Concepts are slippery creatures. Students seem to understand a particular topic one day, and then a week later the same student has apparently forgotten all the deep understanding recently acquired. Especially, it seems, when it comes to mathematics. The videos of Cena and Jonathon are dramatic example of students who seems to understand place value, but on closer examination at a later date it is evident that this concept is still very vague for both students.
The initial knee jerk reaction here is to assume that this is evidence that the initial teaching was in some way flawed and the students probably never really had adequate conceptual understanding.
For me the real question is why children have so much difficulty developing their own understanding a concept that has become effortless for us. If as teachers we have a better appreciation of why some concepts can be so elusive then we are more likely to provide appropriate learning experiences. Current thinking in both neuroscience and artificial intelligence strongly suggests that concepts are the ability to identify patterns and make inferences from those patterns.
Here’s my attempt at explaining why conceptual understanding is fragile, and what can be done to help students acquire conceptual knowledge.
1) Provide many opportunities to manage the concept in various scenarios. The more often a pattern is identified, the less mental effort is required to recall the memory, and the greater the likelihood that the knowledge can be transferred to novel scenarios.
Each student seemed able to explain place value with numbers on the board, but subsequent questioning reveals significant conceptual gaps. This might indicate that either the previous learning was incomplete or partly forgotten, or it could be that the knowledge didn’t transfer to a different environment. Counting activities at the table are probably very familiar to the students, so it’s not so surprising that their previously well-established one-to-one correspondence type counting concept overpowers the newer concept of grouping things into tens and then counting those.
Brains learn by recognising patterns, and patterns don’t exist unless there is sufficient repetition for the pattern to become apparent. The flip side of this is that established learning – the already recognised patterns – is hard to change, which is handy for retaining existing learning but becomes a problem if the initial concept is incomplete or incorrect . When you think about it, it’s only logical that old learning should be hard to change. If it was easy to change, then most learning would only be transitory. Place value requires students to modify their previous learning of one to one correspondence and regard groups of tens and hundreds as special cases of one.
Our own familiarity with the decimal system of counting may make it difficult to appreciate the complexity of the place value concept. I thought I’d try myself out with a few simple addition, subtraction and basic multiplication operations in hexadecimal, and found myself struggling very quickly. In theory counting in hexadecimal or octal is conceptually no more difficult than counting using decimals, but in practice – well it’s something we need to practice before it can be done without considerable effort.
2) Conceptual knowledge usually take time to acquire and consolidate over a period of days or even weeks. Even those lightbulb learning moments can easily fade away from memory if they are not regularly reviewed. Learning seems to require sleep followed by reactivation of the memory of the pattern before it becomes readily available for use with minimal effort. Our experience of remembering jokes is an all too familiar example for most of us of the elusive quality of conceptual knowledge. A good joke requires conceptual understanding of the unexpected inference, innuendo or play on words in the punchline, yet even with this conceptual knowledge most of us struggle to successfully retell the well understood joke a few months later without an occasional review beforehand.
3) Manipulatives and concrete aids provide opportunities to learn in a specific context, while conceptual knowledge requires the ability to generalize from specific contexts. Hands on manipulatives are a great aid to understanding, but not a panacea. The activities and materials need to be carefully managed to provide opportunities to be sure that the focus is on the desired learning experience. Both Cena and Jonathon count accurately and confidently, but their previous experience of counting without grouping into ten seems to overwhelm their developing concept of place value. They both quickly forget about using groups of ten and revert to their already well established of counting sequentially one at a time starting with one. While we easily generalize the concept of place value to any scenario requiring numbers more than ten, to these children it is quite likely counting stars on the board and counters at the table are almost two different concepts that have yet to consolidate into a larger general principle. It would be interesting to know how the students would have answered the questions with the counters if they had had more practice in both counting the counters by groups of ten and creating larger numbers from multiples of ten at the table immediately prior to the session.
4) Once short term memory is overloaded processing information becomes overwhelmingly difficult. It’s virtually impossible to form a concept from connected dots if you can’t keep track of all the required dots. At the board students were accurately answering fairly specific questions about the tens numbers, but the next day when the presenter’s references to the multiples of ten were less direct both students seemed to struggle to keep track of the information. This was particularly evident when students were asked to take away a number that required the subtask of breaking a ten group into ten ones. Both students found it difficult to keep track of the tens and the units at the same time, focusing on one to the exclusion of the other. By switching to using the less mentally taxing counting procedures that they have already mastered they were able to reduce the mental effort.
The take away message from these two videos is that if we to want to develop students’ conceptual understanding, then we need to understand what processes need to occur for that conceptual understanding to develop. The videos also highlight that conceptual understanding is more likely to be gradually developed than acquired through a sudden insight. As Marilyn Burns, the interviewer in these videos, observes, “confusion and partial understanding are natural to the learning process” . (Marilyn Burns is passionate about improving how mathematics is taught in school. She discusses her vision in the second half of this video clip).