It is with envy that some mathematics educators in England look to our colleagues in the Netherlands where the Freudenthal institute has generated a rich, coherent research debate which has been widely implemented in schools. Realistic Mathematics Education offered the antidote to the formalism of the New Maths based on Hans Freudethal’s view that mathematics was not pre-formed. He said; “… the global structure of mathematics to be taught should be understood: it is not a rigid skeleton, but it rises and perishes with the mathematics that develops in the learning process. Is it not the same with the adult mathematician’s mathematics?” So it is very sad to hear that the Commission for Examinations in the Netherlands is considering banning graphing calculators from public examinations. What is it that a calculator does that could be damaging to mathematics developing in the learning process? A machine can do only what a machine can do. If mathematicians continue to fulfill an important role, then clearly they must be able to things that machines cannot do. In his 2001 novel, Uncle Petros and Goldbach’s Conjecture, Doxiadis’ eponymous mathematician dismisses any process a machine could do as ‘shopping maths’. That of course includes anything a computer algebra system (CAS) could do.

So, learners of Freudenthal’s mathematics should have access to the tools to do the shopping maths, to free up the thinking space to engage with real mathematics; solving problems, generating conjectures, developing proof. These are the art of mathematics, not the mechanical grind. Godfrey Hardy acted as the foil to Ramanujan’s genius, but in the ‘apology’ he makes clear how well he understood that Ramajan’s ability for finding extraordinary new relationships that only he could see, was the real mathematical gift. Getting it into a publishable state was the routine work for afterwards.

The excellent Project Euler takes as it’s premise that mathematicians will have access to a high level programming language (Python, which naturally has a powerful CAS) to engage with problems in number theory. The wonderfully named https://brilliant.org/ designed for potential International Maths Olympiad candidates has a whole section of problem solving requiring programming (and hence CAS) available.

Having a machine capable of high level mathematics available in a public examination in mathematics forces examiners to take a considered view of what the maths is that they are examining. It prevents them from asking students to replicate what machines can do and focuses their thinking on the maths that matters. The maths that Hans Freudenthal was so keen to preserve in the Netherlands, against the onslaught of formalism.

This sad situation was brought to my attention through the English translation of a response by *Erik Korthof *to an advert for the new HP Prime graphing calculator. He suggests that the absence of graphing calculators in the past allowed the construction of ‘proper exams’. The task of mathematics education should not be to make the lives of examiners easy. Clearly, asking a student to complete a mechanical task that would be simply done by a machine is very simple. To construct a question knowing that the student has access to such a machine is hard. Specifically so, because the question must demand genuine mathematical thinking and that puts great demands on examiners. In the UK, the most progressive mathematics education project (MEI) for A Level students (age 18) have just had their first cohort complete an examination module with a CAS calculator available. The result is thoughtful, highly mathematical questions of exactly the type University maths courses are excited to see. The link will take you to their answer to Erik Korthof’s question: “Is secondary education served with a Computer Algebra System?”. Clearly they answer a resounding yes and MEI are major players in the future of maths education in England.

As I’ve said elsewhere the existence of tools like HP Prime which allow access to powerful mathematical visualization and calculation tools in the classroom liberates students from the mechanical processes that prevent them thinking deeply about the mathematics. Certainly there will be many lessons where the calculators are put firmly away and students will learn and practice these mechanical processes, like drawing graphs and manipulating algebra, not only because they need to see how they work, but also to give them a better feel for the outcomes. Happily teachers are sophisticated enough to manage this. They can also find secure ways to use exam modes to ensure devices adhere to local regulations. Schools are expert in this. These logistical issues should not be used as an excuse for not allowing students the tools that professionals have access to and reducing what is called maths in schools to a collection of mechanical processes. Especially not from the birthplace of RME and the beautiful, powerful view of mathematics presented to the world by Hans Freudenthal.

As a Dutch math teacher, I feel I need to put my two cents in. I agree with your post on certain levels, but disagree on some points and I feel I need to explain the Dutch situation a bit better, maybe add some nuance. It is in fact all about “this mathematics we are teaching”.

