Academics · Mathematics
How We Teach Math
Most math programs are built around a schedule. Ours is built around how memory and mastery actually work. In partnership with Math Academy — an adaptive learning system built by mathematicians on decades of cognitive-science research — every AHS Online math student gets a personalized course that adapts daily to exactly what they know, supported by our live teachers and daily one-on-one tutoring.
It's the Know and the Do of mathematics, working the way God designed the mind to learn — so that our students become confident, capable problem-solvers. Below are the eight principles behind every math lesson, with the research for parents who want to go deeper.
The problem we set out to solve
In 1984, educational psychologist Benjamin Bloom published a finding that has challenged educators ever since: the average student taught one-on-one with mastery methods performed better than 98% of students taught in a conventional classroom. Tutoring works — but no school can hire a tutor for every child in every subject.
Bloom called it the '2 Sigma Problem,' and it's exactly the problem our math program is designed to answer: an adaptive mastery system that gives every student the precision of a personal tutor, wrapped in real teachers, real tutoring, and a real community.
Mastery Before Moving On Students advance when they've truly learned it — not when the calendar says so.
In a traditional classroom, the class moves on together whether or not every student is ready. Gaps quietly stack up, and a student who missed one idea in September is lost by January. We flip that: every math topic must be genuinely mastered before a student advances, so the foundation under each new concept is solid.
This is the model educational psychologist Benjamin Bloom famously showed could transform ordinary students into top performers — his research found that students taught with one-on-one mastery methods performed better than roughly 98% of students in conventional classrooms.
What this looks like for your student: Your student never gets pushed into Algebra 2 with an Algebra 1 hole. The system checks mastery topic by topic and won't build on sand.
For parents who want the research
Bloom, B. S. (1984). The 2 Sigma Problem: The Search for Methods of Group Instruction as Effective as One-to-One Tutoring. Educational Researcher, 13(6), 4–16.
Kulik, C.-L., Kulik, J. A., & Bangert-Drowns, R. L. (1990). Effectiveness of Mastery Learning Programs: A Meta-Analysis. Review of Educational Research, 60(2), 265–299 — mastery programs raised achievement by about half a standard deviation across 108 studies.
Making Skills Automatic Basics on autopilot free the mind for real thinking.
Working memory is small — cognitive scientists have shown we can only juggle a handful of ideas at once. If a student is still counting on fingers to multiply, there's no room left to think about the actual problem. So we practice core skills until they're automatic, the way a musician drills scales so they can play music.
Once the basics run on autopilot, the mental workspace opens up for reasoning, word problems, and the deeper 'why' behind the math.
What this looks like for your student: Short, frequent practice keeps facts and procedures sharp, so harder problems feel like thinking — not struggling to remember.
For parents who want the research
Miller, G. A. (1956). The Magical Number Seven, Plus or Minus Two. Psychological Review, 63(2), 81–97; and Cowan, N. (2001). The magical number 4 in short-term memory. Behavioral and Brain Sciences, 24(1), 87–114 — working memory holds only a handful of items at once.
The U.S. Department of Education's What Works Clearinghouse practice guide Assisting Students Struggling with Mathematics (WWC 2021006, 2021) gives its highest ‘Strong Evidence’ rating to regularly building math fact fluency — and the National Mathematics Advisory Panel (2008) calls quick recall of facts a pillar of proficiency.
Learning by Doing Students spend their time solving problems, not watching someone else solve them.
Watching a lecture feels like learning; doing the work is learning. The largest analysis ever done on this question — 225 studies across science and math — found that students in active-learning classrooms scored higher and failed dramatically less often than students in traditional lectures.
That's why our math time is built around deliberate practice: short explanations, then straight into carefully chosen problems with immediate feedback — every activity targeted at the specific skill your student is ready to grow.
What this looks like for your student: Most of every session is your student doing math, with feedback in minutes — not sitting through hour-long videos.
For parents who want the research
Freeman, S., et al. (2014). Active learning increases student performance in science, engineering, and mathematics. PNAS, 111(23), 8410–8415 — students in lecture-only courses were 1.5× more likely to fail.
Ericsson, K. A., Krampe, R. T., & Tesch-Römer, C. (1993). The Role of Deliberate Practice in the Acquisition of Expert Performance. Psychological Review, 100(3), 363–406.
Reviewing at the Right Moment Memory is strongest when review arrives just before forgetting.
Cramming works for tomorrow's quiz and fails for life. Since Hermann Ebbinghaus first mapped the 'forgetting curve' in 1885, researchers have confirmed again and again that spacing reviews out over days and weeks locks knowledge into long-term memory far better than repeating it all at once.
Our system tracks every problem your student solves and schedules each topic's review at the moment it does the most good — automatically, personally, and continuously.
What this looks like for your student: Instead of a review week before finals, your student gets a few perfectly timed review problems all year — and walks into tests already knowing the material.
