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

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.

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.

Statistical software for worried students

Statistical software for worried students: Appearances matter

Let’s be honest. Most students of statistics are taking statistics because they have to. I asked my class of 100 business students who choose to take the quantitative methods course if they did not have to. Two hands went up.

Face it – statistics is necessary but not often embraced.

But actually it is worse than that. For many people statistics is the most dreaded course they are required to take. It can be the barrier to achieving their career goals as a psychologist, marketer or physician. (And it should be required for many other careers, such as journalism, law and sports commentator.)

Choice of software

Consequently, we have worried students in our statistics courses. We want them to succeed, and to do that we need to reduce their worry. One decision that will affect their engagement and success is the choice of computer package. This decision rightly causes consternation to instructors. It is telling that one of the most frequently and consistently accessed posts on this blog is Excel, SPSS, Minitab or R. It has been  viewed 55,000 times in the last five years.

The problem of which package to use is no easier to solve than it was five years ago when I wrote the post. I am helping a tertiary institution to re-develop their on-line course in statistics. This is really fun – applying all the great advice and ideas from ”
Guidelines for Assessment and Instruction in Statistics” or GAISE. They asked for advice on what statistics package to use. And I am torn.

Requirements

Here is what I want from a statistical teaching package:

• Easy to use
• Attractive to look at (See “Appearances Matter” below)
• Good instructional materials with videos etc (as this is an online course)
• Supports good pedagogy

If I’m honest I also want it to have the following characteristics:

• Guidance for students as to what is sensible
• Only the tests and options I want them to use in my course – not too many choices
• An interpretation of the output
• Data handling capabilities, including missing values
• A pop up saying “Are you sure you want to make a three dimensional pie-chart?”

Is this too much to ask?

Possibly.

Overlapping objectives

Here is the thing. There are two objectives for introductory statistics courses that partly overlap and partly conflict. We want students to

• Learn what statistics is all about
• Learn how to do statistics.

They probably should not conflict, but they require different things from your software. If all we want the students to do is perform the statistical tests, then something like Excel is not a bad choice, as they get to learn Excel as well, which could be handy for c.v. expansion and job-getting. If we are more concerned about learning what statistics is all about, then an exploratory package like Tinkerplots or iNZight could be useful.

Ideally I would like students to learn both what statistics is all about and how to do it. But most of all, I want them to feel happy about doing statistical analysis.

Appearances matter

Eye-appeal is important for overcoming fear. I am confident in mathematics, but a journal article with a page of Greek letters and mathematical symbols, makes me anxious. The Latex font makes me nervous. And an ugly logo puts me off a package. I know it is shallow. But it is a thing, and I suspect I am far from alone. Marketing people know that the choice of colour, word, placement – all sorts of superficial things effect whether a product sells. We need to sell our product, statistics, and to do that, it needs to be attractive. It may well be that the people who design software are less affected by appearance, but they are not the consumers.

Terminal or continuing?

This is important: Most of our students will never do another statistical analysis.

Think about it :

Most of our students will never do another statistical analysis.

Here are the implications: It is important for the students to learn what statistics is about, where it is needed, potential problems and good communication and critique of statistical results. It is not important for students to learn how to program or use a complex package.

Students need to experience statistical analysis, to understand the process. They may also discover the excitement of a new set of data to explore, and the anticipation of an interesting result. These students may decide to study more statistics, at which time they will need to learn to operate a more comprehensive package. They will also be motivated to do so because they have chosen to continue to learn statistics.

Excel

In my previous post I talked about Excel, SPSS, Minitab and R. I used to teach with Excel, and I know many of my past students have been grateful they learned it. But now I know better, and cannot, hand on heart recommend Excel as the main software. Students need to be able to play with the data, to look at various graphs, and get a feel for variation and structure. Excel’s graphing and data-handling capabilities, particularly with regard to missing values, are not helpful. The histograms are disastrous. Excel is useful for teaching students how to do statistics, but not what statistics is all about.

SPSS and Minitab

SPSS was a personal favourite, but it has been a while since I used it. It is fairly expensive, and chances are the students will never use it again. I’m not sure how well it does data exploration. Minitab is another nice little package. Both of these are probably overkill for an introductory statistics course.

