Pedagogy
How Pythago teaches math.
Pythago combines research-aligned teaching practices with a live conversation and shared canvas. The goal is not to produce answers for the learner. It is to explain clearly, reveal their thinking, adjust the support, and help them build knowledge they can use later.
01
Teach ideas clearly
New material starts with the reason it matters, then develops through definitions, intuition, concrete examples, and visual and symbolic representations.
- Explain essential ideas before requiring the learner to apply them.
- Use worked examples and guided variations before independent practice.
- Connect concrete, visual, verbal, and formal representations.
- Use contrasting cases, common errors, and boundary cases to clarify the idea.
02
Make the learner’s thinking visible
A useful question should reveal a model, prediction, explanation, uncertainty, or choice of method—not merely confirm that an answer looks familiar.
- Ask learners to explain, predict, justify, compare, and teach ideas back.
- Give learners room to think aloud, revise a claim, and ask their own questions.
- Protect the reasoning the learner still needs to produce.
- Use fresh applications and unfamiliar contexts to test transfer.
03
Adjust the next move
The tutor uses the conversation and shared canvas as evidence. It changes the next explanation, representation, question, pace, or hint in response.
- Give the smallest support that unlocks the next step.
- Slow down to repair confusion and deepen the work when the learner is fluent.
- Check a suspected prerequisite gap with a fresh probe instead of inferring it from one mistake.
- Keep incidental arithmetic or bookkeeping from obscuring the main learning target.
04
Require independent evidence
Tutor explanations, praise, and heavily scaffolded answers do not demonstrate understanding. The learner’s own work does.
- Check understanding through explanation, fresh problems, and transfer.
- Give specific feedback and let the learner repair near misses.
- In mastery mode, keep important targets open until the learner provides substantially independent evidence.
- Design each lesson backward from observable completion criteria.
05
Bring important ideas back
Learning continues after the first successful lesson. Pythago returns to prior ideas through retrieval, varied practice, and mixed application.
- Begin review with retrieval before explanation or reminders.
- Use fresh problems, representations, and contexts rather than repeating earlier questions.
- Give fragile or unresolved capabilities more attention.
- Combine related ideas in mixed and unfamiliar transfer tasks.
06
Make progress understandable
When a lesson’s completion criteria are met, Pythago generates a review grounded in the learner’s work, so the next lesson starts from evidence rather than a vague impression.
- Separate what was taught from what the learner demonstrated.
- Identify what changed and what is still developing.
- Preserve concrete evidence from the lesson.
- Name a useful next capability or diagnostic question.
Research
The principles behind the product.
These sources support the teaching practices Pythago uses. They do not replace direct evaluation of Pythago as a complete tutoring system.
Organizing Instruction and Study to Improve Student Learning
U.S. Institute of Education Sciences
Retrieval Practice Consistently Benefits Student Learning
Educational Psychology Review
Distributed Practice in the Classroom
Review of Educational Research
The Impact of Formative Assessment on K–12 Learning
Educational Research and Evaluation
The Power of Feedback Revisited
Frontiers in Psychology
Prompted Self-Explanation in Mathematics
Educational Psychology Review
Effectiveness of Mastery Learning Programs
Review of Educational Research