For parents who want the research
Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis. Psychological Bulletin, 132(3), 354–380 — a synthesis of 839 assessments; spaced practice beat massed practice (47% vs. 37% recall).
Mawson, R. D., & Kang, S. H. K. (2025). The Distributed Practice Effect on Classroom Learning: A Meta-Analytic Review. Behavioral Sciences — a recent meta-analysis of real classrooms (d = 0.54) in which mathematics was the best-represented and most strongly benefited subject.
The U.S. Dept. of Education's IES practice guide Organizing Instruction and Study to Improve Student Learning (Pashler et al., 2007) lists spaced review among its most strongly supported recommendations.
Mixing It Up Blending topics in practice teaches students to recognize which tool to use.
When every problem on the page uses the same method, students stop reading the problem — they just repeat the recipe. Real tests, and real life, don't announce which technique to use. Mixing different kinds of problems together ('interleaving') trains students to recognize each problem type and choose the right approach.
In one randomized classroom study, students who practiced with interleaved problems more than doubled their score on a later test — 77% correct versus 38% for students who practiced the same problems in blocks.
What this looks like for your student: Review assignments deliberately mix earlier topics together, so your student practices choosing the method — not just executing it.
For parents who want the research
Taylor, K., & Rohrer, D. (2010). The effects of interleaved practice. Applied Cognitive Psychology, 24(6), 837–848 — interleaved practice more than doubled test accuracy (77% vs. 38%; Cohen's d = 1.21), mainly by reducing errors in choosing the right method.
Rohrer, D., Dedrick, R. F., Hartwig, M. K., & Cheung, C.-N. (2020). A randomized controlled trial of interleaved mathematics practice. Journal of Educational Psychology, 112(1), 40–52 — a full classroom trial with a large effect (d = 0.83).
Building on What They Know Every new topic quietly strengthens the old ones underneath it.
Math is the most connected subject there is: fractions live inside algebra, algebra lives inside calculus. We use that structure on purpose. When a student learns a new topic, it's chosen so that it exercises the prerequisite skills beneath it — which means earlier learning gets reviewed automatically, in context, while something new is built on top.
Over four years this produces knowledge that is layered and organized instead of a pile of disconnected procedures.
What this looks like for your student: Advancing is reviewing: a student working on new material is continuously reinforcing last month's — and last year's — skills.
For parents who want the research
Arzi, H. J., Ben-Zvi, R., & Ganiel, U. (1985). Proactive and retroactive facilitation of long-term retention by curriculum continuity. American Educational Research Journal, 22(3), 369–388.
Bahrick, H. P. (1984). Semantic memory content in permastore: Fifty years of memory for Spanish learned in school. Journal of Experimental Psychology: General, 113(1), 1–29 — knowledge that is used and built upon can last for decades.
Keeping Similar Ideas Apart Confusable concepts are learned separately so they never blur together.
Some ideas are easy to mix up precisely because they look alike — permutations and combinations, sine and cosine rules, the various factoring patterns. Cognitive psychologists call the resulting confusion 'interference,' and it's one of the oldest documented causes of forgetting.
So instead of marching through a textbook unit that piles look-alike concepts into the same week, our sequence deliberately separates confusable ideas, letting each one settle securely before its cousin arrives.
What this looks like for your student: Your student's next topic is usually refreshingly different from the last one — which keeps concepts distinct and, honestly, keeps math more interesting.
For parents who want the research
Underwood, B. J. (1957). Interference and forgetting. Psychological Review, 64(1), 49–60.
Campbell, J. I. D. (1987–2005, research program). The role of associative interference in learning and retrieving arithmetic facts — interference between similar number facts is a major source of arithmetic errors.
Right-Sized Steps Lessons are broken into steps small enough that no one gets overwhelmed.
When a lesson demands more than working memory can hold, students don't learn less — they learn nothing, and they feel it. Cognitive load theory says the fix is structural: break big ideas into small, sequenced steps, show fully worked examples before asking for independent work, and pair words with pictures.
Our math curriculum is scaffolded at roughly ten times the granularity of a typical textbook, so each step is genuinely within reach, and difficulty rises only as mastery is demonstrated.
What this looks like for your student: A struggling student isn't handed a wall — they're handed a staircase. Worked examples, visual models, and small wins all the way up.
For parents who want the research
Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285.
Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59–89 — the original demonstration of the worked-example effect.
Why Math Academy — and what AHS Online adds
Math Academy operationalizes every principle on this page: fine-grained mastery checks on every topic, review scheduled by a personalized spacing algorithm, deliberately interleaved assignments, and a knowledge graph that layers each new concept onto the ones beneath it.
What the software can't do is what we add: certified teachers who know your student by name, daily one-on-one math tutoring with a real mentor, and a faith-centered community where hard work in mathematics is part of becoming who God intends them to be. The system brings the precision; our people bring the encouragement.