Word problems

An inventory

Ben Wilbrink

This page belongs to the broader issues treated in the mathematics education page. Publications in Dutch are mentioned and annotated in the separate web page wiskundeonderwijs, and in woordproblemen (en contextproblemen).

Nederlandstalige publicaties over redactiesommen, ingeklede vergelijkingen, of hoe ze ook ooit zijn genoemd: vooral geplaatst in woordproblemen (en contextproblemen).

Stephen K. Reed (1999). Word problems: Research and curriculum reform. Erlbaum. [als eBook in KB] Questia preview; reviewed by Lieven Verschaffel.

This is really a remarkable book, because cognitive scientist Stephen Reed, knowing very well the arguments of his colleagues Anderson, Reder & Simon, goes even further than Anderson c.s. already did in believing that constructivism and situationism might have a point, here and there. Well, I don't. Of course, the book now is 15 years old, and Stephen Reed may have changed his opinions as expressed in the book. Nevertheless, I will explore how and why it is that Reed sees some merit in the research on word problems done by members of the constructivist movement.

Chapter 10. Wor(l)d Problems

• This last section of the book focuses on topics that the NCTM standards recommended for increased attention. One of the objectives of the NCTM standards is to promote the transfer of knowledge learned in the classroom to problems outside the classroom. Therefore, it is not surprising that one of the recommended topics for increased attention is the use of real-world problems to motivate and apply theory (NCTM, 1989).

Yes, this is one of the articles of faith that is causing so much trouble and loss of quality in math education. The other article is that math education really should be aimed at learning to think (mathematically). Stephen Reed does not start his exposition with red flagging these notions; he seems to work on the notion ‘Well, let's see how far we can get on these NCTM-notions, there should at least be some scientific merit in them?’. For the record: of course the NCTM issued a revision of the standards in 2000, taking a less extreme constructivist position on math education. Stephen Reed apparently was not yet aware of the (extent of) the standards-revision.

Reed starts off on situated learning, a key protagonist article on the subject being Brown, Collins & Duguid (1989 text) (see f.e. the acknowledgement: Greeno, Lave, Schoenfeld). The term protagonist is mine, indicating that the authors in my opinion have an axe to grind — other than the scientific axe that is.

• Authentic Activity Our case so far rests on an undefined distinction between authentic and school activity. If we take learning to be a process of enculturation, it is possible to clarify this distinction and to explain why much school work is inauthentic and thus not fully productive of useful learning.

  ( . . . )

  Authentic activities then, are most simply defined as the ordinary practices of the culture.

  Brown, Collins & Duguid

I am flabbergasted. Education, as far as I know, is such an ‘ordinary practice of the culture’ — yet it is not ‘authentic’ enough?

• Archetypal school activity is very different from what we have in mind when we talk of authentic activity, because it is very different from what authentic practitioners do.

As not unusual, Whitehead (1929) is misused to make the point of much of school knowledge being inert knowledge. If you find Brown c.s. arguing the blatantly obvious, trust your own assessment, mistrust their claims of research evidence [by Vygotsky, Leontiev, and other activity theorists, see their note 1] for what always has been self-evident to, at the very least, psychological researchers. Controlling situations in the psychological laboratory does not imply situations are deemed inconsequential, quite the contrary.

Reed seems impresssed by the example of Schoenfeld (1985: Problem solving) teaching his college students to think mathematically through problem solving. Of course, students learn a lot this way. The question is, of course, is it a better way than conventional teaching of math? That is an empirical question that needs to be resolved by experiment (unless Schoenfeld concedes his method is not a general substitute for math courses). Where is the empirical evidence? Schoenfeld is a mathematician interested in the frontline of cognitive psychology at the time: work by Newell & Simon, for example. This is a good opportunity to check on Schoenfeld’s psychological thinking on what it is to think mathematically. I expect to be able to prove Schoenfeld a romantic thinker on thinking. I will come back to this issue later on. The question is: why does Reed uncritically go along with Schoenfeld’s thinking on problem solving?

Next stop: Magdalene Lampert (1986). Reed copies text from Brown cs.—with some alterations, of course. Why does he do that? Is he such a busy man that it is too much trouble to get the original article from Cognition and Instruction? Reed must have written his text in a hurry, which might explain why he refrains from critical comments on the thinking of researchers such as Brown, Collins & Guguid. My problem now is: Have Brown c.s. fairly rendered the analysis and results of Lampert (1986), and if they did not do so, did Reed correct the biased picture? Well, having read the closing sentences of the Lampert report 1986 (below), the question now is: why should Reed have cited a believer in construcivism? The Lampert report is a statement of belief, and has nothing whatsoever to do with sientific research. Lampert, speaking for herself:

• Magdalene Lampert (1986). Knowing, doing, and teaching multiplication. Cognition and Instruction, 3, 305-399. get free pdf of concept via ERIC
  • Fourth graders can do and [sic] think mathematically. They have the capacity to gather information, to generate and test hypotheses, and to produce and evaluate solutions. They can talk about what they are thinking, they can listen to and appreciate another’s procedural way of understanding something, and they ca invent problem-solving procedures that are both useful and sensible. What sort of help do children need from adults in order to be able to do these things and to be confident in their ability to do them? They need to be asked questions whose answers can be “figured out” not by relying on memorized rules for mving numbers around but by thinking about what numbers and symbols mean. They need to be treated like sense-makers rather than like rememberers and forgetters. They need to see connections between what they are supposed to be learning. They need to learn to do computation competently and efficiently without losing sight of the meaning of what they are doing and its relation to solving real problems. The lessons I have taught and described here suggest that it is possible to do these things in conventional classrooms.

