea”. Or something like that. It seems to come from psychology.

When teaching mathematical models in entry level operations research/management science we would spend some time clarifying what we mean by a model. I have written about this in the post, “All models are wrong.”

In a simple, concrete incarnation, a model is a representation of another object. A simple example is that of a model car or a Lego model of a house. There are aspects of the model that are the same as the original, such as the shape and ability to move or not. But many aspects of the real-life object are missing in the model. The car does not have an internal combustion engine, and the house has no soft-furnishings. (And very bumpy floors). There is little purpose for either of these models, except entertainment and the joy of creation or ownership. (You might be interested in the following video of the Lego Parisian restaurant, which I am coveting. Funny way to say Parisian!)

Many models perform useful functions. My husband works as a land-surveyor, and his work involves making models on paper or in the computer, of phenomenon on the land, and making sure that specified marks on the model correspond to the marks placed in the ground. The purpose of the model relates to ownership and making sure the sewers run in the right direction. (As a result of several years of earthquakes in Christchurch, his models are less deterministic than they used to be, and unfortunately many of our sewers ended up running the wrong way.)

Our world is full of models:

  • a map is a model of a location, which can help us get from place to place.
  • sheet music is a written model of the sound which can make a song
  • a bus timetable is a model of where buses should appear
  • a company’s financial reports are a model of one aspect of the company

Deterministic models

A deterministic model assumes certainty in all aspects. Examples of deterministic models are timetables, pricing structures, a linear programming model, the economic order quantity model, maps, accounting.

Probabilistic or stochastic models

Most models really should be stochastic or probabilistic rather than deterministic, but this is often too complicated to implement. Representing uncertainty is fraught. Some more common stochastic models are queueing models, markov chains, and most simulations.

For example when planning a school formal, there are some elements of the model that are deterministic and some that are probabilistic. The cost to hire the venue is deterministic, but the number of students who will come is probabilistic. A GPS unit uses a deterministic model to decide on the most suitable route and gives a predicted arrival time. However we know that the actual arrival time is contingent upon all sorts of aspects including road, driver, traffic and weather conditions.

Model as a way of thinking about something

The term “model” is also used to describe the way that people make sense out of their world. Some people have a more deterministic world model than others, contributed to by age, culture, religion, life experience and education. People ascribe meaning to anything from star patterns, tea leaves and moon phases to ease in finding a parking spot and not being in a certain place when a coconut falls. This is a way of turning a probabilistic world into a more deterministic and more meaningful world. Some people are happy with a probabilistic world, where things really do have a high degree of randomness. But often we are less happy when the randomness goes against us. (I find it interesting that farmers hit with bad fortune such as a snowfall or drought are happy to ask for government help, yet when there is a bumper crop, I don’t see them offering to give back some of their windfall voluntarily.)

Let us say the All Blacks win a rugby game against Australia. There are several ways we can draw meaning from this. If we are of a deterministic frame of mind, we might say that the All Blacks won because they are the best rugby team in the world.  We have assigned cause and effect to the outcome. Or we could take a more probabilistic view of it, deciding that the probability that they would win was about 70%, and that on the day they were fortunate.  Or, if we were Australian, we might say that the Australian team was far better and it was just a 1 in 100 chance that the All Blacks would win.

I developed the following scenarios for discussion in a classroom. The students can put them in order or categories according to their own criteria. After discussing their results, we could then talk about a deterministic and a probabilistic meaning for each of the scenarios.

  1. The All Blacks won the Rugby World Cup.
  2. Eri did better on a test after getting tuition.
  3. Holly was diagnosed with cancer, had a religious experience and the cancer was gone.
  4. A pet was given a homeopathic remedy and got better.
  5. Bill won $20 million in Lotto.
  6. You got five out of five right in a true/false quiz.