R and R Commander

R is a useful and versatile statistical language for higher level statistical analysis and learning but it is not suitable for worried students. It is unattractive.

R Commander is a graphical user interface for R. It is free, and potentially friendlier than R. It comes with a book. I am told it is a helpful introduction to R. R Commander is also unattractive. The book was formatted in Latex. The installation guide looks daunting. That is enough to make me reluctant – and I like statistics!

The screenshot displayed on the front page of R Commander

iNZight and iNZight Lite

I have used iNZight a lot. It was developed at the University of Auckland for use in their statistics course and in New Zealand schools. The full version is free and can be installed on PC and Mac computers, though there may be issues with running it on a Mac. The iNZight lite, web-based version is fine. It is free and works on any platform. I really like how easy it is to generate various plots to explore the data. You put in the data, and the graphs appear almost instantly. IiNZIght encourages engagement with the data, rather than doing things to data.

For a face-to-face course I would choose iNZight Lite. For an online course I would be a little concerned about the level of support material available. The newer version of iNZight, and iNZight lite have benefitted from some graphic design input. I like the colours and the new logo.

Genstat

I’ve heard about Genstat for some time, as an alternative to iNZight for New Zealand schools, particularly as it does bootstrapping. So I requested an inspection copy. It has a friendly vibe. I like the dialog box suggesting the graph you might like try. It lacks the immediacy of iNZight lite. It has the multiple window thing going on, which can be tricky to navigate. I was pleased at the number of sample data sets.

NZGrapher

NZGrapher is popular in New Zealand schools. It was created by a high school teacher in his spare time, and is attractive and lean. It is free, funded by donations and advertisements. You enter a data set, and it creates a wide range of graphs. It does not have the traditional tests that you would want in an introductory statistics course, as it is aimed at the NZ school curriculum requirements.

Statcrunch

Statcrunch is a more attractive, polished package, with a wide range of supporting materials. I think this would give confidence to the students. It is specifically designed for teaching and learning and is almost conversational in approach. I have not had the opportunity to try out Statcrunch. It looks inviting, and was created by Webster West, a respected statistics educator. It is now distributed by Pearson.

Jasp

I recently had my attention drawn to this new package. It is free, well-supported and has a clean, attractive interface. It has a vibe similar to SPSS. I like the immediate response as you begin your analysis. Jasp is free, and I was able to download it easily. It is not as graphical as iNZight, but is more traditional in its approach. For a course emphasising doing statistics, I like the look of this.

Data, controls and output from Jasp

Conclusion

So there you have it. I have mentioned only a few packages, but I hope my musings have got you thinking about what to look for in a package. If I were teaching an introductory statistics course, I would use iNZight Lite, Jasp, and possibly Excel. I would use iNZight Lite for data exploration. I might use Jasp for hypothesis tests, confidence intervals and model fitting. And if possible I would teach Pivot Tables in Excel, and use it for any probability calculations.

This is a very important topic and I would appreciate input. Have I missed an important contender? What do you look for in a statistical package for an introductory statistics course? As a student, how important is it to you for the software to be attractive?

The problem with videos for teaching maths and stats

The message of many popular mathematics and statistics videos is harming people’s perceptions of the nature of these disciplines.

I acknowledge the potential for conflict of interest in this post –  critically examining the role of video in learning and teaching mathematics and statistics – when StatsLC has a YouTube channel, and also provides videos through teaching and learning systems.

But I do wonder what message it sends when people like Sal Khan of Khan Academy and Mister Woo are applauded for their well-intentioned, and successful attempts to take a procedural view of mathematics to the masses. Video by its very nature tends towards procedures, and encourages the philosophy that there is one way to do something. Both Khan and Woo, and my personal favourite, Rob Tarrou, all show enthusiasm, inclusion and compassion. And I am sure that many people have been helped by these teachers. In New Zealand various classroom teachers ‘flip” their classrooms, and allow others to benefit from their videos on YouTube. One of the strengths, according to Khan, is that individual students can proceed at their own pace. However Jo Boaler states in her book, Mathematical Mindsets, that “Sadly I have yet to encounter a product that gives individualised opportunities and also teaches mathematics well.”