    Closing sentences of the Lampert 1986 report.

Magdalene Lampert (2001). Teaching problems and the problems of teaching. New Haven: Yale University Press. [PEDAG. 47.b.99] [on problem based learning of 10 year olds, math only, with Magdalene as teacher (during 8 years). Lampert studied math, esp. intuitionism ;-) ] [Interesting study, skimmed it, brought it back. I do not know what to make of it, but I surely do not have the time to study the book in its entirety. There is nowhere a sign to be found that empirical testing of this approach might be in order. There is no reference made to the research literature on learning. And so forth. Disappointing? Yes. I do not think that Lampert specifies empirical data that could be used to evaluate the approach of PBI.]

Then: Carraher, Carraher & Schliemann, 1987. Reed: ‘A classic study of the relation between real-world knowledge and school knowledge investigated how problem context influenced the strategies used by a group of third graders from a poor area in Brazil’. The 1993 book: Street mathematics and school mathematics.

• T. Nunes, A. D. Schliemann & D. W. Carraher (1993). Street Mathematics and School Mathematics. Cambridge University Press. info, publication Analúcia D. Schliemann; [niet online, niet als eBook in KB]

Way too much weight has been given by constructivists/situationists to this small study. One of the problems with it is that many assumptions stay implicit. Another notion that is lacking: of course in a new kind of situation it will take some time to get your math routines in order. Especially in vocational situations. Expertise in school math does not imply that one is expert in math in all kinds of vocational situations. Obvious, isn’t it? Most readers of superficial research like the Carraher, Carraher & Schliemann studies take the reported research conclusions for granted. At least the researchers will have thought carefully about the design en the analysis of their research? Of course they will have done so. That does not mean they have not overlooked any alternative interpretation of their results. So be careful.

Reed mentions a kind of replication study, by Baranes, Perry & Stigler (1989). At least they have been critical of the Carraher, Carraher & Schliemann results.

• Baranes, R., Perry, M., & Stigler, J. W. (1989). Activation of real-world knowledge in the solution of word problems. Cognition and Instruction, 6, 287-318. preview

Kenneth R. Koedinger & Mitchell J. Nathan (2004). The real story behind story problems: Effects of representations on quantitative reasoning. The Journal of the Learning Sciences, 13, 129-164 abstract en pdf (scans). [Via KB reguliere pdf downloaden: JSTOR]

Story problems are believed to be difficult, more difficult than problems given in symbolic representations. Koedinger and Nathan articulate this theme, and present results of their research. Their article gives me hope to be able to solve the problem of context problems in reform math education. If I may say so.

Funny problem, from Hudson (1983):

• For example, Hudson (1983) found that nursery school children were 17% correct on a standard story phrasing, “There are 5 birds and 3 worms. How many more birds are there than worms?” However, performance increased to an impressive 83% when the story is phrased as, “There are 5 birds and 3 worms. How many birds don't get a worm?” The latter phrasing elicits a match-and-count strategy that is more accessible for novice learners than the more sophisticated subtraction strategy elicited by the former, more standard phrasing.

  p. 135-136

What struck me in this example: the first phrasing is not a math problem at all: children are asked to subtract quantities in different ‘units’ from each other, a nonsense operation of course. The wording of the second problem is correct in the linguistic sense, but is this a math problem? What would make it a math problem: something with sets and a mapping of one set on the other? Quite another issue whether it is to be recommended to give problems in such a crude concrete form: read James Flynn on the progress mankind has made from thinking in concrete terms only to the ability to think in abstract terms. Presenting birds-and-worms problems locks children in a concrete mode their parents and great-parents learned to transcend.

• We have not found prior experimental comparisons of solution correctness on matched algebra story problems and equations for students learning algebra.

  p. 131

  In contrast with the idea that equations are transparently understood, a key point of this article is to test the claim that comprehension of algebra equations is not trivial for algebra learners and that, as a consequence, algebra story problems can sometimes be easier to solve than matched equations. The verbal facilitation claim directly contradicts the documented views of many teachers and education re-searchers (Nathan & Koedinger, 2000a, 2000b) and cognitive scientists (e.g., Cummins et al., 1988; Geary, 1994) that algebra story problems are inherently more difficult than matched equations.

  p. 139

Ik maak maar eens uitvoerige aantekeningen bij de uitvoerige discussie van Koedinger & Nathan. Ik heb het gevoel bij dit artikel dat het de eerste publicatie is waarbij ik een beetje grond onder mijn voeten voel, een vast punt van waaruit ik verdere verkenningen kan gaan maken. Natuurlijk, dit onderzoek heeft ook zo zijn beperkingen, al was het maar dat de groep leerlingen nogal specifiek van karakter is. Op zich vind ik de keuze trouwens heel goed: juist bij deze leerlingen is er een goede spreiding in oplossingen te verwachten. Maar die spreiding is dus niet generiek geldig. K & N stippen zelf aan dat verder onderzoek met choice - no-choice methodologie (in de groep van Verschaffel veel gebruikt) scherpere inzichten op kan leveren. Afijn, mijn eigen doelstellingen zijn ook al divers, in ieder geval deze twee: ik moet voor een nieuwe editie van Toetsvragen ontwerpen een adequate behandeling van zowel conventionele woordproblemen als contextproblemen geven; het Wilbrink-Kirschner-voorstel om de cognitieve belasting bij contextproblemen in kaart te brengen moet ook verder worden gebracht.

