Why decimals are difficult

Why decimals are difficult

Recently a couple of primary teachers admitted a little furtively to me that they “never got decimals”. It got me wondering about what was difficult about decimals. For people who “get” decimals, they are just another number, with the decimal point showing. Clearly this was not the case for all.

So in true 21st century style I Googled it: “Why are decimals difficult”

I got some wonderfully interesting results, one of which is a review paper by Hugues Lortie-Forgues, Jing Tian and Robert S. Siegler, entitled “Why is learning fraction and decimal arithmetic so difficult?”, which I draw on in this post.

You need to know

For teachers of statistics, this is important. In particular, students learning about statistics sometimes have difficulty identifying if a p-value of 0.035 is smaller or larger than the alpha value of 0.05. In this post I talk about why that may be. I will also give links to a couple of videos that might be helpful for them. For teachers of mathematics it might give some useful insights.

Whole numbers and rational numbers

Whole numbers are the numbers we start with when we begin to learn maths – 1, 2, 3, 4,… and 0. Zero has an interesting role of having no magnitude in itself, but acting as a place-filler to make sure we can tell the meaning of a number. Without zero, 2001 and 201 and 21 would all look the same! From early on we recognise that longer numbers represent larger quantities. We know that a salary with lots of zeroes is better than one with only a few. $1000000 is more than $200 even though 2 is greater than 1.

Rational numbers are the ones that come in between, but also include whole numbers. All of the following are considered rational numbers: ½, 0.3, 4/5, 34.87, 3¾, 2000

When we talk about whole numbers, we can say what number comes before and after the number. 35 comes before 36. 37 comes after 36. But with rational numbers, we cannot do this. There are infinite rational numbers in any given interval. Between 0 and 1 there are infinite rational numbers.

Rational numbers are usually expressed as fractions (½, 3¾) or decimals (0.3, 34.87).

There are several things that make rational numbers (fractions and decimals) tricky. In this post I focus on decimals

Decimal notation and size of number

As I explained before, when we learn about whole numbers, we learn a useful rule-of-thumb that longer strings of digits correspond to larger numbers. However, the length of the decimal is unrelated to its magnitude. For example, 10045 is greater than 230. The longer number corresponds to greater magnitude. But 0.10045 is less than 0.230. We look at the first digit after the point to find out which number is bigger. The way that you judge which is bigger out of two decimals is quite different from how you do it with whole numbers. The second of my videos illustrates this.

Effect of multiplying by numbers between 0 and 1

The results of multiplying by decimals between 0 and 1 are different from what we are used to.

When we learn about multiplication of whole numbers, we find that when we multiply, the answer will always be bigger than both of the numbers we are multiplying.
3 × 4 = 12. 12 is greater than either 3 or 4.
However, if we multiply 0.3 × 0.4 we get 0.12, which is smaller than either 0.3 and 0.4. Or if we multiply 6 by 0.4, we get 2.4, which is less than 6, but greater than 0.4. This can be quite confusing.

Aside for statistics teachers

In statistics we often quote the R squared value from regression. To get it, we square r, the correlation coefficient, and what is quite a respectable value, like 0.6, gets reduced to a mere 0.36.

Effect of dividing by decimals between 0 and 1

Similarly, when we divide whole numbers by whole numbers, the answer will be less than the number we are dividing. 100 / 5 = 20. Twenty is less than 100, but in this case is greater than 5.  But when we divide by a decimal between 0 and 1 it all goes crazy and things get bigger! 100/ 0.5 = 200. People who are at home with all this madness don’t notice it, but I can see how it can alarm the novice.

Decimal arithmetic doesn’t behave like regular arithmetic

Addition and subtraction

When we add or subtract two numbers, we need to line up the decimal places, so that we know that we are adding values with corresponding place values. This is looks different from the standard algorithm where we line up the right-hand side. In fact it is the same, but because the decimal point is invisible, it doesn’t seem the same.

Method for multiplication of decimals

When you multiply numbers with decimals in, you do it like regular multiplication and then you count the number of digits to the right of the decimal in each of the factors and add them together and that is how many digits to have to the right of the decimal in the answer! I have a confession here. I know how to do this, and have taught how to do this, but I don’t recall ever working out why we do this or getting students to work it out.

Method for division of decimals

Is this even a thing? My immediate response is to use a calculator. I seem to remember moving the decimal point around in a somewhat cavalier manner so that it disappears from the number we are dividing by. But who ever does long division by hand?

