Algebraic Reasoning Research Articles

Mathematical Proficiency and Perseverance in Action: The Case of Maria and Andrew*

In response to an expanding definition of mathematical proficiency, educators must attend to what mathematically proficient students should know as well as what they should do. As students are asked to struggle and wrestle with mathematics, educators should expect that students demonstrate perseverance through engagement with difficult mathematics. This has significant implications for seeing one’s self as a learner and doer of mathematics, particularly for those with historically limited access to mathematics-related opportunities. A case study analysis of a pair of students in a second grade classroom engaged in algebraic reasoning revealed a striking example of perseverance in action that supports mathematical learning. Analysis of classroom discourse across the classroom revealed the social and sociomathematical norms of the classroom that supported such perseverance. Developing mathematically proficient students who persevere has widespread implications as it is those who see themselves as doers of mathematics who have the most access to future educational and economic opportunities. *This study was supported in part by a grant (#R305A100181) from the Institute of Education Sciences of the U.S. Department of Education. The views expressed in this paper are the authors’ alone and do not reflect the views/policies of the funding agency.

Analyzing Qualitative Spatio-Temporal Calculi using Algebraic Geometry

Qualitative spatial reasoning is based on calculi which comprise relations and operation tables that encode operations like relation composition. Designing a calculus involves determining these tables and analyzing reasoning properties—a demanding task that is susceptible to errors if performed manually. This paper is concerned with automating computation of operation tables and analysis of qualitative calculi over real-valued domains like the plane 2. We present an approach to specify qualitative relations using polynomial equations that allows methods from algebraic geometry to be applied. This paper shows how reasoning with qualitative relations can be posed algebraically and demonstrates algebraic reasoning using Gröbner base analysis. We evaluate this approach and describe our implementation, which is freely available as part of the spatial reasoning toolbox SparQ.

Characterizing the Development of Specialized Mathematical Content Knowledge for Teaching in Algebraic Reasoning and Number Theory

This article characterizes the development of a deep and connected body of mathematical knowledge categorized by Ball and Bass' (2003b) model of Mathematical Knowledge for Teaching (MKT), as Specialized Content Knowledge for Teaching (SCK) in algebraic reasoning and number sense. The research employed multiple cases across three years from two content courses designed for elementary and middle-level mathematics specialists. Qualitative data were collected and a grounded theory approach to data analysis was employed. The resulting framework characterizes developmental levels of deep and connected mathematical content knowledge for teaching algebraic reasoning and number theory content. The framework consists of four intertwined components related to a teacher's ability to (1) solve problems and justify his/her reasoning, (2) use multiple representations, (3) recognize, use, and generalize conceptually similar tasks, and (4) pose problems. Implications for mathematics teacher education programs are discussed as well as directions for further research.

Spreadsheets as cognitive tools: A study of the impact of spreadsheets on problem solving of math story problems

In this study, we investigated the impact of computer spreadsheets on the problem solving practices of students for math story problems, and more specifically on the transition from arithmetic to algebraic reasoning, through the construction of algebraic expressions. We investigated the relationships among the students' prior knowledge and skills, the verification processes, and the effectiveness of the problem solving tasks. For identifying the factors involved in the problem solving process and their role, in our analysis we employed the Structural Equation Modeling (SEM) approach. We mainly focused on math story problems and on students of tertiary education with little prior experience on the use of computers and spreadsheets. Analysis of the data indicates that spreadsheets can support the transition from arithmetic to algebraic reasoning and this transition is influenced by prior skills of the students relevant to the interaction with the interface (enter formula skills), and the students' frequency of verification of the solution.

Are patterns important? An investigation of the relationships between proficiencies in patterns, computation, executive functioning, and algebraic word problems.