De moeilijkheid bij het bestuderen van dit artikel van Koedinger & Nathan is dat zij een nogal complex onderzoekdesign hanteren (gelukkig maar), waardoor de mentale belasting van de lezer voortdurend behoorlijk aan de tax is ;-) Veel herlezen dus. En dat zoveel mogelijk zonder literatuurverwijzingen op te volgen, anders komt het nooit klaar. Maar goed, ik zal nu toch ook de genoemde literatuur gaan checken, en daar iets over gaan melden.

Het globale design is gebouwd op de drie hypothesen over relatie tussen probleemformulering (problem representation) en probleemoplossen, vooral bij leerlingen in beginnend onderwijs in algebra.

1e hypothese Symbolisch is faciliterend (ik vertaal die termen maar niet): de symbolische vergelijking is transparanter en dus makkelijker te begrijpen dan het ermee corresponderende woordprobleem. Bij deze leerlingen blijkt dat dus niet het geval te zijn. Best verrassend. Maar ja, het blijkt dan ook uit dit onderzoek dat deze leerlingen de woordproblemen vooral niet-algebraïsch te lijf gaan, deze dus niet eerst reduceren tot een symbolische vergelijking. En dat lijkt me het springende punt te zijn: de empirie is totaal anders dan makkelijk opgeschreven ‘didactische logica’. Waar kwam ik dat vaker tegen in de literatuur? Was dat niet bijvoorbeeld een onderzoek van Joke Torbeyns naar oplossingen bij rekenopgaven die ‘handig’ kunnen maar typisch volgens een algemeen algoritme zijn gevonden? Daarom moeten we ook beslist weten hoe leerlingen, bijvoorbeeld vmbo basis-beroeps, de contextopgaven van de rekentoetsen aanpakken (dat kladpapier is goud waard, mag natuurlijk niet worden vernietigd, welke gekken hebben dat bedacht?). De vraag die zich dan opdringt: hoe zit het met problemen die zich niet anders dan langs de weg van de algebraïsche vergelijking laten oplossen? Of hebben we het dan niet meer over leerlingen die wat betreft de algebra nog naïef zijn?

2e hypothese Situatie/context is faciliterend. Dit is een idee afkomstig uit onderzoek naar situated cognition, waarvoor K & N refereren aan:

• Baranes, R., Perry, M., & Stigler, J. W. (1989). Activation of real-world knowledge in the solution of word problems. Cognition and Instruction, 6, 287-318. preview
• Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1 ), 32-41. Ik noemde dit artikel hierboven als pleitbezorger van situated learning text Ik moet er toch eens afzonderlijk aandacht aan besteden.
• T. N. Carraher, D. W. Carraher & A. D. Schliemann (1987). Written and oral mathematics. Journal for Research in Mathematics Education, 18, 83-97. read online free [tijdschrift van de NCTM] Dit is een artikel in een lijn van onderzoek waarvan ik de publicaties niet kan lezen zonder mijn psychologische tenen te krommen. Het mooie is in ieder geval dat er tenminste empirisch onderzoek wordt gedaan: empirische data zijn altijd nuttig. Maar de kletsika eromheen. Brrrr.
• Nunes, T., Schliemann, A . D., & Carraher D, . W. (1993). Street mathematics and school mathematics. New York: Cambridge University Press. info, publication Analúcia D. Schliemann; [niet online, niet als eBook in KB] [UB Wass.weg Leiden] Meer van hetzelfde.

Het komt er dus op neer dat K & N deze hypothese — de situatie faciliteert — toeschrijven aan onderwijs-ideologisch onderzoek. Dat maakt natuurlijk niet uit: een hypothese is een hypothese. De hypothese wordt deels verworpen: weliswaar zijn in dit onderzoek de woordproblemen makkelijker dan de kale vergelijkingen, maar ook de woordvergelijkingen zijn dat. Het ligt dus niet aan de contexten als zodanig, maar er is meer aan de hand. Niet zo’n gek resultaat dus. Op naar de derde.

3e hypothese Taal/woorden zijn faciliterend. Tenminste, voor deze leerlingen die nog niet zo lang bezig zijn met algebra. Zij zijn meer vertrouwd met verbale dan met symbolische modelformulering. En zo blijkt. Maar ik wil toch wel een beetje los proberen te komen van deze heel specifieke situatie van leerlingen die nog niet goed met algebra vertrouwd zijn. Ik vermoed dat dit er niet echt veel mee te maken heeft: in het dagelijks leven lossen mensen hun problemen niet in eerste instantie met formele methoden op, maar zoeken ze een werkbare oplossing met zo weinig mogelijk gedoe. Herbert Simon noemde dat satisficing. Als je met wat proberen een redelijk antwoord op je probleem krijgt, waarom zou je er dan nog algebra van maken? Zoiets dus. Een enorme lijn van onderzoek, niet alleen als erfenis van Herbert Simon, maar ook van Tversky en Kahneman over decision making. Zodoende worden we teruggeworpen op de vraag: waartoe dient onderwijs? Het idee is dat onderwijs juist niet dient om allerlei gepruts en geklungel te praktizeren (zoals het situationisme wil), maar om toegang te krijgen tot methoden die een wetenschappelijke basis hebben en die in de wereld in feite overal worden gebruikt waar correctheid en berekenbaarheid telt. Dus niet in de winkel en bij de planning van de vacantie, maar wel bij financieel beheer en kwesties die gezondheid en welzijn raken, evenals satellitecommunicatie en ontwerpen van computer games.

