Rethinking school mathematics
Andy Begg
Auckland University of Technology
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Rethinking school mathematics is an ongoing professional process and should be part of our thinking about our pedagogy. It involves getting back to basics, it requires a significant effort, and it is a challenging process. Ideally the process occurs at the personal, school-wide, and national levels. To aid this task this workshop involves consideration of twenty non-trivial questions that focus on aspects of education related to our work. These questions are intended to encourage us to rethink our work as teachers.
Introduction
The conference theme, ‘back to basics; is great, but what are the basics? I’ve heard the call back to basics before, and also forward to fundamentals—but basics and fundamentals can be interpreted in ways that reinforce personal prejudices and do not result in change. Considering what is basic (or fundamental) in our work is important and needs to be done periodically. The task can be approached by considering where we are with our present approach to mathematics education, and wondering what might be changed; or by considering what might be ideal and how we might move in that direction. In this workshop I start with the second approach by looking at mathematics and its relationship with the aims of education, in particular, thinking, and with other curriculum areas. My intention is to make our present practice problematic so that we might consider possible changes in our practice.
Q 1: Mathematics and other subjects
What is mathematics, how do you define our subject, and what is its relationship with other subjects?
My thoughts
This question may seem trivial; but is important. Most people and many dictionaries, define mathematics with words such as arithmetic, geometry, algebra, statistics, … This is evident with dictionary definitions, for example: “Mathematics is a group of related sciences including algebra, geometry, and calculus, which use a specialised notation to study number, quantity, shape and space.” But, is mathematics a series of sub-topics? And, is mathematics separate from other subjects? From my perspective mathematics is a unified subject with algebra, geometry, … being an artificial partitioning of mathematical knowledge.
My definition for mathematics is the study of relations, relations being “sets of ordered pairs”, and all operations are relations where the first element of the ordered pair is itself another ordered pair, e.g., + = {((2, 3), 5), ((1, 4), 5), …}. This definition for mathematics was used by the Maori Language Commission when they chose the name Pāngarau (many relationships); it links with the ‘new maths’ of the 1960s as presented by Papy and Papy (1963/68, 1971); and it unifies mathematics.
Workshop 7.2, NZAMT Conference: back to basics, 7–10 July 2015, Auckland University of Technology.
Further, I suggest that all subjects (mathematics, science, language, … ) are part of an artificial (and western) partitioning of knowledge into disciplines, subjects, topics and sub-topics. But knowledge is holistic, all knowledge is interrelated, and this implies that when teaching a subject we should emphasise the connections and similarities with other subjects rather than the differences. However, I accept that having an integrated curriculum has led to mathematics being de-emphasized in the past so will not advocate for that although I would like to!
Q 2: Educational Aims
What are the aims of education and how do these influence our work in mathematics education?
My thoughts
I believe that our work in mathematics and in other subjects at all levels of education, should be guided by our educational aims. At school level in New Zealand these are currently set out in our curriculum (Ministry of Education, 2007, p. 12) as the key competencies:
thinking
using language, symbols, and texts
managing self
relating to others
participating and contributing
These are relevant to all subjects and at all levels of education; but what do we mean by each of them? I believe that if we consider thinking broadly then it includes the other four competencies—hence my major focus on thinking. A fuller discussion of the key competencies appears in Key competencies for the future (Hipkins, Bolstad, Boyd, & McDowall, 2014).
Q 3: Thinking
What forms of thinking are relevant in mathematics education?
My thoughts
Our national curriculum mentions creative, critical (or logical), and metacognitive thinking. However, thinking involves more than these three forms (Lipman, 2003). Mathematics is often considered as involving logical (critical) thinking but it also involves all the other forms of thinking. My classification of thinking has nine forms that are relevant to our work, these are: empirical, critical, creative, meta-cognitive, caring, contemplative, subconscious, cultural, and systems thinking. While I ask questions about all nine, I acknowledge that the boundaries between them are fuzzy, and for learners of different ages some forms are more important than others.
Q 3.1: Empirical (or sense-based) thinking
Can you remember an example in your teaching when a diagram or a model made things simpler for students than a verbal definition?