So what is the problem then? Millions of students love Khan, Woo, ProfRobBob and even Dr Nic. Millions of people also love fast food, and that isn’t good as a total diet.

In my work exploring people’s attitudes to mathematics, I find that many, including maths educators, have a procedural view of mathematics, which fails to unlock the amazing potential of our disciplines.

Procedural maths

Many people have the conception that to do mathematics is to work out the correct procedure to use in a specific instance and use it correctly in order to get the correct answer. This leads to a nice red tick. (Check mark) That was my view of maths for a very long time. I remember being most upset in my first year of university when the calculus exam was in a different format from the ones I had practised on. I was indignant and feared a C at best, and possibly even a failing grade. I liked the procedural approach. I felt secure using a procedural approach, and when I became a maths teacher, I was pretty much wedded to it. And the thing is, the procedural approach has worked very well for most of the people who are currently high school maths teachers.

Computation was an important part of mathematics

I recently read the inspiring “Hidden Figures”, about African American women who had pivotal roles in the development of space travel. For many of them, their introduction into life as a mathematician was as a computer. They did mathematical computations, and speed and accuracy were essential. I wonder how much of today’s curriculum is still aiming to produce computers, when we have electronic devices that can do all of that faster and more accurately.

Open-ended, lively maths

In parallel to the mass-maths-educators, we have the likes of Jo Boaler and Youcubed, Dan Meyer and Desmos, Bobbie Hunter and Mathematics Inquiry Communities, Marian Small, Tracy Zager, Fawn Nguyen and pretty much the entire Math-Twitter-Blogosphere spreading the message that mathematics is open-ended, exciting and far from procedural. Students work in groups to construct and communicate their ideas. Wrong answers are valued as evidence of thinking and the willingness to take risks. Productive struggle is valued and lessons are designed to get students outside of their comfort zones, but still within their zone of proximal development. Work is collective, rather than individualised, and ability grouping is strongly discouraged.

I find this approach enormously exciting, and believe that it could change the perception of the world towards mathematics.

The problem of the social contract

Thus I and many teachers are keen to develop a more social constructivist approach to learning mathematics at all levels. However, teachers – especially at high school – run into the problem of the implicit social contract that places the teacher as the owner of the knowledge, who is then required to distribute said knowledge to the students in the class. Students want to get the knowledge, to master the procedure and to find the right answers with as little effort or pain as possible. They are not used to working in groups, and find it threatening to their comfortably boring, procedural vision of maths class.

Some years ago I filled in for a maths teacher for a week at a school for girls from privileged backgrounds. I upset one class of Year 12 students by refusing to use up class time getting them to copy notes from the whiteboard. I figured they had perfectly good textbooks, and were better to spend their time working on examples when I was there to help them learn. Silly me! But I was breaking with what they felt was the correct way for them (and me) to behave in maths class. In fact their indignation at my failure to behave in the way they felt I should, actually did get in the way of their learning.

So who is right?

I guess my working theory is that there is a place for many types of learning and teaching in mathematics. Videos can be helpful to introduce ideas, or to provide another way of explaining things. They can help teachers to expand their own understanding, and develop confidence. Videos can provide well-thought-out images and animations to help students understand and remember concepts. They can do something the teacher cannot.  I like to think that our StatsLC videos fit in this category. Talking head or blackboard videos can act as “the kid next door” tutor, who helps a student piece something together.

Just as candy cereal can be only “part of a healthy breakfast”, videos should never be anything more than part of a learning experience.

We also want to think about what kinds of learning we want students to experience. We need our students to be able to communicate, to be creative, to think critically and problem solve and to work collaboratively. These are known as the 4 Cs of 21st Century learning. We don’t actually need people to be able to follow procedures any more. What we need is for people to be able to ask good questions, build models and answer them. I don’t think a procedural approach is going to do that.

The following table summarises some ideas I have about ways of teaching mathematics and statistics.

