Mathematical learning through five areas

For this posting, I've sketched out five distinct descriptions of:

- what I specifically learned;

- pivotal moments of when such learning happened; and,

- how such learning is anticipated to impact my classroom teaching practice.

1) The Seven Mathematical Process Expectations: Problem-solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing, and communicating. These expectations "can be seen as the processes through which students acquire and apply mathematical knowledge and skills" (Ontario Ministry of Education, 2005, p. 11). These expectations are learned as students experience the five mathematics curriculum stands (i.e., Number Sense and Numeration, Measurement, Patterning and Algebra, Geometry and Spatial Sense, and Data Management and Probability) and as students come to acquire multiple ways to understand mathematics - that is, these are not learned in isolation but are "interconnected," (p. 18) thus implying the need for teaching mathematics holistically and implementing the curriculum through inquiry-based methods.

I came to realize the importance of the Seven Process Expectations when going through the Ontario Mathematics Curriculum document. The Ontario Ministry of Education (2005) acknowledges that these processes are "essential to a balanced mathematics program" (p. 5). In my teaching practice, I am beginning to situate my pedagogical thinking in the context of these processes; that is, I'm building knowledge of mathematics for students through sustained inquiry, using a multiplicity of complementary assessments to make connections, construct representations, expand reflective capacities, contextualize algorithmic practice, and link digital technologies not by a series of disconnected events but through a play of interrelated, problem-solving experiences. As a result, my mind shifts to the "big picture" of how mathematics can be taught and learned, with the math processes occupying a seminal role informing my program planning and pedagogical practice.

Another context in which I witnessed the use of the mathematical processes occurred when reflecting upon our Task Analysis assignment. In this effort, a reconsideration of a mathematical problem was used to explore how all seven threads of the process expectations were experienced when solving a mathematical problem. Some colleagues invoked comprehensive algorithmic functions such as the "RISE problem solving method as a framework to solve the problem," while others used more focused measures to arrive at their solutions such as "both mental computation as well as using a chart to write out my thinking."

2) The content of the mathematics curriculum, I came to realize, consists of an intricate web of three main threads: big ideas, strategies, and models. These threads are the brainchild of Cathy Fosnot, who found in her research that there are certain "milestones" that children achieve when faced with mathematical challenges appropriate to maturity and level of experience. According to Fosnot, children experience a "cognitive reorganization" (Landscape of Learning, 2017, Big Ideas) when they can "undo ideas," (Ibid) therefore setting up new ways of building concepts to further their learning in novel and evolving contexts. A more refined articulation of Fosnot's findings are seen in the Ontario Ministry of Education's Report of the Expert Panel on Mathematics Education in Grades 4 to 6 in Ontario (2004). The utilization of big ideas, strategies, and models are viewed through the viewpoint of "characteristics" in that assuming these three factors contributes towards building an "effective mathematics instruction." Specifically, an educator's mindset:

• is focused on having students make sense of mathematics;

• is based on problem solving and investigation of important mathematical concepts;

• begins with the student’s understanding and knowledge of the topic;

• includes students as active rather than passive participants in their learning;

• has students communicate and investigate their thinking through ongoing discussion;

• includes all students, whether in the choice of problems or in the communicating of mathematical ideas;

• incorporates ongoing assessment of student understanding to guide future instruction.

(Based on Hiebert et al., 1997)

I became aware of these three building blocks when reading over Ministerial documents as well as viewing online course videos and collegial contributions in course postings. Offering a blend of these situations to students can be achieved very simply and not done at the cost of ambiguity or increased anxiety. One colleague, for example, suggested that "teaching students how to add and subtract money" is one way to promote connections between theory and practice in the domain of mathematical content. Another colleague remarked that "students communicating their ideas helps everyone to understand that there is more than one way to get an answer." Setting up a balanced mathematics program involves a multiplicity of communication practices, so that all learners -- from at-risk students to those who can demonstrate their knowledge and understanding more easily - may benefit and grow further into becoming mathematical learners. As a growing educator of mathematics, I am more senstitive to the needs and interests of students based on the framework of Big Ideas, Strategies, and Models.

3) The relationship between Theory and Practice has become more nuanced and clairvoyant in my understanding after reading through course readings and communicating with course colleagues. For me, theory refers to the Big Ideas that can be utilized to guide instruction and teaching practice, and thus, as one colleague put it, "it challenges teachers to think of deeper questions and tasks for students to work on as they develop their skills." Theory is arguably the most useful instrument of all, as the Big Ideas function to promote conceptual knowledge with the means to develop procedural know-how. In my teaching practice I enjoy stretching my thinking to include basic concepts along with contextually-relevant examples to support students in their journey in and through mathematics education. In the task to learn adding and subtracting of numbers, for example, students can enrich their learning by using coins, counters, and blocks through a series of questions that challenge them to think critically and collaboratively about how to resolve pertinent issues.

Sample: Know's and Do's, September for Literacy; (Similar Representation can be developed for Mathematics Education)

4) Another area of pedagogical practice I have learned is in the context of mathematical development as such is organized through the school year and through the use of the Big Ideas. While Kindergarten students live in the world of "1 to 10" and are expected to learn "one to one correspondence," "part-part-whole," and "5 and 10 are anchors," these number relationships are not lost as students mature and become more familiar with the territory of mathematics education. In fact, Primary and Junior Students acquire richer and deeper understandings of basic principles already learned previously, such as "abstraction," "magnitude," and "cardinality," with larger series, sequencing, and orders of numbers. Reference to course colleague's postings as well as relevant literature has expanded my understanding of the scope of mathematics learning that transpires throughout the school year, and mapping out this understanding has become a useful tool for my future pedagogical planning. I have already sketched out how my Grade 6 class this year will be exposed to Mathematics education going forward, and how I might be able to integrate this knowledge across the curriculum.

5) Finally, an area that continues to personally evolve and which promotes my own critical thinking and pedagogical planning lies in the activity of questioning. I enjoy learning through question and problem-solving practices, and the course has enabled me to experience how to set up insightful challenges and journey through several steps along the way in taking this method and approach. Early Years, Primary, and Junior students in my care are going to benefit from an educator who is more equipped to teach and relay an expanded repertoire of mathematics ideas, strategies, and models of instruction. My understanding of the mathematical processes and content sprung initially from an interest in the subject matter and by a series of questions, both personally posed and by other course colleagues. After reading through the Process Expectations in the Mathematics Curriculum (2005, pp. 11-17), I noted some ambiguity over the concepts of proof and conjecture, even though both terms were explicitly used in the guide. I then did some investigation on the subject by looking at relevant journals, articles, and publications. I realized that educators generally are not positioned to understand at a complex level the use of proof and conjecture, given the lack of personal expertise and professional training. I wanted to know more, given that students from my experience enjoy going over problems, casting their predictions (i.e., conjectures), and attempting to arrive at a sound explanation (i.e., proof). I believe these behavioural tendencies require teachers to be more informed about how mathematics works, both conceptually and procedurally. Nevertheless, I plan on guiding my teaching practice on evidence-based insights gathered from personal experience in the course as well as by continued readings, conferences, and professional development programs in the subject going forward.