The regular mathematics teacher is now a long way from his or her comfort zone. The numbers have gone, along with the red tick, and there are no correct answers. This is an important aspect of understanding probability – that many things are the result of randomness. But with this idea we are pulling mathematics teachers into unfamiliar territory. Social studies, science and English teachers have had to deal with the murky area of feelings, values and ethics forever.  In terms of preparing students for a random world, I think it is territory worth spending some time in. And it might just help them find mathematics/statistics relevant!

On-line learning and teaching resources

Twenty-first century Junior Woodchuck Guidebook

I grew up reading Donald Duck comics. I love the Junior Woodchucks, and their Junior Woodchuck Guidebook. The Guidebook is a small paperback book, containing information on every conceivable subject, including geography, mythology, history, literature and the Rubaiyat of Omar Khayyam.  In our family, when we want to know something or check some piece of information, we talk about consulting the Junior Woodchuck Guidebook. (Imagine my joy when I discovered that a woodchuck is another name for a groundhog, the star of my favourite movie!) What we are referring to is the internet, the source of all possible information! Thanks to search engines, there is very little we cannot find out on the internet. And very big thanks to Wikipedia, to which I make an annual financial contribution, as should all who use it and can afford to.

You can learn just about anything on the internet. Problem is, how do you know what is good? And how do you help students find good stuff? And how do you use the internet wisely? And how can it help us as learners and teachers of statistics and operations research? These questions will take more than my usual 1000 words, so I will break it up a bit. This post is about the ways the internet can help in teaching and learning. In a later post I will talk about evaluating resources, and in particular multimedia resources.

Context

Both the disciplines in which I am interested, statistics and operations research, apply mathematical and analytic methods to real-world problems. In statistics we are generally trying to find things out, and in operations research we are trying to make them better. Either way, the context is important. The internet enables students to find background knowledge regarding the context of the data or problem they are dealing with. It also enables instructors to write assessments and exercises that have a degree of veracity to them even if the actual raw data proves elusive. How I wish people would publish standard deviations as well as means when reporting results!

Data

Which brings us to the second use for on-line resources. Real problems with real data are much more meaningful for students, and totally possible now that we don’t need to calculate anything by hand. Sadly, it is more difficult than first appears to find good quality raw data to analyse, but there is some available. You can see some sources in a previous post and the helpful comments.

Explanations

If you are struggling to understand a concept, or to know how to teach or explain it, do a web search. I have found some great explanations, and diagrams especially, that have helped me. Or I have discovered a dearth of good diagrams, which has prompted me to make my own.

Video

Videos can help with background knowledge, with explanations, and with inspiring students with the worth of the discipline. The problem with videos is that it takes a long time to find good ones and weed out the others. One suggestion is to enlist the help of your students. They can each watch two or three videos and decide which are the most helpful. The teacher then watches the most popular ones to check for pedagogical value. It is great when you find a site that you can trust, but even then you can’t guarantee the approach will be compatible with your own.

Social support

I particularly love Twitter, from which I get connection with other teachers and learners, and ideas and links to blogs. I belong to a Facebook group for teachers of statistics in New Zealand, and another Facebook group called “I love Operations Research”. These wax and wane in activity, and can be very helpful at times. Students and teachers can gain a lot from social networking.

Software

There is good open-source software available, and 30-day trial versions for other software. Many schools in New Zealand use the R-based iNZight collection of programmes, which provide purpose-built means for timeseries analysis, bootstrapping and line fitting.

Answers to questions

The other day I lost the volume control off my toolbar. (Windows Vista, I’m embarrassed to admit). So I put in the search box “Lost my volume control” and was directed to a YouTube video that took me step-by-step through the convoluted process of reinstating my volume control! I was so grateful I made a donation. Just about any computer related question can be answered through a search.

Interactive demonstrations

I love these. There are two sites I have found great:

The National Library of Virtual Manipulatives, based in Utah.

NRich – It has some great ideas in the senior statistics area. From the UK.

A problem with some of these is the use of Flash, which does not play on all devices. And again – how do we decide if they are any good or not?

