The first Real Algebra blog explained the need for a widely effective solution to the problem of learning the course of mathematics starting with algebra and continuing through calculus. Other blogs discussed the importance of developing conceptual understanding and how to do that, and why it is essential to provide resources that make it easy for students to understand the purpose and use of each math topic. We also considered the fact that all of mathematics, and particularly algebra, is a form of language, and some of the implications of that in terms of applications in the real world.
Along the way we pointed out that existing solutions to the problem of learning high school level math have three common flaws: they do not develop conceptual understanding, they do not make it real to students what each topic of that math is for, and they do not develop mastery.
This blog focuses on mastery.
If we’re going to talk about mastery, we’d better start with understanding what the word means. Oddly enough, the master of mastery, Dr. Benjamin Bloom, chose never to define the word because (according to In Search of a Useful Definition of Mastery) he didn’t want to alienate educators that had different definitions. Fair enough, but that won’t really work for us in developing a product that is designed to enable students to achieve mastery.
My colleague Dr. Anastasia Betts, who worked with me to create Age of Learning’s early learner mastery mathematics programs, defines mastery this way: “a high degree of independent proficiency demonstrated consistently over time that transfers and doesn't decay.”
That gives us a goal to aspire to, but we need be clear about what we mean by proficiency, particularly since there are different types of learning objectives and what we mean by proficiency depends on the type of objective.
For Real Algebra, we have adopted these meanings:
Proficiency in the understanding of a concept means that the student can easily relate and compare the concept to other concepts already understood and can recognize and generate instances of the concept in various contexts.
Proficiency in the understanding of a principle means that the student 1) has a proficient level of understanding of the concepts that are elements of the principle, and 2) the student can recognize and generate examples and counter examples of the principle in various contexts.
Proficiency in the execution of a skill means that the student can carry out the actions of that skill in order to generate the desired (correct) results, can recognize situations in which the skill can be usefully applied, and can explain and generate examples of the execution of the skill.
Proficiency in the demonstration of an ability would include proficiency in understanding the relevant concepts and principles and in executing the skills that add up to being able to apply knowledge in order to solve a problem.
How do we develop such proficiencies so effectively that the student can apply them in different contexts (“transfer”) and they do not decay? The answer is contained in the phrase “productive struggle,” which describes a student’s experience as they actively build knowledge, skills, and abilities by reading and listening for understanding, evaluating information, thinking of examples and counter examples, solving problems, and in general executing tasks which they find engaging, challenging, and achievable.
In Real Algebra, the productive struggle occurs primarily in the process of playing a series of learning games—activities in which the student’s response requires demonstration of the targeted understandings, skills, and abilities. To “win” a game, the student must complete a task successfully several times in a row, independently and fluently. Scaffolding is provided as needed, but to reach the goal mastery the student must demonstrate that they can repeatedly succeed independently.
The essence of this approach is to prompt the student to do something, provide appropriate feedback, and adjust the next prompt as needed based on the student’s response. The theory of learning is straightforward: learners construct knowledge in the process of observing, making decisions, and taking action.
The set of games provided and the sequence in which the student engages in them is guided by a detailed map of the relationships of concepts, principles, skills, and abilities that make up the algebra knowledge space.
These first four Real Algebra blogs have sought to lay out the rationale and general framework of the program. Future articles will fill in details.