My thoughts
Empirical (or sense-based) thinking occurs when we are aware of something through our senses—seeing, hearing, feeling, tasting, or smelling. It is the dominant form of thinking of young children, and the starting point for conscious thinking for us all regardless of our culture.
Empirical thinking, or being aware through one’s senses and remembering is nearly automatic—though by improving one’s noticing skills (Mason, 2002) the process can become much richer. Empirical thinking involves sensation followed by perception (Restak, 2012). Sensation is the detection of information (awareness) with our sense organs, and perception is the interpretation of that information so that it can be remembered and used. Interpretation involves constructing meaning and is influenced by one’s prior experience, so empirical thinking is not direct knowing. (And, sometimes before a sensation is interpreted, our body reacts unconsciously but intelligently to it, e.g., one cuts one’s finger, the body’s cells immediately begin to ‘intelligently’ repair the cut before the brain registers the cutting sensation.)
In mathematics education the main two forms of empirical/sense-based thinking are:
- visual thinking (interpreting, imagining, and using 2 and 3-dimensional diagrams; Venn diagrams, arrow graphs, flow charts, Cartesian and statistical graphs, symbols, signs, and gestures; picturing, modelling ideas; noticing (Mason, 2002)); and
- aural/oral thinking (involving: making sense/interpreting what one hears, and saying what one means).
Q 3.2: Critical/rational/logical thinking
Can you remember a time when learning mathematics your teacher moved to logical (critical) thinking too soon?
My thoughts
Critical (or rational) thinking involves logic, it is important in mathematics, and it depends on a logic system and the system’s initial assumptions. However, one can only use logic when one knows what one is talking about, so pre-logic experience is important.
We take western logic for granted and make assumptions without considering alternatives. Absolute proof is never possible with critical thinking—it depends on assumptions made and the logic system being used. One can gather evidence to support a hypothesis, and if all the assumptions are made explicit then a ‘relative’ proof may be useful—but one counter-example disproves a hypothesis. Words (or symbols) are usually used in critical thinking, and diagrams can be used (e.g., Venn diagrams with sets). Proofs are not always possible with diagrams; but they are useful when exploring problems, but they can mislead (e.g., ‘are two non-intersecting straight lines parallel?’ Drawing examples may lead one to conclude that it is true, … but, not in 3-dimensions.)
Critical thinking is often used without initial assumptions being made explicit. This results in ‘solutions’ to problems without consideration of the consequences (e.g., science problems solved without considering the environment; western economics based on having more, not having enough; western philosophy concerned with individual rights, not community good; geometry problems solved for 2-d rather than 3-d; and in arithmetic, 5 + 10 = 3 [according to my wrist watch].)
Q 3.3: Creative thinking
In your learning of mathematics were you given opportunities to be creative?
My thoughts
Creative thinking occurs in art, music, literature, and in all aspects of life including mathematics when we consider alternatives and ask “what if …” questions, or consider how we might prove a result (verbally, symbolically, or with a diagram).
Our ideas of self, of others, and of things we learn all depend on assumptions and one can be creative by making these explicit, questioning them, and considering other assumptions that could be made. Creative thinking involves making connections, finding new connections, finding different solutions to problems in different contexts, or with different initial assumptions (and often assumptions are culturally specific), imagining possibilities, visualising options/conjecturing, modelling reality, designing things, making and seeing patterns, generalising and specialising, and using analogies.
Q 3.4: Metacognitive thinking (i.e. monitoring one’s thinking)
What prompt questions do you ask your students in class to involve them in metacognitive thinking?
My thoughts
Metacognition involves learning to learn, thinking about thinking, reflecting, and self-assessing. It occurs consciously, unconsciously, and automatically. The more one attends to metacognitive thinking the more one feels in control. Typical questions to ask oneself, are: Have I done enough? Should I do more? What else could I do? What have I assumed? Should I try a different assumption? How might I improve it? And students can be encouraged to ask themselves such questions.
Q 3.5: Caring thinking
Of course as teachers we care, but is our caring made explicit to our students? and, how do they demonstrate caring thinking for each other?
My thoughts
Lipman (2003) wrote about caring thinking, and this fits with aims related to self and others (family, living things, the environment, and culture). Caring is influenced by values and clarifying values help learners become more aware of them. One value is respect (respect for others including those from different cultures). Caring depends on beliefs about ‘being’—one may ask, are we all separate; and can we exist without others? Caring thinking involves ethical, emotional and critical thinking. It involves caring for self, for others, for the community, and for all living things.
