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In this paper, we propose distributed algorithms that solve a system of Boolean equations over a network, where each node in the network possesses only one Boolean equation from the system. The Boolean equation assigned at any particular node is a private equation known to this node only, and the nodes aim to compute the exact set of solutions to the system without exchanging their local equations. We show that each private Boolean equation can be locally lifted to a linear algebraic equation under a basis of Boolean vectors, leading to a network linear equation that is distributedly solvable. A number of exact or approximate solutions to the induced linear equation are then computed at each node from different initial values. The solutions to the original Boolean equations are eventually computed locally via a Boolean vector search algorithm. We prove that given solvable Boolean equations, when the initial values of the nodes for the distributed linear equation solving step are i.i.d selected according to a uniform distribution in a high-dimensional cube, our algorithms return the exact solution set of the Boolean equations at each node with high probability., QC 20220201

A total of 266 Finnish students participated in a flexible equation solving test. By flexibility we under-stand the knowledge of multiple strategies and ability to choose the most mathematically appropriate strategy for a given task. Here we focus on the first aspect, namely knowledge of appropriate alterna-tive, so-called innovative strategies. The test measured students’ capacity and inclination for produc-ing innovative strategies. We consider the relationship between these measures and students’ speed and accuracy in solving equations. We find that students with high capacity for innovation have high speed and accuracy. On the other hand, some low capacity students had high speed or accuracy whereas others had low. Inclination for innovation is not related to speed or accuracy.

Flexibility is an important element in learning mathematics. The purpose of this study was to investigate whether flexibility in linear equation solving predicts future academic achievement in mathematics and other subjects. Participants were 149 Finnish high-school students. Results show that flexibility was related to grades in both tracks of mathematics, chemistry, and mother tongue, as well as the total number of exams taken in the national matriculation examination. However, when controlling for accuracy in equation solving, only basic level mathematics and, to some degree, chemistry grade were related to flexibility. On the other hand, flexibility had an impact on students’ choice to participate in the mathematics and physics exams. A theoretical analysis shows that student selection may mask part of the relationship between flexibility and grade.

Allt fler människor använder sig av Youtube och andra öppna internetmiljöer för att lära sig matematik. Enligt tidigare studier finns det en stor spridning när det gäller bostadsort och ålder bland deltagarna i dessa miljöer. Min undersökning visar att ett urval om 10 Youtubevideor om ekvationslösning och deras kommentarsfält också kännetecknas av en stor spridning av matematikdiskurser bland deltagarna. En studie av 85 dialoger i kommentarsfälten visar att i omkring hälften av dialogerna uttrycker sig olika deltagare på sätt som kännetecknar olika matematikdiskurser, medan i den andra hälften alla deltagare uttrycker sig på sätt som kännetecknar samma matematikdiskurs. De förra dialogerna slutar mer sällan med att deltagarna upplever förståelse, medan lärandet i de senare dialogerna begränsas av vad som går att uttrycka inom ramen för den rådande diskursen. Genom att utveckla begrepp och kategorier för att beskriva matematikdiskurserna lägger studien grunden för fortsatt forskning, både kvalitativa med intervjuer av deltagare och kvantitativa med större material., The use of Youtube and other open internet environments for learning mathematics is becoming increasingly common. According to earlier studies there is a great diversity of ages and nationalities among participants in these environments. My investigation shows that a selection of 10 Youtube videos about equation solving and their comments sections also are characterized by a diversity of mathematics discourses. A study of 85 dialogues in the comments sections shows that in approximately half of the dialogues different participants express themselves in manners characteristic of different mathematics discourses, while in the other half all the participants express themselves in manners characteristic of the same mathematics discourse. The former dialogues more seldom end with experiences of understanding, while learning in the latter group of dialogues is limited by what is possible to express within the dominant discourse. By developing terms and categories for describing the mathematics discourses this study lays the foundation for further research, both qualitative research comprising interviews with participants and quantitative research on greater amounts of data.

