Supporting Mathematical Learning Processes by Means of Mathematics Conferences and Mathematics Language Tools
Abstract
In recent decades, the instructional theory of Realistic Mathematics Education has exerted a powerful influence on mathematics education around the world. The idea of progressive mathematisation has gained international acceptance. In this chapter, we will illustrate the way in which we benefited from the idea of organising the teaching and learning of mathematics in keeping with this guiding principle. After some personal memories of the first author, we start by describing what we consider to be the central elements of the principle of progressive mathematisation. This is followed by a description of two methods, the mathematics conferences and mathematics language tools, for rendering the learning and teaching concepts entailed by the principle of progressive mathematisation—especially its vertical component—even more expedient and fruitful. The contribution concludes with an explanation of how we understand the term ‘realistic’ in Realistic Mathematics Education.
Keywords
Progressive schematisation Mathematics conferences Language and mathematics Individual learning processes Cooperative learning Mathematics language tools13.1 The Santa Claus Problem
It must have been at the end of 1983 that the first author—at the time studying to become a primary school teacher—became aware of Adri Treffers’ paper “Fortschreitende Schematisierung – ein natürlicher Weg zur schriftlichen Multiplikation und Division im 3. und 4. Schuljahr” (Treffers, 1983). Taking the multiplication of large numbers as an example, this paper describes how students can be motivated to apply their individual approaches and develop them further in a purposeful manner. The starting question in the paper is: “Santa Claus has his gifts distributed in the village by eight helpers. Each has 23 parcels. How many parcels do they have altogether?”
Starting from problems like this one, the various solution strategies of the students are discussed, explained, and elaborated in interactions between the students or between the teacher and the students. The students can see how other students work and thereby assess the advantages and disadvantages of different strategies (Treffers, 1991). An individual student’s (mental) actions are as vitally important as his or her interaction with other students. The illustrations included in the paper are taken from a lesson where the students were not shown how to solve this type of problem based on the principles of isolating difficulties and increasing complication—as was still widespread in use in the classrooms of that time—but instead by encouraging them to develop their own approaches and then also develop them further. To put it briefly: from inventions to conventions.
Reading Treffers’ paper was a key event for the first author because he realised that the principle of progressive schematisation—or progressive mathematisation, as it should preferably be called—is by no means only important for learning written calculation algorithms, but could also be considered a comprehensive, generally applicable principle for the organisation of mathematical learning or teaching processes. In the German speaking countries, within the didactics of mathematics for primary schools, progressive mathematisation is nowadays considered a guiding principle (Krauthausen & Scherer, 2007).
The principle of progressive mathematisation has naturally also undergone specific adaptations and further developments in Germany. This chapter is meant to report on them. To do this, it starts in Sect. 13.2 with a description of what the central elements of the principle of progressive mathematisation are in our opinion. Then we describe methods for making teaching/learning processes that follow the principle of progressive mathematisation even more expedient and productive.
As conversations amongst students are often not automatically taskspecific and efficient, it is an important task for the teacher to stimulate and organise exchanges amongst the learners that will promote learning. In this respect, we describe the method of socalled ‘mathematics conferences’ in Sect. 13.3. As students occasionally find it difficult to verbalise the description and justification of mathematical facts and contexts, it is also necessary to provide them with tools for further developing their ability to express themselves in words. In this context, Sect. 13.4 describes the socalled ‘mathematics language tools’. Our chapter concludes with comments on how we understand the term ‘realistic’ in Realistic Mathematics Education (RME).
13.2 The Guiding Principle of Progressive Mathematisation
Mathematics classes and the didactical research and development dedicated to them around the world have been inspired by the conception of RME for over four decades. This development arose from the dissatisfaction with the understanding of teaching and learning predominant in the 1960s. In the Netherlands (and not only there), mathematics was reduced to its formal character in an atomised manner, leading to an overemphasis on its structuralist aspects. Teachers taught the procedures demonstratively stepbystep, whereupon the students exhibited inflexible and reproductionbased knowledge (Van den HeuvelPanhuizen & Drijvers, 2014).
The efforts revolving around RME and its influence on an international level laid one of the cornerstones of the constructivistic informed understanding of teaching and learning mathematics established nowadays internationally (see, e.g., Verschaffel, Greer, & De Corte, 2007; Wittmann, 2005).