Okay teacher friends – I now see why you find decimals difficult.

Answers

The paper talks about approaches that help. The main one is that students need to spend time on understanding about magnitude.

My suggestion is to do plenty of work using money. Somehow we can get our heads around that.

And use a calculator, along with judicious estimation.

Here are two videos I have made, to help people get their heads around decimals.

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There are many good ways to teach mathematics

There are many good ways to teach mathematics and statistics

Hiding in the bookshelves in the University of Otago Library, I wept as I read the sentence, “There are many good ways to raise children.”  As a mother of a baby with severe disabilities the burden to get it right weighed down on me. This statement told me to put down the burden. I could do things differently from other mothers, and none of us needed to be wrong.

The same is true of teaching maths and stats – “There are many good ways to teach mathematics and statistics.” (Which is not to say that there are not also many bad ways to both parent and teach mathematics – but I like to be positive.)

My previous post about the messages about maths, sent by maths and stats videos, led to some interesting comments – thanks especially to Michael Pye who “couldn’t get the chart out of [his] head”. (Nothing warms a blogger’s heart more!). He was too generous to call my description of the “procedural approach” a “straw-person”, but might have some justification to do so.

His comments (you can see the originals here) have been incorporated in this table, with some of my own ideas. In some cases the “explicit active approach” is a mixture of the two extremes. The table was created to outline the message I felt the videos often give, and the message that is being encouraged in much of the maths education community. In this post we expand it to look at good ways to teach maths.

Procedural approach Explicit but active approach Social constructivist approach
Main ideas Maths is about choosing and using procedures correctly Maths is about understanding ideas and recognising patterns Maths is about exploring ideas and finding patterns
Strengths Orderly, structured, safe, cover the material, calm Orderly, structured, safe, cover the material, calm and satisfying Exciting, fun, annoying
Skills valued Computation, memorisation, speed, accuracy Computation, memorisation, (not speed), accuracy + the ability to evaluate and analyse Creativity, collaboration, communication, critical thinking
Teaching methods Demonstration, notes, practice Demonstration, notes, practice, guided discussion and exploration via modelling. Open-ended tasks, discussion, exploration
Grouping Students work alone or in ability grouping Students discuss as a whole class or in mixed-ability groups
Role of teacher Fount of wisdom, guide, enthusiast, coach. Fount of wisdom, guide, enthusiast, coach. Another learner, source of help, sometimes annoyingly oblique
Attitude to mistakes Mistakes are a sign of failure Mistakes happen when we learn. (high percentage of success) Mistakes happen when we learn.
Challenges Boredom, regimentation, may not develop resilience. Boredom, regimentation, could be taught purely to the test Can be difficult to tell if learning is taking place, difficult if the teacher is not confident
Who (of the learners) succeeds? People like our current maths teachers Not sure – hopefully everyone!
Use of worksheets and textbooks Important – guide the learning Develops mastery and provide assessment for learning. Limits gaps in understanding. Occasional use to supplement activities
Role of videos Can be central Reinforce ideas and provide support out of class. Support materials

Comments

We agree that speed is not important, so why are there still timed tests and “mad minutes” .

What is good mathematics teaching?

The previous post was about the messages sent by videos, and the table was used to fit the videos into a context. If we now examine the augmented table, we can address what we think good mathematics teaching looks like.

For WHOM?

The biggest question when discussing what works in education is “for whom does it work?”  Just about any method of teaching will be successful for some people, depending on how you measure success. Teachers have the challenge of meeting the needs of around thirty students who are all individuals, with individual needs.

Introversion/extraversion

I have recently been considering the scale from introvert – those who draw energy from working alone, and extraversion – those who draw energy from other people. Contrary to our desire to make everything binary, current thinking suggests that there is a continuum from totally introverted to totally extraverted. I was greatly relieved to hear that, as I have never been able to find my place at either end. I am happy to present to people, and will “work a room” if need be, thus appearing extraverted, but need to recover afterwards with time alone – thus introverted. Apparently I can now think of myself as an ambivert.

The procedural approach to teaching and learning mathematics is probably more appealing to those more at the introverted end of the spectrum, who would rather have fingernails extracted than work in a group. (And I suspect this would include a majority of incumbent maths teachers, though I am not sure about primary teachers.) I suspect that children who are more extroverted will gain from group work and community. If we choose either one of these modes of teaching exclusively we are disadvantaging one or other group.

