When the Common Core State Standards in mathematics were published, I was thrilled. Finally, here was a coherent, rigorous set of standards that provided a clear vision of mastery in the field. By outlining specific standards in this way, it gave me the ability to articulate insights and big ideas which I was only implicitly aware of in my studies. Using Common Core, I figured the process would become well defined; I would simply take the first standard, teach that, then go on to the next.
If Only It Were That Easy
Over the course of several months of pondering and ill-fated attempts to directly translate the standards into learning objectives, I arrived at an insight which might seem obvious to seasoned teachers, but which struck this rookie with the force of divine revelation. Content standards and curriculum are two very different things, and the latter is far harder to do address than the former.
To be sure, it takes some effort to come up with content standards which are rigorous enough to represent an acceptable level of mastery of mathematics for students leaving high school. Different teachers have different ideas about every aspect of mathematics instruction, whether it be how much students need to know or the level of difficulty of problems which should be solved. Nevertheless, a comparison of mathematics curricula from the countries that perform highest on international mathematics assessments shows a great deal of common ground. Now that the Common Core standards have been published, we can see that it's fairly straightforward to articulate a vision of mastery in the field of mathematics.
It is one thing to specify what mastery looks like in a certain field. It is another, very different thing to lay out a roadmap to mastery, in the form of a sequence of learning objectives that culminate in the mastery defined by the standards.
I remember the day I sat down to begin planning my classes prior to my first full year of teaching. I tried to figure out what exactly I was going to teach and in what order. The stress was overwhelming. There were so many possibilities and so many potential avenues of approach. Should I start with a review of real numbers, or should I dive right into equations? Should I wait to introduce the quadratic formula until I had covered factoring and standard form, or should I introduce standard form first and then go on to factoring? Should I teach straight line functions right away and then go into the details of the definition of a function, or should I begin more abstractly with sets, domain and range, and only then move on to straight line functions? These questions multiplied indefinitely.
What is the Answer?
It is no wonder that new teachers typically follow the order laid out in the assigned textbook. After all, that is what textbook companies charge so much for: the provision of an ordered sequence of topics that, in theory, will lead to mastery if covered one after the other. Unfortunately, most mainstream textbooks are very poorly sequenced. For example, a popular Algebra I textbook begins sensibly enough with a chapter on the basic rules of algebra, covering real number and algebraic operations and properties. But the order of progression is mystifying. The first section introduces various algebraic expressions, involving all four operations. After this dizzying introduction (which according to the pacing guide is only supposed to take a day), the textbook completely ignores algebraic expressions for the next four sections, instead focusing entirely on real number operations, exponents and roots. Following those sections, it finally resumes discussion of algebraic expressions. What makes this sequence so poor is that one of the commonly acknowledged 'big ideas' for Algebra I, according to various state content standards, is the continuity between real numbers and algebraic operations. Students making the conceptual leap to algebra after concrete, real-number arithmetic desperately need to experience that continuity: to realize that adding x and y is no different conceptually than adding 2 and 3. It would be much better for a first chapter of Algebra I to develop real number and algebraic operations in tandem. For example, the first section might focus on real number and algebraic addition, the next on subtraction, and so on until all four functions had been explored individually.
Practicing the Skills
Another big issue concerns the sequencing of the practice problems. Ideally, a problem set accompanying a particular learning objective should begin with the simplest, most basic exercises to get students familiar with the concepts. It should then gradually introduce further levels of complexity and abstraction, so that simply completing the entire problem set would guarantee mastery of that objective. But in most commercial textbooks, the problems at the end of the section are presented in big blocks. Aside from a vague distinction between completely abstract problems and those which feature some contrived 'real-world' application, they are all roughly at the same level of complexity. Students doing guided practice are expected to dive right into working problems at the level of complexity targeted by the relevant content standard, which is usually overwhelming. Therefore, they are reduced to trying to mechanically copy the examples presented in the main text, which themselves exhibit the same deficiency: the first example is already at the level of complexity targeted by the content standard, instead of being a very basic, entry-level example upon which a foundation for further understanding can be built.
Even though the Common Core standards represent a genuine achievement, the real problem with math education is at the level of curriculum: determining a sequence of learning objectives that lead step-by-step to the desired level of mastery envisioned in the standards. Unfortunately for new teachers, little guidance in this area can be expected from most mainstream textbooks. Each teacher must build world-class curricula on the basis of his or her own experience and search carefully for the truly exceptional curricular resources, which unfortunately tend to be the hardest to find.