Although mathematical pattern tasks are often found in elementary school curricula and are deemed a building block for algebra, a recent report (National Mathematics Advisory Panel, 2008) suggests the resources devoted to its teaching and assessment need to be rebalanced. We examined whether children's developing proficiency in solving algebraic word problems is related to their proficiencies in patterns, computational, and working memory tasks. Children (N = 151 10-year-olds) were tested twice, 1 year apart, and were administered tests of updating capacities (2 complex span and 1 running span task), computation (from the Wechsler Individual Achievement Test), patterns (function machine, number patterns), and algebraic word problems. Proficiencies on the patterns and computational tasks predicted algebraic proficiency. Proficiencies on the computational and patterns tasks are, in turn, predicted by updating capacity. These findings suggest that algebraic reasoning may be difficult if the child has poor updating capacity and either poor facility with computation or difficulty in recognizing and generalizing rules about patterns.

Algebra readiness for students with learning difficulties in grades 4–8: Support through the study of number

Developing proficiency in algebra is the focus of instruction in high school mathematics courses and is a minimum expectation for high school completion for all students including those with learning difficulties. However, the foundation for success is laid in grades 4–8 (aged 9–14). In this paper, we assert that students' development of algebraic reasoning skills and knowledge is influenced by their understanding of numeracy. We propose that students in grades 4–8 need targeted instruction on three interconnected numeracy concepts. First, we propose that students need a strong conceptual foundation of number and number systems to facilitate computation, estimation and problem solving across number systems. Second, we identify students' understanding of basic number properties as foundational to lawful application of operations. Finally, we note that procedural proficiency, including computational fluency and conceptual understanding, provides a foundation for generalised application of algorithms for solving arithmetic and algebraic problems. Combined, these skills and knowledge enable students to employ algebraic reasoning by generalising their knowledge for solving abstract problems.

For Your Information – May 2011

From students ready to solve pictorial systems of equations, to teachers looking for a creative way to reinforce the concept of variable and solution, to parents looking for extra mathematics challenges for a precocious child—these three booklets offer entertaining doublesided, tear–out activity sheets for practicing algebraic reasoning.

Technology to develop algebraic reasoning

Try these approaches to boost your students' understanding and higher-order thinking skills.

Patterns to develop algebraic reasoning

Such classic problems as the Painted cube help teachers and students deepen their understanding and expand their algebraic thinking.

The Number Crunch game: a simple vehicle for building algebraic reasoning skills

A newspaper numbers game based on simple arithmetic relationships is discussed. Its potential to give students of elementary algebra practice in semi-ad hoc reasoning and to build general arithmetic reasoning skills is explored.

Automatic reasoning for geometric constraints in 3D city models with uncertain observations

This paper presents a novel approach to automated geometric reasoning for 3D building models. Geometric constraints like orthogonality or parallelity play a prominent role in man-made objects such as buildings. Thus, constraint based modelling, that specifies buildings by their individual components and the constraints between them, is a common approach in 3D city models. Since prototyped building models allow one to incorporate a priori knowledge they support the 3D reconstruction of buildings from point clouds and allow the construction of virtual cities. However, high level building models have a high degree of complexity and consequently are not easily manageable. Interactive tools are needed which facilitate the development of consistent models that, for instance, do not entail internal logical contradictions. Furthermore, there is often an interest in a compact, redundancy-free representation. We propose an approach that uses algebraic methods to prove that a constraint is deducible from a set of premises. While automated reasoning in 2D models is practical, a substantial increase in complexity can be observed in the transition to the three-dimensional space. Apart from that, algebraic theorem provers are restricted to crisp constraints so far. Thus, they are unable to handle quality issues, which are, however, an important aspect of GIS data and models. In this article we present an approach to automatic 3D reasoning which explicitly addresses uncertainty. Hereby, our aim is to support the interactive modelling of 3D city models and the automatic reconstruction of buildings. Geometric constraints are represented by multivariate polynomials whereas algebraic reasoning is based on Wu’s method of pseudodivision and characteristic sets. The reasoning process is further supported by logical inference rules. In order to cope with uncertainty and to address quality issues the reasoner integrates uncertain projective geometry and statistical hypothesis tests. Consequently, it allows one to derive uncertain conclusions from uncertain premises. The quality of such conclusions is quantified in a way which is sound both from a logical and a statistical perspective.

