Students' argumentation schemes in terms of solving tasks with negative numbers

(1)

HAL Id: hal-01281845

https://hal.archives-ouvertes.fr/hal-01281845

Submitted on 2 Mar 2016

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Students’ argumentation schemes in terms of solving tasks with negative numbers

Mathias Hattermann, Rudolf Vom Hofe

To cite this version:

Mathias Hattermann, Rudolf Vom Hofe. Students’ argumentation schemes in terms of solving tasks with negative numbers. CERME 9 - Ninth Congress of the European Society for Research in Math- ematics Education, Charles University in Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic. pp.281-287. �hal-01281845�

(2)

of solving tasks with negative numbers

Mathias Hattermann and Rudolf vom Hofe

Bielefeld University, Faculty of Mathematics, Bielefeld, Germany, mathias.hattermann@uni-bielefeld.de

In the context of an experimental project targeted at im- proving the teaching and learning of negative numbers, we explored the argumentation schemes of students in terms of solving respective tasks. The test items used to record the learning progress of the students were tak- en from the large-scale PALMA-study. While learning gains were above average, the argumentation schemes of our experimental students show both, expected and unexpected patterns. Interestingly, we find well per- forming students with a preference for metaphorical reasoning instead of a mixed argumentation including formal reasoning.

Keywords: Negative numbers, lesson studies, metaphorical reasoning.

THEORETICAL FRAMEWORK

The Theory of Grundvorstellungen (GVs)

The theory of Grundvorstellungen (GVs) has a long tradition in Germany (e.g., vom Hofe, 1995). It reflects on the fact that the intuitive level of thinking is often responsible for the understanding and building up of mathematical knowledge as well as for problems in mathematical thinking in the sense of Fishbein:

It is very well known that concepts and formal statements are very often associated, in a person’s mind, with some particular instances. What is usually neglected is the fact that such particular instances may become, for that person, uni- versal representatives of the respective concepts and statements and then acquire the heuristic attributes of models. (Fishbein, 1987, p. 149f.) In contrast to the opinion that mathematics is pure logic, GVs take social aspects into account as well as the environment in which mathematical concepts are built up. The theory of GVs has its anglo-american

correlate in the term concept image which forms a fundamental part of Tall and Vinner’s theory of mental models:

We shall use the term concept image to describe the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes.

It is built up over the years through experiences of all kinds, changing as the individual meets new stimuli and matures. (Tall & Vinner, 1981, p. 2) The concept image must be differentiated from the concept definition that consists of formal defini- tions dealing with mathematics. The theory of GVs (e.g., Kleine, Jordan, & Harvey, 2005) exemplifies the given idea of concept image in more detail and wid- ens the perspective to descriptive and prescriptive utilization.

GVs as a prescriptive notion describe adequate interpretations of the core of the respective mathematical contents which are intended by the teacher in order to combine the level of formal calculating with corresponding real life sit- uations. In contrast, the term GV in descriptive empirical studies is used also as a descriptive notion to describe ideas and images which students actually have and which usually more or less differ from the GVs intended by mathematical instruction. (vom Hofe et al., 2008, p. 49) GVs can be formulated in different contexts, such as fractions (part of a whole, operator, ratio), subtraction (taking away, supplementing, comparing) or functions. On the one hand the descriptive and prescriptive characters of GVs allow on the one hand the formulation of desirable GVs in different contexts and on the other hand the testing of individuals’ GVs .

(3)

Students’ argumentation schemes in terms of solving tasks with negative numbers (Mathias Hattermann and Rudolf vom Hofe)

282 GRUNDVORSTELLUNGEN AND

METAPHORICAL REASONING

Metaphorical reasoning is fundamental to human thinking and can lead to the construction of mental models (English, 1997). According to Lakoff and Nunez (1997) metaphors can serve as a tool to understand difficult and new concepts in mathematics. They see mathematics on the one hand as a product of human imagination and on the other hand as a product of the embodied mind. Sfard (1994) emphasizes that metaphors allow the translation of bodily experiences into abstract mathematical ideas. This means that a metaphor maps from a source domain (experiences) to a target domain (mathematics) and theses experiences help to understand the abstract ideas in mathematics. Lakoff and Nunez (1997) propose four grounding metaphors to understand basic arithmetic.