In education caring is involved when one gets stuck—when should we intervene? One person steps in so the person is not frustrated; another allows the person time to consider alternatives—both reflect caring thinking.
Q 3.6: Contemplative thinking
Does contemplative thinking have a place in mathematics education?
My thoughts
Contemplative thinking includes having hunches (intuition), noticing, being still, meditating, and developing awareness; and is associated with religious contemplation, from eastern western and numerous indigenous cultures. It is not emphasised now as it was in the past because of science and critical thinking, but numerous scientists, mathematicians, and philosophers acknowledge its importance. Contemplative thinking builds on empirical thinking and complements critical thinking. Developing contemplative thinking means developing noticing skills (Mason, 2002), and sensory awareness and openness using analogical thinking (Buhner, 2014). Teachers want students to be reflective, but when asking students to reflect on something they often mean ‘think critically about it’. Reflecting from a contemplative perspective means holding an idea in one’s mind without processing it.
Q 3.7: Subconscious thinking
Does subconscious thinking have a role in the classroom, if so, what is the role?
My thoughts
Subconscious, unconscious, or bodily thinking is important even though we cannot do much about it. Mlodinow (2012) wrote that we are only aware of 5% of what goes on in our brains; our brains unconsciously handle the other 95%. Thus, subconscious thinking shapes our empirical (sense-based) thinking, ever-changing memories, social interactions, logic, and cultural beliefs; how we think about self, others, the world around us; and the assumptions that influence our conscious thinking. According to Davis, Sumara and Luce-Kapler (2008, p.24) our sense organs register about 10 million bits of information each second but we are only consciously aware of about 20 of these bits; our subconscious ‘thinking’ or bodily knowing occurs within the cells of our bodies (and within the cells of all living things) and these cells ‘know’ what must occur to live.
Intuition is linked with contemplation and involves the subconscious becoming conscious. One example of this emerged when a mathematics professor was asked, ‘how do you go about solving these difficult problems?’ He replied, ‘I read the question carefully before going to sleep, then, when I wake up I write out the solution.’ Thus, mathematics not only involves logical/critical thinking, it also involves intuitive, contemplative, or unconscious thinking.
Q 3.8: Cultural thinking
What are some of the ways that people from different cultures think differently?
My thoughts
Cultural thinking arises with people from different cultures. Nisbett (2003) wrote about the ways that people from the east, the west, and from indigenous groups think differently. These ways are not right or wrong, just different—different starting assumptions, different experiences, different vocabulary, different beliefs, different logic systems, different emphasis on nouns and verbs, and so on. Nisbett gave an example (2003, p. 141) of this involving three pictures—some grass, a hen, and a cow—and asked ‘what goes with the cow?’ Westerners used an animal/vegetable division, hen goes with cow; while easterners used a thematic relationship, cows eat grass.
One example from economics involves the basis of decision making, in some countries this is maximizing profit, in others environmental issues are more important. Another example, self-image, with western thinking I see myself as self-sufficient, yet I cannot exist without the world, the air to breath, and the life forms that provide food; so, am I an individual, or simply a part of a bigger organism?
Q 3.9: Systems thinking
In school maths we are usually working with simple and complicated systems, but are you aware of chaos theory or taught students to draw fractals?
My thoughts
Most of school mathematics is simple or complicated. However, ‘systems’ thinking is based on notions of complexity. A simple system is mechanistic, based on the idea that a cause (A) results in an effect (B). A complicated systems is also predictable, but not so simply (A causes B which causes C which causes … which causes Y which causes Z). Contrasting with these are complex systems. Complexity assumes a web of interrelationships with ideas emerging that are not predictable (A, B, C, … all interact but the result is unpredictable as it emerges from the complexity of the interactions). With systems thinking, small catalytic events that are separated by distance and time can cause significant changes in complex systems. Systems thinking techniques are used to study physical, biological, social, scientific, human, and conceptual systems. Such thinking also helps to explain how students who have attended the same class come away with different learning because of their slightly different initial ideas.