# Calculus is the wrong summit of the pyramid.

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

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

He’s right, you know.

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

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

# Statistics-centric curriculum

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

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

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

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

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

## Calculus-centric curriculum |
## Statistics-centric curriculum |

Logical thinking | Communication |

Abstract thinking | Dealing with uncertainty and ambiguity |

Problem-solving | Probabilistic models |

Modelling (mainly deterministic) | Argumentation, deduction |

Proof, induction | Critical thinking |

Plotting deterministic graphs from formulas | Reading and creating tables and graphs from data |

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

## Implementation issues

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

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

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

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

## Computers make it possible

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

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

## Efficient citizens

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

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

Let’s start redesigning our pyramid.

As a statistician in New Zealand, I must react against your post. I teach statistics to University students and I am horrified. Coming from Europe, where a minimum of maths is absolutely necessary to access University, I have been astonished by the extremely low level of students in simple maths – calculus and algebra. I actually find my job of teaching statistics impossible due to the lack of 80% of student in understanding very basic mathematical concept. The mathematical abstraction is an absolute requirement for any science. Statistics is a branch of mathematics. With the proper mathematics education, statistics become very simple.

Also I would argue that in our age of computer, it is even more imporant to understand mathematics.

The law of mathematics govern the logic of computer science. For one to be at ease in a world where algorithm are in charge of everything from health care system to everyday communication, mathematics should be a common basic language shared with all.

I do not argue against the importance of statistics and the critical thinkig that comes with it. But teach the basic first! Stat comes with maturity (best taught in University in my opinion) once the student has mastered a good practice of english and maths. (I will not go on ranting more on how the extremely poor command of English of my students prevent them from doing statistics, but I could)

Hi grumpy

Thanks for commenting. I find that university academics tend to think that the school system is designed specifically to provide universities with candidates prepared and ready to go. I probably thought so at some point. However there are many more people who do not go to university than do, and there are very, very few students who will end up being statisticians of the type you are talking about. Of course those who want to be statisticians should take calculus as well as statistics at school, for all the reasons you express.

But for most people, the statistics they learn at school will be all they ever do. If we can educate the general population to a better understanding of chance, data and evidence, I think this is a worthy peak to the pyramid.

I quite agree with you in theory (i.e. stats should be understood by everyone). But in practice, the few ones that actually go to university – which we can presume are among the best high school students – are terrible in maths / calculus / algebra to a point that they are incapable of doing any stats. I am sorry to say that for rhe already stat-centered curriculum in NZ, the result that we see in University is terrible. I do not blame at all the stat teachers, but the lack of maths/english skills of my students which prevent them to get the stats. In my class (which is not satistics for statistician but an introductory class), 50% of the students go through the class without really understanding statistics, none of them will become a statistician, and only a few of them will be able to apply what they learnt.

So what I am trying to say is that if you can not get those student, at the end of the secondary curriculum to do stats because of a poor maths and english education, you should not push stat earlier in the curriculum at the expense of even more fundamental subject: Maths an English. Instead accuentating better maths / maths english at a basic level (and making them compulsory to a higher level) should lead more naturally to better citizen, who may more easily pick on the critical thinking way of stat later on

Increasing the statistical literacy of the general population is a good objective, but let’s not advance statistics at the expense of mathematics. For the sake of the statistics profession, we need to be thinking of ways to provide students with a better foundation in mathematics, not further diluting the foundation. It is a false dichotomy to say that we need to advance either statistics or calculus. Why can’t we advance both?

I’ve been hearing a fair amount of opposition to advanced placement (A) statistics in the US:

http://matloff.wordpress.com/2014/09/05/good-for-ti-good-for-schools-bad-for-kids-bad-for-stat/

I’m sympathetic to the idea that learning stat in high school may be the only place some students ever get it, but those voicing complaints (I’m not one of them) argue that it turns some students off majoring in stat. I don’t know. However, having taught for over 25 years, my evidence is that we cannot teach real “critical thinking” without a strong logic basis. The chart doesn’t say if the stat-centric plan means ousting calculus. If so, I think that’s a huge mistake. It’s bad enough that statistics has become largely a matter of computer exercises rather than understanding its theoretical foundations (which involves calculus). There’s enough superficiality in statistics nowadays without watering down the calculus and algebra components further.