Special issue: 2nd World Congress on Formal Methods

FM 2009, the 16th International Symposium on Formal Methods, marked the 10th anniversary of the First World Congress on Formal Methods that was held in 1999 in Toulouse, France. We celebrated this by organizing FM 2009 as the 2nd World Congress in the FM series, aiming to once again bring together the formal methods communities from all over the world. The programme committee had members from 46 countries, authors from 38 countries submitted papers, and authors from 23 countries had papers accepted. The five invited speakers represented four continents. The authors of a selection of the accepted papers were invited to submit extended versions of their papers to special anniversary issues of two reputable journals. This is one of them; the companion anniversary issue is in Formal Aspects of Computing. The version of the paper Pointfree Expression and Calculation: From Quantification to Temporal Logic (by Raymond Boute) submitted to the congress got the Best Paper Award. It presents general techniques for reasoning about temporal logic formulas based on a functional temporal calculus resulting often resulting in shorter proofs. The calculational, algebraic reasoning style also helps to explore expressions in the logic without prior knowledge of the of outcome of the derivation, and without the need for semantic arguments. A number of examples ranging from quantified expressions to valid formulas in Temporal Logic of Actions (TLA) are considered. Verification (and its automation) is the motivation for all the other papers. In Reasoning about Memory Layouts (by Holger Gast), we have a method for mechanically verifying the layout of data structures and their components in a program’s memory space. The data structures include local variables, block pointers, arrays, and C-like data

Problem Solvers: Problem: Crack the Code

Secret codes, puzzles, and ciphers have long held the fascination of children. Unbeknownst to them, such challenges offer the perfect opportunity to apply critical thinking skills and use algebraic reasoning skills. This month, challenge yourself and your students with cryptogram puzzles

Promises and potential pitfalls of a ‘cognitive neuroscience of mathematics learning’

The present commentary discusses the papers of the special issue on ‘cognitive neuroscience and mathematics learning’ with respect to methodological and theoretical constraints of using neuroscientific methods to study educationally relevant processes associated with mathematics learning. A special focus is laid on the relevance of subject populations, methodological limitations of current neuroimaging methods and theoretical questions concerning the relationship between the well-studied neural correlates of numerical magnitude processing and the less-investigated neural processes underlying higher level mathematical skills, such as algebraic reasoning.

Algebraic tableau reasoning for the description logic [formula omitted

Semantic web applications based on the web ontology language (OWL) often require the use of numbers in class descriptions for expressing cardinality restrictions on properties or even classes. Some of these cardinalities are specified explicitly but quite a few are entailed and need to be discovered by reasoning procedures. Due to the description logic (DL) foundation of OWL those reasoning services are offered by DL reasoners which employ reasoning procedures that are arithmetically uninformed and substitute arithmetic reasoning by “don't know” non-determinism in order to cover all possible cases. This lack of information about arithmetic problems dramatically degrades the performance of DL reasoners in many cases, especially with ontologies relying on the use of nominals ( O ) and qualified cardinality restrictions ( Q ). In this article we present a new algebraic tableau reasoning procedure for the DL SHOQ that combines tableau procedures and algebraic methods, namely linear integer programming, to ensure arithmetically better informed reasoning procedures. SHOQ extends the standard DL ALC (which is equivalent to the multi-modal logic K m ) with transitive roles, role hierarchies, qualified cardinality restrictions, and nominals, and forms an expressive subset of the web ontology language OWL 2. Although the proposed algebraic tableau (in analogy to standard tableau) is still double exponential in the worst case, it deals with cardinalities in a very informed way due to its arithmetic component and can be considered as a novel foundation for informed reasoning procedures addressing cardinality restrictions.

