Assessing conceptual knowledge through solving arithmetic word problems

We propose to assess conceptual knowledge of mathematical notions by having recourse to isomorphic word problems. We assumed that failing to solve isomorphic problems is an indicator of lack of conceptual knowledge. To reach these conclusions, two experiments were conducted among 4th and 5th grade students. In experiment 1, each student had to solve one isomorphic word problem allowing two solving strategies. Results from experiment 1 showed that semantic context promoted a one-sided perspective, constraining the strategies used. The second experiment questioned if the strategy choices by students reflected a lack of procedural knowledge or difficulty in conceptual knowledge. Students solved several problems with the explicit task of proposing two strategies. Results showed that students struggled to identify the conceptual structure of the problems: they changed their strategy depending on semantic contexts, and did not succeed in adopting two encodings for the same problem. This lack of flexibility does not reflect weak procedural knowledge but difficulties in conceptual knowledge. Once conceptual knowledge is characterized as the ability to get over dependence on semantic context, using word problems as assessment tools for conceptual development opens up educational perspectives.

[1]  B. Rittle-Johnson,et al.  Developing Mathematics Knowledge , 2017 .

[2]  Katherine M. Robinson,et al.  The Understanding of Additive and Multiplicative Arithmetic Concepts , 2017 .

[3]  A. Baroody,et al.  An Alternative Reconceptualization of Procedural and Conceptual Knowledge. , 2007 .

[4]  Michael Schneider,et al.  Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. , 2011, Developmental psychology.

[5]  B. Rittle-Johnson,et al.  Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. , 2007 .

[6]  Michael Schneider,et al.  Developing Conceptual and Procedural Knowledge of Mathematics , 2015 .

[7]  Lieven Verschaffel,et al.  Strategy flexibility in children with low achievement in mathematics , 2007 .

[8]  Susanne Prediger,et al.  Taking away and determining the difference—a longitudinal perspective on two models of subtraction and the inverse relation to addition , 2012 .

[9]  T. P. Carpenter,et al.  Thinking Mathematically : Integrating Arithmetic and Algebra in Elementary School , 2003 .

[10]  A. Miyake,et al.  Individual differences in executive functions are almost entirely genetic in origin. , 2008, Journal of experimental psychology. General.

[11]  Emmanuel Sander,et al.  Arithmetic word problem solving: a Situation Strategy First framework. , 2010, Developmental science.

[12]  Kurt Reusser,et al.  From text to situation to equation: cognitive simulation of understanding and solving mathematical word problems , 1990 .

[13]  J. Hiebert,et al.  Conceptual and Procedural Knowledge in Mathematics: An Introductory Analysis , 1986 .

[14]  M. Alibali,et al.  The development of arithmetic principle knowledge: How do we know what learners know? , 2009 .

[15]  Efraim Fischbein,et al.  Tacit Models and Mathematical Reasoning , 2016 .

[16]  Emmanuel Sander,et al.  Transfer of strategy use by semantic recoding in arithmetic problem solving , 2010 .

[17]  Katherine M Robinson,et al.  Adults' understanding of inversion concepts: how does performance on addition and subtraction inversion problems compare to performance on multiplication and division inversion problems? , 2003, Canadian journal of experimental psychology = Revue canadienne de psychologie experimentale.

[18]  Jeffrey Bisanz,et al.  On “Understanding” Children's Developing Use of Inversion , 2009 .

[19]  Lieven Verschaffel,et al.  Influence of situational and conceptual rewording on word problem solving. , 2007, The British journal of educational psychology.

[20]  Meixia Ding,et al.  Transition from concrete to abstract representations: the distributive property in a Chinese textbook series , 2014 .

[21]  Danièle Coquin-Viennot,et al.  Highlighting the role of the episodic situation model in the solving of arithmetical problems , 2003 .

[22]  B. Rittle-Johnson,et al.  Developing Conceptual Understanding and Procedural Skill in Mathematics: An Iterative Process. , 2001 .

[23]  G. Lakoff,et al.  Where mathematics comes from : how the embodied mind brings mathematics into being , 2002 .

[24]  Miriam Bassok,et al.  Adding Apples and Oranges: Alignment of Semantic and Formal Knowledge , 1998, Cognitive Psychology.

[25]  R. Brissiaud,et al.  Teaching and development: Solving “missing addend” problems using subtraction , 1994 .