Hi Mayo

Thanks for the link. My reading of it was that the problem was TI calculators and the monopoly thereof, which leads to a boring curriculum. Totally agree. The AP statistics is very 20th century and needs a good dose of GAISE. In my post I suggest that most people follow the stats-centric path, and that those who wish to be scientists, statisticians, engineers would take calculus as well. It isn’t really a big shift, particularly in NZ where statistics is already a separate subject in the final year of school.

I take issue with the terms “superficiality” and “watering down”. I think what you mean is a different focus. It seems to me that many mathematical statisticians believe that the maths is the important part, and for them it certainly is. However as George Cobb, a leading light in statistical education, said, “Mathematical understanding is not the only understanding.” I have seen instances of people who are very good mathematicians, but have never really got their hands dirty with real data, or worked with data in context. Context is THE most important thing in real life statistics. I know that there are whole textbooks of mathematical statistics without a single context in them. That is suitable for the mathematical focus.

On the other hand, being able to link the real world context with the output from a statistical package is far from superficial and requires high level critical thinking, communication skills and judgement. To me THAT is statistics and that is what is useful for all students.

Responding to:

“In my class (which is not satistics for statistician but an introductory class), 50% of the students go through the class without really understanding statistics, none of them will become a statistician, and only a few of them will be able to apply what they learnt.”

(love the “Freudian” typo)

Same here. And that was at Humboldt University, Berlin, in the 90ies.

I have come to the conclusion, that “statistics” as a way of thinking is not immanent of our logic (I am a behavioral biologist, so I am referring to inbuilt capabilities of conscious reasoning; i.e. our well-known oblivion to catch probability; and maybe the latter is a mirage, in the first place, but that is another pair of shoes). And that can be taught without ANY calculus.

Care should be taken to separate “statistics as a way of thinking” and “statistics as an applied method in mathematics”. Many years ago I enjoyed the book on “Probability and Random Processes”, noting that what I would call statistics in the calculus sense was incorporated on one and a half pages (maybe different nowadays, I saw there is a third edition). And that was with an implicit eye twinkle for “people who think that that may have meaning” (paraphrase).

So, please, don’t despair of teaching rationale reasoning to pupils at any level. And don’t despair of having students not mastering the calculus. The supposition of educating is something like “putting knowledge into containers” is, of course, misled. If you can get them to have an interest in the “way of thinking”, some of them will master the calculus in no time. For those who can’t grasp it, there are more things that matter in this world then getting an R-script right:)

Appreciating your dedication so much,

SVEN.

Thanks Sven. I love the expression, “people who think that that may have meaning”. I would apply that to using hand calculation for finding a standard deviation – a good exercise for those people who think that that may have meaning. For everyone else… hmmm.

I also really like the idea of statistics as a way of thinking.

And thanks for your encouragement.

Together with grumpy young statistician I’m often surprised by the little understanding of algebra students bring to university. I would also question NZ having “a world-class education system”, particularly with a kid going through the school system right now. Simultaneously, I accept that the role of school goes beyond producing new entrants for university and I do not think more of the same will improve the situation much.

I don’t see much of a role (if any) for calculus in stats for beginners. In fact, calculus doesn’t have much of a role with computers either. The biggest population who really need calculus is engineering students and we should not gear the whole education system for that.

I can see a statistics-centered curriculum still providing many of the skills we expect from math (logical thinking, problem-solving, etc) but with much higher emphasis on making sense of the real world. Perhaps it would also reduce the number of people “damaged” by mathematical irrelevance; those people that could never find math useful, at least enough to care about learning it. They are the same people taking stats at university because “they have to”, often with a very negative attitude towards the subject.

Software can change the way we use to teach computation, but I still find most statistical software lacking for developing understanding. When teaching at uni my emphasis is on students getting a gut feeling for the problems and then putting that together with the formulas. Software should support that process but often becomes a barrier. I’m still looking at new ways of making this connection.

I appreciate your thoughtful post.

Luis