Een heel andere benadering is mogelijk langs de lijn van self-explanation: onderzoek van Michelene Chi (zie haar website voor toegang tot pdf's van recente artikelen en hoofdstukken). Het laat zich immers denken dat die zelf-uitleg typisch faciliterend werkt bij de presentatie van problemen al dan niet in context/woorden/symbolen. Ergo: onderzoek van Michelene Chi moet ook antwoorden kunnen geven op de vele vragen die Koedinger & Nathan in hun artikel opwerpen, en nog veel meer vragen die zij hadden kunnen opwerpen maar dat niet deden (waarom eigenlijk niet?).

In dit experiment zijn drie typen strategieën van belang: terugredeneren (unwind), uitproberen (trail and error), en vergelijking oplossen. Bij het oplossen van vergelijkingen worden relatief veel fouten gemaakt (dat hoeft in andere situaties met andere proefpersonen natuurlijk niet zo te zijn). Terugredeneren is relatief robuust tegen fouten, en wordt makkelijk toegepast juist bij contextopgaven. Uitproberen is onhandig, maar levert wel oplossingen op, tenminste bij de in dit onderzoek gebruikte eenvoudige problemen. Dat in psychologisch onderzoek hiermee wordt gewerkt, zonder veel reflectie op wat een en ander betekent voor het onderwijs, valt wel te begrijpen. K & N benadrukken wel dat choice - no-choice een wenselijke volgende onderzoeksstap is, en dat is waarschijnlijk ondertussen ook wel gedaan (ik moet nog op zoek naar meer recent onderzoek). Wat hier op de achtergrond speelt is een ideologie dat het onderwijs er vooral is om leerlingen te leren denken, dat zelf ontdekken de preferentie manier van leren is, dat meerdere strategieën bedenken een goede zaak is. Kortom, een reeks geloofsartikelen — K & N sommen ze op in de vorm van publicaties en organisaties p. 154 — die niet allemaal even goed wetenschappelijk zijn onderbouwd to say the least. K & N trekken in hun onderzoek ook nog eens een steen uit dat empirisch fundament:

Heffernan & Koedinger (1998). A developmental model for algebra symbolization: The results of a difficulty factors assessment. In Proceedings of the Twentieth Annual Conference of the Cognitive Science Society (484-489). pdf scan.

De sectie Instructional implications gaat kort langs bij vier mogelijke instructieve benaderingen. Het onbevredigende hiervan is dat die benaderingen niet op solide theoretische gronden berusten, dus wat hebben we er dan aan? Als springplank naar empirische datavergaring, om alsnog een begin te maken met theorieontwikkeling? Maar dan gaan we het wiel opnieuw uitvinden. Dat de naam van Jan de Lange valt, in deze sectie, geeft te denken. Het bevalt me dus totaal niet wat ik hier lees, en ik zal het misschien nog een paar keer moeten lezen voordat mij duidelijk is wat precies de tekortkomingen zijn in deze instructietheorietjes van de koude grond. Een enkel idee heb ik al wel. Zo lijkt het me van eminent belang om het verschil helder te krijgen tussen symbolisch systeem, en een semantiek die erop past. Sterker nog: op ieder symbolisch systeem passen waarschijnlijk oneindig veel semantieken. Met andere woorden: algebra als een wiskundig systeem, leer je niet beheersen door er telkens semantiek in te mengen, maar juist door dat niet te doen. Het is dan aan de zaakvakken om specifieke semantiek over een bepaalde wiskunde heen te leggen, en zo een helder representatief systeem te bouwen. Ik begrijp dus niet hoe Koedinger en Nathan hier zo naïef kunnen zijn over semantiek en abstracte systemen. Expert blind spot? Eerder het omgekeerde: gebrek aan wiskundige expertise. Afijn, helemaal helder is het natuurlijk nog niet, maar iedereen kan in het gepruts van de ‘realistische wiskunde’ zien hoe ongelukkig die vermenging van abstractie en semantiek uitpakt. Voor abstracte systemen: kwestie van logica, grondslagen, Patrick Suppes dus [Patrick Suppes (2002). Representation and invariance of scientific structures. TheMIT Press. isbn 1575863332 http://cslipublications.stanford.edu/pdf/1575863332.rissbook.pdf ]. Semantiek: ik vermoed toch dat de literatuur over simulaties en vooral over modelleren van natuurverschijnselen (inclusief gedrag) daar helder genoeg over is. Modellen zijn waarschijnlijk altijd specifiek voor de betreffende systemen, ook al zijn ze door en door wiskundig en in die zin algemeen. Beheersing van wiskunde geeft niet vanzelfsprekend de mogelijkheid om snel voor allerlei fenomenen even modellen op te stellen (nog daarvan afgezien dat je die modellen ook weer empirisch moet testen). Ik kom iedere keer weer op hetzelfde uit: modelleren wordt door rekendidactici als iets simpels beschouwd, maar dat is volkomen ten onrechte. Wel degelijk een voorbeeld van expert blind spot, al is het wel een bijzonder voorbeeld, en heeft blinde vlek meer met misvattingen over psychologie te maken. Nou, dat is dus waar ik spijkerharde theorie en empirie bij wil hebben, en die moet al lang en breed beschikbaar zijn, schat ik zo in. Al in de middeleeuwen was de logica zeer ver ontwikkeld.

