Going Deep on the
– The Course –
Best-in-World Algebra Pedagogy
How does Mark manage to receive such high praise from his students and their parents? The answer is his pedagogy, and his ongoing dedication to refining and improving it.
“Pedagogy” means the subject of how to effectively teach a subject. Here, pedagogy means the subject of teaching algebra. In other words, it means the science AND art of teaching algebra. Mark is a master at this—he has deep knowledge of algebra pedagogy and extraordinary skill in applying that knowledge.
But no person or book taught him the pedagogy he uses. He created it himself!
The components of Mark’s Algebra Pedagogy
The subject matter. This means the algebra topics and kinds of algebra problems to teach. Mark views “traditional algebra one” as the topics and kinds of problems set forth in pre-1960 first-year algebra textbooks, which he views as providing a decent definition of the subject matter (before educators introduced the so-called “new math” in the early 1960s, which was a decade before they injected “pre-algebra”). Mark began there and then substantially revised and improved the subject matter.
The teaching sequence. The MACRO sequence is the flow of broad subjects within algebra, which must be based on which things must be learned before which other (and there are some alternatives—e.g., whether to teach polynomials and factoring before or after teaching linear equations in two variables, which topics are almost entirely independent). Given a workable macro sequence, the MICRO sequence is also crucial (i.e., the step-by-step presentation of concepts and techniques within a single lesson). Textbooks are filled with BAD micro sequences, which is a setup for failure. Mark is a master of the micro sequence.
Text materials. Mark has never seen an algebra textbook he liked. Most of them are abysmal, and it’s getting worse and worse. Long ago, he abandoned existing textbooks and began writing all his own text materials. This includes both explanations and problem sets—both are crucial to effective teaching and learning. AlgebraVictory! students have access to all these materials.
The manner of solving problems. Students should be taught efficient methods to solve problems. Yet, educators often mess this up by inserting unnecessary work and superfluous writing, which kids then keep doing for YEARS. Getting the right balance is sometimes tricky. Sometimes Mark will teach kids an inefficient method for easier understanding, but he will label it “the long, slow method” so that later when he introduces the short-cut, they will DROP the inefficient method. A classic example from arithmetic is “borrowing” when subtracting by crossing out digits in the top number and writing new numbers—the old “hat” method (used before educators introduced “borrowing”) involves less writing and is easier!
How to teach the concepts, including motivation. Given what to teach, how do you teach it? How do you present a great lecture? This is a HUGE topic, and it very much depends on the exact subject matter. Each lesson and each topic must be carefully and expertly crafted, balancing many factors. And lectures should be presented with enthusiasm and joy. The AlgebraVictory lecture videos are encapsulating Mark’s lifetime-best effort at this.
How to teach solving the problems. Long ago Mark discovered the effectiveness of his topic approach. He lectures for a few minutes, introduces some concepts, including how to do some kind of problem. Sometimes it’s a “stepping-stone problem” that will not itself appear on any exam but is used to build a skill needed within other problems. Then he stops lecturing, and the students to two or three problems. Then, he resumes lecturing on the next topic. This works very well in the classroom and is anticipated to work well for online students. After several topics in a lesson, the students have acquired some substantial skills, and they move on to the lesson “assignment” of 20 or 30 problems.
The problem sets. There is a deep art to writing an effective problem set. Many factors come into play. Done well, and with good topic lectures, students just naturally move through the problems. The AlgebraVictory! problem sets are Mark’s lifetime-best effort at writing problem sets—he is making a major upgrade from his last version used in the classroom.
Tutoring a properly taught student who can’t do a problem. When students have been properly taught how to do a type of problem (as they will have been in AlgebraVictory!), but they get stuck—i.e., don’t know how to start, or get part way through and don’t know what to do next, or they complete the problem but get the wrong answer and can’t find their mistake—they need individualized help. Mark calls this “tutoring” even when he does it in the classroom. He is an expert algebra tutor. One of his longstanding DREAMS is to build a system that automates his tutoring expertise. That will be a major software and database development project for Version 2 of AlgebraVictory! (and beyond). Importantly, the best thing to say to a student who is stuck is usually NOT to tell them what to do! Instead, Mark will give lots of EMOTIONAL encouragement with ZERO math content by saying something like, “Okay… so, what do you need to do next? You can do it. Go ahead!” It is amazing how often that’s all a student needs. On the other hand, sometimes they need content help. How to do tutoring well is a huge topic, and, in fact, will be introduced into AlgebraVictory! in layers spanning several product versions.
Tutoring an improperly taught student. When students in the Middle 80% have been badly taught, the solution is to TEACH them, using all of the above! There is really no viable shortcut. However, with the Math-Smart 10%, it often works to simply show them how to do the problems, with a light explanation.
Pedagogical knowledge needed to teach algebra to the Math-Smart 10%
Why is teaching the Math-Smart 10% so much easier than the Middle 80%?
The simplistic answer to this question is that, well… it’s just easier to teach smart kids! That’s true. But why? The real answer lies in the AMOUNT of pedagogical knowledge and skill the teacher must have, relative to the content of the subject being taught.
The following diagram gives a rough idea of how much the teacher must know to teach the Math-Smart 10%. The yellow circle is the entire content of Algebra One—all the algebra that the students must learn to pass the final exam. The purple circle is the pedagogical knowledge the teacher must have to successfully teach them—which, of course, must include what the students must learn, hence the yellow circle is inside the purple.
Thousands of math teachers HAVE this amount of pedagogical knowledge and skill. They are GOOD TEACHERS for the Math-Smart 10%!
Pedagogical knowledge needed to teach algebra to the Middle 80%
But what about the Middle 80%? They require vastly more pedagogical knowledge and skill, as shown on the following diagram:
The big purple circle includes a deep understanding of the entire system of learning, including all of the above discussion of algebra pedagogy—not merely following a textbook. It also includes all the myriad of mistakes, misunderstandings and confusions students encounter with respect to the myriad of different kinds of algebra problems; PLUS how to lead them out of those difficulties.
Extremely few teachers have this level of pedagogical knowledge and skill. And there are NO materials or training programs to empower math teachers to acquire it.
THIS is the underlying cause of the Math Education Crisis!!
Resolving the Math Education Crisis requires empowering math teachers with:
Deep professional training inside the purple circle (i.e., a later AlgebraVictory! Teacher Training Course), for which teachers should receive continuing education credits that increase their salaries; and
Awesome in-classroom tools for their students to use within the purple circle (i.e., a later AlgebraVictory! Classroom version, with full tutoring power).
This is another of Mark’s DREAMS!
In the meantime…
That dream is a few years away. First, with the help of YOUR investment, we must complete AlgebraVictory! Version 1 and deliver it to homeschoolers. The revenues it generates will finance the development of the later versions.
Indeed, the big purple circle is the pedagogical structure that underlies AlgebraVictory! Trail. Version 1 of the Trail will be a bit bumpy because the tutoring system will be missing—yet, a bumpy tail will be vastly better than NO trail! As we develop subsequent versions of AlgebraVictory! that incorporate the tutoring knowledge in the purple circle, the Trail will get smoother and easier to hike.
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