On-line textbooks

Why would you buy a textbook when you can get one on-line. I routinely directed my second-year statistical methods for business students to “Concepts and Applications of Inferential Statistics”. I’ve found it just the right level. Another source is Stattrek. I particularly like their short explanations of the different probability distributions.

Practice quizzes

There aren’t too many practice quizzes  around for free. Obviously, as a provider of statistical learning materials, I believe quizzes and exercises have merit for practice with immediate and focussed feedback. However, it can be very time-consuming to evaluate practice quizzes, and some just aren’t very good. On the other hand, some may argue that any time students spend learning is better than none.

Live help

There are some places that provide live, or slightly delayed help for students. I got hooked into a very fun site where you earned points by helping students. Sadly I can’t find it now, but as I was looking I found vast numbers of on-line help sites, often associated with public libraries. And there are commercial sites that provide some free help as an intro to their services. In New Zealand there is the StudyIt service, which helps students preparing for assessments in the senior high school years. At StatsLC we provide on-line help as part of our resources, and will be looking to develop this further. From time to time I get questions as a result of my YouTube videos, and enjoy answering them ,unless I am obviously doing someone’s homework! I also discovered “ShowMe” which looks like a great little iPad app, that I can use to help people more.

This has just been a quick guide to how useful the internet can be in teaching and learning. Next week I will address issues of quality and equity.

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.

Conceptualising Probability

The problem with probability is that it doesn’t really exist. Certainly it never exists in the past.

Probability is an invention we use to communicate our thoughts about how likely something is to happen. We have collectively agreed that 1 is a certain event and 0 is impossible. 0.5 means that there is just as much chance of something happening as not. We have some shared perception that 0.9 means that something is much more likely to happen than to not happen. Probability is also useful for when we want to do some calculations about something that isn’t certain. Often it is too hard to incorporate all uncertainty, so we assume certainty and put in some allowance for error.

Sometimes probability is used for things that happen over and over again, and in that case we feel we can check to see if our predication about how likely something is to happen was correct. The problem here is that we actually need things to happen a really big lot of times under the same circumstances in order to assess if we were correct. But when we are talking about the probability of a single event, that either will or won’t happen, we can’t test out if we were right or not afterwards, because by that time it either did or didn’t happen. The probability no longer exists.

Thus to say that there is a “true” probability somewhere in existence is rather contrived. The truth is that it either will happen or it won’t. The only way to know a true probability would be if this one event were to happen over and over and over, in the wonderful fiction of parallel universes. We could then count how many times it would turn out one way rather than another. At which point the universes would diverge!

However, for the interests of teaching about probability, there is the construct that there exists a “true probability” that something will happen.

Why think about probability?

What prompted these musings about probability was exploring the new NZ curriculum and companion documents, the Senior Secondary Guide and nzmaths.co.nz.

In Level 8 (last year of secondary school) of the senior secondary guide it says, “Selects and uses an appropriate distribution to solve a problem, demonstrating understanding of the relationship between true probability (unknown and unique to the situation), model estimates (theoretical probability) and experimental estimates.”

And at NZC level 3 (years 5 and 6 at Primary school!) in the Key ideas in Probability it talks about “Good Model, No Model and Poor Model” This statement is referred to at all levels above level 3 as well.

I decided I needed to make sense of these two conceptual frameworks: true-model-experimental and good-poor-no, and tie it to my previous conceptual framework of classical-frequency-subjective.

Here goes!

Delicious Mandarins

Let’s make this a little more concrete with an example. We need a one-off event. What is the probability that the next mandarin I eat will be delicious? It is currently mandarin season in New Zealand, and there is nothing better than a good mandarin, with the desired combination of sweet and sour, and with plenty of juice and a good texture. But, being a natural product, there is a high level of variability in the quality of mandarins, especially when they may have parted company with the tree some time ago.

