What characterises the heuristic approaches in mathematics textbooks used in lower secondary schools in Norway?
DOI:
https://doi.org/10.7146/nomad.v16i4.148407Abstract
In this paper I present findings of an analysis of how mathematics textbooks treat heuristic approaches. The aim of this analysis is to give a characterisation of the occurrence of nine well-known heuristic approaches by analysing 740 examples presented in six ninth grade textbook series. The findings show that many of the problems in the examples are being solved by using one or more heuristic approaches, but the characteristics of the examples and the textbooks’ lack of discussion of the approaches themselves make it challenging to teach and learn these in school. The heuristic approaches seem to be used rather incidentally, which is supported by the fact that none of the textbooks explicitly treat or mention problem solving.
References
Alseth, B. (1995). Undervisning i problemløsningsstrategier (Teaching problem solving strategies). Nordic Studies in Mathematics Education, 3 (3), 7-26.
Alseth, B., Breiteig, T. & Brekke, G. (2003). Endringer og utvikling ved R97 som bakgrunn for videre planlegging og justering - matematikkfaget som kasus. (Changes and development for R97 as background for further planning and adjustment - the mathematics subject as case). Notodden: Telemarksforsking.
Apple, M. (1992). The text and cultural politics, Educational Researcher, 21 (7), 4-11. https://doi.org/10.2307/1176356
Bierhoff, H. (1996). Making the foundations of numeracy: a comparison of primary school textbooks in Britain, Germany and Switzerland. Teaching Mathematics and its Applications, 15 (4), 141-160. https://doi.org/10.1093/teamat/15.4.141
Björkqvist, O. (2003). Matematisk problemløsning. (Mathematical problem solving). In B. Grevholm (Ed.), Matematikk for skolen (pp. 51-70). Bergen: Fagbokforlaget Vigmostad & Bjørke AS.
Blomhøj, M. & Jensen, T. H. (2007). What's all the fuss about competencies? Experiences with using a competence perspective on mathematics education to develop the teaching of mathematical modelling. In W. Blum, P. Galbraith, H. Henn & M. Niss (Eds.), Applications and modelling in mathematics education: The 14th ICMI Study (pp. 45-56). New York, USA: Springer. https://doi.org/10.1007/978-0-387-29822-1_3
Boesen, J. (2006). Assessing mathematical creativity: comparing national and teacher-made tests, explaining differences and examining impact (PhD thesis). Umeå University.
Borgersen, H. E. (1994). Open ended problem solving in geometry. Nordic Studies in Mathematics Education, 2 (2), 6-35.
Branca, N. (1980). Problem solving as a goal, process and basic skill. In S. Krulik & R. Reys (Eds.), Problem solving in school mathematics 1980 Yearbook, (pp. 3-8). Reston: NCTM.
Brändström, A. (2005). Differentiated tasks in mathematics textbooks: An analysis of the levels of difficulty (Licentiate Thesis). Luleå University of Technology.
Bryman, A. (2008). Social research methods (Third edition). Oxford University Press.
Chávez, O. (2003). From the textbook to the enacted curriculum: textbook use in the middle school mathematics classroom (Unpublished doctoral theses). University of Missouri, Colombia.
Charles, R. & Lester, F. K. (1984). An evaluation of a process-oriented mathematical problem-solving instructional program in grades five and seven. Journal for Research in Mathematics Education, 15, 15-34. https://doi.org/10.5951/jresematheduc.15.1.0015
Cohen, L., Manion, L. & Morrison, K. (2011). Research methods in education (Seventh Edition). London: Routledge.
Danielsen, I. J., Skaar, K. & Skaalvik, E. M. (2007). De viktige få: analyse av elevundersøkelsen 2007. (The important few: analysis of the pupil survey 2007). Kristiansand: Oxford Research.
Doyle, W. (1988) Work in mathematics classes: the content of student thinking during instruction, Educational Psychologist, 23 (2), 167-180. https://doi.org/10.1207/s15326985ep2302_6
Ernest, P. (1998). Mathematical knowledge and context. In A. Watson (Ed.), Situated cognition and the learning of mathematics (pp. 13-31). Oxford: Centre for Mathematics Education Research.
Fan, L. & Kaeley, G. S. (2000). The influence of textbook on teaching strategies: an empirical study. Mid-Western Educational Researcher, 13 (4), 2-9.