Learning is a (re)constructive activity stimulated by concreteness; teaching involves the use of problems that can be realised by students (thus ‘realistic’ does not necessarily mean reallife).

Learning is a longterm process moving from concreteness to abstraction; teaching involves globally guiding students from their informal, contextbound strategies to formal mathematics.

Learning is facilitated by reflection on one’s own thought processes and those of others; teaching involves encouraging students to look back and to reflect on the teaching/learning process.

Learning is always embedded in a sociocultural context; thus, teaching involves opportunities for communication and cooperation as in small group work or wholeclass discussion.

Learning is the construction of knowledge and skills to a structured entity; teaching involves intertwining different learning strands.
Besides these five overarching characteristics, the principle of progressive mathematisation has two interlocking components, that is, vertical and horizontal mathematisation: “In the horizontal component the way towards mathematics is paved via model formation, schematising, symbolising. The vertical sketch is concerned with mathematical processing and level raising in the structuring of the problem field under consideration” (Treffers, 1987, p. 247).
Horizontal mathematisation is hence described as a bridge from the real world to formal symbolic mathematics, while vertical mathematisation concerns activities within the formal symbolic realm. It is meanwhile of decisive importance that a student will only be enabled to reach a higher level of mathematics (Gravemeijer & Doorman, 1999) by way of vertical mathematisation which, amongst other aspects, includes activities devoted to reorganising, economizing and linking numerical structures (see Van den HeuvelPanhuizen, 2002).
The distinction between horizontal and vertical mathematisation, also known as “twoway mathematisation” (Van den HeuvelPanhuizen, 2010, p. 3), fundamentally contributes to our understanding of the RME concept while also distinguishing it from other learning and teaching approaches (see, e.g., Streefland, 1991; Treffers, 1987; Van den HeuvelPanhuizen, 2010). There should nevertheless be no sharp distinction between horizontal and vertical mathematisation activities. That horizontal and vertical mathematisation processes can dovetail is part and parcel of the RME theory (Treffers, 1993). “The distinction between horizontal and vertical mathematizing depends on the specific situation, the person involved and his environment” (Freudenthal, 1991, p. 42).
Although RME hence attaches great value to the theoretical equality of horizontal and vertical mathematisation, there have also been phases in the development of RME where there was a tendency to focus on engaging with questions of horizontal mathematisation (see Treffers 1993; Van den HeuvelPanhuizen, 2002). Even today, the research into vertical mathematisation appears to be paid less attention in international mathematical didactics than is the case with horizontal mathematisation (see Glade & Prediger, 2017). Therefore, the focus in this chapter will be on aspects of vertical mathematisation.
Nevertheless, it should be emphasised that the principle of progressive mathematisation in its horizontal and vertical component is paid a great deal of attention in Germany, as mentioned above. Progressive mathematisation is repeatedly referred to in research and development papers published by staff of the Institute for Development and Research in Mathematics Education in Dortmund (IEEM) (Akinwunmi, 2012; Glade, 2016; Link, 2012), and also in projects such as PIKAS^{1} and KIRA,^{2} which contribute to the professionalisation of teachers. This is also where further developments and adaptations to German education take place. In this chapter, we report about two adaptations of vertical mathematisation: mathematics conferences and mathematics language tools. The examples have been taken from the PIKAS project.
13.3 Using Mathematics Conferences
13.3.1 Learning to Subtract in the Number Domain up to 1000
Based on Sundermann and Selter (2012) and as a further development of Selter (1998), a concrete implementation for subtraction in the domain up to 1000 is to be described first. The third graders involved in the PIKAS project were, from their experiences in the previous school year, already familiar with various ways of calculating in the domain up to 100. Now they were challenged in three activities of each several teaching hours to expand the number domain to 1000.
Possible calculation strategies
Pair of problems  Standard calculation  A clever calculation could be 