There are two possible outcomes for my future event. The mandarin will be delicious or it will not. I will decide when I eat it. Some may say that there is actually a continuum of deliciousness, but for now this is not the case. I have an internal idea of deliciousness and I will know. I think back to my previous experience with mandarins. I think about a quarter are horrible, a half are nice enough and about a quarter are delicious (using the Dr Nic scale of mandarin grading). If the mandarin I eat next belongs to the same population as the ones in my memory, then I can predict that there is a 25% probability that the mandarin will be delicious.

The NZ curriculum talks about “true” probability which implies that any value I give to the probability is only a model. It may be a model based on empirical or experimental evidence. It can be based on theoretical probabilities from vast amounts of evidence, which has given us the normal distribution. The value may be only a number dredged up from my soul, which expresses the inner feeling of how likely it is that the mandarin will be delicious, based on several decades of experience in mandarin consumption.

More examples

Let us look at some more examples:

What is the probability that:

  • I will hear a bird on the way to work?
  • the flight home will be safe?
  • it will be raining when I get to Christchurch?
  • I will get a raisin in my first spoonful of muesli?
  • I will get at least one raisin in half of my spoonfuls of muesli?
  • the shower in my hotel room will be enjoyable?
  • I will get a rare Lego ® minifigure next time I buy one?

All of these events are probabilistic and have varying degrees of certainty and varying degrees of ease of modelling.

Easy to model Hard to model
Unlikely Get a rare Lego ® minifigure Raining in Christchurch
No idea Raisin in half my spoonfuls Enjoyable shower
Likely Raisin in first spoonful Bird, safe flight home

And as I construct this table I realise also that there are varying degrees of importance. Except for the flight home, none of those examples matter. I am hoping that a safe flight home has a probability extremely close to 1. I realise that there is a possibility of an incident. And it is difficult to model. But people have modelled air safety and the universal conclusion is that it is safer than driving. So I will take the probability and fly.

Conceptual Frameworks

How do we explain the different ways that probability has been described? I will now examine the three conceptual frameworks I introduced earlier, starting with the easiest.

Traditional categorisation

This is found in some form in many elementary college statistics text books. The traditional framework has three categories –classical or “a priori”, frequency or historical, and subjective.

Classical or “a priori” – I had thought of this as being “true” probability. To me, if there are three red and three white Lego® blocks in a bag and I take one out without looking, there is a 50% chance that I will get a red one. End of story. How could it be wrong? This definition is the mathematically interesting aspect of probability. It is elegant and has cool formulas and you can make up all sorts of fun examples using it. And it is the basis of gambling.

Frequency or historical – we draw on long term results of similar trials to gain information. For example we look at the rate of germination of a certain kind of seed by experiment, and that becomes a good approximation of the likelihood that any one future seed will germinate. And it also gives us a good estimate of what proportion of seeds in the future will germinate.

Subjective – We guess! We draw on our experience of previous similar events and we take a stab at it. This is not seen as a particularly good way to come up with a probability, but when we are talking about one off events, it is impossible to assess in retrospect how good the subjective probability estimate was. There is considerable research in the field of psychology about the human ability or lack thereof to attribute subjective probabilities to events.

In teaching the three part categorisation of sources of probability I had problems with the probability of rain. Where does that fit in the three categories? It uses previous experimental data to build a model, and current data to put into the model, and then a probability is produced. I decided that there is a fourth category, that I called “modelled”. But really that isn’t correct, as they are all models.

NZ curriculum terminology

So where does this all fit in the New Zealand curriculum pronouncements about probability? There are two conceptual frameworks that are used in the document, each with three categories as follows:

True, modelled, experimental

In this framework we start with the supposition that there exists somewhere in the universe a true probability distribution. We cannot know this. Our expressions of probability are only guesses at what this might be. There are two approaches we can take to estimate this “truth”. These two approaches are not independent of each other, but often intertwined.