Alan H. Schoenfeld (1985). Mathematical Problem Solving. Academic Press. isbn 0126288704

The book still stands as his major publication (see his Berkeley webpage). As far as I know, the first edition from 1985 is also the last one. In 2012 Schoenfeld looked back on the book, in Seoul, Korea : How we think: A theory of human decision-making with a focus on teaching. pdf 5 Mb. In a 2013 publication (see below) he did some more looking back; quite revealing, I think. For example: it is an ego-document, in the sense that Schoenfeld presents his thinking as originating with Schoenfeld, apparently with no debt to other thinkers/experimenters on the subject. Amazing. Makes one think of Hans Freudenthal.

• ( . . . ) Mathematical Problem Solving offered a framework, not a theory. Above and beyond pointing out what is important — that is what a framework does — a theory should provide rigorous explanations of how and why things fit together. That is what my current work is about. What I have been working on for the past 25 years is a theoretical approach that explains how and why people make the choices they do, while working on issues they care about and have some experience with, amidst dynamically changing social environments.

• Alan H. Schoenfeld (2013). Reflections on problem solving theory and practice. The Mathematics Enthusiast TME, vol 10, nos.1&2, p. 9 pdf.
  Abstract: In this article, the author reflects on the current state of mathematical problem solving, both in theory and in instruction. The impact of the book Mathematical Problem solving (Schoenfeld, 1985) is also discussed, along with implications of problem solving today with the advent of 21st century technologies.
• Alan H. Schoenfeld (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws: Handbook for Research on Mathematics Teaching and Learning (pp 334-370). pdf

The word problem in primary education is an old format already known in Babylon.

Scientific research is one thing, teaching word problems another. The following publication brings them togther in an interesting way: it shows how it is possible for a teacher to get to grips with word problems in sixth grade. I located the article by googling on the title of the Verschaffel c.s. book, of course.

Shigehiro Kinda (2012). Generating multiple answers for a word problem with insufficient information. Instructional Science, 40, 1021-1031. preview

Onderwijs-ideologisch gekleurd? De eerste zinnen van het artikel wijzen wel in die richting.

Santiago Vicente, Jose Orrantia & Lieven Verschaffel (2007). Influence of situational and conceptual rewording on word problem solving. British Journal of Educational Psychology, 77, 829-848. abstract

contexten en ‘realistisch rekenen’

Ik heb dit deel verplaatst naar een eigen webpagina contexten.htm. Zie overigens ook die voor het werk van Adri Treffers (hier), de Nederlandse auctor intellectualis van contexten (zijn proefschrift gaat er in zijn geheel over, overigens zonder ook maar een snipper empirisch onderzoek).

Lieven Verschaffel, Brian Greer and Erik de Corte (2000). Making sense of word problems. Lisse: Swets & Zeitlinger.

• Moeilijk vindbaar; de KB heeft een exemplaar ter inzage
• Besproken door Bert van Oers in Pedagogische Studiën, 78 nummer 3. (niet online beschikbaar).
• review Christoph Selter;

  ---

Lieven Verschaffel, Brian Greer, Wim Van Dooren and Swapna Mukhopadhyay (Eds.) (2009). Words and Worlds. Modelling Verbal Descriptions of Situations. Sense Publishers. preview: Introduction, Chapters 1, 2

Lieven Verschaffel, Wim Van Dooren, Brian Greer and Swapna Mukhopadhyay (2010). Reconceptualising Word Problems as Exercises in Mathematical Modelling, Die Rekonzeptualisierung von Textaufgaben als Übungen in mathematischer Modellierung. Journal für Mathematik-Didaktik, 31, 9-29. (From the issue entitled Empirical Research on Mathematical Modelling) abstract

• From the abstract: During the last 20 years, many scholars have argued in various ways that the (traditional) practice of word problems in school mathematics does not foster in students, indeed inhibits, a genuine disposition towards mathematical modelling and applied problem solving. In this article we give a brief review and discussion of this research, including a summary of earlier work culminating in the book by Verschaffel et al. (2000 ) and with special attention to the more recent empirical work.
• Rolf Biehler & Dominik Leiss (2010). Empirical research on mathematical modelling. Journal für Mathematik-Didaktik, 31, 5-8. [Introduction to the special issue]
• Dominik Leiss, Stanislaw Schukajlow, Werner Blum, Rudolf Messner & Reinhard Pekrun (2010). The Role of the Situation Model in Mathematical Modelling — Task Analyses, Student Competencies, and Teacher Interventions. Journal für Mathematik-Didaktik, 31, 119-141. abstract
  From the abstract: The results show that: (1) strategies for constructing an adequate situation model have a significant influence on modelling competence, (2) mathematical reading competence and intra-mathematical competence can explain almost one third of the variance of the performance on the modelling test, (3) teacher interventions may encourage students to adopt strategies facilitating the construction of situation models, but an increase of modelling competence requires separate strategy training.
• Luzia Zöttl, Stefan Ufer & Kristina Reiss (2010). Modelling with Heuristic Worked Examples in the KOMMA Learning Environment. Journal für Mathematik-Didaktik, 31, 143-165. abstract
  From the abstract: . . . first results of a large-scale field study examining the effectiveness of example based learning for initial skill acquisition in the field of modelling. The results presented here take into account short-term as well as longterm effects on the learning of 8th-grade students. They suggest, that the participants’ modelling competencies increased significantly during the training but that long-term effects were much smaller.

