“Novice learners may benefit most from well-guided low-paced instructional procedures, while more knowledgeable learners may benefit more from minimally guided forms of instruction.” -Slava Kalyuga

The Example that Led to Reflection

I never cease to be amazed at the level of knowledge that my teachers keep bringing to the table in my class. Last week we were discussing probability trees, and one student was leading the activity with the following tree (probabilities of drawing a yellow, green or black ball without replacement):

After the student was finished answering a couple of questions we had about the tree, I posed the challenge “Create a question where the final answer is 2/5.” I asked this question because I wanted them to get more comfortable with conditional probability. For example, the probability that we will draw a black ball, given that the first ball is yellow is 2/5, so P(B|Y) = 2/5.

Much to my surprise, the first answer given was “Determine the probability of drawing a black or green ball, given that the first ball drawn was black.” I had to sit back and try to figure out where this answer was coming from since I had not anticipated it (this is both the joy and challenge of allowing students to lead the discussion)!

Since the events of drawing a black ball and drawing a green ball are mutually exclusive, we can calculate

P(B or G | B) = P(B|B) + P(G|B) = 1/5 + 1/5 = 2/5.

Can you determine the branches used to create this question? After doing some of what Michael Jacobs calls “Maths C.S.I.” I had successfully determined how the student was thinking.

Are All of Our Students Really Novices?

Over the weekend I began pondering about how there is a lot of talk that mathematics students need to be treated like novices, especially in elementary school. For example, in Anna Stokke’s C.D. Howe Report, she states

To be effective, instructional techniques must cater to the limitations of a person’s working memory, which can hold only a limited amount of new information. This is particularly important for novice learners who have difficulty focusing on new concepts when their working memory is overwhelmed.

I don’t necessarily *disagree* with the statement above – one which is taken from Kirschner, Sweller & Clark, and heavily founded in Cognitive Load Theory – it is important for us as teachers to understand when learners may have limitations, and how to effectively combat these limitations. I do, however, think it is important for us to also reflect on how often we treat our students as novice learners, and realize their potential as non-novice learners. Those who argue in favour of CLT often view their learners as novices, effectively by-passing the expert-reversal effect. Stated briefly, the expert-reversal effect states methods that typically work well to elicit learning in novice learners are not necessarily the best methods to elicit learning in non-novice learners. For example, as one progresses in their knowledge of mathematics, worked examples become less conducive to learning.

In lieu of this thought, I pose some questions:

1) Are all of our students *actually* novice learners? Is it possible that our students are *sometimes* non-novices?

2) If we agree that at least some of our students are non-novices, what methods should we utilize to elicit learning in these individuals? Must it still be direct instruction and worked examples?

3) If we believe that our students are novice learners, will we *ever* see them as non-novice learners? Does this belief we hold affect their learning?

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