These four grounding metaphors (Motion along a Path (numbers as point locations or movements), Object Collection (bringing together, taking away), Object Construction (combining, decomposing), Measuring stick (comparing) link structures from every-day life to mathematics. But these grounding metaphors are insufficient to handle operations with negative numbers, so Lakoff and Nunez (1997) propose to stretch these metaphors to be useful for the explanation of the enlarged number domain including zero and negative numbers, aware of the fact they get more contrived the more they are stretched. In this respect metaphors can be used to build up GVs for different contexts in mathematics. However, it is well known that the use of metaphorical reasoning also has dis- advantages since it is less efficient compared to facts or algorithms. Thus, it is important to investigate the relation between prescriptive GVs of an individual that is based on metaphorical reasoning (in our case a card game called Plus-Minus-Game) and the compe- tence of the individual to solve computational tasks.

THE PALMA-PROJECT

The Project for the Analysis of Learning and Achievement in Mathematics (PALMA) focused on

students’ development in mathematics throughout grades five to ten in Bavaria, Germany (vom Hofe et al., 2008), with probands of eleven to sixteen years of age. The theoretical framework stressed the differ- ences between the performance of algorithmic operations and the activation of basic concept images/

GVs. Because of its longitudinal design, the PALMA

study allowed new insights into the achievement development of students as well as into the impact of modelling competences that require the utilization of basic concept images/GVs. PALMA included an- nual assessments between 2002 and 2007. About 2000 students from 83 classes and 42 schools participated in the study so that the development of achievement was observed over a period of 6 years (vom Hofe et al., 2008).

CURRENT RESEARCH ON NEGATIVE NUMBERS The genesis of negative numbers in mathematical his- tory is of historical-cultural significance (Schubring, 1986). Since decades negative numbers have been part of the curriculum in many countries and for Germany it was Freudenthal who pointed out that dealing with them must be studied in detail (Freudenthal, 1973).

He talks about didactic models of negative numbers, a problem that is still relevant today. Nearly all exist- ing models dealing with everyday life and negative numbers contain specific insufficiencies, so that an ideal model for their teaching and instruction does not exist. Fishbein takes reference to the fundamental article by Glaeser (1981) and refers to this issue as follows:

The difficulty of accepting the negative numbers as meaningful mathematical entities derives from the difficulty of identifying a good intuitive, familiar model which would consistently satis- fy all the algebraic properties of these numbers, says Glaeser. As a matter of fact, such a model does not exist. One may create some models, but only by using a system of artificial conventions.

(Fishbein, 1987, p. 100)

Current research in Germany concerning negative numbers is insufficient. Nevertheless, schoolbooks around the world (with only a few exceptions) pro- mote the utilization of didactic models such as temper- atures, balance-debt-models or elevator-models. With regard to our focus of interest, two publications are of peculiar interest. Chiu (2001) analyzes the utilization of metaphors (in the sense of Lakoff and Nunez 1997 standing in line with the discussed theories of GVs and concept images) for the explanation of computations with negative numbers in a quantitative study and with the help of additional clinical interviews.

He concludes

(4)

…that both novices and experts have the same arithmetic metaphors but use them differently.

[…]. Experts used metaphors less often in favor of more efficient methods. Both used metaphors when they faced difficulties. However, novices had more difficulties and used metaphors more often. (Chiu, 2001, p. 113)

In her work with 99 students Kilhamn (2008) also ex- amines forms of metaphorical reasoning. 23 probands justified their answers to the task “-3 − (-8) = ” by metaphorical reasoning using the thermometer, money debts or movements along a number line in their explanations. The result of the study is surprising because all students who solely referred to metaphorical reasoning failed (n = 14), whereas every student who used both metaphorical reasoning and arithmetic rules succeeded (n = 9) in solving the task -3 − (-8) = (Kilhamn, 2008, p. 6). In spite of the small amount of probands, it seems as if both, metaphorical reasoning and formal computations are substantial elements when dealing with tasks related to negative numbers.

RESEARCH PROJECT: DESIGN, QUESTIONS AND METHODOLOGY Design

Our current project focuses on the generation of specific GVs and argumentation schemes based partly on metaphorical reasoning with regard to negative numbers in a long term-study. As the cited studies show, GVs as well as metaphorical reasoning are very relevant for the effective learning of negative numbers, even though a formalized understanding must complete the GVs. It is known that the stretching of the grounding metaphors (see the theoretical part) to handle negative numbers is problematic and yields not always to the desired results, so we will use a card game to build up GVs and to research students’ argumentation schemes for the addition/subtraction of integers. Our project, lasting from 2012 to 2015, is based on a collaboration between Bielefeld University and the Laboratory School Bielefeld. The latter is inno- vative with respect to both, its educational profile as well as its Teacher-as-Researcher Model (Hollenbach

& Tillmann, 2009). This model allows teachers to eas- ily perform educational or subject-related research that compensates for parts of their teaching obliga- tions. Three teachers and three researchers have collaborated to plan a 12-weeks-unit of instruction for negative numbers in grade 7 (n = 21 students), in

which GVs play a fundamental role. The unit is divided into three parts: 1. introduction, 2. addition/subtraction, 3. multiplication/division of negative numbers.

