And when, towards the end of the video, an obviously capable student did try to express what he already understood in numerical terms and in English, we saw him hesitate and stumble. The students were on the verge of needing to deal with the variable-symbolization problem. Throughout the video, the letters R and B were used as labels of rings, while at the same time, the discussion and analyses focused on numbers of rings of various kinds, and how these numbers were related. The key is to treat R as standing for the number of red rings, B as the number of black rings, the adjacency between the 2 and the letter it is adjacent to as standing for multiplication, and equality as equality between quantities - between numbers rather than between sets of objects or their labels. We should note that the students in the video, both speaking and writing in their tables, did an admirable job of repeatedly mentioning that they were dealing with numbers of rings.
Thought processes underlying a common misconception.
Focus on learning problems in mathematics. On the meaning of variable. Quantitative reasoning, complexity, and additive structures. Conceptions of school algebra and uses of variables. Conservation of equation and function under transformation of variable.