# Interesting findings on concepts, procedures and problem-solving

A new paper has been published by Bethany Rittle-Johnson and colleagues. It is interesting for two reasons.

I am not yet able to understand all of the statistical analysis that the group conducted but, on the face of it, the experiment seems fair and well controlled. Elementary school students were randomised to one of four conditions. The first group got a double dose of ‘conceptual instruction’ – more later – followed by problem solving. The second received a dose of conceptual instruction followed by procedural instruction followed by problem solving. The design ensured that the double dose of conceptual instruction was the same length as the conceptual-procedural instruction.

The other two conditions where the same except that the order of problem solving and instruction was reversed so that the problem solving came first.

The topic being taught was the principle of equivalence. This is the idea that the equals sign – “=” – in an equation means that the left hand side has the same value as the right hand side. Many students interpret the equals sign as meaning ‘write your answer here’ and continue to persist in thinking of it this way. The principle of equivalence is a key concept for students to grasp before they can hope to handle anything like algebra. As such, it has been widely researched and techniques like using the analogy of a balance scale have shown promise.

Rittle-Johnson et. al. found few significant effects of the four different conditions. Interestingly, the order of problem-solving versus instruction had no effect. This is a contrast to the results of ‘productive failure’ and ‘preparation for future learning’ studies that have shown that problem solving first is superior to instruction first. There could be many reasons for this. Rittle-Johnson and colleagues suggest that it could have something to do with the ages of the students – their study was of elementary students whereas most productive failure studies involve teenagers. I would add that some preparation for future learning studies vary more than one thing at a time and that productive failure does not always utilise what I would regard as a strong control. So there could be a number of issues at play.

Interestingly, although results on an immediate post-test showed no significant effects of the condition, the students with a double dose of conceptual instruction performed better on a later test than they did on the post-test whereas those who had received conceptual-procedural instruction simply maintained the same performance on the later test. The difference was significant.

This is odd because we tend to think of learning as decaying over time rather than improving. As the authors suggest, this improvement may be related to the principle of equivalence being something that elementary students encounter daily in maths lessons – it is a threshold concept that opens up a new landscape to them.

As a maths teacher, I have to admit that I struggle with the idea of separating instruction into procedural versus conceptual in the way that this study and an earlier study by two of the same researchers sought to do. I like to think that I do both simultaneously. I never demonstrate a procedure without explaining why it works and yet this is what the researchers did in the procedural component of the instruction:

“In the current study, procedural instruction built on the language and ideas introduced in the conceptual instruction, but it did not explicitly integrate the two types of instruction or fully explain why the procedure worked.”

So I am imagining something like, “You take this and you put it here then you take away this…”

This causes me to wonder about a few issues. Do elementary teachers who do not possess specialist mathematics training tend to teach procedures without explaining why they work? That would seem to be a big problem. One way to approach this might be to furnish them with good textbooks that offer quality explanations. When I first teach a new topic I often turn to the textbook explanation. I later find myself improving on it – usually by adding intermediary steps – or abandoning it completely. Yet a good text might be a great resource for a non-specialist.

And does this relate to the great North American mathematical false choice? I have observed that explicit instruction tends to be characterised as ‘telling’ in the U.S. and Canada and that the only alternative offered seems to be to ask students to figure things out for themselves. It is as if explaining things to students is not considered to be an option.

### 5 Comments on “Interesting findings on concepts, procedures and problem-solving”

1. gregashman says:

Reblogged this on The Echo Chamber.

2. Dylan Wiliam says:

You asked “Do elementary teachers who do not possess specialist mathematics training tend to teach procedures without explaining why they work?” I can’t track it down at the moment, but I think I remember a study from the TIMSS work that showed that, in US middle schools, 80% of concept statements such as y^n x y^m = y^(m+n) were unelaborated. In other words most teachers did not even do something as simple as providing an example such as:

y^3 x y^2 =( y x y x y) x (y x y) = y x y x y x y x y = y^5

Scary if true.

• gregashman says:

Yes – that is very scary.

I teach Grade 12 Maths Methods in Australia which is roughly equivalent to A Level maths in the UK and AP Calculus in the U.S. My students have been manipulating indices using the index laws since Grade 8. However, I still go through proofs and numerical examples when we start looking at exponentials and logarithms.

It seems that the fact that x^0=1 is something that my students have to recognise repeatedly in lots of different contexts. This is much easier to remember – in my view – if you can construct something like 2x2x2/2x2x2 as an example of 2^0 but that requires an understanding of what indices are and how the laws are derived.

Liping Ma (1999; 2012 2nd). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Lawrence Erlbaum. reviewed by Roger Howe.

Somewhat related, and available online:

Liping Ma (2013). A critique of the structure of U.S. elementary mathematics. Notices of the AMS, 60 nr 10, 1282-1296. pdf

3. J.D. Fisher says:

Reprinting a comment I made here (https://goo.gl/OXOXTK) with a few minor edits:

“Does this relate to the great North American mathematical false choice?”

That has been my assessment, although I might frame it (somewhat tongue-in-cheek) in terms of the cost of professional development: which is cheaper, to admit that a huge number of professional teachers do not have adequate basic knowledge about a subject and send them all back to school, or to nurture a philosophy which makes *not* transmitting what you don’t have the highest moral ground you can aspire to?﻿