One is a model estimate, based on theory, such as that the probability of a single outcome is the number of equally likely ways that it can occur over the number of possible outcomes. This accounts for the probability of a red brick as opposed to a white brick, drawn at random. Another example of a modelled estimate is the use of distributions such as the binomial or normal.

In addition there is the category of experimental estimate, in which we use data to draw conclusions about what it likely to happen. This is equivalent to the frequency or historical category above. Often modelled distributions use data from an experiment also. And experimental probability relies on models as well.  The main idea is that neither the modelled nor the experimental estimate of the “true” probability distribution is the true distribution, but rather a model of some sort.

Good model, poor model, no model

The other conceptual framework stated in the NZ curriculum is that of good model, poor model and no model, which relates to fitness for purpose. When it is important to have a “correct” estimate of a probability such as for building safety, gambling machines, and life insurance, then we would put effort into getting as good a model as possible. Conversely, sometimes little effort is required. Classical models are very good models, often of trivial examples such as dice games and coin tossing. Frequency models aka experimental models may or may not be good models, depending on how many observations are included, and how much the future is similar to the past. For example, a model of sales of slide rules developed before the invention of the pocket calculator will be a poor model for current sales. The ground rules have changed. And a model built on data from five observations of is unlikely to be a good model. A poor model is not fit for purpose and requires development, unless the stakes are so low that we don’t care, or the cost of better fitting is greater than the reward.

I have problems with the concept of “no model”. I presume that is the starting point, from which we develop a model or do not develop a model if it really doesn’t matter. In my examples above I include the probability that I will hear a bird on the way to work. This is not important, but rather an idle musing. I suspect I probably will hear a bird, so long as I walk and listen. But if it rains, I may not. As I am writing this in a hotel in an unfamiliar area I have no experience on which to draw. I think this comes pretty close to “no model”. I will take a guess and say the probability is 0.8. I’m pretty sure that I will hear a bird. Of course, now that I have said this, I will listen carefully, as I would feel vindicated if I hear a bird. But if I do not hear a bird, was my estimate of the probability wrong? No – I could assume that I just happened to be in the 0.2 area of my prediction. But coming back to the “no model” concept – there is now a model. I have allocated the probability of 0.8 to the likelihood of hearing a bird. This is a model. I don’t even know if it is a good model or a poor model. I will not be walking to work this way again, so I cannot even test it out for the future, and besides, my model was only for this one day, not for all days of walking to work.

So there you have it – my totally unscholarly musings on the different categorisations of probability.

What are the implications for teaching?

We need to try not to perpetuate the idea that probability is the truth. But at the same time we do not wish to make students think that probability is without merit. Probability is a very useful, and at times highly precise way of modelling and understanding the vagaries of the universe. The more teachers can use language that implies modelling rather than rules, the better. It is common, but not strictly correct to say, “This process follows a normal distribution”. As Einstein famously and enigmatically said, “God does not play dice”. Neither does God or nature use normal distribution values to determine the outcomes of natural processes. It is better to say, “this process is usefully modelled by the normal distribution.”

We can have learning experiences that help students to appreciate certainty and uncertainty and the modelling of probabilities that are not equi-probable. Thanks to the overuse of dice and coins, it is too common for people to assess things as having equal probabilities. And students need to use experiments.  First they need to appreciate that it can take a large number of observations before we can be happy that it is a “good” model. Secondly they need to use experiments to attempt to model an otherwise unknown probability distribution. What fun can be had in such a class!

But, oh mathematical ones, do not despair – the rules are still the same, it’s just the vigour with which we state them that has changed.

Comment away!

Post Script

In case anyone is interested, here are the outcomes which now have a probability of 1, as they have already occurred.

  • I will hear a bird on the way to work? Almost the minute I walked out the door!
  • the flight home will be safe? Inasmuch as I am in one piece, it was safe.
  • it will be raining when I get to Christchurch? No it wasn’t
  • I will get a raisin in my first spoonful of muesli? I did
  • I will get at least one raisin in half of my spoonfuls of muesli? I couldn’t be bothered counting.
  • the shower in my hotel room will be enjoyable? It was okay.
  • I will get a rare Lego minifigure next time I buy one? Still in the future!