In the second part we introduced the Plus-Minus- Game which is explained in the next paragraph. It is designed to build up GVs and to foster metaphorical reasoning for the addition and subtraction of negative numbers. Teachers and researchers worked together on the methodological basis of Lesson Studies (e.g., Hart, Alston, & Murata, 2011). This collaborative meth- od has proven to be effective with regard to teachers’

professional development.

The Plus-Minus-Game

The game consists of a dice with a green, a blue, a red, a yellow, a black and a white face, 11 green and 11 red cards. The 11 green cards are labelled with the numbers from 0 to 10. The 11 red cards are labelled with the numbers from -10 to 0. The game is played by three to four players. After having shuffled the cards, you put a deck of green cards and a deck of red cards on the table. Every player draws a card and puts it on the table without covering it. The youngest player begins to throw the die. For each colour the die shows, there are specific instructions what to do.

Green: Take a green card from the deck. Blue: Give any green card to your left neighbor. Red: Take a red card from the deck. Yellow: Give any red card to your left neighbor. Black: Give the red card with the highest absolute value to your left neighbor. White: Give the red card with the lowest absolute value to your left neighbor. The aim of the game is to get the highest score in total. The game is over when all cards are collected. After playing the game, we introduced the notation presented in Figure 1 to motivate the utilization of brackets as claimed by Malle (2007). We used brackets to differentiate between a card value and a score. In an equation, we interpret the first summand as a score (without brackets), the second summand as a card value (with brackets) and the result as a score (without brackets). Figure 1 focuses on the notation in addition and subtraction tasks. The interpretation of the minuend as a score in a subtraction task is crucial.

Example: 3 − (-6) = 9 Interpretation: The player has several cards (e.g. (5), (-6) and (4) which are added up to 3). He has to give away the card (-6), so the player’s new score is 9. If someone interprets the minuend as a card value instead of a score, the interpretation will fail because it is not possible to give away the card (-6) while holding only the card with value (3). Problems resulting from such misinterpretations are analyzed

(5)

284 in detail in Hattermann (2013). For more information

concerning teaching experiences see Hattermann, vom Hofe and Viehmeister (2014).

Research questions

In this paper we focus on three selected research questions of the project. Since the students who took part in the PALMA study did not participate in les- sons focusing on the development of GVs, as it was the case with our experimental group, we formulate the first question: 1. How does the experimental class from the Laboratory School perform in answering the PALMA-items on negative numbers in comparison to the PALMA-students after the instruction unit on negative numbers? 2. What argumentation schemes (GVs) can be identified in the students’ explanations for the tasks “(-5) + (-7) =“ and “(-9) − (-4) =” after having fin- ished the course of addition/subtraction of negative numbers? 3. Are there any particular interrelations between both, the argumentation schemes identified and the students’ success by solving computational tasks?

Methodology

After the second and the third part of the course the students took part in an evaluation in which they computed different tasks and answered questions that sought to identify underlying conceptions and GVs. In the first evaluation students had to answer the following question:

Imagine your classmate Jacob was ill last week and he was not able to attend class. Write a short

letter in which you explain to him what has to be done in the following tasks and explain why:

a) (-5) + (-7) = and b) (-9) − (-4) =. You may use all devices for an explanation that we used in class.

The written letters were analyzed with regard to the argumentation schemes used by the students by means of a qualitative text analysis. In a first analysis the research group analyzed together five letters to create categories of argumentation schemes. In the following two researchers analyzed independently students’ letters to Jacob and decided for one of these categories. In a last step inconsistent classifications were discussed and the categories were revised. In a second step the identified individual argumentation schemes were confirmed with semi-standardized interviews. In these interviews tasks as “(-3) + (-7) =”

occurred (as in the letter to Jacob) and students’ were motivated to solve these tasks and to explain their argumentation explicitly. They could use a number line, the plus-minus-game or other explanations. The video analysis was carried out by two researchers. They assigned independently argumentation schemes to students’ statements in the interview. Finally, the argumentation patterns of the students were compared to their performance in solving 14 computational tasks e.g. “(-12) + (-7) =”. Our students worked on the PALMA items on negative numbers at the end of the school year (like the students in PALMA).

