“To learn efficiently, students must be engaged in activities that are appropriate in level of difficulty and otherwise suited to their current achievement levels and needs. It is important not only to maximize content coverage by pacing the students briskly through the curriculum, but also to see that they make continuous progress all along the way, moving through small steps with high (or at least moderate) rates of success and minimal confusion or frustration. If lessons are to run smoothly without loss of momentum and students are to work on assignments with high levels of success, teachers must be effective in diagnosing learning needs and prescribing appropriate activities.Their questions must usually (about 75% of the time) yield correct answers and seldom yield no response at all; their seat work activities must be completed with 90-100% success by most students.”

Teacher Behavior and Student Achievement, Jere Brophy and Thomas Good, April 1984

One of the key findings of the process-product education research of the 1950s-1970s was that more effective teachers seek and obtain a high success rate from their students. In a sense, this may seem like a tautology. Success is an outcome measure associated with effective teaching. So it’s a little like suggesting a comedian be funny or a runner be fast. Nonetheless, it’s not hard to see why such a factor would be of interest to researchers – nothing is easier to observe and quantify in a classroom than the proportion of questions that students answer correctly.

The question arises as to whether this is of any consequence to teachers. Despite the potential circular logic, I do think the exhortation to obtain a high success rate *is* useful, with one or two caveats. First, I will explain what I think a bad interpretation of this would be and then I will explain how I think it can be of use.

One conclusion we may draw from the need to obtain a high success rate is that we should reduce the demand of the curriculum. Instead of asking students to grapple with Pythagoras’ theorem, we could get them to do an investigation where they measure the lengths of the sides of various triangle and make a poster about this. I do not think this is a useful way to interpret this research finding because there is no strategy for meeting important goals. We want to enable students to jump over the bar where it sits, not permanently lower the bar.

So, instead, I would suggest two ways of obtaining a high success rate that are not entirely distinct. The first is a building-up strategy. To extend the metaphor, when we are building up, we gradually increase the height of the bar, providing feedback and reteaching at any point where students cannot clear it. Eventually, they are able to clear the high bar. In scaffolding, we keep the bar where it is but provide lifts and supports that we gradually take away until the student can clear it for themselves. The role of formative assessment is obvious because, without it, we cannot work out whether the bar has been cleared or not.

Building-up is classic explicit instruction. By November, my Year 12 mathematics students will need to be able to answer some pretty complex probability questions. The process of building up to this began way back in primary school with simple probability concepts. They are now at the stage where they can draw tree diagrams and have used the various probability relationships in a narrow range of contexts, but they are not yet generally able to answer the more advanced questions that they will face at the end of the year.

So part of the space between now and then involves building up through simpler examples to more complex ones.

However, I have also started scaffolding. This is when I take a full, complex question and structure it in such a way that I can obtain a high success rate even at this early point.

Here is an example. First, I project the opening blurb of a problem before any specific questions have been asked.

The probability distribution of a discrete random variable, X, is given by the table below.

x 0 1 2 3 4 Pr(X=x) 0.2 0.6p ^{2}0.1 1-p 0.1

At this point, I ask students to show me, on their mini-whiteboards, the value that the probabilities must sum to. This is not something that is asked in the question but it is something the students need to know in order to be able to answer it. If I obtain a high success rate here, I can move on to ask them what are the allowable values of p. Here, I stress the important point that 0.6p^{2} and 1-p must be numbers between 0 and 1, including 0 and 1. I ask them to annotate their copy of the question with these two conditions.

I then project the first part of the question:

a. Find p

At this point, I would ask the students to sum the probabilities, set them equal to one and collect like terms in order to get a quadratic equation. They should be able to do this. I would then ask how to deal with the quadratic equation, given that it contains of decimals. What strategy can we use? I would want to see ‘x 10’ or, better still ‘x 5’ on the mini-whiteboard. We would then discuss this.

Finally, I would ask them to factorise the quadratic equation and solve for p. The answers should be p=2/3 or p=1. I would then ask them to test that both answers satisfy the condition that 0.6p^{2} and 1-p must be numbers between 0 and 1, including 0 and 1. In this case, they do.

This way, a question that far fewer than 75% of them would have been able to answer cold, will likely have been answered correctly by virtually all of the students.

It’s easy to see why. I have stopped students from going off in the wrong direction. I have reminded them of relevant steps. In this case, both of the calculated answers are allowed, but I have reminded students that they need to consider this point in such questions. In terms of motivation, I have given them a sense of success, enabling them to tackle a question they would have previously found daunting.

Ultimately, ‘obtain a high success rate’ comes from an observation of what effective teachers do. But it is not entirely tautological. I think there is a message in there for us.