Why learning objectives are so important

Update in 2017

This is one of the most popular posts on this blog. You may also be interested in a case study of what happens when students do not get learning objectives: Why people hate statistics.

Original post:
The most useful thing I learned in my teacher training at Auckland College of Education in 1985 was to write learning objectives. Not many years, and two babies later, I began lecturing at the University of Canterbury in Management Science/Operations Research. I was the only academic staff member to have formal teacher-training. My first task, when put in charge of MSCI210, Statistical Methods for Management, was to write learning objectives. This was revolutionary, but the idea infiltrated through other courses over the years.

A learning objective states specifically what a student should be able to do.

Here are some examples of good learning objectives:

Students will be able to:

  • Identify different levels of data in new scenarios.
  • Explain in context a confidence interval.
  • Determine which probability distribution out of binomial, poisson or normal is most appropriate to model in an unfamiliar situation.
  • Compare two time series models of the same data and evaluate which is more appropriate in a given context.

Learning objectives need to be specific and measurable.

Here are some things that people might think are learning objectives, but are not:

  • Students will understand the central limit theorem. (The term “understand” is not measurable)
  • Students will learn about probability trees (“learn” is not measurable, and does not specify the level. Do students need to be able to interpret or create probability trees?)

There are vast numbers of resources on learning objectives online.

Here is one I liked, with Bloom’s taxonomy of levels of learning. These are higher and lower levels of learning objectives, ranging from being able to state principles, through to synthesis and evaluation.

http://teachonline.asu.edu/2012/07/writing-measurable-learning-objectives/

And here are some useful verbs to use when writing learning objectives;

http://www.schreyerinstitute.psu.edu/pdf/SampleVerbs_for_LearningObjectives.pdf

It is not difficult to find material on developing learning objectives.

Not just learning objectives

A course is more than the set of its learning objectives. The learning objectives specify the skills, but there are also attitudes and knowledge to be considered. The starting point for course design is the attitudes. What do we want the students to feel about the topics? What changes do we wish them to contemplate in their thinking? Then the skills and knowledge are specified, often starting at a quite general level, then working down to specifics.

For example, we might wish to teach about confidence intervals. We need to determine whether students need to be able to calculate them, interpret them, estimate or derive them.  We need to decide which confidence intervals we are interested in – for means alone, or proportions and slopes as well? Sometimes I find there are concepts I wish to include in the learning objectives, but they don’t really work as objectives. These I put as “important concepts and principles”.

I have put an example of learning objectives and concepts and principles at the end of this post.

Learning objectives tell students what is important

Without learning objectives it is difficult for students to know what they are supposed to be learning. In a lecture, a teacher can talk extensively about a case, but unless she states explicitly, it can be difficult for the students to know where to direct their attention. Do they need to know the details of that specific case or what principles are they supposed to glean from the example? Or was it just a “war-story” to entertain the troops? Students can waste a great deal of time studying things that are not necessary, to the detriment of their learning as a whole. The uncertainty also causes unnecessary anxiety.

Learning objectives enable good assessment development

Each year as we wrote our assessments we would go through the learning objectives and make sure they were assessed.  This way the assessment was fair and applied to the course. If we found it difficult to write a question to assess a learning objective we would think again about the learning objective, and what it is we really want the students to be able to do. It made it easier to write fair, comprehensive assessments.

Learning objectives encourage reflection and good course design and development

As instructors write and review the learning objectives in a course, they can identify the level of learning that is specified in each. At an entry-level course, it is acceptable to have a number of lower level learning objectives. However, there needs to be some serious thinking done if a post-graduate course is not mainly made up of higher level learning objectives. I have seen tests in stage 2 and 3 papers that tested mainly recall and common-sense. It was evident that the instructor had not thought clearly about the level of learning that was expected.

