Approach and evidence

Where the design comes from

• Kiselev’s Arithmetic. Grades 5 and 6 follow the systematic course of A. P. Kiselev’s school textbook, in his order except that divisibility comes in Grades 4 and 6: each operation defined, its laws proved, and each rule given with its reason.
• Davydov’s research on number as measuring. V. V. Davydov’s work on teaching number through measuring quantities shapes Kindergarten to Grade 4: each new kind of number arises from a new way of measuring. We use his idea, not his classroom results, which are weak evidence.
• The practice of the Soviet-era textbooks. The year is planned to at least the amount of practice the Soviet-era textbooks gave a child of that age, counting only the child’s own work.
• The child first. Every rule comes from something the child has already done by hand. A long problem is planned in writing and, where it can be, solved two ways and checked.

What the research supports

The design rests on findings that the What Works Clearinghouse’s 2021 practice guide on helping children who struggle with mathematics rates as strong: moving from objects to drawings to numbers, and solving word problems by drawing how the quantities relate. It rests too on spaced review and retrieval practice, which an Institute of Education Sciences practice guide rates as moderate and strong evidence (Pashler and others, 2007), and on a meta-analysis of rewards and children’s own interest (Deci, Koestner and Ryan, 1999), which is why the books give no rewards. This is a reason to expect the course to work, not a result. A one-page note on this research, and where the course departs from it, is sent on request.

Where the course knowingly departs

It never times a child, though the same 2021 practice guide recommends regular timed activities to build fluency, with strong evidence; the course builds fluency by daily untimed practice instead. And in Grades 2 to 5 a new idea often opens with a short guided try with the objects before the explanation, where the strongest review of the question (Sinha and Kapur, 2021) favors explanation first at that age. The Teacher’s Edition still gives every step and the whole reason. The classroom trial below checks both.

Results

Edition 2.0 has not yet been taught in a classroom. We have no test results, pilot results or efficacy studies of our own, and we will not show any until they exist.

The course is designed for a classroom trial whose tests are fixed in writing before it starts, including whether a child who works half again as slowly finishes the core in the lesson’s minutes, and whether untimed practice builds the fluency the state expects.

If a district’s adoption committee pilots programs, we join on its own criteria and terms, the same as any other program. Our terms for a pilot are in writing and sent on request. The classroom trial above is a separate study, which we would welcome with a district, its measures agreed with the district and fixed in writing before it starts.