Hi Beatrice, difficult questions, thanks! First, I made a new Scratch model of this overnight that better achieves what I wanted, and with less code. It updates in real time as you move the sliders: https://scratch.mit.edu/projects/187449420/.
To your question, why I wrote “in contrast to what the physical manipulative suggests”, maybe there is some presumption on my behalf about this; however, some Montessori experts, and math professors, have agreed with me that the apparatus implies a fixed proportionality. I researched the Montessori trinomial for over a year for my graduate thesis. Primarily, I studied the language and word choices used by children (Montessorians and non-Montessorians, before and after engagement with the apparatus) to describe and make sense of the apparatus; I also asked about the relationship of the manipulative to the accompanying algebraic notation. Generally, I found that children who had familiarity with the apparatus had difficulty describing and perhaps noticing its properties as physical substance; instead they spoke about what one does with it, often demonstrating as they spoke, and sometimes speaking, as if about doctrine, of its utility as equipment. Non-Montessorians, lacking familiarity with the apparatus, would speak about its properties as physical substance, e.g. “it’s blocks, it’s red/blue/yellow, it’s cold, it’s wood”, etc. (I have ~80 pages of transcribed interviews on the topic). Alongside my research I recorded my evolving questions. One of my first questions was Could you change the value of one variable - thus the proportions between the three variables - and still build a solid cube that expresses the trinomial? Today I can visualize this in my mind, but at first, like many of the children I interviewed, I couldn’t relate to the algebraic notation (is it a formula? a tautology?), but only to the static, physical apparatus. And the analogies suggested by that apparatus will always be fixed at 4 : 3 : 2. Does that make sense?
Absolutely agree with you that these Scratch projects wouldn’t easily find their way into an orthodox Montessori approach