Sometimes we find we are assessing things we have not specifically taught the students. The use of learning objectives, linked with assessment design, helps us to identify the background knowledge that we assume students have. One colleague was frustrated that the students did not seem able to apply the statistical results to a managerial context. However, nowhere had she specified that students would be required to do so, and nowhere had she actually taught students how to do this. She also assumed a level of understanding of business,  that was probably not appropriate in undergraduate students.

Like it or not, assessment drives learning

I spoke recently to a maths advisor who informed me that teachers should be teaching to the curriculum not to the assessments. I felt he was idealistic, and told him so. My experience is that university students will learn what is assessed, and nothing else. I don’t know at what age this begins, but I suspect National Testing, the bane of good education, has lowered the age considerably. How wonderful it would be if our students learned for the sheer joy of learning! Where there are assessments looming, I fear this is unlikely.

When we write exams we are also writing learning materials for future students. One of the most common ways to prepare for an assessment is to do exercises from previous assessments. So when we feel that students were not really coming to grips with a concept, we include questions in the assessment, that can then be used by future students for review.

Information promotes equity and reduces unnecessary stress

The use of learning objectives can help reduce the “gaming” aspects that can proliferate in the absence of clear information. This is apparent at present in the world of Year 13 Statistics in New Zealand. The information regarding the external standards for 2013 is still sketchy (1 July 2013). The exams are written by external examiners and will take place in November of this year. However there is still only vague and sometimes incorrect information as to exactly what may or may not be included in the exams. Because of this, teachers are trying to detect, from what is or isn’t in the formula sheet and the (not totally correct) exemplars what might be in the finals, and what to include in the school practice exams.  I suspect that some teachers or areas have more information than others. The way to make this fairer is to specify what is included in the material that may be included, as learning objectives. Let us hope that some clarity comes soon, for the sake of the teachers and the students.

Your learning

So what were the learning objectives for this post?

As a result of reading this post, readers will

  • Reflect on their methods of course development and assessment with respect to using learning objectives.
  • find further resources on the internet regarding learning objectives.
  • Make comments on the good and bad aspects of this post! (oops – I didn’t teach that one)

Coda – Example of some learning objectives

Here is a set of learning objectives for the final section of a service course in quantitative methods for management. It is based on Excel and traditional (normal-based) methods of statistical analysis.  They are far from perfect, including several ideas in many of them.

Evidence Section Learning Objectives

E1.   Explain the process underlying hypothesis tests.
E2.   Interpret a p-value in context for a given set of hypotheses.
E3.   Formulate a null and alternative hypothesis in words for problems involving means, proportions, differences of two means and differences of two proportions.
E4.   Use Excel to perform a hypothesis test on one or two means and interpret the results.
E5.   Use Excel to perform a hypothesis test on one or two proportions and interpret the results.
E6.   Use Excel and PivotTables to perform a Chi-sq test on table data, and interpret the results.
E7.   Explain the concept of Type I and Type II errors and identify which (or neither) has occurred in a given situation.
E8.   Use Excel to plot bi-variate data, find the correlation; interpret the output.
E9.   Use Excel to fit a linear regression line; interpret the output.
E10. Evaluate the validity of statements about the nature of statistical thinking, including the concepts of causation, sample size, models, experimentation, statistical significance, effect size and subjectivity.
E11. Determine which test is most appropriate in a given situation, from: test for a mean or a proportion, difference between proportions, difference of two means: independent samples or paired data, chi-sq test for independence, regression.

 Important concepts or principles

E12. Inferential statistics uses information collected in a sample to make predictions or judgements about the population from which the data is drawn.
E13. An effect is statistically significant when there is evidence from the sample to reject the null hypothesis.
E14. The p-value for a hypothesis test of a claim about a population parameter is the probability of getting, by chance, a sample as least as extreme as the observed one if the null hypothesis is true.

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.