Not all Equals are Equal: Decoupling Thinking Processes and Results in Mathematical Assessments

One of the greatest challenges in mathematics education is in fostering an understanding of what mathematicians would recognise as “mathematical thought.” We seek to encourage students to develop the transferable skills of abstraction, problem generalization and scalability as opposed to simply answering the specific question posed. This difference is perhaps best illustrated by the famous – but likely apocryphal – tale of Gauss’s school days and his approach to summing all positive integers up to and including 100, rather than just summing each sequentially. Especially with the rise of technology-enabled marking and results-focussed tutoring services, the onus is on the educator to develop new types of question which encourage and reward the development of mathematical processes and deprioritise results alone. Some initial work in this area is presented here.

Weekly 10-minute-tasks to Promote Students Solving Equations in a Content-oriented Manner

When solving equations in school, students often rely on routines and do not consider alternative ways of solving. Even basic equations which could be solved quite fast using common sense are regularly solved in a complicated way. To overcome this reliance on routine, a study with 17 classes of grade 10 students was carried out. Weekly 10-minute-tasks, which contained appropriate subtasks to enhance content-oriented solving, were solved by students over the course of one school year. These tasks were designed with the purpose of reducing the dominance of routines and the aim of using insight in the solving of equations.

Mathematical Modeling Thinking: Laying the Foundation for Mathematical Modeling Competency

Mathematical modeling competency requires frequent practice and sufficient time to derive experience solving open-ended contextual problems. Specific ways of thinking necessary in modeling are identified by contrasting Pólya’s general problem-solving framework, which may be familiar worldwide. These ways of thinking are developed through mathematical activities that promote dispositions for eventual success in modeling. We posit that mathematical modeling thinking (MMT) is necessary for building modeling competency. This paper describes MMT and illustrates how it can be developed through a well-known problem of universal human cultural greeting exchange. While connecting to world cultures, we examine ways to promote MMT practices such as making useful simplifications, looking for patterns, utilizing multiple representations, mathematizing the situation, and reflecting on the solution. We conclude with practical ways to effect MMT as the foundation for developing mathematical modeling competency.

The Mathematical Culture of Ojibwe Students – An Ethnographic Study

Human beings have an innate capacity to communicate, count, detect patterns, locate, and create. With these capacities we invent, design, play, and explain. Regardless of academic background, we also have the innate capacity to use mathematics in meaningful ways. However, in spite of this innate capacity, there is a large disconnect between innate function and success in academic mathematics. Our research is based on interviews of 14 Ojibwe-identifying tribal college students. The instrument was constructed based on Bishop’s (1988) set of six universals or activities people have always done. We present the development of the instrument, interview process, and initial findings. Findings include common ethnomathematical threads found among the interviewed students. Our goal is to use this research to improve ourpreK-12 professional education teacher program and positively impact Ojibwe student learning.

Solving Mathematical Problems in the Context of Some Obstacles between Teachers and Students

Great mathematical discoveries are mostly based on huge knowledge of their explorers and long, solid work leading slowly to the finding. There are also well known cases of the “accidental” discoveries that happened quickly, intense and their founders did not even realize the range of the discovery, because they were working on something else at the time. Nevertheless, each finding requires energy, devotion and concentration of its discoverer. Solving mathematical problems demands quite the same things, thus teachers may find some opportunities to create curious, open-minded young discoverers. It is not an easy job to do though, because there is a great risk of killing pupils’ enthusiasm by teacher’s skepticism, there is a large chance to nip pupils’ energy in the bud by routine operations and there is a huge possibility to discourage pupils’ endeavors by giving them wrong-chosen problems to solve.

Online Strategies Enhancing Mathematics Teacher Knowledge for the Digital Age: Discourse and Critical Reflection

This study designed online graduate courses to enrich inservice mathematics teachers’ Technological Pedagogical Content Knowledge (TPACK). The effort identified key experiences to engage teachers in discourse and critical reflections for relearning, rethinking, and redefining teaching and learning as they know and learned it, transforming their TPACK with respect to teaching with digital technologies. The experiences modeled inquiry tasks merging content, technology and pedagogy as described in TPACK, connecting teachers with experiences as students learning about and with technologies. Critical reflections on the experiences as learners and as teachers combined with the online community of learners’ discourse, transforming their teacher knowledge. The collection of strategies involving discourse and critical reflection did enhance the participants’ TPACK, providing recommendations for designing online inservice teacher education courses.

Developing High-School Students´ Competences through Math Research Workshops – the M&L Project

Mathematics is or it should be about problem solving and math thinking. However, what mathematics students learn in schools is more about procedures for solving different types of math exercises and problems. In many cases, students learn by heart algorithms and words (math concepts) and use them for solving different math tasks. School math is very far from what mathematicians do and, in many cases, doesn’t motivate students for learning math. This paper presents the way we organized the assessment of the students’ skills developed through math research workshops and some of the assessment results. Even though we didn’t assess all the competences the students develop through the math research workshop, the findings show that the students certainly develop their problem-solving skills.

Do Future Mathematics Teachers Need the Course „Integration of Digital Technologies in Teaching Mathematics“, and if so, what exactly can it help them with?

In the current research we analysed our teaching experience in the course “Integration of digital technologies in teaching mathematics”. The students were mathematics student teachers. The main goal of the course was to demonstrate the potential of digital technologies in teaching mathematics and to provide the students with basic skills in the intellectual use of these technologies. During the course the students, after getting acquainted with various mathematical software packages, build and present their own teaching units. We were interested to analyse the students’ attitudes towards the course. A multiple-choice questioner was formulated, and the collected data were analysed. We observed that most of the students found the course being helpful for their future teaching. The obtained results indicated that the described course provided them a didactic model to emulate.

Engage Students More Hopscotch Mathe has Students Jumping for Joy

The goal was to create a system to teach children deep thinking skills, as well as problem solving skills which they could later use in tomorrow’s innovation economy. The by-product is they learn the Times Table. We cover more in less time…under 5 hours, we go up to 20x20, and introduce the children to complex algebraic equations, too. Guess what? They love it – and ask for more! The times table represents the problem to be solved. Each intersection represents a smaller aspect of the problem. They learn various techniques. No dumb sing-song melodies. They build on what they know. We do not go linearly through the table. We jump around…and cover whatever we can. When we are through I show them that if they only knew 7x4 = 28, they have the problem solving skills where they can solve the whole table. The idea behind Kinestetic Math is to get into their world, and reach them at their level. Children like to run, jump, colour and move around – so do we. We use our fingers, our knuckles, and our legs to learn the Times Table. This paper covers a small section of the program, Magic Squares and Hopscotch Math, as an introduction to a different kind of thinking and how innovative thinking can be applied to teaching. I introduce the program with a 10x10 grid representing the times table. Every time we solve one of the blocks on the table, they get to color the block however they want.

Collaborative Solution Discovery: A Problem Solving Process

Loyola Marymount University (LMU) has developed a new approach to problem solving, Collaborative Solution Discovery (CSD), to help practitioners in a school system leverage their individual passions in a way that grows students’ positive math identity through mathematical thinking, problem solving, and self-regulation. By focusing on how students and teachers interact with each other in real-time in an ideal classroom, practitioners take ownership of a process to guide their students in growing their positive math identity and thus taking ownership of their own math learning. Practitioners measure progress along the way through metrics that are created, defined, used, and continually refined by themselves to attain their ideal math learning environment. The entire CSD process results in a system that owns ist improvement efforts—improvement efforts that are flexible, adaptable, and sustainable.