[26]  Stephanie M. Carlson,et al.  How specific is the relation between executive function and theory of mind? Contributions of inhibitory control and working memory , 2002 .

[27]  B. Rittle-Johnson,et al.  Not a One-Way Street: Bidirectional Relations Between Procedural and Conceptual Knowledge of Mathematics , 2015, Educational Psychology Review.

[28]  Ann Dowker,et al.  The development of arithmetic concepts and skills: Constructing adaptive expertise , 2003 .

[29]  Susan B. Empson,et al.  A longitudinal study of invention and understanding in children's multidigit addition and subtraction , 1998 .

[30]  Martha W. Alibali,et al.  Defining and measuring conceptual knowledge in mathematics , 2014 .

[31]  Lieven Verschaffel,et al.  Analyzing and Developing Strategy Flexibility in Mathematics Education , 2011 .

[32]  Esther Ziegler,et al.  Consistent advantages of contrasted comparisons: Algebra learning under direct instruction , 2016 .

[33]  G. Deák,et al.  Cognitive flexibility in young children: General or task-specific capacity? , 2015, Journal of experimental child psychology.

[34]  Jeffrey Bisanz,et al.  Chapter 3 Understanding Elementary Mathematics , 1992 .

[35]  P. Pintrich.  The Role of Metacognitive Knowledge in Learning, Teaching, and Assessing , 2002 .

[36]  Malcolm Swan,et al.  Choice of operation in verbal problems with decimal numbers , 1981 .

[37]  Miriam Bassok,et al.  OBJECT-BASED REPRESENTATIONS : TRANSFER BETWEEN CASES OF CONTINUOUS AND DISCRETE MODELS OF CHANGE , 1995 .

[38]  A. Treffers,et al.  The empty number line in Dutch second grades: Realistic versus gradual program design , 1998 .

[39]  K. H. Canobi,et al.  Concept-procedure interactions in children's addition and subtraction. , 2009, Journal of experimental child psychology.

[40]  E. Corte,et al.  Influence of rewording verbal problems on children's problem representations and solutions , 1985 .

[41]  K. Holyoak,et al.  Teaching the Conceptual Structure of Mathematics , 2012 .

[42]  B. Rittle-Johnson,et al.  Flexibility in Problem Solving: The Case of Equation Solving. , 2008 .

[43]  W Kintsch,et al.  Understanding and solving word arithmetic problems. , 1985, Psychological review.

[44]  Mary S. Riley,et al.  Development of Children's Problem-Solving Ability in Arithmetic. , 1984 .

[45]  K. M. Robinson,et al.  Children's additive concepts: Promoting understanding and the role of inhibition , 2013 .

[46]  Kurt Reusser,et al.  Every word problem has a solution—The social rationality of mathematical modeling in schools , 1997 .

[47]  B. Greer.  Modelling Reality in Mathematics Classrooms: The Case of Word Problems. , 1997 .

[48]  Lieven Verschaffel,et al.  Children's understanding of the addition/subtraction complement principle. , 2016, The British journal of educational psychology.

[49]  E. Sander,et al.  Learning to be an opportunistic word problem solver: going beyond informal solving strategies , 2020 .

[50]  Jon R. Star,et al.  The nature and development of experts’ strategy flexibility for solving equations , 2009 .

[51]  J. Oakhill,et al.  The Strategic use of Alternative Representations in Arithmetic Word Problem Solving , 2005, The Quarterly journal of experimental psychology. A, Human experimental psychology.

[52]  K. Begolli,et al.  Teaching Mathematics by Comparison: Analog Visibility as a Double-Edged Sword. , 2016 .

[53]  Tom Hudson,et al.  Correspondences and Numerical Differences between Disjoint Sets. , 1983 .

[54]  Noriyuki Inoue,et al.  The Realistic Reasons behind Unrealistic Solutions: The Role of Interpretive Activity in Word Problem Solving. , 2005 .

[55]  J. Thibaut,et al.  When masters of abstraction run into a concrete wall: Experts failing arithmetic word problems , 2019, Psychonomic Bulletin & Review.

[56]  J. Star.  Reconceptualizing procedural knowledge. , 2005 .

[57]  L. Verschaffel,et al.  Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education , 2009 .

[58]  Lucy Cragg,et al.  The role of cognitive inhibition in different components of arithmetic , 2015 .

[59]  Robert S. Siegler,et al.  The relation between conceptual and procedural knowledge in learning mathematics: A review , 2021, The Development of Mathematical Skills.