Different cultures

In New Zealand we are finding that children from cultures where a more social approach is used for learning do better when part of learning communities that value their cultural background and group endeavour. In Japan it is expected that all children will master the material, and children are not ability-grouped into lowered expectations. Dominant white western culture is more competitive. One way for schools to encourage large numbers of phone calls from unhappy white middle-class parents is to remove “streaming”, “setting”, or “ability grouping.”

Silence and noise

I recently took part in a Twitter discussion with maths educators, one of whom believed that most maths classes should be undertaken in silence. One of the justifications was that exams will be taken in silence and individually. This may have worked for him, but for some students the pressure not to say anything is stifling. It also removes a great source of learning, their peers. Students who are embarrassed to ask a teacher for help can often get help from others. In fact some teachers require students to ask others before approaching the teacher.

Moderation

As is often the case, the answer lies in moderation and variety. I would not advocate destroying all worksheets and textbooks, nor mandate frequent silent individual work. Here are some of suggestions for effective teaching of mathematics.

Ideal maths teaching includes:

  • Having variety in your approaches, as well as security
  • Aiming for understanding and success
  • Trying new ideas and having fun
  • Embracing your own positive mathematical identity (and getting help if your mathematical identity is not positive)
  • Allowing children to work at different speeds without embarrassment
  • Having silence sometimes, and noise sometimes
  • Being competent or getting help – a good teaching method done poorly is not a good teaching method

Here are links to other posts related to this:
The Golden Rule doesn’t apply to teaching

Educating the heart with maths and statistics

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

And thank you again to those who took the time to comment on the previous post. I’m always interested in all viewpoints.

The Central Limit Theorem – with Dragons

To quote Willy Wonka, “A little magic now and then is relished by the best of men [and women].” Any frequent reader of this blog will know that I am of a pragmatic nature when it comes to using statistics. For most people the Central Limit Theorem can remain in the realms of magic. I have never taught it, though at times I have waved my hands past it.

Sometimes you don’t need to know.

Students who want that sort of thing can read about it in their textbooks or look it up online. The New Zealand school curriculum does not include it, as I explained in 2012.

But – there are many curricula and introductory statistics courses that include The Central Limit Theorem, so I have chosen to blog about it, in preparation to making a video. In this post I will cover what the Central Limit does. Maybe my approach will give ideas to teachers on how they might teach it.

Sampling distribution of a mean

First let me explain what a sampling distribution is. (And let me add the term to Dr Nic’s long list of statistics terms that cause unnecessary confusion.) A sampling distribution of a mean is the distribution of the means of samples of the same size taken from the same population. The distribution of the means will be different from the distribution of values in the original population.  The Central Limit Theorem tells us useful things about the sampling distribution and its relationship to the distribution of the values in the population.

Example using dragons

We have a population of 720 dragons, and each dragon has a strength value of 1 to 8. The distribution of the strengths goes from 1 to 8 and has a population mean somewhere around 4.5. We take a sample of four dragons from the population. (Dragons are difficult to catch and measure so it will just be 4.)

We find the mean. Then we think about what other values we might have got for samples that size. In real life, that is all we can do. But to understand what is happening, we will take multiple samples using cards, and then a spreadsheet, to explore what happens.

Important aspects of the Central Limit Theorem

Aspect 1: The sampling distribution will be less spread than the population from which it is drawn.

Dragon example

What do you think is the largest value the mean strength of the four dragons will take? Theoretically you could have a sample of four dragons, each with strength of 8, giving us a sample mean of 8. But it isn’t very likely. The chances that all four values are greater than the mean are pretty small.  (It’s about a 6% chance). If there are equal numbers of dragons with each strength value, then the probability of getting all four dragons with strength 8 is 0.0002.

So already we have worked out that the distribution of the sample means is going to be less spread than the distribution of the original population.

Aspect 2: The sampling distribution will be well-modelled by a normal distribution.

Now isn’t that amazing – and really useful! And even more amazing, it doesn’t even matter what the underlying population distribution is, the sampling distribution will still (in most cases) look like a normal distribution.

If you think about it, it does make sense. I like to see practical examples – so here is one!

Dragon example

We worked out that it was really unlikely to get a sample of four dragons with a mean strength of 8. Similarly it is really unlikely to get a sample of four dragons with a mean strength of 1.
Say we assumed that the strength of dragons was uniform – there are equal numbers of dragons with each of the strengths. Then we find out all the possible combinations of strengths from samples of 4 dragons. Bearing in mind there are eight different strengths, that gives us 8 to the power of 4 or 4096 possible combinations. We can use a spreadsheet to enumerate all these equally likely combinations. Then we find the mean strength and we get this distribution.

