By Carsten Henriksen

It is definitely useful to be able to add and subtract when checking your receipt at the grocery store or when sweating over your tax return. The mathematical skills are also crucial to our ability to understand and work with numbers, and when we need to orient ourselves in time and space. But today this ability is being challenged by the technological aids.

Do we even have something today we can call 'basic mathematical skills' and how are we to understand them and the role they are playing? The municipal primary and lower secondary school still distinguishes between basic arithmetical skills and problem-solving, but are you better off for knowing how to solve a mathematical problem than for knowing how to do basic arithmetic? Quarterly has asked Lena Lindenskov who is a professor with special responsibilities within the didactics of mathematics and natural science at the School of Education, University of Aarhus. 

She says that the demands made on mathematical skills have always been changing, and still are. For instance, the shop assistant used to be required to be able to add and subtract mentally in order to be able to give change to the customers. Today, if your purchases at the retail outlet FAKTA total DKK 63.75 and you pay with a DKK 100.00 bill, you place the bill in a tray and the machine starts calculating, adding and subtracting and finally spits out DKK 36.25 in another tray.

"The shop assistant has had nothing to do with it. But not all shops are that sophisticated. Normally, the machine calculates the amount of change due to the customer, but the shop assistant still needs to count the money, and if the customer asks if he can withdraw an extra DKK 500 from his credit card at the same time, the shop assistant needs to be able to assess whether there is enough cash in the till and count it out. So you see, we are still required to have some number skills", says Lena Lindenskov.

Actually, we need to include more skills among the basic mathematical skills today, compared with earlier, she says. For instance, the ability to calculate percentages has now become more important, as has the ability to assess information in diagrams and tables because information is conveyed as percentages or in the form of diagrams and tables in more situations and trades now than earlier.

"Today, you practically keep running into technology with built-in quantity determination, but you cannot use it if you cannot understand the various displays on the machines! So the expansion of the relevant mathematical skills is due to technology, but is also institutionally determined. In the general upper secondary school system, you would say that the solving of equations and the understanding of the concept of variables are basic mathematical skills that must be mastered at a certain level before the pupils start the upper secondary school, and today an increasing number of young people finish upper secondary school", she says.

Too posh for basic arithmetical skills

''Homespun philosophy' we say, slightly patronising people who are in to reasoning about self-evident things - tedious things. We could say the same about the calculator, except that it is not a person but a machine that has taken over the tedious calculation work. But while it can be difficult to object to the trivial work in industry being taken over by robots and machines, the case is rather more complex when it comes to calculation - because do we not need to be able to put two and two together mentally?

"We find ourselves in a brand-new situation where we are still confused as to how to feel about technology and the type of skills required to use it", says Lena Lindenskov.

"It used to be simple: you had to do it all by hand and in your head. Now we have this calculator technology which can do fractions with numerators and denominators and they are not even expensive. The previous generations of calculators could only handle decimal figures.

But many people still feel a bit embarrassed about taking out the calculator at the supermarket and I have met adults at adult vocational training and general adult education centres who were distressed about never having learned to divide. Even though they were good at solving problems and understanding relations, they needed the basic arithmetical skills, like e.g. dividing when they were out with their friends and needed to split the bill. But the teacher at the centre did not grant their request, i.a. because of the formal programme requirements. But we should have more open education programmes where people can learn basic arithmetic when they themselves feel that they need it", says Lena Lindenskov.

When the preparatory adult education programme in mathematics was discussed around year 2000, a proposal was made to rename the subject 'arithmetic'. But the Danish Ministry of Education would not approve, and the didactic teachers of the university world also have a hard time coming to terms with that name. They are worried that it will exclude conceptual understanding and spatiality.

"For many people 'arithmetic' has connotations of simple, hard and fast, so using this term does involve certain risks. But still, in our work with learning disabilities in mathematics we have chosen to use the metaphor "arithmetic gaps" and not "mathematics gaps". Among other things because we want to appeal to the parents, and there is still a generation of parents who regard arithmetic to be more useful than mathematics", says Lena Lindenskov.

If the elementary skills are not ranking as high as they used to, the reason is that they are - elementary.

"Skills are associated with mediocrity, and that is why we become reluctant to deal with the concept. If we look at those who find it difficult to learn mathematics and ask: What can they in fact learn, and what do they have to learn, the general answer is: the four basic arithmetical operations. Well, those are the skills and anybody can learn them, you think - it is merely a matter of practice. The rest, on the other hand - all the things you cannot learn, including the personal competencies - is then regarded as being more valuable. Unfortunately, that attitude prevents us from seeing why the skills are necessary, constructive and - valuable", says Lena Lindenskov.

Mathematical baggage

According to Lena Lindenskov, mastering the four basic arithmetical operations does not suffice, however. Basic mathematical skills go beyond that today, and she is of the opinion that we should have some criteria determining what everybody needs to have in their mathematical baggage.

"We have final targets, attainment targets and Joint National Targets, but we have no joint decision about what all pupils need to be able to do. I would like us to have that. And now would be a great time to start establishing such criteria, considering the labour market challenges we are faced with. From a social point of view, we need everybody to do well", she days.
 
Lena Lindenskov is of the opinion that we must aim for ensuring that everybody acquires the mathematical skills necessary to get on in everyday life and be able to get through the educational system after having completed the basic general education. But also that we are routed for lifelong learning in terms of competencies and attitudes. And what skills are we talking about then?