 Procedural approach Social constructivist approach Main ideas Maths is about choosing and using procedures correctly Maths is about exploring ideas and finding patterns Strengths Orderly, structured, safe, cover the material, calm Exciting, fun, annoying Skills valued Computation, memorisation, speed, accuracy Creativity, collaboration, communication, critical thinking Teaching methods Demonstration, notes, practice 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. Another learner, source of help, sometimes annoyingly oblique Attitude to mistakes Mistakes are a sign of failure Mistakes happen when we learn. Challenges Boredom, regimentation, may not develop resilience. Can be difficult to tell if learning is taking place, difficult if the teacher is not confident. Who succeeds? People like our current maths teachers Not sure – hopefully everyone! Use of worksheets and textbooks Important – guide the learning Occasional use to supplement activities Role of videos Can be central Support materials

If we are to have a world of mathematicians, as is our goal as a social enterprise, then we need to move away from a narrow procedural view of mathematics.

I would love to hear your thoughts on this as mathematicians, statisticians, teachers and learners. Do we need to be more careful about the messages our resources such as textbooks and videos give about mathematics and statistics?

Dr Nic, Suzy and Gina talk about feelings about Maths

This hour long conversation gives insights into how three high achieving women feel about mathematics. Nicola, the host, is the author of this blog, and has always had strong affection for mathematics, though this has changed in nature lately. Gina and Suzy are both strongly negative in their feelings about maths. As the discussion progresses, listen for the shift in attitude.

And here is a picture of the three of us.

Dr Nic, Gina and Suzy.

Here are some of the questions we discuss over the hour:

1. Tell me about your relationship with maths.
2. How do you think your feelings about maths have affected your life?
3. If you saw this as an opportunity to talk to people who teach mathematics, what message would you like to give them?
4. How do you feel about the idea that you could change how you feel about maths?

Study elections in mathematics because it is important

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

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

Elections are mathematically and statistically interesting and relevant

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

Feedback

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

Suggestions for lessons

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

Physical resources

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

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

From NZ maths  – On the Campaign Trail (CL 4)

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

From NZ maths – Whose News (CL 4)

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

Graphical representations

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

How many people will be employed on election day?

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

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

Fairness of booth provision

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

The mathematics of MMP

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

Overhang and scenario modelling

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

Small party provisions

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

Gerrymandering

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

Statistical analysis of age and other demographics

Statistics should be analysed in response to a problem, rather than just for the sake of it.
Suggested Scenario: A new political party is planning to appeal to young voters, under 30 years of age. They wish to find out which five electorates are the best to target. You may also wish to include turn-out statistics in your analysis.

Statistical analysis of turn out

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

Class size matters to teachers

Class size is a perennial question in education. What is the ideal size for a school class? Teachers would like smaller classes, to improve learning. There is evidence of a small positive effect size due to reducing class size from meta-analysis published in John Hattie’s Visible Learning. But it makes sense, teachers argue – fewer children in the class means more opportunities for one-to-one interactions with the teacher. It makes for easier crowd control, less noise and less stress for teachers and pupils. And in these days of National Standards, it makes the assessment load more realistic.

Educational Research is difficult

I’d just like to point out that educational research is difficult. One of my favourite readings on educational research is an opinion piece by David Berliner, Educational Research: The hardest science of all,  where he explains the challenge of educational research. It was written in response to a call by the US Government for evidence-based practices in education. Berliner reminds us of how many different factors contribute to learning. And measuring learning is itself an inexact science. At one point he asks: “It may be stretching a little, but imagine that Newton’s third law worked well in both the northern and southern hemispheres—except of course in Italy or New Zealand—and that the explanatory basis for that law was different in the two hemispheres. Such complexity would drive a physicist crazy, but it is a part of the day-to-day world of the educational researcher.”

So with this in mind, I decided to ask the experts. I asked NZ primary school teachers who are just gearing up for the 2017 school year. These teachers were invited via a Facebook group to participate in a very short poll using a Google Form. There were just eight questions – the year level they teach, the minimum, maximum and ideal size for a class at that level, how many children they are expecting in their class this year and how long they have been teaching. The actual wording for the question about ideal class size was: “In your opinion what is the ideal class size that will lead to good learning outcomes for the year level given above?” There were also two open-ended questions about how they had chosen their numbers, and what factors they think contribute to the decision on class-size.

Every time I do something like this, I underestimate how long the analysis will take. There were only eight questions, thought I. How hard can that be…. sigh. But in the interests of reporting back to the teachers as quickly as possible, I will summarise the numeric data, and deal with all the words later.