68 − 25 and 568 − 325  68 − 25 68 − 20 − 5 Jump strategy or 60 − 20 = 40 8 − 5 = 3 40 + 3 = 43 Split strategy  568 − 325 500 − 300 = 200, so 243 Analogy strategy 
72 − 46 and 872 − 546 95 − 32 and 795 − 432  Jump strategy or Split strategy  Analogy strategy 
61 − 26 and 761 − 226  Jump strategy or Split strategy  60 − 25 or 760 − 225 Adjustment strategy 
71 − 68 and 471 − 468 92 − 87 and 792 − 587  Jump strategy or Split strategy  68 + _ = 71 or 587 + _ = 592, +200 Determining the difference 
142 − 99 and 642 − 299 171 − 98 and 871 − 398  Jump strategy or Split strategy  142 − 100 + 1 or 871 − 400 + 2 Auxiliary problem 
The task given was: “Calculate as cleverly as possible! Write down your calculation strategies so that other children can understand them.” The students were furthermore encouraged to explain their calculation strategy and justify why they had done it this way. Finally, they were asked to give their calculation strategy a name, because this would raise their awareness of how different the various calculation strategies are, and ease their communication about the various strategies.
The students initially worked on the task on their own, with the teacher providing individual support. Then they were encouraged to form mathematics conferences, a method we will describe in greater detail below, for communicating about their calculation strategies. The results of these conferences were finally presented to the entire class, attended by a discussion of why certain strategies can be cleverer than others, depending on the numerical values.
The second activity, titled “We calculate the way other children calculated”, was aimed at sensitizing the students to the variety of possible calculation strategies. First, the students were involved in actively applying the various clever strategies of other students, and then they were asked to rate the strategies. The goal of the activity was not that children master all strategies. However, they should have the opportunity to encounter each of them.
Problems for Activity 3
Problem  Standard calculation  A clever calculation could be 

864 − 243  864 − 200 − 40 − 3 Jump strategy or 800 − 200 = 600 60 − 40 = 20 4 − 3 = 1 600 + 20 + 1 Split strategy  – 
546 − 198  Jump strategy or Split strategy  546 − 200 + 2 Auxiliary problem 
917 − 458  Jump strategy or Split strategy  – 
672 − 668  Jump strategy or Split strategy  668 + _ = 672 Determining the difference 
All the numbers (i.e., digits) of the first number are greater than those of the second number, which is why one can easily subtract the hundreds from the hundreds, the tens from the tens and the ones from the ones.
The one number is very close to the next hundred, which is why one can very easily apply the changetrick.
She indicated with arrows how she converted the 546 − 198 problem into 548 − 200.
Afterwards, the students were asked to communicate in mathematics conferences with their classmates about their calculation strategies and give reasons why they thought their calculation strategy was clever. In the end, the students wrote for one or several problems a note about a socalled ‘particularly clever strategy’. This note was meant to be discussed in class in the reflection phase. In this phase, individual students or group of students could visualise, explain and justify their results by putting their notes on the board underneath the corresponding problem.
13.3.2 TaskRelated Exchange with the Help of Mathematics Conferences
In the activity previously described, the taskrelated discussion between the students in mathematics conferences acted as a central activity for joint learning (see also, Anders & Oerter, 2009; Götze, 2007; Sundermann, 1999). A mathematics conference is understood as a meeting of students in which in small (heterogenic) groups individual solution strategies of students are presented and reflected on (see Sundermann & Selter, 1995). However, this does not mean that all teamwork can immediately be called a mathematics conference. Important is that the students are challenged to describe and justify their approach to solving a problem, to explain their discoveries, and to follow the thought processes of the other students.
This taskrelated exchange benefits all students, so the mathematics conference has a dual function. On the one hand the students who act as an author (authorstudent) benefit by verbalising their own thought processes and attempting to present them understandably. On the other hand, students who are the listeners (listenerstudent), are simultaneously actively involved as well by being asked to trace and compare the approaches of the students who describe their strategies. In this way, they can provide the authorstudents criterialed feedback.
In contrast to having only a reflection phase in a wholeclass setting, this form of cooperation in small groups steps up the verbal involvement (and engagement with the posed problem) of the individual student and also offers weaker and less communicative students a chance to speak. Of course, organising mathematics conferences will not render the reflection phase in the wholeclass setting superfluous. Discussions in wholeclass remain important. The mathematics conferences with the small groups can be a particularly good preparation for them. By having the backing of their small group quieter students will possibly be encouraged to articulate their thoughts here as well.
Mathematics conferences can already be introduced from the first year of school. The essential requirement for making this method successful is the quality of the problem that is used. The problem needs to be demanding enough and should permit various ways of thinking and solving it, so that an exchange is also meaningful from the perspective of the students.
13.3.3 Tools for Organising Mathematics Conferences
In Phase 1 (the Iphase) the students have enough time for their individual work and for describing their own solution strategy, so that they will be able to engage in an exchange about their approach afterwards. They write down their thoughts about solving the problem and then try to present them in a manner that the other students can understand. When a student has completely solved and understandably explained the given problem from his or her perspective or possibly wants to have support from the other students, he or she registers for the mathematics conference by writing down his or her name in a list that is displayed in the classroom on paper or on the blackboard. As soon as three students—as a rule—have registered, they convene for a mathematics conference. The exchange can begin as soon as the group has come together in a quiet place.