Desoete, A., Roeyers, H., & De Clercq, A (2003). Can offline metacognition enhance mathematical problem solving? Journal of Educational Psychology, 95, 188-200. abstract

• Genoemd als voorbeeld van hoe goed onderzoek wel degelijk mogelijk is in Wilbrink, Hulshof & Pfaltzgraff (2012) html van concept-versie

H. O. Pollak (1969). How can we teach applications of mathematics? Educational Studies in Mathematics, 2, 393-404. preview [Tijdschrift is ook toegankelijk via abonnement op de KB, behalve de meest recente 5 jaren]

• Examples of the regular kinds of nonsense in word problems. Amusing. Wordt de laatste jaren geciteerd: zie hier

Patrick Suppes, Elizabeth F. Loftus, Max Jerman (1969). Problem-Solving on a Computer-Based Teletype. Educational Studies in Mathematics, 2, 1-15. Publicaties van Patrick Suppes zijn integraal online beschikbaar, zoals ook deze.

Torulf Palm (2008). Impact of Authenticity on Sense Making in Word Problem Solving. Educational Studies in Mathematics, 67, 37-58 [nog geen pdf kunnen downloaden] [pdf via KB JSTOR] abstract

Kan interessant zijn vanwege empirisch materiaal over hoe leerlingen in klas 5 (groep 7) ‘realistische’ woordproblemen aanpakken. Daar isimmers niet zo gek veel informatie over beschikbaar.

Murad Eld Jurdak (2006). Contrasting perspectives and performance of high school students on problem soving in real world situated, and school contexts. Educational Studies in Mathematics, 63 #3, 283-301.

Olive Chapman (2006). Classroom practices for context of mathematics problems. Educational Studies in Mathematics, 62, 211-230

Lieven Verschaffel, Brian Greer and Erik de Corte (2000). Making sense of word problems. Lisse: Swets & Zeitlinger.

• This is quite an impressive review of research in Leuven and Belfast, and therefore also of the field as a whole. On a personal note: the book meticulously spells out the bad practices in artithmetics and mathematics education, at least as far as these damned word problems are concerned. As such, it provides lots of arguments to stop bad practices, in classrooms as well as in national tests and examinations. I will use this work to make my criticisms of, for example, the Nationale Rekentoets html more to the point.
• Regrettably, the book or its chapters are not available online. Happily, many of the sources mentioned in this book are; you will (in due time) find them in this page.
• As the title indicates, the book is about word problems as such, emphasizing their mathematical or educational sense or nonsense. Other research, such as Van Essen or Verdwaald (see below). sees upon aspects of word problems that are problematic in, for example, a linguistic sense.
• 7. Mathematical modeling of aspects of reality.
  • The authors would like to see word problems used as good problem solving exercises or even projects. I do not quite see the natural link between empirical research into the abuses of the word problem in education, and the proposal to use word problems in mathematical problem solving. The latter activity could do very well without the bad connotations of the word problem.
  • This chapter necessarily is rather philosophic, and in places also psychologistic (misplaced use of psychological concepts).
  • First the philosophic part. The chapter's subject is 'the relationship between aspects of reality and mathematical structures.' The naive conception behind the abuse of word problems as researched by Verschaffel and others is 'that all application problems can be appropriately responded to by simple mathematical formulations, while ignoring complications in the real-life situations described.' They try to replace this naive notion by some les naive notions. I suspect, however, that the subject still is treated superficially, ignoring fundamental work on mathematical modeling - more often than not this being measurement - by Patrick Suppes, among others. Some of the others are: Luce, Krantz, and Tversky (for references see the measurement page). An example is the measurement of length: one piece of cloth has length a, another similar piece of cloth has length b, what is the length of the two pieces cloth lined up? Why would it be appropriate in this case to take that length to be a + b? I did expect Verschaffel a.o. to treat this case in some detail, instead they refer to a bunch of references not knit together very tightly. I have yet to study this literature, however. In the mean time, I must thank Verschaffel and others for the lead in the direction of the work on mapping aspects of reality into numbers (as one possible mathematical structure to map into). p. 125:"We propose a characterization of mathematics as having two faces, one being its role in describing aspects of the world, and the other the constructio of abstract structures; modeling forms the link between these two aspects."
  • I am not particularly happy with the above characterization. My intuition is that it is an outright faulty one. Using mathematics to describe aspects of the world is mathematical modeling, is mapping aspects of the world into an abstract structure. Maybe this is a fine point without repercussions on the research project of Verschaffel and others; even so, it would be better to establish the basics in an appropriate way.
  • John has 7 apples and Mary has 3. How many do they have altogether?
    
    How many cookies are there in 3 boxes with 4 cookies in each box? [p. 126]
    By the way, Verschaffel a.o. never seem to comment on the style problem of using numbers '3,' '7' instead of 'three,' 'seven' in word problems. Such a breach of style emphasizes that the numbers themselves as the crucial part of the word problem.
    Looks simple, yet understanding these operations is psychologically complex, as well as philosophically. Wittgenstein is called on for the philosophical part, a number of other authors for the psychological one: "e.g. Carpenter, Moser, & Romberg, 1982; Fuson, 1992; Greer, 1992; Harel & Confrey, 1994; Hoebert & Behr, 1989; Verschaffel & De Corte, 1993, 1996." Presumably, lots of others could have been mentioned here as well. I'd like to see the work of e.g. Susan Carey mentioned (or other people in her line of research). I will look into the Verschaffel a.o. references.
  • Carpenter, Moser, & Romberg, 1982
• 8. Socio-cultural contexts of word problems.
  • Cooper and Dunne 1998, see below

Cathy Claman (2003). 10 Real SAT’s. New York: College Entrance Examination Board.

