National Reflections on the Netherlands Didactics of Mathematics

National Reflections on the Netherlands Didactics of Mathematics is a monograph derived from ICME-13.  It contains 17 chapters that originated from the Thematic Afternoon session on “European Didactic Traditions” held in Hamburg in 2016.  France, Italy, Germany, and the Netherlands all present their own unique approaches to teaching and learning mathematics in schools and in research.  This key session in mathematics didactics formed a greater awareness of Dutch mathematics education and teaching and learning in the Netherlands by looking at the Dutch domain-specific instruction theory of Realistic Mathematics Education (RME).

 

As pointed out in the Preface, all 28 authors reflect from the inside, meaning, they all describe the Dutch approach.  There are many chapters that discuss the theoretical underpinnings of the Dutch approach that started about 50 years ago and have become commonplace in the Netherlands classroom landscape, which is known as RME.  Several other chapters dive back further in time or use history in the teaching of mathematics.  These chapters focus on the changes due to the use of technology.  In addition, there are chapters that look at the relationship between Dutch mathematicians and mathematics education, a review into the process of innovation, how a particular Dutch institute has worked on various reforms, and how teacher education and testing are arranged in the Netherlands.

 

In Chapter 1: A Spotlight on Mathematics Education in the Netherlands and the Central Role of Realistic Mathematics Education by the editor, Marja Van den Heuvel-Panhuizen, provides the reader with a detailed history of the issues in mathematics education in the Netherlands and the role of RME.  The goal of the Thematic Afternoon session at ICME-13 was to examine what the four countries have in common given the diversity found in cultures, history, and in politics.  As pointed out on page 2, common characteristics that came to the forefront and can be considered the key elements of European didactics were: “a strong connection with mathematics and mathematicians, the key role of theory, the key role of design activities for learning and teaching environments, and a firm basis in empirical research” (Blum et al., 2019, p. 2).  Heuvel-Panhuizen goes on to discuss the Focus on a Particular Type of Tasks, Usefulness as a Key Concept, Common Sense and Informal Knowledge, Mathematical Content Domains Subject to Innovation, The Systemic Context of Dutch Education, The Implementation of RME, and The Context of Creating a New Approach to Mathematics Education.

 

Some of the highlights of the book that I really enjoyed were the context of a helix of a propeller used to introduce the concept of the sine function on page 17, Fig. 2.1.  For Façade Flags on page 46, students are asked to make a drawing of the flags and calculate the lengths of sides c and d, and the area.  In Chapter 10: Digital Tools in Dutch Mathematics Education: A Dialectic Relationship, Figure 10.1 presents a sample of an assignment question used on the national examination on pages 181-182.  One of the questions was to find the value of n in:

 

\( x(t)=\left( 1 +\frac{1}{n} \sin(nt) \right) \cos(t) \)

 

\(y(t)=\left(1 + \frac{1}{n} \sin(nt) \right) \sin(t) \)

 

so that the graph of \( (x(t), y(t)) \) is the ‘curved circle.’  The very next example addresses the issue of how technology can sometimes be limited in looking at graphs.  The example

 

\( f(x)=\frac{x^{2}+x-1}{x-1} \)

 

is presented along with a misleading graph, Figure 2, of the vertical asymptote.

 

Other highlights of examples include An Observation: Two Students as Surveyors presented on pages 243-250.  Here, two students, Stefan and Marco, were asked to measure the height of their school’s gym.  The importance of mental calculations is discussed on pages 91-93 using the textbook series Nieuw Rekenen.  These mental calculations provide a didactical tool to motivate number sense, insightful arithmetic, and applications.  Pages 92 gives five examples, one of which is:

Calculate:

 

\( 5 \times 98 = 5 \times 90 + 5 \times 8 = \)

 

but also 

 

\( 5 \times 100 - 5 \times 2 = \)

 

and half of 

 

\( 10 \times 98 = \)

 

Calculate in different ways.

 

\( 7 \times 98 \) \( 4 \times 98 \) \( 12 \times 25 \)

 

Another great example is on pages 274-278 where the experimental unit (Kindt, 1997) is presented with the operators of \( \Delta \) and \( \Sigma \).  For me, this was especially great to see as students have a great deal of trouble when it comes down to sequences.  Here, the theorem of Leibniz is formed as:

 

\( \sum_{k=0}^{n-1} \Delta F(k) = F(n) - F(0) \)

 

which is the discrete counterpart of:

 

\( \int_{0}^{a} dF(x) = F(a)-F(0) \)

 

The example goes on to point out that one can find the formulas for partial sums of a sequence by looking at differences and by expanding

 

\( (k+1)^{n}-k^{n}  \)

 

for \( n = 2, 3, 4.\)

 

I personally fully enjoyed reading this monograph and learning about the multitude of examples and diverse perspectives in Dutch mathematics education.  The book presents examples and history of the ever-evolving changes in technology through primary and secondary school levels.  For me, it was a very eye-opening experience to gain this international insight and to become further self-aware of various ideas of what other countries are doing in mathematics education.  Those teachers who would like to have the same enriching experience, try something different in the classroom, or have new topics of discussions in faculty or departmental meetings or focus groups,  should give this monograph consideration.

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Deborah Gochenaur is an associate professor of mathematics at Shippensburg University.