Open Graphs and Computational Reasoning

We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of half-edges (edges which are drawn with an unconnected end) and enjoy rich compositional principles by connecting graphs along these half-edges. In particular, this allows equations and rewrite rules to be specified between graphs. Particular computational models can then be encoded as an axiomatic set of such rules. Further rules can be derived graphically and rewriting can be used to simulate the dynamics of a computational system, e.g. evaluating a program on an input. Examples of models which can be formalised in this way include traditional electronic circuits as well as recent categorical accounts of quantum information.

The Effects of Schema-Broadening Instruction on Second Graders' Word-Problem Performance and Their Ability to Represent Word Problems with Algebraic Equations: A Randomized Control Study.

The purpose of this study was to assess the effects of schema-broadening instruction (SBI) on second graders' word-problem-solving skills and their ability to represent the structure of word problems using algebraic equations. Teachers (n = 18) were randomly assigned to conventional word-problem instruction or SBI word-problem instruction, which taught students to represent the structural, defining features of word problems with overarching equations. Intervention lasted 16 weeks. We pretested and posttested 270 students on measures of word-problem skill; analyses that accounted for the nested structure of the data indicated superior word-problem learning for SBI students. Descriptive analyses of students' word-problem work indicated that SBI helped students represent the structure of word problems with algebraic equations, suggesting that SBI promoted this aspect of students' emerging algebraic reasoning.

Dynamic consistency in process algebra: From Paradigm to ACP

The coordination modelling language Paradigm addresses collaboration between components in terms of dynamic constraints. Within a Paradigm model, component dynamics are consistently specified at various levels of abstraction. The operational semantics of Paradigm is given. For a large, general subclass of Paradigm models a translation into process algebra is provided. Once expressed in process algebra, relying on a correctness result, Paradigm models are amenable to process algebraic reasoning and to verification via the mCRL2 toolset. Examples of a scheduling problem illustrate the approach.

Criteria for uniqueness and extrema of moments in continuous end spans with differential settlement

This paper algebraically analyses an end span of a continuous beam which experiences differential settlement under a traveling point load or, separately, a uniformly distributed load (UDL). The beam has two distinct section flexural stiffnesses along the span, with a given ratio between the stiffnesses and with given support stiffnesses. The analysis outputs a novel result that, for the UDL or for any given location of the point load, there is a specific location of stiffness change along the span at which the moment diagram is unique irrespective of the ratio between section stiffnesses. This result punctures the common assumption that stiffening parts of a continuous beam always serves to attract higher moments to those parts. Then, differential calculus is applied to the algebraic analysis to reveal another novel result that, again for the UDL or for any given location of the point load, the moment diagram is an extremum when the point of stiffness change along the span matches a salient point along the moment diagram. For the point load, an important restriction on the location of stiffness change that triggers the moment extremum is identified. Also for the point load, the extremum criterion is further differentiated to arrive at an expression for the specific ratio between the two section stiffnesses which just ensures that any given allowable moment at the inner support is not exceeded as the load travels along the span. Design and research uses of these results are discussed. The paper illustrates how algebra, plots, calculus and physical reasoning may be combined to enhance understanding of mathematically complex systems.

Tartaglia's ragioni: A maestro d'abaco's mixed approach to the bombardier's problem

AbstractIn La nova scientia (1537), Niccolò Tartaglia analyses trajectories of cannonballs by means of different forms of reasoning, including ‘physical and geometrical reasoning’, ‘demonstrative geometrical reasoning’, ‘Archimedean reasoning’, and ‘algebraic reasoning’. I consider what he understood by each of these methods and how he used them to render the quick succession of a projectile's positions into a single entity that he could explore and explain. I argue that our understanding of his methods and style is greatly enriched by considering the abacus tradition in which he worked. As a maestro d'abaco in sixteenth-century Venice he had access to a great variety of mathematical and natural-philosophical works. This paper traces how Tartaglia drew elements from a vast spectrum of sources and combined them in an innovative manner. I examine his use of algebra and geometry, consider what he knew about Archimedes and suggest a reading of his enigmatic phrase ‘Archimedean reasoning’, which has eluded satisfactory interpretation.