How has the authorstudent solved the problem?

Why did he or she proceed in this way?

Is the attempt at explanation by the authorstudent understandable?

Is the selected approach clever?

Who has chosen another route? What is different about it?

….
If asking these kinds of guiding questions has become a natural habit of the students, this habit can contribute to structuring the conversation and hence to students’ learning from each other. The questions can deliberately direct the students’ attention from their individual approaches to other ways of looking at things, and can stimulate a critical and constructive questioning of the solution strategies amongst the students.
Although the responsibility for a mathematics conference is largely in the hands of the students, adequate support by the teacher is nonetheless of vital importance. The teacher, in the role of moderator, can keep the conversation going and cognitively activate the students by way of targeted interventions. In this way, the teacher contributes to the constructive progress of the mathematics conference.
If the students are not yet used to discuss their solution strategies in mathematics conferences, it can be helpful for students to reflect upon this method with other students at a meta level. An advantageous way to do this is the socalled ‘fishbowl’. This means that a group of volunteers who were just about to start a mathematics conference moves to the centre of a circle of chairs. In addition to the chairs provided for the three students, there is also another empty chair with three smileys, Open image in new window , lying on it. These can be used after the conference by the observing students in the outer circle to indicate constructive (methodrelated or contentrelated) feedback (“I liked that everyone was allowed to finalise his/her speaking” or “I find that your solution strategy is not so clever because….”) and/or for giving tips (“If you also used arrows or colours in the description of your discovery packages, the other students could probably understand that better”). To ensure that discussion rules are complied, the student providing feedback is going to sit down on the empty chair.
In Phase 3 (the Wephase) the evaluation of the process and the reflection upon the results of the discussion can be finally presented in the wholeclass setting. The questions formulated in the overview poster (see Fig. 13.6) about the organisation of this presentation will urge the students to prepare it in a as structured and targetoriented manner as possible.
13.4 Learning to Describe and Explain by Using Mathematics Language Tools
This section is dedicated to describing a second adaptation of the RME approach, the socalled ‘mathematics language tools’. Before starting this description, we should make clear the importance that mathematics language has in German mathematics teaching as reflected in the German mathematics curricula. In the nationwide education standards of the Standing Conference of Ministers of Education and Cultural Affairs (KMK, 2004), start from the assumption that learning mathematics involves more than the acquisition of knowledge, such as knowing multiplication tables by heart, and skills, such as completely mastering the standard way of written addition.
13.4.1 Mathematics, More Than Calculating
The first three tasks of Variant B are also included in Variant A, but Variant B focuses on more. In Task 4 of this worksheet, the students are asked to examine the impact of the different arrangements of 3, 4 and 6 on the other numbers in the pyramid. In Task 5, they are asked to create pyramids with the target number of 20. And in Task 6, they are finally invited to freely invent number pyramids. This touches upon contentrelated as well as processrelated competences.
The development of processrelated competences is hence a central objective in mathematics education. It is, however, also observable that some students, for example, found it difficult to discover interdependencies between the bricks of a pyramid, describe these interdependencies or give reasons why the bricks are interdependent.

A learning environment should be characterised by a challenging yet easily understandable problem definition(s) to ideally render the assignments accessible for every student.

The challenging problem should lend itself to various ways of solving it (i.e., not only by means of a single approach) so that students at different levels of learning can address it in keeping with their individual skills and capabilities.

