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Refinement of Mathematics

In addition to focused studies on Christian and classical education, my recent studies have also included a consideration of the curriculum and methods of leading international ministries of education. I believe that every public, private, and home educator would do well to contemplate the findings of recent international research. In this article, I offer, first, a characterization of the strengths and weaknesses of mathematics instruction in the United States and second, possible refinements of our instructional methods in the home.

International Perspective on Mathematics Instruction in American Schools

The 2007 Trends in International Mathematics and Science Study (TIMSS) reported mathematics programs in the United States to produce students with lesser understanding than students educated by the programs of the following eight countries: China, Republic of Korea, Singapore, Hong Kong, Japan, England, and the Russian Federation. The TIMSS report is one among many efforts to identify and learn from the strengths and weaknesses of international mathematics programs as well as determine the reasons for the mediocre understanding possessed by students in the United States.

The following statements summarize characterizations made by researchers and serves as a partial explanation for the ongoing mediocre performance of students in the United States, relative to those of other nations.

In the United States:

  • Different teachers omit different topics. Not all students make the transition to the next grade level with the same set of prerequisite skills.
  • Spiral curricula incorporate repetition and review without enrichment and extension.
  • Teachers are overworked. They have neither the time nor energy to devote to professional development, research, collaboration, curriculum design, etc.
  • Students are under-challenged because their teachers subscribe to an ability-based model, which affords excuses, over-values efficiency, and minimizes errors. An effort-based model that promotes hard work and resilience and considers errors to be a natural and necessary part of the learning process would be more effective.
  • Content is presented in a more piecemeal and prescriptive way. The current (public) curriculum is unnecessarily modular, lacking in depth, and excessive in breadth.
  • Teacher’s subject matter knowledge is fragmented and often procedural rather than cohesive and built upon fundamental conceptual understanding.
  • Basic skills are learned without a supporting and motivating context.
  • Students misapprehend the nature of mathematics and separate it from other disciplines.
  • Methods of education reform are misguided document-based attempts to teacher-proof curricula through prescriptive standards. They are often received with pessimism. Implementation is generally incomplete and ineffective.

U.S. reform efforts characterized by the prescribing of new content standards, new textbooks, firing and rehiring of teachers and administrators, implementation of after-school programs, and increasing of accountability through external standardized testing have not directly addressed the nine weaknesses listed above. Even the more recent scrutinizing of teacher tenure policy and reconsidering of school-choice / voucher programs (although having the potential to weed out bad teachers and motivate others by a concern for individual job security as well as federal funding based on attendance) fail to address the most needed component of meaningful reform: refining the educational philosophy and pedagogy of individual teachers.

Refining Mathematics Instruction in the American Home

Surely the indictments against mathematics programs in the United States are not necessarily reflective of the condition of mathematics instruction in the American home. Nevertheless, home-schooling parents would do well to examine their own curriculum, philosophy, and instructional methods in light of the characterizations above. Refinements to your instruction might include a decision to:

  • Correlate your decisions to include / exclude particular content items with the recommendations made by research-based publications of content standards such as the most recently published Common Core State Standards for Mathematics available in PDF form at
  • Devote some portion of your annual preparation to readings that will strengthen your understanding of specific mathematical content and refine your instructional methods. Meet with other parents to share what you have learned. For your first read, consider Knowing and Teaching Elementary Mathematics by Liping Ma.
  • Stretch your child with more challenging mathematical problems, so that he/she has the opportunity to regularly experience, respond to, and learn from moments of failure. Teach your children to view incorrect solutions as opportunities to refine understanding and build resilience.
  • Build into your annual curriculum formal assessment pieces requiring an understanding of concepts spanning across multiple conceptual units, composed of problems that require your child to show work and/or write justifications. Give your child the opportunity to demonstrate understanding in a way that is not afforded to them in the form of traditional, frequent, modular, and extremely predictable tests.
  • In addition to answering the “When?”, “What?”, and “How?” questions, answer your child’s “Why?” and “What for?” questions. Know your mathematical content well enough to answer questions like: Why must fractions have the same denominator in order for them to be added to or subtracted from one another? Why does the sum of two like fractions have the same denominator but not the same numerator?
  • Look for opportunities to share with your child real applications of the content he/she is studying (association and contextualization increases likelihood of retention). Consider both ancient and modern applications.
  • Join your child in studying the major characters and events within the history of mathematics – a history that naturally intersects with other disciplines like science, art, and religion. Let this kind of contextualization paint a picture of mathematics that extends the discipline beyond the lines of prescribed algorithms.

Notice that all of the refinements listed above do not require the replacing or supplanting of any part of an already existing approach to mathematics instruction but rather a fine-tuning of the classical philosophy and pedagogy, the time-tested framework we have all come to love.


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