Around the world, mathematics is highly valued and great importance is placed on learning mathematics. Private tutors in non-Asian countries serve a remedial purpose, whereas in Asia, everyone has a tutor for providing an increased knowledge base and skill development practice. Many students in Asia enroll in programs like “Kumon,” which focus on practicing skills (which has its place) and “doing” math rather than “doing and understanding” math. When you ask students who are well rehearsed in skills to problem solve and apply their understanding to different contexts, they struggle. The relationship between the facts, skills, and conceptual understandings is one that needs to be developed if we want our students to be able to apply their skills and knowledge to different contexts and to utilize higher order thinking.
Why Do We Need to Develop Curriculum and Instruction to Include the Conceptual Level?
According to Daniel Pink (2005), author of A Whole New Mind, we now live in the Conceptual Age. It is unlike the Agricultural Age, Information Age, or the Industrial Age because we no longer rely on the specialist content knowledge of any particular person. The Conceptual Age requires individuals to be able to critically think, problem solve, and adapt to new environments by utilizing transferability of ideas. “And now we’re progressing yet again—to a society of creators and empathizers, of pattern recognizers and meaning makers” (Pink, 2005, p. 50).
Gao and Bao (2012) conducted a study of 256 college-level calculus students. Their findings show that students who were enrolled in concept-based learning environments scored higher than students enrolled in traditional learning environments. Students in the concept-based learning courses also liked the approaches more. A better grasp of concepts results in increased understanding and transferability.
With the exponential growth of information and the digital revolution, success in this modern age requires efficient processing of new information and a higher level of abstraction. Frey and Osborne (2013) report that in the next two decades, 47% of jobs in the United States will no longer exist due to automation and computerization. The conclusion is that we do not know what new jobs may be created in the next two decades. Did cloud service specialists, android developers, or even social marketing companies exist 10 years ago?
How will we prepare our students for the future? How will our students be able to stand out? What do employers want from their employees? It is no longer about having a wider knowledge base in any one area.
Hart Research Associates (2013) report the top skills that employers seek are the following:
- Critical thinking and problem solving,
- Collaboration (the ability to work in a team),
- Communication (oral and written), and
- The ability to adapt to a changing environment.
How do we develop curriculum and instruction to prepare our students for the future?
We owe our students more than asking them to memorize hundreds of procedures. Allowing them the joy of discovering and using mathematics for themselves, at whichever level they are able, is surely a more engaging, interesting and mind-expanding way of learning. Those “A-ha” moments that you see on their faces; that’s why we are teachers.
David Sanda, Head of Mathematics Chinese International School, Hong Kong
The Structure of Knowledge and the Structure of Process
Knowledge has a structure like other systems in the natural and constructed world. Structures allow us to classify and organize information. In a report titled Foundations for Success, the U.S. National Mathematics Advisory Panel (2008) discussed three facets of mathematical learning: the factual, the procedural, and the conceptual. These facets are illustrated in the Structure of Knowledge and the Structure of Process, developed by Lynn Erickson (2008) and Lois Lanning (2013).
The Structure of Knowledge is a graphical representation of the relationship between the topics and facts, the concepts that are drawn from the content under study, and the generalization and principles that express conceptual relationships (transferable understandings). The top level in the structure is Theory.
Theory describes a system of conceptual ideas that explain a practice or phenomenon. Examples include the Big Bang theory and Darwin’s theory of evolution.
The Structure of Process is the complement to the Structure of Knowledge. It is a graphical representation of the relationship between the processes, strategies, skills and concepts, generalizations, and principles in process-driven disciplines like English language arts, the visual and performing arts, and world languages.
For all disciplines, there is interplay between the Structure of Knowledge and the Structure of Process, with particular disciplines tipping the balance beam toward one side or another, depending on the purpose of the instructional unit. The Structure of Knowledge and the Structure of Process are complementary models. Content-based disciplines such as science and history are more knowledge based, so the major topics are supported by facts. Process-driven disciplines such as visual and performing arts, music, and world languages rely on the skills and strategies of that discipline. For example, in language and literature, processes could include the writing process, reading process, or oral communication, which help to understand the author’s craft, reader’s craft, or the listener’s craft. These process-driven understandings help us access and analyze text concepts or ideas.
Both structures have concepts, principles, and generalizations, which are positioned above the facts, topics, or skills and strategies. Figure 1.1 illustrates both structures. Figure 1.1 can also be found on the companion website, to print out and use as a reference.
The Structure of Knowledge and the Structure of Process for Functions
Mathematics can be taught from a purely content-driven perspective. For example, functions can be taught just by looking at the facts and content; however, this does not support learners to have complete conceptual understanding. There are also processes in mathematics that need to be practiced and developed that could also reinforce the conceptual understandings. Ideally it is a marriage of the two, which promotes deeper conceptual understanding. Figure 1.2 illustrates the Structure of Knowledge for the topic of functions.
Topics organize a set of facts related to specific people, places, situations, or things. Unlike history, for example, mathematics is an inherently conceptual language, so “Topics” in the Structure of Knowledge are actually broader concepts, which break down into micro-concepts at the next level.
Figure 1.1: Side by Side: The Structure of Knowledge and the Structure of Process
© 2014 H. Lynn Erickson and Lois A. Lanning
Transitioning to Concept-Based Curriculum and Instruction, Corwin Press Publishers, Thousand Oaks, CA.
As explained by Lynn Erickson (2007), “The reason mathematics is structured differently from history is that mathematics is an inherently conceptual language of concepts, subconcepts, and their relationships. Number, pattern, measurement, statistics, and so on are the broadest conceptual organizers” (p. 30).
More about concepts in mathematics will be discussed in Chapter 2.
Facts are specific examples of people, places, situations, or things. Facts do not transfer and are locked in time, place, or a situation. In the functions example seen in Figure 1.2, the facts are y = mx + c, y = ax2 + bx + c, and so on. The factual content in mathematics refers to the memorization of definitions, vocabulary, or formulae. When my student knows the fact that y = mx + c, this does not mean she understands the concepts of linear relationship, y-intercept, and gradient.
According to Daniel Willingham (2010), automatic factual retrieval is crucial when solving complex mathematical problems because they have simpler problems embedded in them. Facts are the critical content we wish our students to know, but they do not themselves provide evidence of deep conceptual understanding.
Figure 1.2: The Structure of Knowledge for Functions
Adapted from original Structure of Knowledge figure from Transitioning to Concept-Based Curriculum and Instruction, Corwin Press Publishers, Thousand Oaks, CA.
Formulae, in the form of symbolic mathematical facts, support the understanding of functions. This leads to a more focused understanding of the concepts of linear functions, quadratic functions, cubic functions, exponential functions, variables, and algebraic structures in Figure 1.2. The generalization “Functions contain algebraic structures that describe the relationship between two variables based on real-world situations” is our ultimate goal for conceptual understanding related to the broad concept of functions. Please take a look at the companion website for more examples of the Structure of Knowledge and the Structure of Process on the topic of linear functions. See Figures M1.1 and M1.2.
Concepts are mental constructs, which are timeless, universal, and transferable across time or situations. Concepts may be broad and abstract or more conceptually specific to a discipline. “Functions” is a broader concept, and the micro-concepts at the next level are algebraic structures, variables, linear, quadratic, cubic, and exponential. Above the concepts...