In our education system there are four different math courses. I say courses, and not levels (which I think is the case in GB). The subjects in these math courses know as A, B, C and D differ greatly and are designed to prepare students for their future studies.

* Maths A prepares students for studies in such fields as economics and health. They receive a broad but basic education in functions and calculus, some trig and probability theory and statistics.

* Maths C is designed for the social sciences and hardly has any calculus, focussing on probability and statistics.

* Maths B is for students who will be going on to the exact sciences and they will cover a lot of hardcore math: calculus at an advanced level, trigonometry, geometry.

* Maths D is a supplement to math B and contains subjects I did not see untill my first year at university (it is for the real fanatics and it is a JOY to teach).

I should add that the subjects for these courses have been researched en mainly determined by the wishes of the Dutch universities, who were allowed to have a say in their future students knowledge of maths.

I do believe being able to solve equations algebraically is a skill worth learning. It takes a systematic way of thinking, it takes accuracy and requires dealing with a certain level of abstraction. But it is not for everyone, let’s be clear on that too! In my opinion, there should be distinction between the Math courses here. Future engineers and mathmaticians will benefit from the experience being able to solve equations algebraically, since they will encounter lots of abstract thinking and programming in their future. Also, these are the students that enjoy doing rigorous mathematics! In my Maths B classes I find that they are used to plotting graphs on their calculator but when I show them basic graph forms and how to do translations and transformations, they love it! They love being able to figure out how stuff works, the challenges of algebraic problemsolving and abstract thinking. They are as you would want future scientists to be.

Of course, the students in courses A and C are a different cup of tea. They struggle with grasping the abstraction needed to do algebra. Give them any realistic situation and their common sense will kick in, but adding an x distracts them somehow. Just today, one of my students preparing for his exams was repeating some exercises and had to find the derivative of y(t) = t^4 + e^4. He asked me why the correct answer doesn’t mention the e^4. Didn’t I teach him that “an e stays the same” in the derivative? He does not understand how e^x is different from e^4. x can be any number right? Why not 4? Moreover, and this student won’t remember, but it was the third time he asked me a similar question.

I am convinced that there are students who despite willpower and best efforts, are not able to do algebra. I often give formulas in words instead of the appropriate symbols. The symbols seem dead to them, completely meaningless. These are the students that would greatly benefit from a calculator with CAS, as it will allow them to solve problems they would otherwise be unable to do. Problems they do not need to do algebraically, given their future path. I think CAS could help these students to gain a better understanding of mathematics, a general understanding of the possibilities math offers, that is, which is basically all they need.

I see students in my classes who come from universities, saying their teachers (who are researchers at univerisities) do not understand programs such as SPSS. They have learned what to do to achieve results, they know only the steps to follow. What if the results different than expected? How to know what to expect? How to get an inkling that something may be wrong with the data? They have no idea. But what use is it to know the steps to the salsa if you don’t have any sense of rhythm or the passion that is associated of the dance? Mathematics to me is the same way; there’s mechanics which are fulfilling if you can master them completely, but the feel of mathematics, the logic and deductive reasoning is more important to me. What good is is to go through the motions, learn the mechanics, without understanding? Our children might as well be machines, if they do not get a chance to understand what is the heart and soul of mathematics.

In my experience as a teacher, there are students who are not able to reach the level of abstraction asked of them. They simply do not see it. If I have to choose between teaching these or any other students for that matter, the steps without understanding, or vice versa, I’d chose the latter in a heartbeat. And I’d like to think that any teacher who loves math would do the same.

I agree absolutely. I am often amazed when teachers let their students use calculators in lessons where mental work is being developed. I just ask them who is in charge! CAS is an excellent tool and a danger, so teachers need to take control of which tools their students can use. I find Escalante’s success inspiring and disheartening. As a classroom teacher I always failed to get to the real maths with some students, but Escalante did it with all his and he really did and then replicated it with thousands more over the ten year’s the project was really going.