CCTM- Diagnosing Secondary Students' Conceptual Gaps

presented by Glenn Bruckhart and Bernice German of the consulting company, Peak Achievement
By far the most power presentation. And who doesn't respect Glenn Bruckhart and his work?
The team of Peak Achievement is working on helping teachers better assess their students, interpreting these assessments to identify conceptual gaps and providing professional development on how to fill them. The documentation they provided was from a 9th grade pilot with previous low unsatisfactory scores on CSAP. After 5 months of working with Peak and receiving grade level instruction and peak's intervention program, over 45% of students moved one or more grade levels. The class averaged a gain of 45 points when the state's average was a 14 point gain.
The biggest conceptual gap that they identified in these 9th graders was subtraction. Everyone was shocked that it went back that far. The big takeaways:

• The assessments they used to see where the students are should not measure what they can do, rather, how they think. The example assessment not only asked student to solve a addition or subtraction problem, but to explain their thinking in words or pictures. (contact me for copies of the material from the presentation)
• Students had trouble with subtraction and division which may be connected to the fact that they are the non-commutative operations.
• Students start math at counting. They start with concrete examples. Ideally, it moves from counting concrete examples to using numbers to represent these concrete objects. This representation is then used in operations which goes into abstract concepts. If we don't use this sequence properly to move students from concrete to representational to abstract thinking, they don't really understand the concept and may or may not learn the algorithm. That is not enough to build on with higher math.
• The example we did at our tables is take 46 popsicle sticks and bundle groups of tens with rubber bands. The four bundles represented the four in the tens column and the 6 loose sticks were the 6 ones in the ones column. We were asked to take away 19. This required unbundling at least one of the groups of tens. Unbundling them clearly kept the idea that the bundle was a still ten popsicle sticks. We were representing the problem 46-19 with concrete examples. The words "take away" 19 is directional which may help students understand that starting with 19 and taking away 46 is different. It would be better to use this directional and concrete language rather then minus, subtract or even borrow (from the ten's column).
• Students should be able to recognize what numbers are referring to. The assessment item asked what is 12 apples + 25 oranges. Students who answered 37 should be asked 37 of what? This is directly relatable to algebra when student should know that 2 + 3x means two of something and three of something different. This also can help students to identify useful and non-useful numbers in word problems. It is assessing how well students have bridged from concrete to representational thinking. If they don't understand what numbers are standing for, they can't really do the math correctly, unless they get lucky.

I did speak with with both presenters and Bernice would really like to work with our district. I explained that our PL set up is moving away from inviting outside experts come work with us. They are working with another district to provide PD for some staff and having them go back and relay it to their colleagues, so they may consider that option for us, if we ever get to that point.

First, I know that the concentration of most efforts and energy right now is trying to set-up the curriculum to avoid such gaps and I agree that it is important to get out of reactive mode and into proactive, quality first instruction mode. After that is complete, we can work on such interventions. However, I really think that we could use the experts within our district to provide a similar intervention. The key is admin buy-in and are we ready as a school/district/community to acknowledge such gaps? What do you think about some of the ideas in the presentation? What type of gaps would you guess some of our struggling students have? Do you think something like this would be worthwhile? How do you think the students and community would react if we found similar gaps and provided similar interventions? What types of gains would you predict?