Or we could take some samples of four dragons and see what happens. We can do this with our cards, or with a handy spreadsheet, and here is what we get.

Four samples of four dragons each

The sample mean values are 4.25, 5.25, 4.75 and 6. Even with really small samples we can see that the values of the means are clustering around some central point.

Here is what the means of 1000 samples of size 4 look like:

And hey presto – it resembles a normal distribution! By that I mean that the distribution is symmetric, with a bulge in the middle and tails in either direction. A normal distribution is useful for modelling just about anything that is the result of a large number of change effects.

The bigger the sample size and the more samples we take, the more the distribution of the means (the sampling distribution) looks like a normal distribution. The Central Limit Theorem gives mathematical explanation for this. I put this in the “magic” category unless you are planning to become a theoretical statistician.

Aspect 3: The spread of the sampling distribution is related to the spread of the population.

If you think about it, this also makes sense. If there is very little variation in the population, then the sample means will all be about the same.  On the other hand, if the population is really spread out, then the sample means will be more spread out too.

Dragon example

Say the strengths of the dragons occur equally from 1 to 5 instead of from 1 to 8. The spread of the means of teams of four dragons are going to go from 1 to 5 also, though most of the values will be near the middle.

Aspect 4: Bigger samples lead to a smaller spread in the sampling distribution.

As we increase the size of the sample, the means become less varied. We reduce the effect of one extreme value. Similarly the chance of getting all high values in our sample or all low values gets smaller and smaller. Consequently the spread of the sample means will decrease. However, the reduction is not linear. By that I mean that the effect achieved by adding one more to the sample decreases, depending on how big the sample is in the first place. Say you have a sample of size n = 4, and you increase it to n = 5, that is a 25% increase in information. If you have a sample n = 100 and increase it to size n=101, that is only a 1% increase in information.

Now here is the coolest thing! The spread of the sampling distribution is the standard deviation of the population, divided by the square root of the sample size. As we do not know the standard deviation of the population (σ), we use the standard deviation of the sample (s) to approximate it. The spread of the sampling distribution is usually called the standard error, or s.e.

 

Implications of the Central Limit Theorem

The properties listed above underpin most traditional statistical inference. When we find a confidence interval of a mean, we use the standard error in the formula. If we used the sample standard deviation we would be finding the values between which most of the values in the sample lie. By using the standard error, we are finding the values between which most of the sample means lie.

Sample size

The Central Limit Theorem applies best with large samples. A rule of thumb is that the sample should be 30 or more. For smaller samples we need to use the t distribution rather than the normal distribution in our testing or confidence intervals. If the sample is very small, such as less than 15, then we can still use the t-distribution if the underlying population has a normal shape. If the underlying population is not normal, and the sample is small, then other methods, such as resampling should be used, as the Central Limit Theorem does not hold.

Reminder!

We do not take multiple samples of the same population in real life. This simulation is just that – a pretend example to show how the Central Limit Theorem plays out. When we undergo inferential statistics we have one sample, and from that we use what we know about it to make inferences about the population from which it is drawn.

Teaching suggestion

Data cards are extremely useful tools to help understand sampling and other aspects of inference. I would suggest getting the class to take multiple small samples(n=4), using cards, and finding the means. Plot the means. Then take larger samples (n=9) and similarly plot the means. Compare the shape and spread of the distributions of the means.

The Dragonistics data cards used in this post can be purchased at The StatsLC shop.

Why people hate statistics

This summer/Christmas break it has been my pleasure to help a young woman who is struggling with statistics, and it has prompted me to ask people who teach postgraduate statistical methods – WTF are you doing?

Louise (name changed) is a bright, hard-working young woman, who has finished an undergraduate degree at a prestigious university and is now doing a Masters degree at a different prestigious university, which is a long way from where I live and will remain nameless. I have been working through her lecture slides, past and future and attempting to develop in her some confidence that she will survive the remainder of the course, and that statistics is in fact fathomable.

Incomprehensible courses alienating research students

After each session with Louise I have come away shaking my head and wondering what this lecturer is up to. I wonder if he/she really understands statistics or is just passing on their own confusion. And the very sad thing is that I KNOW that there are hundreds of lecturers in hundreds of similar courses around the world teaching in much the same way and alienating thousands of students every year.

And they need to stop.