Allt fler människor använder sig av Youtube och andra öppna internetmiljöer för att lära sig matematik. Enligt tidigare studier finns det en stor spridning när det gäller bostadsort och ålder bland deltagarna i dessa miljöer. Min undersökning visar att ett urval om 10 Youtubevideor om ekvationslösning och deras kommentarsfält också kännetecknas av en stor spridning av matematikdiskurser bland deltagarna. En studie av 85 dialoger i kommentarsfälten visar att i omkring hälften av dialogerna uttrycker sig olika deltagare på sätt som kännetecknar olika matematikdiskurser, medan i den andra hälften alla deltagare uttrycker sig på sätt som kännetecknar samma matematikdiskurs. De förra dialogerna slutar mer sällan med att deltagarna upplever förståelse, medan lärandet i de senare dialogerna begränsas av vad som går att uttrycka inom ramen för den rådande diskursen. Genom att utveckla begrepp och kategorier för att beskriva matematikdiskurserna lägger studien grunden för fortsatt forskning, både kvalitativa med intervjuer av deltagare och kvantitativa med större material., The use of Youtube and other open internet environments for learning mathematics is becoming increasingly common. According to earlier studies there is a great diversity of ages and nationalities among participants in these environments. My investigation shows that a selection of 10 Youtube videos about equation solving and their comments sections also are characterized by a diversity of mathematics discourses. A study of 85 dialogues in the comments sections shows that in approximately half of the dialogues different participants express themselves in manners characteristic of different mathematics discourses, while in the other half all the participants express themselves in manners characteristic of the same mathematics discourse. The former dialogues more seldom end with experiences of understanding, while learning in the latter group of dialogues is limited by what is possible to express within the dominant discourse. By developing terms and categories for describing the mathematics discourses this study lays the foundation for further research, both qualitative research comprising interviews with participants and quantitative research on greater amounts of data.

Over the last decades, several representation systems have been used to improve the learning of the manipulations of algebraic expressions. In addition, the introduction of new technologies in Mathematical Education allowed the simultaneous use of different representation systems. Some of these applications use a physical representation of the equation, as for example, balances or areas, to give some meaning to the algebraic manipulations. However, the use of real models has the difficulty of translating the actions performed in the physical environment to the algebraic language. In this work we analyse the effect of using the program DragonBox Algebra© in a school context. In particular, we study the students’ competence in solving equations when the students return to the world of algebra. We compared the results of two tests taken by the students before and after an intervention in an Initial Professional Qualification Program group. The tests consisted of a set of equations with one or more letters. The results show a significant increase in the post test scores., Numerosos estudios han analizado el papel del uso de distintos sistemas de representación como una forma de favorecer el aprendizaje de la manipulación de las expresiones algebraicas. La introducción de las nuevas tecnologías en el campo de la educación matemática ha posibilitado la utilización simultánea de distintos sistemas de representación. Han aparecido diversas aplicaciones que utilizan representaciones físicas de la ecuación como podría ser balanzas o áreas para dar significado a las manipulaciones algebraicas. Sin embargo, el uso de modelos reales plantea la dificultad de trasladar las acciones que realizamos en el soporte físico al lenguaje algebraico. En el trabajo que presentamos analizamos el efecto que tiene usar en una situación escolar el juego DragonBox Algebra©. En concreto, se estudia cómo varía la competencia de resolver ecuaciones cuando regresan al mundo del álgebra. Con este fin, comparamos los resultados de dos pruebas que se administraron antes y después de la intervención en un grupo de un Programa de Cualificación Profesional Inicial. Los cuestionarios estaban formados por ecuaciones que incluían una o más letras. Los resultados muestran un incremento significativo en las puntuaciones obtenidas en el post test.

Algebra involves various activities including transformational activitiesthat are mainly about solving equations. Fluency (or flexibility) in theseactivities is important. Several researchers have proposedconceptualisations of flexibility in equation solving. This paper makes areflection about flexibility in equation solving that contributes to theextension of Star and Seifert’s operationalisation. Examples are used ascontext for the reflection. The need for another property of flexibility,namely making connections, is suggested to deepen investigations ofstudents’ flexibility in equation solving and its provision in teaching.