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Introduction to order of operations | Arithmetic properties | Pre-Algebra | Khan Academy

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The Structure of Process is made up of strategies and skills that exhibit what students are able to accomplish. They follow these categories to achieve measurable end goals in their learning process. For example, a student will perform a skill, which Wathall describes as "operations or actions that are embedded in strategies"(31). This skill could be subtraction or addition that students implement to achieve the end goal of solving a problem. The Structure of Process is heavily reliant on the actions students take to apply their knowledge and it can lead them towards a conceptual understanding. Oftentimes, however, the student may rely too heavily on these strategies and skills, but fail to understand why we use them when we do. Key components of the Structure of Process are skills, processes, and algorithms.

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Investigation is closely related to the process of problem-solving. It means "being able to explore unfamiliar mathematical situations" (49). The idea behind this is to delve further into topics and concepts by researching relevant ideas to better your understanding. This process can cause a student's creativity to flourish because they are now using different methods to solve problems that seem challenging and unfamiliar. Similar to problem-solving, investigation allows students the opportunity to discover and make mistakes and learn from them.

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The process of creating representations focuses on the idea of visually representing mathematical concepts to aid in student understanding. By creating a visual representation, students are given the ability to communicate mathematical concepts to other learners. It is also a great tool for teachers when discussing a new difficult topic. A good example of this being a successful resource for student understanding is using graphs to depict average velocity versus instantaneous velocity. Here, a student can visually see the changes which would be much harder to explain if the teacher only gave an equation or a real-life example. The two different types of representation Wathall discusses are pictorial and schematic.

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To summarize Wathall's process of making connections, this process refers to the student's ability to make connections between facts and symbols and how things relate to each other in the real world. It also describes the student's ability to connect past problems to future problems and relate different concepts (45). It is important for students to be able to make connections as it deepens their understanding of overarching patterns and allows them to apply these ideas to their lives. An example that Wathall provides as a method for students to make connections are graphic organizers such as Venn diagrams.

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The process of communication refers to the student's ability to talk and write about mathematics (38). If students are able to clearly communicate their ideas and steps they took to evaluate a problem they will be more able to understand the concept and explain it to others. This can take place in a group setting where there are opportunities for student-student interaction. Wathall gives an example of this called number talks which are characterized by "class discussion, classroom environment and community, the teacher's role, mental mathematics, and purposeful, computation problems" (39). This is a good resource for enabling student discussion and communication within a classroom setting. They allow students to demonstrate their fluency on a topic both orally and written.

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The process of reasoning and proof is paired down to "the ability to make generalizations and provide explanations and justification for arguments" (38). This process is a key method for students to use that aids them in understanding a concept. It builds on their critical thinking skills and allows them to make mistakes in a controlled setting in which they can learn from and then apply. This process is often guided by inductive reasoning and allows students to form their own generalizations and principles

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The principles for problem-solving are understand the problem, devise a plan, carry out the plan, and review and extend your knowledge as explained by Wathall (37). Problem-solving is often referred to as the "fundamental building block of mathematics" (37). Most students will use this process when they are faced with an unfamiliar and challenging problem. It is a great way to get students to think outside of the box and apply their knowledge of previous concepts where they are able.

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As defined by Wathall, skills are "small operations of actions that are embedded in strategies"(31). Examples of these skills are operations such as parentheses, exponents, multiplication, division, addition, and subtraction. These operations are used by students in a methodical way to support their learning of a topic. They often are the base level when learning mathematics and are used as building blocks to get to higher and more complex concepts.

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As defined by Wathall, algorithms are "skills that allow learners to follow a set of rules leading to a specific outcome"(35). A good example of an algorithm is PEMDAS, which is a guide for students to follow when they are exposed to problems that have multiple operations. Algorithms are extremely useful for students in understanding broader concepts such as the order of operations. A common pitfall for students when using algorithms is that they become so heavily reliant on the algorithm, but do not understand the concept and why they are applying it. This is commonly seen when students use cross multiplication.

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Wathall describes mathematical processes to be "complex, sophisticated performances" (32) and "broad techniques that students draw upon when learning mathematics and support the understanding of the concepts in a unit"(32). There are generally six different mathematical processes that are widely accepted when learning mathematics. They are Problem Solving, Reasoning and Proof, Communicating, Making Connections, Creating Representations, and Investigating. These are all used by students, usually in conjunction with one another, to assist in grasping more difficult concepts.

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Key Distinctions Between Processes, Algorithms, and Skills

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Key Distinctions Between Processes, Algorithms, and Skills

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Mathematical Processes are the overarching umbrella for the Structure of Process. Students use these processes as a general and broad technique to comprehend mathematical concepts. When students get into each of the different categories within mathematical processes they have the opportunity to apply the skills and processes they have learned to further their understanding and solve problems.