RESULTS

Exemplary results from the experimental class in comparison to PALMA

To answer our first research question, we compared the solution rates with regard to PALMA-items (grade 7) on negative numbers of the representative PALMA-sample and with our experimental group at the Laboratory School Bielefeld. For example, the task termed Justifying from PALMA had a solution frequency of only 19%, the lowest solution rate of all items on negative numbers in the PALMA-sample. It read: “If you add two negative numbers, you get a negative number again. Is this statement correct? Justify your answer.” (Figure 2)

Figure 2 displays the task in German together with the solution of Leo (pseudonym) from our experimental group. Leo writes: “Yes, it is right, because if you add more negative numbers it becomes less.” The overall solution rate of 43% of our experimental group for

Figure 1: Notation and interpretation for addition and subtraction tasks with integers

(6)

this exemplary task provides evidence for our success in building up GVs for mathematical concepts.

Altogether, the students worked on 20 PALMA-items dealing with negative numbers. The solution rates of the experimental class were at least 10% higher compared to the PALMA-rates for six tasks. One example of these tasks is the following: “The water level of a water reservoir declines to 8cm. On the next day the water level rises about 3cm. How does the water level change in these two days.” In contrast to that, the experimental group solution rate was at least 12% lower than for the PALMA-group for four items. These are the following items: (+9) · (-8) = ; (-27) + (+3) = ; (-6) · (-8)

= In the fourth task the probands had to determine a temperature on a thermometer and the solution rates were 71% (experimental group) respectively 86%

(PALMA). We explain these results by the fact that the plus-minus-game was not used to explain the multiplication of integers. Furthermore the solution rates of the thermometer task are high in comparison to the other solution rates. This gives evidence that this task is one of the easiest tasks and mistakes occur by care- lessness. For the ten tasks remaining, the difference of solution rates did not exceed 9%. One example of these tasks is the following: “Mr. Knodel has 450€ on his bank account. He transfers an invoice amount. The actual balance is -300€. What amount did he trans- fer?” Despite of the small amount of probands in our experimental class, the results show that the average achievement of our experimental group is compara- ble to the achievement of the PALMA-probands, who came from Bavaria being one of the high-performance regions in Germany.

Exemplary results from the evaluation and the interview-study

In the letters that were written to Jacob during the first evaluation, we identified four consistent argumentation schemes that were verified in the following interviews. The first one is called formal reasoning: e.g.

“In both tasks you can use the additive inverse of the second number and change the arithmetic operator (-3) − (-5) is the same as (-3) + (+5) = (+2).”

The second scheme is called metaphorical reasoning.

In our case, the students’ respective justifications were based on the number line or the Plus-Minus- Game: e.g. “Dear Jacob, in a) you have -5, the + means that you get a card and because this card is -7 you have 7 less so -12. “

The third scheme is called mixed argumentation. It contains parts of both formal and metaphorical rea- soning: e.g. “In task b) your score is -9 and then you give your [card] -4 so you get -5 you get the same by calculating -9 + additive inverse (+4).” This example shows both metaphorical reasoning and formal reasoning. Whereas the explanation with the card game is identified as a kind of metaphorical reasoning, the identification of +4 as the additive inverse of -4 hints to a formal understanding.

Figure 2: Task for justification from Leo from the experimental class

Figure 3: Example for formal reasoning

Figure 4: Example for metaphorical reasoning

(7)

286 In the fourth category no justification can be identi-

fied, e.g.: “Because I learned it this way.” In a last step we compared the students’ argumentation scheme(s) with their results on 14 computational tasks on negative numbers as for example “(-12) + (-7) =” (Figure 6).

We show only the results of these 17 out of 21 students who wrote a letter to Jacob and took part in the evaluation.

As Figure 6 displays, 13 out of 17 students score at least 11 out of 14 points. Four students score less than 11 points whereas three of these students are not able to justify or explain their strategies in the letter to Jacob or to the interviewer. We find very good results (13 or 14 points) of students who used mixed argumentation, formal reasoning or metaphorical reasoning, whereas the mixed argumentation seems to be dominant with high achievers. This result is in line with those of Kilhamn (2008). In contrast to her study, however, we find students with results of 11 and 13 points in spite of their preference for only metaphorical reasoning.

An explanation for this result is the structure of the Plus-Minus card game, which undoubtedly repre-

sents a form of metaphorical reasoning, while it also represents the mathematical structure very clearly and omits other contexts, such as those of everyday life. In addition, however, we find three students who used metaphorical reasoning only and scored only 9 and 11 points of 14. One reason for this might be the fact that the Plus-Minus-Game represents addition and subtraction tasks within a maximal number range from -55 to +55, whereas the students had to solve tasks such as -530 + (-210) in the computation part as well.