Here is the approach: You have approximately eight weeks, made up of four hour sessions, in which to teach your masters students everything they could possibly need to know about statistics. So you tell them everything! You use technical terms with little explanation, and you give no indication of what is important and what is background. You dive right in with no clear purpose, and you expect them to keep up.

Choosing your level

Frequently Louise would ask me to explain something and I would pause to think. I was trying to work out how deep to go. It is like when a child asks where babies come from. They may want the full details, but they may not, and you need to decide what level of answer is most appropriate. Anyone who has seen our popular YouTube videos will be aware that I encourage conceptual understanding at best, and the equivalent of a statistics drivers licence at worst. When you have eight weeks to learn everything there is to know about statistics, up to and including multiple regression, logistic regression, GLM, factor analysis, non-parametric methods and more, I believe the most you can hope for is to be able to get the computer to run the test, and then make intelligent conclusions about the output.

There was nothing in the course about data collection, data cleaning, the concept of inference or the relationship between the model and reality. My experience is that data cleaning is one of the most challenging parts of analysis, especially for novice researchers.

Use learning objectives

And maybe one of the worst problems with Louise’s course was that there were no specific learning objectives. One of my most popular posts is on the need for learning objectives. Now I am not proposing that we slavishly tell students in each class what it is they are to learn, as that can be tedious and remove the fun from more discovery style learning. What I am saying is that it is only fair to tell the students what they are supposed to be learning. This helps them to know what in the lecture is important, and what is background. They need to know whether they need to have a passing understanding of a test, or if they need to be able to run one, or if they need to know the underlying mathematics.

Take for example, the t-test. There are many ways that the t-statistic can be used, so simply referring to a test as a t-test is misleading before you even start. And starting your teaching with the statistic is not helpful. We need to start with the need! I would call it a test for the difference of two means from two groups. And I would just talk about the t statistic in passing. I would give examples of output from various scenarios, some of which reject the null, some of which don’t and maybe even one that has a p-value of 0.049 so we can talk about that. In each case we would look at how the context affects the implications of the test result. In my learning objectives I would say: Students will be able to interpret the output of a test for the difference of two means, putting the result in context. And possibly, Students will be able to identify ways in which a test for the difference of two means violates the assumptions of a t-test. Now that wasn’t hard was it?

Like driving a car

Louise likes to understand where things come from, so we did go through an overview of how various distributions have been found to model different aspects of the world well – starting with the normal distribution, and with a quick jaunt into the Central Limit Theorem. I used my Dragonistics data cards, which were invented for teaching primary school, but actually work surprisingly well at all levels! I can’t claim that Louise understands the use of the t distribution, but I hope she now believes in it. I gave her the analogy of learning to drive – that we don’t need to know what is happening under the bonnet to be a safe driver. In fact safe driving depends more on paying attention to the road conditions and human behaviour.

Assumptions

Louise tells me that her lecturer emphasises assumptions – that the students need to examine them all, every time they look at or perform a statistical test. Now I have no problems with this later on, but students need to have some idea of where they are going and why, before being told what luggage they can and can’t take. And my experience is that assumptions are always violated. Always. As George Box put it – “All models are wrong and some models are useful.”

It did not help that the lecturer seemed a little confused about the assumption of normality. I am not one to point the finger, as this is a tricky assumption, as the Andy Field textbook pointed out. For example, we do not require the independent variables in a multiple regression to be normally distributed as the lecturer specified. This is not even possible if we are including dummy variables. What we do need to watch out for is that the residuals are approximately modelled by a normal distribution, and if not, that we do something about it.

You may have gathered that my approach to statistics is practical rather than idealistic. Why get all hot and bothered about whether you should do a parametric or non-parametric test, when the computer package does both with ease, and you just need to check if there is any difference in the result. (I can hear some purists hyperventilating at this point!) My experience is that the results seldom differ.

What post-graduate statistical methods courses should focus on

Instructors need to concentrate on the big ideas of statistics – what is inference, why we need data, how a sample is collected matters, and the relationship between a model and the reality it is modelling. I would include the concept of correlation, and its problematic link to causation. I would talk about the difference between statistical significance and usefulness, and evidence and strength of a relationship. And I would teach students how to find the right fishing lessons! If a student is critiquing a paper that uses logistical regression, that is the time they need to read up enough about logistical regression to be able to understand what they are reading.They cannot possibly learn a useful amount about all the tests or methods that they may encounter one day.

If research students are going to be doing their own research, they need more than a one semester fly-by of techniques, and would be best to get advice from a statistician BEFORE they collect the data.

