## Experiment

**Our hypotheses**: Math requires a solid foundation. There are a few critical building blocks in that foundation.

Students who have those critical building blocks (CBBs) are in position to learn at grade level and beyond. They can use critical thinking to solve grade level and beyond problems.

**Experiment 1**. Find the critical building blocks for grades 6 and above

We developed a short constructed-response diagnostic assessment, and correlated scores on our assessment with two grade level indicators. Specifically, we had data from 6^{th} and 9^{th} graders in Colorado’s state test at the time, CSAP, and the NWEA MAPS tests. Our first correlation coefficients were in the 0.6 range. Through an iterative process, we improved the assessment, and achieved correlation coefficients ranging from 0.8 to 0.9.

Our assessment of 15 questions includes questions like: 4 – 5 = ? and simple fraction addition and multiplication. All of our questions come from from grades 1 through 5. The assessment takes about ten minutes.

**Our conclusion**: The high correlation between our diagnostic assessment and large-scale assessments of mathematics indicates that there is a strong association between student math gaps and their performance on state and national assessments.

**Experiment 2.** Strengthen and fill gaps in the critical building blocks and measure student growth

We provided lessons in the CBBs in double dose math classes. The first class period was a grade level class, and the second was the Peak Achievement math lessons. We provided weekly professional development for the teachers to use our lessons. Our lessons are based on the transition from hands-on objects, through pictures, and finally abstract notation. Problem sets included critical thinking, in a supportive setting.

Typical student growth was three grade levels in 5 ½ months of classroom work. This was true even for students who were at grade level at the beginning of the project. In one of the pilots, students were given homework help instead of Peak Achievement’s lessons for the second math class; their growth was half that of students using Peak’s lessons to strengthen the CBBs.

**Our conclusion**: Filling gaps in CBBs and strengthening understanding of them in a double dose math class leads to more student growth than any other program.