Thoughts on Mathematics Education in the Home

Welcome to another post in my homeschooling-thoughts series! As you may remember, I've been exploring subject-specific home education strategies on the blog in 2024. Even in toddler years (Child #1), one can set up children for success in later work by nurturing their love of learning and discovery, as well as being mindful of the destination ahead (a well-rounded, well-educated mind in a healthy body). This week, I will explore one of my weaker subjects, geography. Here are links to my other posts on health education, life sciences, history, and geography.

Basic Background for Mathematics

Sources for this section inclue 3 glossaries (ThoughtCo, Math is Fun, Story of Mathematics) and Encyclopedia Britannica. I know I'm trying to summarize an impossibly large amount of information in an impossibly small space, but I'll try anyway.

General Areas and Some Terms

According to the encyclopedia, mathematics is "the science of structure, order, and relation" which has been especially important since the 1600s. This will help to frame our discussion . . . math is not limited to numbers! 

Most math curricula and standards I've encountered tend to group lessons and assigned problems into several conceptual areas. Any one of the glossaries linked above has 100-200 terms, which can be fitted into these conceptual areas. (I don't think it makes complete sense to alphabetize terms, as they do, when discussing a non-alphabet-related subject. But I digress.)

The "numbers" category contains such terms as decimal, integer, fraction, absolute value, and order of operations (aka PEMDAS = parentheses, exponents, multiplication, division, addition, subtraction).

The "shapes and spaces" category contains such terms as angle, area, unit of measurement, circumference, chord, perimeter, and polygon.

The "populations and samples" category contains such terms as bell curve, plots/charts, mean/median/mode, and normal distribution. (This category is my wheelhouse in one course I teach.)

Finally, the "tools" category contains such terms as abacus, logic, algorithm, trigonometric function, and equation.

Another way to look at mathematics is by which areas are generally taught early in a K-12 (or beyond) curriculum and which are generally taught later. Granted, there is considerable overlap in the following order because multiple areas can be introduced early and then reinforced in a spiral fashion as students progress or age.

• Arithmetic
• Analysis
• Algebra
• Geometry
• Set theory
• Statistics
• Trigonometry
• Probability theory
• Combinatorics
• Game theory
• Number theory
• Numerical analysis
• Optimization

What other areas would you include, other than Britannica's list above?

Historical Sketch

Major cultures, continents, and countries where mathematics developed include Mesopotamia, Egypt, Greece, Islam, and later Europe. Mesopotamia is a region where the earliest civilization developed, around southwestern Europe, from before written history through AD 600s.

In Mesopotamia and ancient Egypt, focus areas of mathematics were practical counting, astronomy, geometry for construction, and decimals. Many of the earliest historical sources of evidence we have for these developments include clay/stone tablets and papyri. Check out the history of the Ancient Near East (ANE) for more information.

In Greece, some centuries later, major developments took place before and after Euclid. These included geometry (Euclid's major contribution), logical reasoning and proofs, early algebra, optics, mechanics (how physical substances move and act), and astronomy. Other important people were Archimedes and Apollonius (also a pagan philosopher).

In the pre-15th-century Islamic world (then including portions of the Middle East and northern Africa), contributions to mathematics were also significant, though most remain untranslated into other languages. Primary areas of contribution were in algebra and geometry.

By the time of the Scientific Revolution, many contributors to mathematical subfields were in Europe. Major areas included physics, astronomy (including revised planetary and stellar models), analytic geometry, and calculus which was tremendously important for scientific investigation. Mathematics education shifted from individuals to universities. Major figures included Newton and Leibniz (a Bibliovore favorite).

Later on, furthering of what had already been developed focused on fields including differential equations, linear algebra, and non-Euclidean geometry. Major figures included Gauss, Riemann, and Cantor.

How is Math Taught in Schools?

The primary source for this section is from National Council of Teachers of Mathematics (NCTM), spelling out the Common Core standards used in most United States public school systems. These standards are informed by methods and standards used in well-performing school systems in Hong Kong, Signapore, and Korea. 

Overall, the "ideal" student of mathematics should, throughout their K-12 education, exhibit several characteristics. These include (1) perseverance in solving problems, (2) abstract and quantitative reasoning abilities, (3) ability to design and critique logical arguments, (4) ability to select the best tools for approaching problems, (5) attention to precision and detail, (6) attention to structure, and (7) repeated use of successful reasoning strategies. You can see how these might be applicable to life in general, not just mathematics!