Final word

So here is my take-home message:

Stop making graduate statistical methods courses so outrageously difficult by cramming them full of advanced techniques and concepts. Instead help students to understand what statistics is about, and how powerful and wonderful it can be to find out more about the world through data.

Your word

Am I right or is my preaching of the devil? Please add your comments below.

Talking in class: improving discussion in maths and stats classes

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

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

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

Prizes can help motivate students to give answers in unfamiliar settings

Prizes can help motivate students to give answers in unfamiliar settings

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

It is important to talk in maths and stats classes

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

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

Classroom atmosphere

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

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

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

Google Image is awash with classroom posters outlining “Talk moves”. I have been unable to trace back the source of the term or the list, and would be very pleased if someone can tell me the source,  to be able to attribute this structure.

Good questions

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

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

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

Multiple answers and open-ended questions

Where possible we need to ask questions that can have a number of acceptable answers. A discussion about what to do with outliers will seldom have a definitive answer, unless the answer is that it depends! Asking students to make a pictorial representation of an algebra problem can lead to interesting discussions.

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

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

Whole class and group discussion

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

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

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

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

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!

 

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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Understanding Statistical Inference

Inference is THE big idea of statistics. This is where people come unstuck. Most people can accept the use of summary descriptive statistics and graphs. They can understand why data is needed. They can see that the way a sample is taken may affect how things turn out. They often understand the need for control groups. Most statistical concepts or ideas are readily explainable. But inference is a tricky, tricky idea. Well actually – it doesn’t need to be tricky, but the way it is generally taught makes it tricky.

Procedural competence with zero understanding

I cast my mind back to my first encounter with confidence intervals and hypothesis tests. I learned how to calculate them (by hand  – yes I am that old) but had not a clue what their point was. Not a single clue. I got an A in that course. This is a common occurrence. It is possible to remain blissfully unaware of what inference is all about, while answering procedural questions in exams correctly.

But, thanks to the research and thinking of a lot of really smart and dedicated statistics teachers, we are able put a stop to that. And we must. Help us make great resourcces

We need to explicitly teach what statistical inference is. Students do not learn to understand inference by doing calculations. We need to revisit the ideas behind inference frequently. The process of hypothesis testing, is counter-intuitive and so confusing that it spills its confusion over into the concept of inference. Confidence intervals are less confusing so a better intermediate point for understanding statistical inference. But we need to start with the concept of inference.

What is statistical inference?

The idea of inference is actually not that tricky if you unbundle the concept from the application or process.

The concept of statistical inference is this –

We want to know stuff about a large group of people or things (a population). We can’t ask or test them all so we take a sample. We use what we find out from the sample to draw conclusions about the population.

That is it. Now was that so hard?

Developing understanding of statistical inference in children

I have found the paper by Makar and Rubin, presenting a “framework for thinking about informal statistical inference”, particularly helpful. In this paper they summarise studies done with children learning about inference. They suggest that “ three key principles … appeared to be essential to informal statistical inference: (1) generalization, including predictions, parameter estimates, and conclusions, that extend beyond describing the given data; (2) the use of data as evidence for those generalizations; and (3) employment of probabilistic language in describing the generalization, including informal reference to levels of certainty about the conclusions drawn.” This can be summed up as Generalisation, Data as evidence, and Probabilistic Language.

We can lead into informal inference early on in the school curriculum. The key Ideas in the NZ curriculum suggest that “ teachers should be encouraging students to read beyond the data. Eg ‘If a new student joined our class, how many children do you think would be in their family?’” In other words, though we don’t specifically use the terms population and sample, we can conversationally draw attention to what we learn from this set of data, and how that might relate to other sets of data.

Explaining directly to Adults

When teaching adults we may use a more direct approach, explaining explicitly, alongside experiential learning to understanding inference. We have just completed made a video: Understanding Inference. Within the video we have presented three basic ideas condensed from the Five Big Ideas in the very helpful book published by NCTM, “Developing Essential Understanding of Statistics, Grades 9 -12”  by Peck, Gould and Miller and Zbiek.

Ideas underlying inference

  • A sample is likely to be a good representation of the population.
  • There is an element of uncertainty as to how well the sample represents the population
  • The way the sample is taken matters.

These ideas help to provide a rationale for thinking about inference, and allow students to justify what has often been assumed or taught mathematically. In addition several memorable examples involving apples, chocolate bars and opinion polls are provided. This is available for free use on YouTube. If you wish to have access to more of our videos than are available there, do email me at n.petty@statslc.com.