• In a negative sense, this kind of publication/test is an important example of the main problem of word problems: their situated character.
• On p. 142 figures a table of concordance between words and math operations. For example:
• More than, older than, farther than, greater than, sum of:   +   Addition.
• The sum of two integers is 36.     x + y = 36.
• My comment: this is a direct invitation for students not to think, but to act on associative memory. For example: A ship transports 25 sheep and 5 less goats than sheep. How old is the captain? The College Board would like you to do a subtraction operation here.

Tolar, T. D., Fuchs, L., Cirino, P. T., Fuchs, D., Hamlett, C. L., & Fletcher, J. M. (2012, June 18). Predicting Development of Mathematical Word Problem Solving Across the Intermediate Grades. Journal of Educational Psychology. Advance online publication. doi: 10.1037/a0029020

Noriyuki Inoue (2005). The realistic reasons behind unrealistic solutions: the role of interpretive activity in word problem solving. Learning and Instruction, 15, 69-83. [I have not yet seen this article. I do not have access.

• abstract It is known that school-aged children have a strong tendency to solve mathematical word problems by mechanically calculating numbers even if their calculational answers seem unrealistic. The present study found that undergraduate students also demonstrate this tendency, but many of them could justify their ‘unrealistic’ responses with sensible rationales. In-depth clinical interviews revealed that some of the ‘unrealistic’ responses stemmed from simply an unanticipated, but realistic understanding of the problem situations, while others stemmed from intentionally conforming to the culture of schooling. It is suggested that cognitive functioning in problem solving is highly dependent on an individual's contextual interpretation of the activity.

Carlo Dapueto and Laura Parenti Source (1999). Contributions and Obstacles of Contexts in the Development of Mathematical Knowledge. Educational Studies in Mathematics, Vol. 39, No. 1/3, also: Teaching and Learning Mathematics in Context (1999), pp. 1-21 [Genoa Group]

T. P. Carpenter, J. M. Moser and T. A. Romberg (Eds) (1982). Addition and subtraction: A cognitive perspective. Erlbaum. Not available in questia.com

• Other titles are, however. Searching on the title of this book results in a barrage of relevant titles, some of them, not available to Verschaffel a.o. 2000:
  • Arthur J. Baroody and Ann Dowker (Eds) (2003). The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise. Erlbaum. questia [als eBook in KB]
  • Peter Gates (2001). Issues in mathematics teaching. Routledge. questia
  • John Mason and Sue Johnston-Wilder (2004). Fundamental constructs in mathematics education. RoutledgeFalmer. questia
  • Paul Cobb, Erna Yackel, Kay McClain (2000). Symbolizing and communication in mathematics classrooms: Perspectives on discourse, tools, and instructional design. Erlbaum. questia
  • Robert J. Sternberg and Talia Ben-Zeev (Eds) (1996). The nature of mathematical thinking (pp. 55-80), Mahwah, NJ: Lawrence Erlbaum. questia
  • Liping Ma (1999/2010). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Erlbaum. [als eBook in KB te leen]
  • Handbook of International Research in Mathematics Education Book by Lyn D. English, Maria Bartolini Bussi, Graham A. Jones, Richard A. Lesh, Dina Tirosh; Lawrence Erlbaum Associates, 2002 questia

Gerard van Essen (1991). Heuristics and arithmetic word problems : the effects of instructing heuristic strategies upon the ability of elementary school children to solve arithmetic word problems. Dissertation University of Amsterdam.

• Highly relevant research. A line of research not mentioned in Verschaffel a.o. (2000). [I have seen the dissertation, I have yet to study it thoroughly]

Arnoud Verdwaald (1998). Relational transformations in the process of discourse representation of simple word problems. Dissertation Catholic University of Nijmegen.

• Just as in the case of the Van Essen dissertation: Highly relevant research. A line of research not mentioned in Verschaffel a.o. (2000). [I have seen the dissertation, I have yet to study it thoroughly]

Yeap Ban Har, Ho Siew Yin, Berinderjeet Kaur & Lee Ngan Hoe (2002?). Children making sense during word problem solving. pdf [no longer available? 2-2008] paper. The pictured non-standard word problem is from this research.

• from the abstract ... derived from an investigation into children' responses to standard and non-standard mathematics word problems before and after an intervention programme. Standard word problems can be solved by identifying the correct operation and performing the necessary computation. The story context does not affect the solution. In solving non-standard word problems the story context is important in obtaining a correct solution.

Erik De Corte , Lieven Verschaffel , and Brian Greer (2000). Connecting mathematics problem solving to the real world. In: Proceedings of the International Conference on Mathematics Education into the 21st Century: Mathematics for living (pp. 66-73). Amman, Jordan: The National Center for Human Resource Development. pdf

David H. Jonassen (2003). Designing Research-Based Instruction for Story Problems. Educational Psychology Review, 15, 267-296. pdf [not available any more 2-2008]

• from the abstract Despite current research efforts in mathematics and science problem solving that emphasize situated and socially mediated approaches to solving authentic, complex problems, story problems remain the most common form of problem solving in K-12 schools and universities. Story problems typically present a quantitative solution problem embedded within a shallow story context. Most often, students use a procedural approach to their solution, directly translating story values into solvable algorithms. (...)This research review shows that solving story problems requires that learners construct a conceptual model of the problem that integrates the situational (story) content with an understanding of the semantic structure of the problem based on the principles of mathematics or science being practiced in the problem.