In kindergarten, recommended focus areas include primarily work with whole numbers up to 100 classified into object sets (i.e., manipulatives like marbles or acorns). Other areas to spend time on include simple descriptions of spaces and shapes, to develop children's awareness and vocabulary for later, more detailed, work.

In first grade, children may learn addition and subtraction facts from 0-20, place value (tens and ones), linear measurement with whatever units are common to the geographic area, and more attributes of shapes.

In second grade, children expand their knowledge of base-10 notation, addition and subtraction, units of measurement, and attributes of shapes.

In third grade, children expand their knowledge of arithmetic operations to include multiplication and division up to 100. Additionally, they may learn about fractions and unit fractions (1/2, 1/3, 1/4, etc.), rectangular arrays of data, and more formal naming/description of 2-dimensional shapes.

In fourth grade, children extend their arithmetic skills to multiple-digit multiplication and division, operations with fractions, and classification of geometric shapes.

In fifth grade, children perform more operations with fractions, extend division skills to 2-digit divisors and work with decimals, and explore volume of 3-dimensional geometric shapes.

In sixth grade, children initiate problem-solving with ratios and rates (e.g., recipe proportions and velocities of objects), negative numbers, division using fractions, equations, and elementary statistical thinking.

In seventh grade, children may learn about proportional relationships, rational numbers, operations with linear equations (straight lines that can be plotted on a 2-dimensional graph), scale drawings, problem-solving with 2- and 3-dimensional shapes, and making inferences about populations based on samples from those populations.

In eighth grade, students extend their equation-solving skills to those involving 2 variables, use functions as descriptors of motion, and continue analysis of 2- and 3-dimensional space and figures using, among other things, the Pythagorean Theorem.

In high school, the amount of mathematical training depends on whether the student is intending to go on to take college-level math. Required focus areas include number, quantity, algebra, functions, modeling, geometry, statistics, and probability. College math courses (some possibly taken for credit during high school) include calculus, differential equations, and linear algebra.

How can Christian Home Educators teach Math?

I'd like to start off with some autobiographical information. In my experience of learning math as a homeschooled child, I can see many advantages to what my parents did. What did we use in our Charlotte Mason-style home education experience?

• In early elementary and preschool years, we focused on real-life applications such as following recipes, counting and sorting toys, measuring siblings' growth, and generally developing numerical familiarity. I think my mother might have introduced some word problems from Ray's Arithmetic (which I inherited by request) as well.
• Around fourth or fifth grade, we transitioned to Saxon Math from 5th-grade material up through Advanced Math (i.e., precalculus and bits of trigonometry). Since the curriculum does not have a separate geometry text, I used and enjoyed what we stumbled across: the University of Chicago School Mathematics Project's text.
• Each child was allowed to go at his or her own pace as long as we completed at least the minimum work per school year in a given subject at grade level. Thus, I have a clear memory of working through advanced algebra (Saxon's Algebra 2) in the car and various offices around 8th grade, several lessons per day, during a season where my family was making frequent trips to grandparents.
• We also participated in math team starting in 6th or 7th grade, through high school. The local homeschool organization had a mother who took point to organize weekly group practices using old math tests from competitions with other schools in our size division. We therefore got a lot of experience in explaning our thought processes to others, mentoring weaker students, and engaging in timed live competitions with public and private schools' teams every couple of months. Over time, the teams got stronger and ended up with a streak of state victories in our division.

What's the common thread? An expected and manifested love of learning shared across family members and support group members. Thus, none of us experienced math anxiety.

Some more general tips that can help the typical homeschool family set themselves up for math success (from ProdigyGame) include these:

• Graded complexity of presentation
• Multiple modes of teaching
• Address fear of failure
• Give plenty of practice
• Cope with own math anxiety (12 tips)

From a Christian and Charlotte Mason perspective, I'd recommend two more things. First, since God is a God of order, connect the math you and your children are learning to a fleshed-out doctrine of creation. Second, use biography! That way, you can connect mathematics to history, especially with these pre-1800s famous mathematicians!

• Einstein
• Newton
• Bigollo (Fibonacci)
• Thales (1st, Greek)
• Pythagoras
• Descarte (though do be aware of the implications of his problematic, Cartesian philosophy)
• Archimedes
• Pascal
• Euclid
• Aryabhata
• Ptolemy
• Lovelace
• Turing
• Banneker
• Khayyam
• Eratosthenes
• Fermat
• Napier
• Leibnitz - one of the Bibliovore's favorites
• Bernoulli
• Pacioli
• Exceptions: Cantor, Boole

What do you think of these strategies? What others do you have? Feel free to share in the comments!