Reform mathematics gets a makeover

In 1989, the National Council of Teachers of Mathematics in the U.S. published the first version of its Principles and Standards for School Mathematics. It was a pivotal moment for mathematics education both in America and across the world. Despite the relatively poor performance of the U.S. in comparison to other countries and states in international tests such as PISA or TIMSS, people look to America for sexy new ideas.

The standards came to represent a movement known as ‘reform’ mathematics. The antecedents of this movement can be traced to the constructivism of Piaget and Vygotsky from earlier in the 20th century and further back to progressive education in general. John Dewey, for instance, promoted the idea of learning through experience and Paolo Freire opposed the ‘banking model’ of education where teachers transmit facts and procedures to their students. Reform mathematics is generally supportive of experiential learning and skeptical of transmission.

A chapter, written in 1996 by Catherine Twomey Fosnot, a professor of education and director of mathematics in New York, and Randall Stewart Perry, a polymath, describes the features of constructivist teaching. It could equally be a description of reform mathematics:

“Learning… requires invention and self-organization on the part of the learner. Thus teachers need to allow learners to raise their own questions, generate their own hypotheses and models as possibilities, test them out for viability, and defend and discuss them in communities of discourse and practice.

Disequilibrium facilitates learning. “Errors” need to be perceived as a result of learners’ conceptions, and therefore not minimized or avoided. Challenging, open-ended investigations in realistic, meaningful contexts need to be offered which allow learners to explore and generate many possibilities, both affirming and contradictory. Contradictions, in particular, need to be illuminated, explored, and discussed…

Dialogue within a community engenders further thinking. The classroom needs to be seen as a “community of discourse engaged in activity, reflection, and conversation” (Fosnot, 1989). The learners (rather than the teacher) are responsible for defending, proving, justifying, and communicating their ideas to the classroom community. Ideas are accepted as truth only in so far as they make sense to the community and thus they rise to the level of “taken-as shared.””

Despite its long history, there is little evidence to support reform (or constructivist) maths teaching. When trials are conducted, it is often quite different models that perform the best. For instance, in Project Follow Through, Engelmann and Becker’s Direct Instruction program, which breaks mathematics down into its component parts and then directly teaches and trains students in those parts before bringing them back together, outperformed other teaching methods in both students’ learning of procedural skills and in more complex problem solving. Those models most similar to reform mathematics – the ‘cognitive’ models – often under-performed control conditions.

Project Follow Through is not definitive but we don’t have to stop there. There is a wealth of evidence that demonstrates similar outcomes, some suggestive and correlational and some from well-controlled experiments.

This is what we might expect to find if we study the relevant cognitive science. Our working memories are very limited and approaches that break learning down into manageable, memorable chunks are more easily processed by learners than those that expect students to grapple with complex problems from the outset.

This raises the question: where do advocates of reform mathematics go from here? Should they… er… reform it? Perhaps. Alternatively, the time might be right for a relaunch.

This appears to be what Jo Boaler has embarked upon with her new book, “Mathematical Mindsets,” and her accompanying internet campaign. Here, reform mathematics is linked to the educational idea of the moment: Carol Dweck’s mindset theory. Mindset has good data to support it but the way that it is often operationalised in schools is deeply worrying and the link to reform maths is tenuous.

Neatly sidestepping issues of effectiveness, this link relies on the idea of maths anxiety, a well known effect where maths makes some people anxious. Boaler argues that this is a result of traditional maths teaching that emphasises performance under timed conditions.

Reform maths supposedly offers an alternative that is friendlier to students: “Let’s role some dice, form hypotheses and discuss our thinking – here’s a beanbag to sit on – watch that you don’t trip over my kaftan – we’re all learning this together.”

It is not at all clear that this works. Certainly, in the short term, diversion away from activities that students find stressful will reduce anxiety but would you suggest curing someone of a fear of mice by ensuring that they avoid mice? It is also likely that the kind of problem solving that we might find in reform maths classes might offer its own pressures.


How would you cure mouse anxiety?

Longer term interactions actually seem to show a different effect. It is a lack of mathematical achievement that leads to later maths anxiety. The logic of this should lead to us attempting to reduce maths anxiety by choosing methods that are the most effective for teaching mathematics to the greatest number of students. Those timed tests have a role in establishing the easy retrieval of math facts, leading to better problem solving. In fact, if we return to Project Follow Through, it was the Direct Instruction students who showed the greatest growth in self-esteem. So we’re back to square one.

In a new article in The Conversation, reform mathematics is referred to as a “mindset-approach”. I suppose that this was inevitable and either represents a reinvention or an attempt at a reinvention of the idea. It’s curious that flawed educational ideas have developed this habit of latching on to fashionable ones. It’s an attempt to invoke the halo effect but advocates of Mindset theories more generally should watch that they don’t start to grow horns.


6 Comments on “Reform mathematics gets a makeover”

  1. Greg – Piaget with his ideas about assimilation and accommodation was, in his time, a cognitivist. It’s pure schema theory. At a certain moment, his thoughts were co-opted by constructivism.

    • gregashman says:

      Thank you. It is important not to confuse people’s actual writings with later interpretations. A useful correction.

    • Raf Feys says:

      Piaget werd in de jaren ‘1970 het boegbeeld van de formalistische en hemelse New Math (omwille van zijn formalistische schema’s e.d.) In de jaren tachtig werd hij merkwaardig genoeg het boegbeeld van het andere extreem, de ‘aardse’ en ‘contextuele’ constructivistische wiskunde. Ik schreef een scriptie over Piaget in ’68-’69, een 450 pagina’s. Piaget was wel een vat vol tegenstrijdigheden, maar kan m.i. toch als constructivist bestempeld worden.

  2. Janita says:

    Spot on, Greg! The “mindset” idea seems to this onlooker to be another step along the path of converting teaching to social work.

  3. […] The flaws in this argument: It is reasonable to note the number of students demonstrating a fixed mindset with respect to mathematics. However, it is a non sequitur to conclude we need to use implicit methods. Where is the evidence that this works any better? In the absence of this evidence, does the argument even make sense? We may engage students in investigatory activities but if they don’t actually learn the maths then this could set them up for an even bigger fall later. If students are anxious about timed tests then do you remove timed tests or do you help them cope with the timed tests? How would you help someone overcome a fear of mice? […]

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s