Early results

There were about 200 useable responses. There was a wide range of experience within the teachers. A third of the teachers had been teaching for five years or shorter, and 20% had been teaching for more than twenty years. There was no correlation between the perceived ideal class size and the experience of the teacher.

The graph below displays the results, comparing the ideal class-size for the different year levels. Each dot represents the response of one teacher. It is clear that the teachers believe the younger classes require smaller classes. The median value for the ideal class size for a New Entrant, Year 1 and/or Year 2 class is 16. The median value for the ideal class size for Year 3/4 is 20, for Year 5/6 is 22 and for year 7/8 is 24. The ideal class size increases as the year level goes up. It is interesting that even numbers are more popular than odd numbers. In the comments, teachers point out that 24 is a very good number for splitting children into equal-sized groups.

These dotplot/boxplots from iNZight show each of the responses, and the summary values.

It is interesting to compare the maximum class size the teachers felt would lead to good learning outcomes. I also asked what class size they will be teaching this year.  The table below gives the median response for the ideal class size, maximum acceptable, and current class size. It is notable that the current class sizes are all at least two students more than the maximum acceptable values, and between six and eight students more than the ideal value.

 Median response Year Level Number of respondents Ideal class size Maximum acceptable Current New Entrant Year 1/2 56 16 20 22 Year 3/4 40 20 24.5 27.5 Year 5/6 53 22 25 30 Year 7/8 46 24 27 30

Financial considerations

It appears that most teachers will be teaching classes that are considerably larger than desired. This looks like a problem. But it is also important to get the financial context. I asked myself how much money would it take to reduce all primary school classes by four pupils (moving below the maximum, but more than the ideal)? Using figures from the Ministry of Education website, and assuming the current figures from the survey are indicative of class sizes throughout New Zealand, we would need about 3500 more classes. That is 3500 more rooms that would need to be provided, and 3500 more teachers to employ. It is an 18% increase in the number of classes. The increase in salaries alone would be over one hundred million dollars per year. This is not a trivial amount of money. It would certainly help with unemployment, but taxes would need to increase, or money would need to come from elsewhere.

Is this the best way to use the money? Should all classes be reduced or just some? How would we decide? How would it be implemented? If you decrease class sizes suddenly you create a shortage of teachers, and have to fill positions with untrained teachers, which has been shown to decrease the quality of education. Is the improvement worth the money?

My sympathies really are with classroom teachers. (If I were in charge, National Standards would be gone by lunchtime.) I know what a difference a few students in a class makes to all sorts of things. At the same time, this is not a simple problem, and the solution is far from simple. Discussion is good, and informed discussion is even better. Please feel free to comment below. (I will summarise the open-ended responses from the survey in a later post.)

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.

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.

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

Has the Numeracy Project failed?

The Numeracy Development Project has influenced the teaching of mathematics in New Zealand. It has changed the language people use to talk about mathematical understanding, introducing the terms “multiplicative thinking”, “part-whole” and “proportional reasoning” to the teacher toolkit. It has empowered some teachers to think differently about the teaching of mathematics. It has brought “number” front and centre, often crowding out algebra, geometry, measurement and statistics, which are now commonly called the strands. It has baffled a large number of parents. Has the Numeracy Development Project been a success? If not, how can we fix it?

I have been pondering about the efficacy and side-effects of the Numeracy Project in New Zealand. I have heard criticisms from Primary and Secondary teachers, and defense and explanation from advisors. I have listened to a very illuminating podcast from one of the originators of the Numeracy Project, Ian Stevens, I have had discussions with another educational developer who was there at the beginning. I even downloaded some of the “pink booklets” and began reading them, in order understand the Numeracy Project.

Then I read this article from the US organisation, National Council of Teachers of Mathematics, Strategies are not Algorithms,  and it all started to fall into place.
The authors explain that researchers analysed the way that children learn about mathematics, and the stages they generally go through. It was found that “Students who used invented strategies before they learned standard algorithms demonstrated better knowledge of base-ten number concepts and were more successful in extending their knowledge to new situations than were students who initially learned standard algorithms.” They claim that in the US “(t)he idea of “invented strategies” has been distorted to such a degree that strategies are being treated like algorithms in many textbooks and classrooms across the country.” I suspect this statement also applies in New Zealand.

