The Dance of Number, in two parts (Part 1, Part 2), two volumes per part with detailed solutions manuals, scope and sequence, quizzes and exams, is estimated to be published in the Autumn/Winter of 2017.
The Dance of Number is a sequenced and tightly integrated curriculum involving four textbooks totaling 1741 pages (Grades 7-10). It takes the student on a journey from Arithmetic to Prealgebra to Algebra I, Algebra II, and Trigonometry (with Geometry integrated throughout).
The texts are not named Algebra I or Algebra II, etc., because the four-volume sequence tells an coordinated story engaging the student in the nature of the structure of number, the development of its history, and its interpenetration with science. As the student takes this journey, all the topics of arithmetic, high school algebra, most of geometry, and a complete study of trigonometry are unfolded.
Because of this harmonized approach, these texts are different than most of what is on the market. The author wants the student to see how the ideas/branches of mathematics interpenetrate (e.g., you are doing algebraic operations and geometrical procedures as you are learning the elements of trigonometry). Our current textbook structure is not that successful at doing this.
After this sequence is completed, the student would be ready either for a complete course in Geometry (if desired) or PreCalculus.
The series will be published through Amazon's Createspace. The author page is here.
As a high school teacher of mathematics, I have observed far too many students who can't do algebraic work because they have not mastered Arithmetic. When it comes to dividing by a fraction, many cannot remember what to invert, a symptom of a deeper conceptual issue: "What does it mean to divide a number by a fraction, and what should my answer approximately be?" I have also noted that very few students possess what I call the joy of number sense, i.e., a feel for numbers, the pulse of their patterns - how they work, how they interact, and how they reveal rational order.
Why? Our digital world of calculators and computers that may be at fault. (Note: I am not an opponent of technological progress.) Too often students succumb to pushing buttons instead of doing an addition or multiplication problem in writing. In our world of digital cash registers, when was the last time that you saw a store clerk return change to you by counting backward, one of the lost arts of mental Arithmetic? Maybe our educational system, by experimenting with all kinds of pedagogical fads, has lost its way and lost far too many students to math illiteracy as a result.
The mastery of arithmetic operations is the foundation for success in Algebra, Geometry, and Trigonometry. We can understand Algebra, as Sir Isaac Newton (1642-1727) observed, as generalized Arithmetic. In Algebra, instead of working with numerals - symbols of the number concept - you are working with symbols of numerals, which are symbols of symbols. These symbols consist of letters that mathematicians combine with numerals. Because mathematics is consistent, you add, subtract, multiply, and divide algebraic symbols in the same way you add, subtract, multiply, and divide numerals in Arithmetic. Algebra, combined with principles of Geometry and Trigonometry, forms the backbone for understanding Calculus, the mathematics of motion at an instant, a method that Newton invented, along with Gottfried Wilhelm Leibniz (1646-1716), when trying to solve scientific problems. The historian Arnold Toynbee (1889-1975) notes the significance of Calculus:
Looking back, I feel sure that I ought not to have been offered the choice [whether to study Greek or calculus - JN] … calculus ought to have been compulsory for me. One ought, after all, to be initiated into the life of the world in which one is going to live. I was going to live in the Western World … and the calculus, like the full-rigged sailing ship, is … one of the characteristic expressions of the modern Western genius.
[Arnold Toynbee, Experiences (New York: Oxford University Press, 1969), pp. 12-13.]
Calculus is the rigging of the ship of modern science and technology. It is also the foundation of higher branches of mathematical study, subjects you will study in University. From the mountaintop of higher mathematics, you can behold a multitude of rational wonders. To climb to this height is a worthy and achievable goal, but you must master the rudiments of counting and algebraic operations as your first steps in the journey upwards.
The pages that follow examine the dance of number, explorations that will enable you to discern the rhythm of numerical and spatial patterns, both within mathematics and in the physical creation. These investigations begin with the premise that the Triune God of Scripture, the interpenetrating dance of His being, is the ground for those rational patterns. The more you are at home with these patterns, the better you can understand and use the good gifts that God has granted to you in the physical world. If you want to understand the nature and structure of the physical creation, you must master arithmetic and algebraic processes. May the living God bless you with delight and understanding as you endeavor to learn the dance of these patterns.
The reason for the popularity of some math curricula is its ease of use for both students and teachers/parents. These curricula, especially those designed specifically for homeschoolers, are primarily formula- or skill-based. In these curricula, there is little serious interpenetration with history, philosophy, theology, science, or learning to read and write real mathematics, i.e., its syntax.
The only way most math textbooks can achieve popularity is by stooping to the lowest level, i.e., they give the student math formulas or techniques that help the student to get answers to specific types of problems. This approach limits the development of conceptual thinking by trumping the why question with the how question, i.e., methods eclipse justification. Understanding and doing math requires the interplay of two thought components: (1) inductive reasoning, the search for patterns or the derivation of a way to solve a problem and (2) deductive reasoning, the establishment of the absolute truth of certain propositions or the derivation of the correct steps to get the correct answer to a problem. Instead of dictating methods, the student should learn how to discover them via induction and justify them via deduction. Many math curricula sacrifice what is essential in mathematics, problem-solving and critical thinking skills, under the facade of making math easier to learn.
The goal of this math curricula is to develop mathematical utility, the how, in the context of mathematical exploration, how the how is obtained, and mathematical justification, the why.