Scholars Online Mathematics Course Sequence

Available Courses

Discovering and Mastering Mathematics

Scholars Online currently offers a five-year sequence of mathematics courses, beginning with the Discovering Mathematics Series — Discovering Algebra, Discovering Geometry and Discovering Advanced Algebra — which provides a comprehensive algebra and geometry curriculum. Advanced students who have already mastered algebra and geometry can work with our teachers on a similar Pre-Calculus and Calculus series authored by Paul Foerster. These courses meet the mathematics standards of both the NCTM and Washington State.

Our math teachers believe that learning happens by doing — by doing problems and by creating models, virtually or actually. Teachers incorporate the use of the TI-84 calculator and The Geometer's Sketchpad, and Tinkerplots, requiring students to engage in hands-on exploration of many areas of mathematics. Students are challenged to go beyond the computational answer to a deeper conceptual understanding. Knowing that the answer is correct is not enough. Knowing that the answer is correct is important; knowing why it is correct is the key that enables students to construct their own core knowledge base and develop their own deep understanding. In pursuing understanding, the textbook is a flexible tool rather than scheduling taskmaster. Teachers may extend time spent in basic areas and selectively use enrichment materials from different parts of the text to ensure complete understanding, rather than pursuing rote completion of the assigned text.

Balancing comprehension and execution

For a number of decades, there have been strong disagreements about how to teach math. The Traditionalist and Modernist opponents in these “Math Wars” trade phrases like “drill-and-kill” and, “It doesn’t matter if the answer is right or wrong, if you understand what you’re doing”. The “Traditionalist” school places a lot of value on specifics, on details, and on correctness, often using drills to achieve a certain fluency with the symbols.

The “Modernist” school places a lot of value on understanding, often trying to stimulate discovery on the part of the student, rather than mere duplication and rote memorization. “New math” can ask more, not less, of the students. This is also insufficient for some students, especially the more concrete thinkers.

We believe this is a false dichotomy, sensed, even if only unconsciously, by people on both sides of the “Math Wars”. The Traditionalists are not opposed to understanding and discovery, nor are the Modernists opposed to fluency and correctness. How can we reconcile the two approaches, and reap the individual benefits each can offer?

We can start by remembering that the final goal of education is learning, mastery on the part of the student, not teaching. We suggest that math learning proceeds in three steps:

• Imitation
• Illumination
• Inspiration

We can all hope for inspiration, but that can neither be scheduled nor taught.

The origins of mathematics are physical, sensory, and tangible: a shepherd has twenty sheep; if one is sold, then he has nineteen. A farmer in the time of the prophets has three ephahs of olive oil, and sells two, leaving one. These don’t change — and though the units for liquid measure might, the quantity of olive oil just doesn’t depend on how we measure it. The fundamental rules of algebra reflect the behavior, the invariant behavior, of physical things. For this reason, many parts of mathematics have an almost inevitable aspect to them. People say, “But of course!” when a mathematical concept is well expressed.

“Show me how to…” is a phrase every parent, most older siblings, friends, mentors, and teachers have heard. After being shown, the learner tries it independently. Sometimes the learner’s attempts work, sometimes they don’t. We learn “debugging skills” at an early age, in exploring the world around us, in trying to do things successfully. Humans learn by doing, by “trying things out” — so a teacher has to “carve out” a period of time from everyday life when the student can focus on an activity and practice it. This is both class-time and homework-time. Whether it’s playing the piano, playing football, or mathematics, proficiency requires practice, and learning requires repetition.

After doing a new activity a number of times, many people have an “Aha!” experience, as they figure out “what’s really going on here.” We can reasonably call that “Aha!” experience “illumination”. Once seen in the bright light of understanding, at least some things will never look the same again. This “Aha!” experience can lead to further exploration and to real discovery on the part of the learner — and whether or not that discovery is original or not, it’s a good thing, as the Modernists emphasize. Unfortunately, that doesn’t always happen. After all, it took many very intelligent people hundreds of years to figure out what should be taught in a one-year math course.

But we’d like even more – more than rote learning, more even than understanding and application. We’d especially like a level of inspiration, an enthusiasm, that leads some students into becoming professional mathematicians. There’s only one way here: the learner must be at least a bit independent of the teacher, because the learner intends to surpass the teacher. Even the touchstone of inspired living, Jesus Christ Himself, said that His followers would do greater things than He Himself — changing the model for the relationship between teacher and student forever. And he did it by living it, by demonstrating it.

Echoing Jesus Christ, an effective mathematics teacher

• demonstrates,
• asks questions,
• requires practice from the student,
• guides, and
• corrects a student’s misunderstandings.

Computers and the internet are only a means. Behind any internet learning platform there must be an effective teacher who can provide the illumination and inspiration needed for the fullest expression of the good that God created in us.

A note about placement

Scholars Online tutors use a variety of methods to assess student readiness for courses. Math course placement is determined by a combination of the following:

1. Successful completion of all pre-requisite course work.
2. Conversations with the previous instructor.
3. Conversations with the parents.
4. Conversations with the student.

Topics that may be covered in these conversations include but are not be limited to

• Academic strengths & weaknesses
  • What do they enjoy studying and learning about?
  • What are they still a work in progress?
• Mathematical skills and challenges
  • What are they good at?
  • With what skills might they still need support?
• Level of support at home
  • How much mathematical support is there at home?
  • How much will the parents be there to encourage and support us with this student?
• Students wishes
  • Do they really want to proceed to this level course?

Not only do these conversations help us place the student, but they also help us determine how we cn best support the student at any level of instruction, including one that may help the student stretch beyond their own self-assessed readiness.