Fan, L. & Zhu, Y. (2000). Problem solving in Singaporean secondary mathematics textbooks. The Mathematics Educator, 5 (1-2), 117-141.
Fan, L. & Zhu, Y. (2007a). Representation of problem-solving procedures: a comparative look at China, Singapore and US mathematics textbooks. Educational Studies in Mathematics, 66, 61-75. https://doi.org/10.1007/s10649-006-9069-6
Fan, L. & Zhu, Y. (2007b). From convergence to divergence: the development of mathematical problem solving in research, curriculum, and classroom practice in Singapore. ZDM - The International Journal on Mathematics Education, 39 (5-6), 491-501. https://doi.org/10.1007/s11858-007-0044-1
Freeman, D. J. & Porter, A. C. (1989). Do textbooks dictate the content of mathematics instruction in elementary schools? American Educational Research Journal, 26 (3), 403-421. https://doi.org/10.3102/00028312026003403
Garner, R. (1992). Learning from school texts. Educational Psychologist, 27, 53-63. https://doi.org/10.1207/s15326985ep2701_5
Grevholm, B. (2004). Mathematics worth knowing for a prospective teacher. In B. Clarke, D. M. Clarke, G. Emanuelsson, B. Johansson, D. V. Lambdin, et al. (Eds.). International perspectives on learning and teaching mathematics (pp. 519-536). Gothenburg: National Centre for Mathematics Education.
Grønmo, L. S., Bergem, O. K., Kjærnsli M., Lie S. & Turmo A. (2003). Hva i all verden har skjedd i realfagene? Norske elevers prestasjoner i matematikk og naturfag i TIMSS 2003. (What on earth is going on in science? Norwegian pupils' performance in mathematics and science in TIMSS). Institutt for lærerutdanning og skoleutvikling, Universitet i Oslo.
Grønmo, L. S. & Ostad, T. (2009). Tegn til bedring. Norske elevers prestasjoner i matematikk og naturfag i TIMSS 2007 (Signs of improvement. Norwegian pupils' performance in mathematics and science). Oslo: Unipub.
Hatfield, L. L. (1978). Heuristic emphases in the instruction of mathematical problem solving. In L. L. Hatfield & D. A. Bradbard (Eds.), Mathematical problem solving: Papers from a research workshop, (pp. 21-42). Columbus: ERIC Information Analysis Center for Science, Mathematics and Environmental Education.
Hensey, L. K. (1996). An examination of elementary mathematics textbooks problem-solving items during the nineties, and possible influence on the NCTM standards on such items (Doctoral dissertation). University of Iowa.
Hembree, R. (1992). Experiments and relational studies in problem solving: a meta-analysis. Journal for Research in Mathematics Education, 23 (3), 242-273. https://doi.org/10.5951/jresematheduc.23.3.0242
Johansson, M. (2005). The mathematics textbook. From artefact to instrument. Nordic Studies in Mathematics Education, 10 (3-4), 43-64.
Johansson, M. (2006). Textbooks as instruments. Three teachers' way to organize their mathematics lessons. Nordic Studies in Mathematics Education, 11 (3), 5-30.
Jakobsson-Åhl, T. (2008). Word problems in upper secondary algebra in Sweden over the years 1960-2000. Nordic Studies in Mathematics Education, 13 (1), 7-28.
Kantowski, M. G. (1977). Processes involved in mathematical problem solving. Journal for Research in Mathematics Education, 8, 163-180. https://doi.org/10.5951/jresematheduc.8.3.0163
Kantowski, M. G. (1981). Problem solving. In E. Fennema (Ed.), Mathematics education research: implications for the 80's (pp. 111-126). Reston: National Council of Teachers of Mathematics.
Kilpatrick, J. (1967). Analyzing the solution of word problems in mathematics: an exploratory study (Unpublished doctoral dissertation, Stanford University). Dissertation Abstracts International, 28, 4308A. (Eric, ED027182)
KUF. (1987). Mønsterplanen for grunnskolen. Oslo: Aschehoug.
KUF. (1996). Læreplanverket for den 10-årige grunnskolen. Oslo: Nasjonalt læremiddelsenter.
