Why is learning mathematics frightening for some?

When I tell people I'm a maths teacher, I have become accustomed to a somewhat standard response: an intake of breath,  raised eyebrow and "oh, wow!". It is then not unusual to hear: "oh, I was never any good at maths at school".  Why is this? Why when adults reflect back on whether they were ‘good at’ or 'enjoyed' school mathematics is it not uncommon to hear they have negative memories? Reasons can often be expressed as little more than something like: ‘I don’t know why, I just hated it, it scared me’. So, the issues surrounding this perception are clearly far reaching with feelings maintaining vehemence long into adult life. This raises questions to me about the psychology and motivation of mathematics learners. Given there still seems to be a steady stream of  secondary school leavers being frightened of maths, I ask: Is mathematics so frightening for some?

Richard Skemp's classic book The Psychology of Learning Mathematics was one of the first mathematics teaching texts I read during my PGCE year. And,  I actually read read it (as apposed to academically reading it... skimming it). In it Skemp states that since the 1960’s there has been a continued increase in concern about the teaching of mathematics (and student motivation) and that consequently learners looking back have negative retrospective views. He sees this is because for many “mathematics at school is a collection of unintelligible rules which, if memorised and applied correctly, lead to ‘the right answer’ (the criterion for which is a tick by the teacher)”. This statement made me think a lot. When recalling my school and university days the statement rang true. Often I had no idea what was going on in some of my undergraduate mathematics lectures, but there was from somewhere a desire and motivation to understand it.  As a student I was very happy collecting a shopping list of magical rules and using them to collect my ticks. It is only since teaching I have been driven to try and understand where these rules come from and question why they work. This seems like a glowing reference for teaching with no understanding whatsoever. However, for some reason I seem to be at best in a minority (perhaps even an oddity). My career as a teacher has made me aware that, for most, true motivation and desire comes from somewhere else. Given I don't know where mine came from I began to think about what drives mathematics learners.

Various studies have provided evidence to support what Skemp described.  A 1988 study by Dossey, Mullis, Lindquist & Chambers’ examined National Assessment data from the USA in the late 1980’s. It showed that 83% of seventh grade (12 to 13 years old) and 81% of eleventh grade (16 or 17 years old) students agreed or strongly agreed with the belief that ‘there is always a rule to follow in mathematics’. I suspect this is not uncommon. What then for the tasks that are multi-step or more complicated? What happens when the mythical ‘rule to follow’ cannot be promptly found. It is reasonable to guess that it is at this point motivation and performance decreases, and the seeds of anxiety towards problem and subject are sown. I can't remember the first person I heard the following quote from regarding 'problem solving' in mathematics, but it has tuck with me: "everything's a problem if you can't do it".

How is mathematics learning (as an endeavour) viewed?

In 1974 Richard Lazarus coined the term 'mathophobia' to describe mathematics learners issues with belief, motivation, and consequent anxiety. More generally this has been considered the 'affective domain', where ‘affect’ relates to how students feel when learning (McLeod, 1988). Origins of this can be traced to 1956 when Bloom divided learning  broadly into three areas or “domains”: Affective, Cognitive and Psychomotor.  ‘Psychomotor’ related to manual or physical skills.  ‘Cognitive’ related to mental skills and knowledge. While the ‘affective’ related to emotion. Whilst no perfect, and relatively old I think it still provides a good place to start discussing why some students might be frightened when learning mathematics.

Reyes (1984) suggested learning mathematics is a primarily ‘cognitive endeavour’, with Skemp (1986) suggesting a subdivision of 'instrumental' (to describe algorithmic  learning), and ‘relational’ (learning based on understanding mathematical processes) . Both concede that as in other cognitive fields, 'affect' can play an important role in students' decisions about  how they actually approach cognitive mathematical study.  Goldin (2002) goes a step further and sees the role of the 'affective domain' as essential, and not just supplementary, to cognitive processes, while Leder & Grootenboer (2005) suggest student 'affectivity'  is directly linked to cognition and stability when learning mathematics.

It seems reasonable to surmise that if things aren't going to plan for a learner they may become anxious and worried. This is not unique to mathematics and the idea of fostering resilience more generally is perhaps for another blog. Specifically to mathematics Mason (2003), and Garofalo (1989) identify the following as classic misconceptions that can knock resilience and raise anxiety when learning mathematics:

• The difficulty of a mathematics problem is related to the size and quantity of the numbers.

• All problems can be solved by performing one arithmetical operation, in rare cases two.

• The operation to be performed is determined by the keywords of the problem, usually introduced in the last sentence or in the question, thus it is not necessary to read the whole text of the problem.

• The decision to check what has been done depends on how much time is available.

There are countless others which I'm sure spring to mind for all mathematics teachers. But, let us now assume that mathematics learning can be an emotion pursuit, and also that for a variety of reasons sometimes this emotion manifests itself negatively. How do we reverse this or, more usefully, stop it occurring in the first place.