Strategies taught as algorithms

Whitacre and Wessenberg refer to a paper by Carpenter et al, A Longitudinal Study of Invention and Understanding in Children’s Multidigit Addition and Subtraction. I was able to get access to read it, and found the following:
“Although we have no data regarding explicit instruction on specific invented strategies, we hypothesize that direct instruction could change the quality of children’s understanding and use of invented strategies. If these strategies were the object of direct instruction, there would be a danger that children would learn them as rote procedures in much the way that they learn standard algorithms today.” (Emphasis added)

Were they right? Are the strategies being taught as rote procedures in some New Zealand classrooms? Do we need to change the way we talk about them?

How I see the Numeracy Development Project (NDP)

The NDP started as a way to improve teacher pedagogical content knowledge to improve outcomes for students. It was intended to cover all aspects of the New Zealand Mathematics and Statistics curriculum, not just number. Ian Stevens explained: “Numeracy was never just Number. We decided that in New Zealand numeracy meant mathematics and mathematics meant numeracy.”

The Numeracy Development Project provided a model to understand progression of understanding in learning mathematics. George Box once said “All models are wrong and some models are useful.” A model of progression of understanding is useful for identifying where we are, and how to progress to where we would like to be, rather like a map. But a map is not the landscape, and children differ, circumstances change, and models in education change faster than most. I recently attended a talk by Shelley Dole, who (I think) suggested that by emphasising additive thinking in the early school years, we may undo the multiplicative and proportional thinking the students had already. If all they see is adding and subtracting, any implication towards multiplicative and proportional thinking is stifled. It is an interesting premise.
The Numeracy Project (as it is now commonly called) suggested teaching methods, strongly based around group-work and minimising the use of worksheets. Popular invented strategies for arithmetic operations were described, and the teaching of standard algorithms such as vertical alignment of numbers when adding and subtracting was de-emphasised.
An unintended outcome is that the Numeracy Project has replaced the NZ curriculum in some schools, with “Number” taking centre stage for many years. Teachers are teaching invented strategies as algorithms rather than letting students work them out for themselves. At times students are required to know all the strategies before moving on. Textbooks, worksheets and even videos based around the strategies abound, which seems anathema to the original idea.

Where now?

So where do we go from here?

To me empowerment of teachers is pivotal. Teachers need to understand and embrace the beauty of number theory, the practicality of measurement, the art and challenge of geometry, the detective possibilities in data and the power of algebra to model our world. When mathematics is seen as a way to view the world, and embedded in all our teaching, in the way literacy is, maybe then, we will see the changes we seek.

Why Journalists need to understand statistics – Sensational Listener article about midwifery risks

The recent article in the Listener highlights again the need for all citizens to  be statistically literate. In particular I believe statistical literacy should be a compulsory part of all journalists’ training. I have written before about this. I was happy to see letters to the Editor in the 22 October issue of the Listener condemning the sensationalist cover, which was not supported in the article, and even less supported in the original research. I like the Listener, and subscribe, but this was badly done!

The following was written by a fellow statistician, John Maindonald and published here with his permission.

Midwife led vs Medical led models of care

A just published major observational study, comparing midwife led with medical led models of care has attracted extensive media attention.  The front cover of the NZ Listener (October 8) presented the “results” in particularly sensationalist terms (“ALARMING MATERNITY RESEARCH …”).

http://www.listener.co.nz/archive/october-8-2016-2/

Much more alarming is what this sensationalist cover page has made of results that are at an optimistic best suggestive.

Adjustments, inevitably simplistic, were made for 8 factors in which the groups differed.  There is, with so many factors operating, no good way to be sure that the inevitably simple forms of adjustment were adequate.  Additionally, there will have been differences in mothers’ circumstances that the deprivation index used was too crude to capture.  Substance abuse was not taken into consideration.

Here are further links:

http://www.otago.ac.nz/news/news/otago622928.html

(Otago U PR)

http://journals.plos.org/plosmedicine/article?id=10.1371/journal.pmed.1002134

(the paper)

I am disappointed that in its response to criticism of its presentation in Letters to the Editor, the Listener (October 22) continues to defend its reporting.

John Maindonald.

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