The students should be supported to be able to adequately present their discoveries orally and in writing.
13.4.2 Sums of Consecutive Natural Numbers

the assignment can be realised by students

the assignment enables learning processes moving from concreteness to abstraction

the students are continuously stimulated to reflect on their own thought processes and those of others

opportunities are created for communication and cooperation in small groups or wholeclass discussions, and

the assignment supports the construction of knowledge and skills into a structured entity.
Numbers up to 25 that can be written as the sum of consecutive natural numbers
Sum  Addition with  

2 summands  3 summands  4 summands  5 summands  6 summands  
1  
2  
3  1 + 2  
4  
5  2 + 3  
6  1 + 2 + 3  
7  3 + 4  
8  
9  4 + 5  2 + 3 + 4  
10  1 + 2 + 3 + 4  
11  5 + 6  
12  3 + 4 + 5  
13  6 + 7  
14  2 + 3 + 4 + 5  
15  7 + 8  4 + 5 + 6  1 + 2 + 3 + 4 + 5  
16  
17  8 + 9  
18  5 + 6 + 7  3 + 4 + 5 + 6  
19  9 + 10  
20  2 + 3 + 4 + 5 + 6  
21  10 + 11  6 + 7 + 8  1 + 2 + 3 + 4 + 5 + 6  
22  4 + 5 + 6 + 7  
23  11 + 12  
24  7 + 8 + 9  
25  12 + 13  3 + 4 + 5 + 6 + 7 
Strategies for finding numbers that can be written as the sum of consecutive natural numbers
Strategy  Description  Example  