Lester, F. K. (1994). Musings about mathematical problem-solving research 1970-1994. Journal for Research in Mathematics Education, 25, 660-675. https://doi.org/10.5951/jresematheduc.25.6.0660
Lester, F. K. (1996). Problemlösningens natur (The nature of problem solving). In G. Emanuelsson, K. Wallby, B. Johansson & R. Ryding (Eds.), Nämnaren Tema Matematik - ett kommunikationsämne (pp. 85-91). Göteborg: Kompendiet
Lester, F. K. & Lambdin, D. V. (2004). Teaching mathematics through problem solving. In B. Clarke, D. M. Clarke, G. Emanuelsson, B. Johansson, D. V. Lambdin, et al. (Eds.), International perspectives on learning and teaching mathematics (pp. 189-203). NCM, Göteborg University.
Li, Y. (1999). An analysis of algebra content, content organization and presentation, and to-be-solved problems in eight-grade mathematics textbooks from Hong Kong, Mainland China, Singapore, and the United States (Doctoral Dissertation). University of Pittsburgh.
Love, E. & Pimm, D. (1996). "This is so": a text on texts. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.) International handbook of mathematics education (Vol. 1, pp. 371-409). Dordrecht: Kluwer Academic Publishers.
Lucas, J. (1974). The teaching of heuristics problem-solving strategies in elementary calculus. Journal for Research in Mathematics Education, 5, 36-46. https://doi.org/10.5951/jresematheduc.5.1.0036
Mason, J & Davis, J. (1991). Fostering and sustaining mathematics thinking through problem solving. Geelong: Deakin University.
Mayer, R. E., Sims, V. & Tajika, H. (1995). A comparison of how textbooks teach mathematical problem solving in Japan and the United States. American Educational Research Journal, 32 (20), 443-460. https://doi.org/10.3102/00028312032002443
MOE. (2007). Secondary mathematics syllabus. Singapore: Curriculum Planning and Development Division. Retrieved January 10, 2011 from http://www.moe.gov.sg/education/syllabuses/sciences/files/maths-secondary.pdf
Ng, L. E. (2002). Representation of problem solving in Singaporean primary mathematics textbooks with respect to types, Polya's model and heuristics (Unpublished master's thesis). Singapore: National Institute of Education.
Nibbelink, W. H., Stockdale, S. R., Hoover, H. D. & Mangru, M. (1987). Problem solving in the elementary grades: textbook practice and achievement trends over the past thirty years. Arithmetic Teacher, 35 (1), 34-37. https://doi.org/10.5951/AT.35.1.0034
Niss, M. & Jensen, T. H. (2002). Kompetencer og matematiklæring. Ideer og inspiration til udvikling af matematikundervisning i Danmark. København: Undervisningsministeriets forlag. Retrieved January 10, 2010 from http://pub.uvm.dk/2002/kom/
Niss, M. (2003). Den matematikkdidaktiske forskningens karakter og natur. (Aspects of the nature and the state of research in mathematics education). In B. Grevholm (Ed.), Matematikk for skolen (pp. 335-364). Bergen: Fagbokforlaget Vigmostad & Bjørke AS.
Pehkonen, E. (1995). On pupils' reactions to the use of open-ended problems in mathematics. Nordic Studies in Mathematics Education, 3 (4), 43-57.
Pehkonen, E. (2007). Problem solving in mathematics education in Finland. Retrieved May 15, 2009 from http://www.unige.ch/math/EnsMath/Rome2008/ WG2/Papers/PEHKON.pdf
Pepin, B. & Haggarty, L. (2001). Mathematics textbooks and their use in English, French and German classrooms: a way to understand teaching and learning cultures. Zentralblatt für Didaktik der Mathematik, 33 (5), 158-175. https://doi.org/10.1007/BF02656616
Pepin, B. & Haggarty, L. (2002). An investigation of mathematics textbooks and their use in English, French and German classrooms: who gets an opportunity to learn what? British Educational Research Journal, 28 (4), 567-590. https://doi.org/10.1080/0141192022000005832
Polya, G. (1945). How to solve it. Princeton: Princeton University Press. https://doi.org/10.1515/9781400828678
Reys, B. J., Reys, R. R. & Chávez, O. (2004). Why mathematics textbooks matter. Educational Leadership, 61 (5), 61-66.
Robitaille, D. F. & Travers, K. J. (1992). International studies of achievement in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics education (pp. 687-709). New York: Macmillan.
