Student Bite Size vs Curriculum Portion Size

by Justin Skycak (x.com/justinskycak) on

Students eat meals of information at similar bite rates when each spoonful fed to them is sized appropriately relative to the size of their mouth. (Note that equal bite rates does not imply equal rates of food volume intake.)

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Thinking deeply about the role of instruction in supporting learners, it’s possible to arrive at the following misconception.

The Misconception: If Instruction is Done Perfectly, Won’t All Students Learn at the Same Rate?

Higher math is heavily g-loaded, which creates a cognitive barrier for many students. The goal of instructional scaffolding, guidance, and review is to help boost students over that barrier.

But if the purpose of scaffolding, guidance, and review is to help students overcome cognitive blockers, then wouldn’t a theoretical learning environment with infinite scaffolding, guidance, and review completely factor out cognitive differences, causing students to learn at the same rate?

Sure, the speed at which students learn (and remember what they’ve learned) varies from student to student. It has been shown that some students learn faster and remember longer, while other students learn slower and forget more quickly (e.g., Kyllonen & Tirre, 1988; Zerr et al., 2018; McDermott & Zerr, 2019).

But perhaps these studies are simply reflective of unfavorable learning conditions, and people would learn at the same rate in an optimally favorable learning environment?

The Resolution: Under Favorable Learning Conditions, Student Bite Size Equals Curriculum Portion Size

Continuing to think deeply about this thought experiment, one will eventually realize that infinite scaffolding, guidance, and review is not synonymous with optimally favorable learning conditions.

Sure, students will eat meals of information at similar bite rates when each spoonful fed to them is infinitesimally small. However, “eating at the same rate” would be a ceiling effect.

Faster learners would be capable of learning faster, but the curriculum would be too granular relative to their generalization ability and/or provide too much review relative to their forgetting rate, thereby creating a ceiling effect that prevents fast learners from learning at their top speed.

A maximally-favorable learning environment would require that the curriculum’s granularity is equal to the student’s bite size and the rate of review is equal to the student’s rate of forgetting.

The amended metaphor: Students eat meals of information at similar bite rates when each spoonful fed to them is sized appropriately relative to the size of their mouth.

This type of learning environment would maximize each student’s individual potential, free of ceiling effects. Critically, however, students would not progress through it at the same rate: equal bite rates does not imply equal rates of food volume intake.

Consistency with Observations

This framing of favorable learning conditions (“student bite size equals curriculum portion size”) is consistent with the phenomenon that math becomes hard for different students at different levels. The following factors affect students differentially as they move up the levels of math:

  • Combinatorial explosion in the problem space -- lowers the "bite size" more for students with lower generalization ability, or, equivalently, reduces the perceived granularity of the curriculum. (This may be a contributing factor in cases when, e.g., students do fine in math but struggle in physics.)
  • Large body of knowledge to maintain -- increases the amount of review more for students with higher forgetting rates. Also reduces effective "bite size" since an increasing portion of each bite will consist of reviewing fuzzy prerequisite material.

It is also consistent with the concept of soft and hard ceilings on the highest level of math that one can reach:

  • Say we have a student with low generalization ability and high forgetting rate. Then a favorable curriculum takes more time to work through (as compared to a favorable curriculum for an average student) due to increased granularity and review, and that multiplier increases as they go up the levels of math.
  • At some point "it requires lots of practice to learn" becomes synonymous with "can't learn" -- first in a soft sense of "the benefits of engaging in this much practice do not outweigh the opportunity costs of neglecting to develop my skills in other domains that I find easier," and then in a hard sense of "the amount of practice required exceeds the sum of waking hours over the remainder of my life."

Further Reading

For extensive elaboration, see the following posts:


References

Kyllonen, P. C., & Tirre, W. C. (1988). Individual differences in associative learning and forgetting. Intelligence, 12(4), 393-421.

McDermott, K. B., & Zerr, C. L. (2019). Individual differences in learning efficiency. Current Directions in Psychological Science, 28(6), 607-613.

Zerr, C. L., Berg, J. J., Nelson, S. M., Fishell, A. K., Savalia, N. K., & McDermott, K. B. (2018). Learning efficiency: Identifying individual differences in learning rate and retention in healthy adults. Psychological science, 29(9), 1436-1450.


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