1. Extension at the end  The addition is extended by the consecutive summand  3 + 4 + 5 → 3 + 4 + 5 + 6  
2. Extension at the front  The addition is extended by the previous summand  3 + 4 + 5 → 2 + 3 + 4 + 5  
3. Reduction at the end  The last summand is left out  3 + 4 + 5 → 3 + 4  
4. Reduction at the front  The first summand is left out  3 + 4 + 5 → 4 + 5  
5. Increasing all  All summands are increased by one  3 + 4 + 5 → 4 + 5 + 6  
6. Decreasing all  All summands are decreased by one  3 + 4 + 5 → 2 + 3 + 4  
7. Starting with the next natural number  The first summand is the consecutive natural number of the last summand from the previous addition  2 + 3 + 4 → e.g., 5 + 6 + 7  
8. Starting with the precursor  The last summand of the new addition is the ‘precursor’ of the previous addition  7 + 8 + 9 → 2 + 3 + 4 + 5 + 6  
9. Starting with the last summand  The first summand is same as the last summand of the previous addition  3 + 4 + 5 → e.g., 5 + 6  
10. Starting with the addition  The first summand is the sum of the previous addition  3 + 4 + 5 → e.g., 12 + 13  
11. Analysing the first summands  Looking for which number has not been used as the first summand  3 + 4 + 5 = 12, e.g., 2 + 3 + 4 = 9 5 + 6 + 7 = 18  
12. Combinations of different strategies  Consecutively use of (different or equal) strategies, e.g., “extension at the end” and afterwards “reduction at the front”  3 +  4 + 5  
→  3 +  4 + 5 + 6  
→  4 + 5 + 6 
13.4.3 Mathematics Language Tools
A number of students will need support for describing how to find numbers that can be written as the sum of consecutive numbers and giving reason to be sure that they found them all. Even if the students are able to find all the additions and can develop a systematic approach for finding them all, this does not necessarily mean that they can always express their way of working and their ‘proof’ in a manner which is comprehensible to their classmates and the teacher.
These difficulties are normal and can, among other things, be traced back to the fact that students resort to the language they primarily use in everyday life. Just as most students will not learn multiplication tables or standard algorithms without any further support, they will also not automatically acquire competences in describing strategies and giving reasons why using this results in all possible additions.
In our opinion, it is therefore helpful to offer the students language structures in agreement with the socalled ‘scaffold’ approach (see for more information, Gibbons, 2002, 2006), which can serve to support the description of things they noticed (Götze, 2015). The promotion of a specific technical language can be approached at two levels. Besides the communication in the teaching situation (microscaffolding) is essential, elements for advancing the technical language are in the planning of the lessons (macroscaffolding) are vital as well. The possibilities of macroscaffolding will be addressed below by way of the sample lesson, titled “Sums of consecutive natural numbers”.
Agreement on a common language is already decisive in the introductory stage. What suggests itself for this is the creation of socalled ‘lexical stores’ which include frequently used terms that can then be applied when working on an assignment. This is in no way meant to prescribe a normatively defined use of language to the students. The objective rather implies establishing a consensus for mutual communication that both is based on the students’ previous language skills and also is technically adequate.
These formulation aids can be used to support students when they describe their discoveries and the causal relationships in technical terms. It is, for example, conceivable that a student who repeatedly applies the increaseall strategy in the addition of consecutive natural numbers with three summands initially discovers changes in the sum, and will then base his or her description on the offered mathematics language tools. This could help to establish a reason like “If all three summands are increased by 1 each, the sum of them will increase by 3.” Applying the decreaseall strategy could conversely enable the statement: “The sum is decreased by 3 if every summand is decreased by 1.”
Another option is having the students creating descriptions by their own for a series of assignments that they analyse in partner work. Further ideas for the creation and embedding of mathematics language tools can be found on the website of the PIKAS project.^{4} One can say in summary that the creation of a lexical store can on the one hand be helpful because the students are provided with an optional orientation for description and argumentation when working on assignments concerning the sum of consecutive natural numbers. And on the other, it can also support the communication amongst the students, as well as the dialogue with the teacher. Advancing the ability to express oneself can productively stimulate the taskrelated exchange in the classroom.
13.5 Numbers Can Be Realistic Too
Therefore, in RME, problems presented to students can come from the real world but also from the fantasy world of fairy tales, or the formal world of mathematics, as long as the problems are experientially real in the student’s mind. (Van den HeuvelPanhuizen & Drijvers, 2014, p. 521)
The equal status of reallife and formal problems is also repeatedly underscored in the RME literature (see, e.g., Freudenthal 1991; Van den HeuvelPanhuizen & Drijvers 2014). RME is nonetheless occasionally accused of attaching greater (or a too great) value to references to real life, which means that the ideas of RME’s inventors are not always adequately understood (see Van den HeuvelPanhuizen, 2002; Wittmann, 2005).
A search for possible reasons why RME is often reduced to approximations of real contexts could rely on various explanation patterns. The name alone suggests a proximity to reallife issues in various languages, as described above. The German translation of the term ‘realistic’ also tends to be more signified by a relation with real life than it is the case in the Dutch language.
The mechanistic and structuralistic teaching and learning concepts so rightly criticised by RME will, at best, place references to contexts at the end of the learning process, in order to apply the structures that have been formally learned before. RME attaches a different value to the role of contexts. References to contexts are additionally characterised as starting points for the process of learning mathematics. The initially acquired ‘real’ models serve to support the mastery of mathematical problems on a formal and symbolic level (see Gravemeijer & Doorman, 1999; Treffers & De Moor, 1996; Van den HeuvelPanhuizen 2008). To distinguish itself from traditional teaching and learning concepts, RME has attached very great importance to the doubtlessly essential meaning of references to daily life.
But the Wiskobas^{5} Bulletins, surely one of the sources of RME, or the publications of the TAL^{6} project, for example, not only show that the vertical component is absolutely present, but also that mathematics (with no relation to reality) is regarded as a context of its own, as we tried to show in our contribution. RME could possibly highlight this aspect even more strongly.
As mentioned earlier on, what counts for students is the existence of a context that makes sense. The context does not have to be a reallife one. Pure numerical contexts can also be quite meaningful for students (e.g., Steinweg, 2001), or, to put it differently, ‘numbers can be realistic, too!’
Footnotes
 1.
pikas.dzlm.de (website in German).
 2.
kira.dzlm.de (website in German).
 3.
For more information on the various calculation strategies, see kira.dzlm.de/062.
 4.
See pikas.dzlm.de/304 (website in German).
 5.
Wiskunde op de Basisschool (Mathematics in Primary School). Wiskobas Bulletin is the journal published by the Wiskobas project from 1971 to 1981.
 6.
Tussendoelen Annex Leerlijnen (Intermediate Attainment Targets and Learning Lines). In the TAL project, the socalled ‘teachinglearning trajectories’ for primary mathematics education have been developed (see, e.g., Van den HeuvelPanhuizen, 2008).
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