By explaining, teachers attempt to ‘transfer’ pre-developed concepts and methods to pupils. This approach to teaching mathematics is typically used as a strategy to ensure that students acquire knowledge and skills that are considered a prerequisite by society for participation in institutions of higher learning.
Encouraging students to think for themselves, however, strengthens their productive capacities to engage in their own tentative explorations of the unknown, with dead-ends and errors along the way that serve as compost for the growth of fertile ideas and for the unfolding of an inner light, out of which mathematical structures crystallise in an inner space of imagination. The reality of mathematical structures can only be experienced through inner production. “Producing is easier than receiving, because reception always involves two points of view, while in production only one's own point of view counts, at least at first. Hence, learners should be allowed to begin by being productive.” (1) Productivity is more effective in developing genuine mathematical understanding than receptivity and frees the spirit.
“Mathematics is regarded as a demonstrative science. Yet this is only one of its aspects. Finished mathematics presented in a finished form appears as purely demonstrative, consisting of proofs only. Yet mathematics in the making resembles any other human knowledge in the making. You have to guess a mathematical theorem before you prove it; you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies; you have to try and try again. The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.” (2)
As is well known, Rudolf Steiner proposed that mathematics should begin from the whole that then gets analysed into parts (e.g. 12 = ?), as opposed to starting from parts that then get assembled into a whole (e.g. 5 + 7 = ?, the more traditional and still very common teaching approach). By taking the whole as a starting point, mathematical questions or problems become open questions. Rather than directing students to one specific answer, this approach creates an opening to many possible answers and solutions.
Modern didactics of mathematics education has proposed that working with open questions is generally more invigorating than giving students closed questions with only one correct answer. Open questions stimulate playing with a problem and, on a reflective journey of invention, lead to diverse answers that can be compared and discussed.
‘Opening Mathematics’ wants to encourage Waldorf teachers around the world to entrust students with more mathematical production through the use of open-ended mathematics problems in the classroom and to reflect upon their use. For more resources, please refer to the website opening-mathematics.net.
‘Opening Mathematics’ encourages you to participate by sharing your experience of working with open questions throughout the Waldorf mathematics curriculum, i. e. from class 1 to 12 (or 13).
By sharing you can help to inspire and encourage others in developing their teaching practices. Your sharing of experiences will create a broad view of practices and understandings of mathematics education in Steiner Waldorf schools. ‘Opening Mathematics’ Team will be collecting and studying your experiences and reflections with the purpose of deepening our understanding of existing practices and of exploring possibilities to improve these.
1. Please send in examples of open questions you’ve worked with successfully (stating the class level you used them in).
2. Briefly describe experiences of working with these open questions and report on success as well as obstacles and failures you experienced.
In particular, ‘Opening Mathematics’asks you to reflect on to the following questions:
3. How did you arrive at the questions?
4. Were the open questions tailored to the needs of a student cohort that spanned a wide range of abilities?
5. Did you observe that a wide range of answers emerged and if so, please describe if and how this elicited peer learning among the students.
6. To what extent did you connect your subsequent teaching to the students’ answers? Did this in any way lead to some new, or unexpected understandings for either you or the students?
7. Did you encourage your students to reflect on their learning pathway? If so, please describe how this played out.
Please send your response to waldorf100. Please respond in English, German, French, Italian, Spanish or Dutch. ‘Opening Mathematics’team is also interested to learn about what further support you would like to see in future. @opening-mathematics.net
Responses will be evaluated by a team around Aziza Mayo (Professor of Value(s) of Waldorf-Steiner Education at Hogeschool Leiden), Detlef Hardorp (mathematician and former Waldorf teacher of mathematics) and Daniel Jaeger (former teacher of mathematics and Waldorf class teacher).
Results will be published online at opening-mathematics.net. If you do not explicitly state otherwise, you agree to have whatever you sent to ‘Opening Mathematics’cited in the report. This will happen anonymously, unless you give ‘Opening Mathematics’explicit permission to mention your name and your school.
(1) Peter Gallin (2011) „Mathematik als Geisteswissenschaft. Der Mathematikschädigung dialogisch vorbeugen“. In: Helmerich M., Lengnink K., Nickel G., Rathgeb M. (eds) „Mathematik Verstehen“, Vieweg+Teubner. Online: www.gallin.ch/Gallin_MathAlsGeistesw.pdf(German) and www.gallin.ch/ArtikelMathGeisteswGallinEnglish.pdf (English: “Mathematics belongs to the humanities or Preventing mathematics injury through dialogue”
(2) George Polya, “Induction andAnalogy in Mathematics”, Princeton 1954