Röj-Lindberg, A-S. (1999). Läromedel och undervisning i matematik på högstadiet. En kartläggning av läget i Svenskfinland. (Textbooks and mathematics teaching at junior secondary level. A survey of the situation in Swedish Finland).Vasa: Svenskfinlands läromedelscenter.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando: Academic Press.
Schoenfeld, A.H. (1988). When good teaching leads to bad results: the disasters of "well-taught" mathematics courses. Educational Psychologist, 23 (2), 145-166. https://doi.org/10.1207/s15326985ep2302_5
Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334-370). New York: Macmillan.
Schmidt, W. H., McKnight C. C., Valverde G. A., Houang R. I. & Wiley D. E (1996). Many visions, many aims: A cross-national investigation of curricular intentions in school mathematics. London: Kluwer Academic Publishers. https://doi.org/10.1007/978-94-011-5786-5
Schmidt, W. H., McKnight, C. C., Houang, R. T., Wang, H., Wiley, D. E. et al. (2001). Why schools matter: a cross-national comparison of curriculum and learning. San Francisco: Jossey-Bass.
Schroeder, T. & Lester, F. K. (1989). Developing understanding in mathematics via problem solving. In P. Trafton & A. Schulte (Eds.), New directions for elementary school mathematics (pp. 31-42). Reston: NCTM.
Skolverket. (2003). Nationella kvalitetsgranskningar 2001-2002. Lusten att lära - med fokus på matematik, 221. (National quality study 2001-2002. The desire to learn - with focus on mathematics). Stockholm: Skolverket.
Smith, J. P. (1973). The effect of general versus specific heuristics in mathematical problem solving tasks (PhD Dissertation, Columbia University). Dissertation abstracts international, 34, 2400A.
Stacey, K. (2005). The place of problem solving in contemporary mathematics curriculum documents. Journal of Mathematical Behaviour, 24, 341-350. https://doi.org/10.1016/j.jmathb.2005.09.004
Stanic, G. M. A. & Kilpatrick, J. (1989). Historical perspectives on problem solving in mathematics curriculum. In R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 1-22). Reston: NCTM.
Stiff, L. V. (1988). Problem solving by example. School Science and Mathematics, 88 (8), 666-675. https://doi.org/10.1111/j.1949-8594.1988.tb11872.x
Stockdale, S. R. (1991). A study of the frequency of selected cue words in elementary textbook word problems. School Science and Mathematics, 91 (1), 15-21. https://doi.org/10.1111/j.1949-8594.1991.tb15560.x
Törnroos, J. (2001). Mathematics textbooks and students' achievement in the 7th grade: What is the effect of using different textbooks? In J. Novotna (Ed.), European Research in Mathematics Education II (pp. 516-525). Prague: Faculty of Education, Charles University.
UFD. (2005a). Kunnskapsløftet [Knowledge Promotion]. Retrieved August 1, 2010 from http://www.udir.no/upload/larerplaner/Fastsatte_lareplaner_for_Kunnskapsloeftet/Grunnskole_og_gjennomgaende/matematikk_010810.rtf
UFD. (2005b). Kunnskapsløftet [Knowledge Promotion]. Retrieved August 1, 2010 from http://www.udir.no/upload/larerplaner/Fastsatte_lareplaner_for_Kunnskapsloeftet/english/Mathematics_subject_curriculum.doc
Utdanningsdirektoratet (2005). Kartlegging av læremidler og læremiddelpraksis. Retrieved January 7, 2007 from http://www.skolenettet.no/moduler/templates/Module_Article.aspx?id=37694&epslanguage=NO
Wilson, J. (1967). Generality of heuristics as an instructional variable (Doctoral Dissertation, Stanford University). Dissertation abstracts international, 28, 2575A.
Wyndhamn, J. & Säljö, R. (1997). Word problems and mathematical reasoning - a study of children's mastery of reference and meaning in textual realities. Learning and Instruction, 7 (4), 361-382. https://doi.org/10.1016/S0959-4752(97)00009-1
Zhu, Y. (2003). Problem solving in China, Singapore and US mathematics textbooks: A comparative study (Unpublished doctoral dissertation). Singapore: National Institute of Education.
Zhu, Y. & Fan, L. (2006). Focus on the representation of problem types in intended curriculum: a comparison of selected mathematics textbooks from Mainland China and the United States. International Journal of Science and Mathematics Education, 4 (4), 609-626. https://doi.org/10.1007/s10763-006-9036-9
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
