Each summer, as schools get ready for a new school year, the question returns, “Should we be teaching algebra to our children?” it seems to have been started by Andrew Hacker who has tried to argue and continues to argue that algebra is both difficult and an unnecessary burden for most of our students, and thus should be dispensed with. He continues to ask and the echo continues to resound, but they focus on the wrong question. The question should not be “Whether algebra?” but instead must be “What algebra?” For there are many algebras, but in particular we should focus on two, the algebra of solving equations and the algebra of functions.

Most of the algebra we teach our students is the algebra of solving equations for unknown quantities. Its origin dates back almost 1300 years to the Muslim flowering. It is the algebra we are all familiar with using x as the sign for the unknown quantity and the collection of techniques for finding out what x is equal to. Its fundamental methods, balancing and completion, are found in the title of the great work from Al Khwarizmi from which its name derives. Today it has become little more than the practice of manipulating symbols for the problems that it was designed to solve are no longer the primary problems of the business community which stimulated its practice.

The other algebra that we trace directly back only 500 years to the late 1630’s is the algebra of functions. This algebra does not solve equations, it builds tables and graphs; it is not about unknown quantities, it is about variables, quantities that can and do take on multiple values; it relies not on a series of paper algorithms, but on dynamic engines that graph and change. It is, in sum, not a static algebra but a dynamic algebra and thus the algebra of science. And of most importance, it is the algebra of spreadsheets making it today, the algebra of business too.

The algebra of functions need not and does not rely on the algebra of equations as a foundation, particularly in this time of computers where spreadsheets automatically compute. It does not need the mechanical skills of solving specialized forms of equations, nor the conceptual confusion of this x thing called by the algebra of equations “a variable” but in reality a placeholder for an unknown number. Introduced in schools today as a modest extension of the old algebra for most students, the algebra of functions only makes a serious appearance for those planning to take calculus. Despite its importance in business today, the algebra of functions and their spreadsheet tools are not a significant part of our standard math curriculum even in college algebra courses where the paper and pencil mechanics of functions are studied but not, not their applications. It is no wonder that our students complain so bitterly that the math they are forced to learn is abstract, meaningless, and utterly useless.

We should not be asking whether we should be teaching algebra, we should be asking “What algebra should our students be learning?”

To learn more you may want to read my paper, The Idea that Changed the World.

These two words have caught the imagination of educators and parents. They were designed to be the frames for talking about the value of digital learning. They were to replace the bland “individualized learning.” They were meant to symbolize a focus on the student, student-centered, and on that great American tradition, individual freedom. What surprises me is how we can attach so importance to this idea of personalization of learning and at the same time accept a common core curriculum, so dominating that it today takes up most of the school day.

The core of an apple is maybe 1/5th of its width, and if we square it to get its area 1/25th or cube it to get its volume, 1/125th of the whole. So if there is core knowledge, which it is argued, has to be at the foundation of what every student should know, and if we believe that learning should be individualized, then shouldn’t that core knowledge be a very small ratio to the whole? Shouldn’t core learning make up only a tiny portion of the school day? Shouldn’t we redefine core curriculum if we have any hope of making learning a personalized affair?

Because personalized means that a student has choices. It means that what, as well as how, a student learns and certainly why a student learns, must be the student’s choice. It must fit their interests as well as their abilities. We have always known that students have to be motivated to learn, they have to care to pay attention, pay attention to concentrate, and concentrate to learn. So, in fact, learning can’t be forced or required. It can’t be mandated. Learning has to be wanted which is the reason, the fundamental reason, it has to be personalized.

So let’s go back to work and define a core curriculum that represents the true core of the learning apple. The core curriculum need not follow the apple 1/125th model, that would most certainly be too stringent. But we should expect it to be reasonable. Core knowledge in the 21st century is changing every day and is easily searched for. Core skills on the other hand, 21st century skills, are sure to be necessary for as far into the future as we can see. Shouldn’t our curriculum reflect that? Shouldn’t the Math Common Core flip the importance and thus the time and priority between the Standards of Practice and the Standards of Content. For only then will we engage our students, personalize this their dreams, and really prepare them for 21st century work and life.

It was an appropriate title for the movie about Jaime Escalante and it is an appropriate title for the role that teachers continue to play. We all too frequently see our role in both K-12 and in college as an actor standing and delivering. As problematic as that vision may be for our physical classrooms today, it is even more of a problem for digital learning classrooms. It is the reason that the most common refrains we hear about teaching online is how much harder it is, how much more time it takes, how difficult it is to keep connected with students. For we have taken the stand and deliver classroom model and transmogrified it into the online model.

Stand and deliver teaching puts the educational burden on the teacher. Students are the recipients of the knowledge in the head of the teacher. I am reminded of this old Egyptian image of Akhenaten’s god. In the paper classroom the teacher’s ability to motivate, to tell a story, to organize, and to simplify the textbook’s knowledge was nearly all of the content available to students. Stand and deliver was a reasonably efficient way to bring the content to the student. Eye contact, proximity, raised hand signals, and easy verbal interaction made this model sufficiently flexible, engaging, and rewarding.

But stand and deliver in the digital classroom without eye contact, proximity, or easily recognized hand signals requires us to rely on other means to hold the engagement of students or to recognize their learning signals. A number of tweaks have been tried to make the model work. MOOCs make their video lectures less than 7 minutes long and separate them with student activities. Teachers make themselves available 24/7 to talk online with individual students. New communication and presentation formats have been tried to enable students to engage with each other as well as with the teacher. Teachers have to provide a wide variety of additional materials and different formats to accommodate student needs. But if technology is to enable a more valuable and efficient learning experience then we must think anew about its use and let go of the thousand-year-old stand and deliver model.