"For some people learning mental and written arithmetic is as easy as falling off a log, but that is not something we all need to be able to do. But we do all need to develop an elementary number sense so that we can make relatively accurate estimates and assess whether the calculator shows the correct total. And the education should provide the pupils with skills sufficient to enable them to make correct and optimum use of aids considering the needs and skills of the individual", she says.

Lena Lindenskov also points to the handling of formulae.

"You would hardly have found this important earlier, but knowing how you can work with formulae and the questions they can answer is a very important skill today because they are used in many different education programmes and in all kinds of subjects besides mathematics", she says.

Technology requires new skills

It is not only the calculator that changes the picture of the basic mathematical skills. As different technologies as the stop watch, GPS navigation equipment and the Excel spreadsheet are all potential threats against our ability to orient ourselves in time and space and understand and process numbers and data.

"The development of our sense of space has changed humongously with technology. It used to be an important skill for many people to be able to orient themselves on a map and understand it in relation to reality. Internet routeplanners change the concept of distance, as distance can now also be counted in hours and minutes, and while you need to be able to relate to the map distance between the points, they still require an elementary sense of space. For instance, you have to have a physical sense of what 300 metres is.

Now we are surrendering ourselves to the GPS navigation equipment and I sometimes worry whether a bleep and an arrow on our wrist watch or our cell phones directing us to go to the right will limit our sense of orientation in space. Wrong-way drivers are the caricature of what happens when we no longer master the skill: to orient ourselves in space", says Lena Lindenskov.

She points out that with the spreadsheet Excel anyone can produce instant bar charts, which you used to have to spend hours carefully preparing yourself. But you do not get instant understanding of their output.

"Therefore, technology requires a brand-new type of basic skills; skills which the educational system still regards with immense uncertainty, but which it is required to appraise and accommodate", says Lena Lindenskov.

The first step on the abstraction ladder

The basic mathematical skills form part of a far more fine-meshed net of competencies than indicated by the distinction made by the Danish primary and lower secondary school between basic arithmetical skills and problem-solving.

"The assignments aimed at obtaining general and basic arithmetical skills are intended to allow the pupils to practice their basic skills, but they also have a higher purpose, namely to give them intimate insight into the basic concepts in order to allow them to move towards increasingly higher levels of abstraction. Technology allows us to skip the first levels, but when we used to spend so much time doing sums in the old days, we did not just do it to find that 2+9=11, but to get a physical understanding of the numbers and form some structures in the brain allowing us later to see and understand algebra and its variables as generalisations on the concrete experiences with numbers.

This is the kind of learning, you may fear is forgotten if we do not prioritise the basic skills. How else can we make abstractions? Of course, you can describe mathematics as a formal system with rules at symbol level, but I prefer to look upon mathematics as something which is constantly generalised from lower levels and which begins with an acquired certainty about numbers and space", says Lena Lindenskov.

Mathematics as a sheet anchor

From a historical point of view, mathematics has always had a metaphysical function. From the idealism of Plato to the ideals of modern natural science, mathematics has been praised for its purifying properties. It has served a higher cognitive purpose as the impenetrable bulwark of rationality against phantasm, pipe dreams and delusions. According to Plato, the logic and non-contradictory reasoning of mathematics was intended to train the pure thought to tackle the highest truths in the liberation from the treacherous conveyance of the senses of half truths and whole lies.

With the birth of modern natural science in the 17th century, the poles switched. Mathematics came into favour as the means of converting facts and empiricism into rules and regularity. Nature is a book written in the language of mathematics and if you want to make science about it, you will need to measure, count and weigh, Galilei said.

In the bestseller written by the German author Daniel Kehlmann from 2005 'Measuring the world', the scientist Humboldt dreams, towards the end of his life, of writing a book consisting only of facts based on counts and measurements. Ultimately, the cosmos is to be calculated by means of a number of arithmetical operations. That is the only way to obtain a sheet anchor in the fickleness of life.

There is a long way from being able to add, subtract, multiply and divide to reflecting on the intellectual-historical importance of mathematics. But the mathematical skills of the average citizen actually have an educative function, Lena Lindenskov says.

"Mathematics has some powerful means of description and analytical tools which recur across different cultures and which can be used in all kinds of connections. Mathematical skills constitute a key competency compared to learning other things, e.g. using other and more than 'I feel' arguments in the democratic discussion. In reality, mathematics is the most democratic school subject there is because what is true and false is not merely defined by human authorities", says Lena Lindenskov.

About facts and numbers

"Facts, Humboldt repeated, he still had facts, he would write them all down, a vast work full of facts, every fact in the world, contained in a single book, all facts and nothing but facts, the entire cosmos all over again, but stripped of error, fantasy, dream, and fog; facts and numbers, he said in an uncertain voice, they were maybe what could save one. If he thought, for example, that they had been travelling for twenty-three weeks, that they'd covered fourteen thousand five hundred versts, visited six hundred and fifty-eight stopping points, and, he hesitated, used twelve thousand two hundred and twenty four horses, then the chaos became graspable and one felt better."

Daniel Kehlmann: Measuring the world (2005)

Facts

Lena Lindenskov, professor with special responsibilities within the didactics of mathematics and natural science at the School of Education, University of Aarhus.