Jacobse, Annemieke E. and Harskamp, Egbert G.(2009) 'Student-controlled metacognitive training for solving word problems in primary school mathematics', Educational Research and Evaluation, 15: 5, 447 - 463

Archimedes The cattle problem. A quite complicated word problem which Archimedes solved in epigrams, and which he communicated to students of such matters at Alexandria in a letter to Eratosthenes of Cyrene. html site by Chris Rorres.

• The answer is a number of 206545 digits. The site develops the solution, as well as one programmed in Maple. Even Archimedes might be amazed.
• A fine example of a word problem as mathematical amusement.

A snail was asked for dinner by a swallow. The swallow's nest was a mile away. The snail could travel only a twelfth of a foot a day. Say now how many days it took the snail to appear at the dinner.

This word problem dates from approximately 800, it is by Alcuin, Europe's (Charles the Great's) first 'school' teacher, and it already illustrates one characteristic of many word problems: it really is an absurd question, demanding an absurd answer. Assume a mile to be 7000 feet ..... .

Why wasn't it absurd to Alcuin? Well, the goal of posing this kind of questions was to sharpen the boys' minds. In the restricted sense of exercising arithmetics only, of course. The means were seen as adequate to reach this goal .... . The pupils, of course, were not in a position to question the master's questions, as is the case for today's pupils in elementary education.

A medieval textbook writer using realistic word problems more often than slightly absurd ones is Fibonacci (see Sigler's (2003) translation of his voluminous work, mentioned below).

Anton J. H. Boonen (2015). Comprehend, visualize & calculate: Solving mathematical word problems in contemporary math education. Dissertation msterdam: Free University. downloads

Discussie gestart op Twitter twitter thread

• Today Anton Boonen defended his dissertation on solving mathematical word problems in contemporary math education.

  Jos Beishuizen questioned Anton on the omission of Freudenthal's name, & and on what arithmetics is: pure, or embedded in context problems? If I heard him correctly, Anton believes arithmetics should be its application in non-routine contexts (my words). I thoroughly disagree. ‘Routine’ word problems might be part of math educ, as far as I am concerned, granted they do not take up a lot of time. They fall squarely into the rubric of problem solving I have treated here (in Dutch, but mark the English literature used) ‪http://benwilbrink.nl/projecten/toetsvragen.7.htm‬. ‘Non-routine’ word problems imo do not deserve a place in math education, except maybe for illustrative purposes: they test for intelligence. Therefore, I am interested in the results of Anton’s research in this way: do the results support use in math education, or not? The argument hinges on the meaning of what Anton calls intellectual skills (and therefore teachable): aren't they just intelligences? Why ‘non-routine’ problems do not belong in summative tests: A.D. de Groot (yes, him) 1970 http://www.benwilbrink.nl/publicaties/70degroot.htm. A revolutionary article.

Catherine Thevenot and Pierre Barrouillet (2015). Arithmetic Word Problem Solving and Mental Representations 1pp 58-179 abstract In Roi Cohen Kadosh & Ann Dowker (Eds.) (2015). The Oxford Handbook of Numerical Cognition. Oxford University Press. [UB Leiden PSYCHO C6.-172]

Tinne Dewolf, Wim Van Dooren, Emre Ev Cimen & Lieven Verschaffel (2013). The Impact of Illustrations and Warnings on Solving Mathematical Word Problems Realistically The Journal of Experimental Education, 1-18. researchgate.net

Discussion of the reults also in terms of cognitive load theory. (split-attention effect)

• Contrary to the authors' expectation, findings from both studies showed that neither the illustration nor the warning, or even the combination of both manipulations, had a positive impact on the number of realistic reactions.

  The Impact of Illustrations and Warnings on Solving Mathematical Word Problems Realistically. Available from: https://www.researchgate.net/publication/271623013_The_Impact_of_Illustrations_and_Warnings_on_Solving_Mathematical_Word_Problems_Realistically?showFulltext=true [accessed Oct 17, 2015].

Menso Folkerts (1978). Die älteste mathematische Aufgabensammlung in lateinischer Sprache: Die Alkuin zugeschriebenen Propositiones ad acuendos iuvenes. Überlieferung, Inhalt, Kritische Edition. Österreichische Akademie der Wissenschaften, Mathematisch-naturwissenschaftliche Klasse, Denkschriften 116, 1978, S. 13–80.

Alcuin. On internet more accessible presentations of the 53 or 56 puzzles, such as J J O'Connor and E F Robertson: Propositiones ad acuendos iuvenes by Alcuin webpage Lieven Verschaffel, Brian Greer and Erik de Corte (2000). Making sense of word problems. Lisse: Swets & Zeitlinger.

Summarizes research by the authors, while integrating the research of others on the subject of word problems.

Barry Garelick (March 17, 2017). Ed Speak, Dept. blog

‘To problem solve’: lingo of progressivism.

  http://www.benwilbrink.nl/projecten/wordproblems.htm http://goo.gl/JaVJ7