Tim Psaradelis’s working instructional framework for his classes at Tower School.
A studio for
mathematical judgment.
Students learn to formulate questions, build models, calculate accurately, challenge claims, and defend decisions.
The central change
A chapter describes the mathematics to cover. A studio describes what students do with it. Keep a coherent content progression underneath; let worthwhile questions drive the experience students see.
A lesson can begin with an unresolved decision, an elegant mathematical puzzle, a surprising pattern, a design constraint, or a claim that needs testing. Students make a first attempt; the teacher uses that attempt to decide what to teach next.
Questions before procedures
Create a reason to need a method. Then teach it explicitly, connect it to student representations, and practice it until it becomes usable.
Representations before answers
Move among diagrams, language, tables, equations, and graphs. Ask students to explain what each reveals and conceals.
Judgment alongside fluency
Ask “Is this reasonable?”, “Which assumption matters?”, and “When does this fail?” Accurate arithmetic remains essential.
Revision as visible work
Keep the initial model and its revision. Assess what changed and why, alongside an independent demonstration of understanding.
What changes across the course
| Design decision | Tower pilot | Evidence to look for |
|---|---|---|
| Curriculum organization | Recurring ideas: quantity, change, structure, space, uncertainty. Chapter lessons remain available as tools. | A student uses an earlier idea in a new situation. |
| Student agency | Choice of representation, test case, and extension within a shared mathematical target. | Every student can explain the target idea independently. |
| Support | Brief prerequisite groups and teacher-selected hints; regroup as needs change. | Students enter the same investigation without permanent ability tracks. |
| Practice | Short retrieval and mixed practice, selected from observed errors. | Accuracy and retention improve on new examples. |
| Assessment | Independent transfer, a brief oral defense, and a revised artifact. | Students can defend an answer without a tool supplying the reasoning. |
| Technology | Teacher-projected models first; calculators, spreadsheets, or approved AI when they serve the question. | Students can explain what the tool did and verify the output. |
A 50-minute starting rhythm
- 0–5 minutes: Retrieve a prerequisite and estimate the investigation’s outcome.
- 5–13: Independent first attempt. Teacher listens for useful approaches and barriers.
- 13–23: Partner modeling. Give a targeted hint or brief prerequisite lesson when needed.
- 23–33: Compare two contrasting approaches; formalize the mathematics with explicit instruction.
- 33–43: Focused practice, a changed condition, or a claim to critique.
- 43–50: Individual transfer check and a short revision note.
This is a pilot rhythm, not a required daily script. Some investigations deserve multiple classes. Challenge work stays in class; manageable fluency practice can be homework.
Design for different entry points
For the lower-level Grade 6 group, begin with concrete quantities, fraction strips, teacher modeling, and a smaller numerical case. Do not wait for perfect prerequisite fluency before offering a meaningful question. Teach the missing skill in a short guided group and return students to the shared problem.
Students ready for more change assumptions, generalize a pattern, prove a boundary, or construct a counterexample. Extension means deeper structure, not simply more items or faster advancement.
Use AI to create intellectual friction
Begin with human estimation and a model. Then compare a generated claim with evidence. Ask for a counterexample, find a hidden assumption, or critique a plausible error. Finish with an independent explanation.
The studio’s claim cards are deliberately written fictional examples. There is no live AI integration, student account, or submission system here. SchoolAI is our student AI tool. Students use it with teacher guidance through school-approved activities. Students identify tool assistance; the teacher checks mathematical feedback and makes assessment decisions.
How the site can drive the change
Built now
A Studio entry point; three original investigations; progressive support cards; claim critique; revision and transfer prompts; the research and pilot design.
Next, after classroom evidence
A concept map connecting investigations to prerequisites; teacher-selected practice; a portfolio of models and revisions; approved student-created problems.
A later portfolio would require deliberate account and storage design. The current site is a classroom canvas and resource hub; it does not track individual mastery.
Test the design for four weeks
Week 1: run one studio in each grade and collect an independent baseline. Week 2: revise the access and support cards based on student attempts. Week 3: run a second investigation that reuses the same core idea. Week 4: give an unfamiliar transfer problem and a delayed fluency check.
Compare accuracy, explanations, transfer, participation, and student questions with recent classroom work. Review samples from every student, especially those who needed substantial support. This is local improvement evidence, not a causal study. Keep a feature only if it makes the mathematics more accessible or the learning more visible.
Possible signals to revise: a group’s polished solution hides individual confusion; students repeatedly wait for hints; discussion displaces needed practice; tool output replaces independent thinking.
Research, change, and uncertainty
Research supports parts of this design; it does not establish this entire model as superior.
- Middle school: The IES practice guide rates reflection and visual representations as supported by strong evidence, and multiple strategies and articulation of concepts as moderate evidence. Read the guide.
- College learning: A 2014 meta-analysis of 225 undergraduate STEM studies found better examination performance and lower failure rates with active learning. This is relevant context, not direct proof for middle school. Read the study.
- College and AI: MAA’s 2026 guidance emphasizes reasoning, reflection, communication, procedural fluency, and modeling. It cautions that unmonitored work can be unreliable evidence of individual understanding. Read the statement.
- Statistics: ASA’s GAISE work organizes inquiry around questions, data, analysis, and interpretation. See the investigative process.
My design inference: middle school should prepare students to choose models, question quantitative claims, understand uncertainty, and defend reasoning—while maintaining the number and algebra fluency that future courses require.