Motivation

Just as Skemp started me to question what mathematics learning was for what might be wrong with it, so McClelland has more recently made me think about how you can foster change. For those unfamiliar with the Ice-Berg Model, see the diagram below. It appeals to me primarily because of its simplicity I think, and also its obviousness. The basic premise is that things at the top of the ice-berg are visible and easily changeable (like the tip of an ice-berg). Whereas the 'hidden base' is firstly hidden, but also the thing that gives stability. Changing these things is very difficult. Right at the base underlying all else it motivation. What motivates a person? What motivates a learner?

McClelland's Iceberg Model

Markku Hannula (2004 & 2006)  concluded that motivation is very difficult to assess , understand or change (unsurprisingly I suppose due to it's hidden nature). It is internal to the person concerned and not directly observable. Specifically it is defined as “a potential to direct behaviour that is built into the system that controls emotion. This potential may be manifested in cognition, emotion and/or behaviour”. For example, the motivation to solve a mathematics task might be manifested in beliefs about the importance of the task, but also in persistence (behaviour), and in sadness or anger if failing (emotion).

Research by McCleod (1992)  showed that many students believe that mathematical ability is inherited and that learning mathematics is related to ability rather than effort. Carol Dweck's work (2000) examined this notion and focused on the relationship between motivation, intelligence and confidence and introduced two contrasting theories of intelligence. ‘Entity Theory’ suggests people’s intelligence is fixed, stable and that people are born with a specific amount. ‘Incremental Theory’ suggests that intelligence is malleable, it is not fixed and hence students can cultivate intelligence through learning. Dweck also proposes that although when initially approaching a task confidence is certainly a good thing, it often doesn’t help learners when they meet with difficulty. This can quickly lead to anxiety. This idea has been termed ‘learned helplessness’, occurring when students are move from misplaced confidence, to being discouraged, then turning off, and then giving up trying to learn mathematics (Yates, 2009). Debellis and Goldin (2006) describe this as people simply freezing doing whatever they can to avoid situations that involve the use of, or discussion of, mathematics. Some subjects in these studies described mathematics classrooms as threatening, frightening, bewildering and places which lead to guilt, embarrassment and shame.

My ideas discussion points and things to disagree with

1. Mathematics is a less rich subject when “mathematics at school is a collection of unintelligible rules which, if memorised and applied correctly, lead to ‘the right answer’ (the criterion for which was a tick by the teacher)”. In fact, this to me is not mathematics. I do think there is a place for memorising some rules (though possible facts is a better word), and I have no problem with students getting ticks for them if they are short factual recall questions. But, they need to be intelligible and built on understanding where students are able to think back to construct where they came from. I'm certainly not going to consider each time I am asked 7x8 that is built from repeated addition of units which can be thought of as 7 blocks of 8 (or that 8 blocks of 7 produce the same result). But, I could if I needed to. Where the line is drawn for these facts is, I suppose, dependent on the individual learner and the trick of a good maths teacher is to pick up on where the finger-tip facts end and the thinking and applying begins.

2. There is no correct way of doing things. I get frustrated by any new fangled one-size-fits all approach. This includes the ones that try to make themselves look like they are not one-size fits-all. Ultimately for me there is no substitute for spending time with students, talking to other maths teachers, and trying things out. Any wall chart, poster, set of procedures or 'map' is missing the point. Plan, do, review.

3. Building resilience in learners is very tough - so much so to try and come up with any one solution seems unlikely. I want my students to give problems a go, and when they go wrong to be happy with that, and then try again. However, at some point they might need a quick win. They might need to memorise some facts. They might even need to remember a rule. For example, I've played around for a few years with how I introduce Pythagoras. Either as a thing to discover, or a thing to prove (i.e. here is a formula that works, remember it - it's useful, mostly for exams. Answer some questions using it, you'll get them right. Now, you're comfortable with it, lets explore it, dissect it and understand it). I've also tried the same with trigonometry. For years I was a stickler for drawing triangles, measuring side lengths, explaining we were simply dividing and finding ratios. And, with some classes I still do this. But, with others I let them use the rules first, then go back and learn where they came from.

4. Maths teachers need to be considered as learners and it is a problem when we are put on a pedestal (which I think happens often through no fault of our own). I think the cloak of the 'clever maths teacher' that 'knows it all' still exists, and that doesn't help anyone. This isn't our fault, but it is something we can do something about. There is no way my subject knowledge or pedagogical understanding is as good as it could be, or needs to be - but I am very happy that I am unhappy about it. We can all improve. I can learn from my peers, NQTs, trainees, students, parents. No problem.

5. Doing challenging maths is important. If we're not challenged and excited about doing maths then I'm pretty sure our students will pick up on this.

6. Don't dress up mathematics to be something it's not.