PERSPECTIVES

In the near future, we will revise the unit on negative numbers and teach it again in more classes. We will use several models, such as the balance-debt-model, the Plus-Minus-Game and a model dealing with the number line to identify more argumentation schemes that are specific for these models. Our aim is to gain more insight into individual problems, when dealing with specific models. Furthermore we will work on an effective combination of models and underlying argumentation schemes for addition/subtraction and multiplication of negative numbers.

Figure 5: Example for mixed argumentation

Figure 6: Argumentation schemes and achieved results in computational tasks

(8)

REFERENCES

Chiu, M. (2001). Using metaphors to understand and solve arithmetic problems: Novices and experts working with negative numbers. Mathematical thinking and learning, 3(2–3), 93–124.

English, L.D. (1997). Analogies, metaphors, and images:

Vehicles for mathematical reasoning. In L. D. English (Ed.), Mathematical reasoning, analogies, metaphors and im- ages (pp. 3–18). Mahwah, New Jersey: Lawrence Erlbaum Associates.

Fishbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht, The Netherlands: Reidel.

Freudenthal, H. (1973). Mathematik als pädagogische Aufgabe.

Stuttgart, Germany: Klett.

Glaeser, G. (1981). Epistémologie des nombres relatifs.

Recherches en Didactique des Mathématiques, 2(3), 303–346.

Hattermann, M., Hofe, R. v., & Viehmeister, F. (2014). Rote Karte?

Ich hab‘ grün! Mathematik lehren, 183, 15–19.

Hattermann, M. (2013). Einführung und erste Rechenoperationen mit ganzen Zahlen: Ein

Erfahrungsbericht. In G. Greefrath, F. Käpnick, & M. Stein (Eds.), Beiträge zum Mathematikunterricht 2013 (pp. 420–

423). Münster, Germany: WTM.

Hart, L. C., Alston, A. S., & Murata, A. (2011). Lesson study re- search and practice in mathematics education: Learning together. Dordrecht, The Netherlands: Springer.

Hofe, R. v. (1995). Grundvorstellungen mathematischer Inhalte.

Texte zur Didaktik der Mathematik. Heidelberg, Germany:

Spektrum Akad. Verl.

Hofe, R. v., Jordan, A., Hafner, T., Stölting, P., Blum, W., & Pekrun, R. (2008). On the development of mathematical modelling competencies: The PALMA longitudinal study. In M.

Blomhoj & S. Carreira (Eds.), Mathematical applications and modelling in the teaching and learning of mathemat- ics. Proceedings from Topic Study Group 21 at the 11th International Congress in Mathematical Education in Monterrey (pp. 47–60), Monterrey, Mexico: ICMI.

Hollenbach, N. & Tillmann, K.-J. (2010). The teacher-researcher model at the Laboratory School: initial concept and today’s practice. In N. Hollenbach & K.-J. Tillmann (Eds.), Teacher Research and School Development. German approach- es and international perspectives (pp. 201–208). Bad Heilbrunn, Germany: Klinkhardt.

Kilhamn, C. (2008). Making sense of negative numbers through metaphorical reasoning. In MADIF - 6 The sixth Swedish Mathematics Education Research Seminar. Retrieved September 2, 2014, from http://www.mai.liu.se/SMDF/

madif6/Kilhamn.pdf.

Kleine, M., Jordan, A., & Harvey, E. (2005). With a focus on

‘Grundvorstellungen’ Part 1: a theoretical integration into current concepts. ZDM - The International Journal on Mathematics Education, 37(3), 226–233.

Lakoff, G., & Nunez, R. (1997). The metaphorical structure of mathematics: Sketching out cognitive foundations for a mind-based mathematics. In L. D. English (Ed.), Mathematical reasoning, analogies, metaphors and imag- es (pp. 21–89). Mahwah, New Jersey: Lawrence Erlbaum Associates.

Malle, G. (2007). Die Entstehung negativer Zahlen: Der Weg vom ersten Kennenlernen bis zu eigenständigen Denkobjekten.

Mathematik lehren, 142, 52–57.

Schubring, G. (1986). Rupture dans le statut mathématique des nombres négatifs. Petit X, 12, 5–32.

Sfard, A. (1994). Reification as the birth of metaphor. For the Learning of Mathematics, 14(1), 44–54.

Tall, D., & Vinner, S. (1981). Concept Image and Concept Definition in Mathematics with particular reference to Limits and Continuity. Educational Studies in Mathematics, 12, 151–169.