Changing Attitudes Toward Math Instruction

For some time now, it seems that when American education is addressed in the media, it is reported in a less than flattering light. Compared with other countries, schools in the United States are supposedly lagging behind in performance, test scores and so on. While the accuracy and relevance of such comparisons can and should be debated, there is one thing that I personally have found during my pursuit of an education degree: many people struggle with the subject of mathematics. I have witnessed college students – potential teachers – who are unable to perform basic math functions. I have also spoken with quite a few students that have problems with the subject and have formed some strong conclusions based on my observations.

I believe that schools in our country have perpetuated a cycle of sub-par math instruction that has resulted in poor math performance and a general acceptance of the fact that some people simply “can’t do math.” This unfortunate statement becomes alarming when individuals that struggle with math become elementary teachers; what quality of math instruction can be expected from teachers that have a shaky understanding of the subject themselves? However, teachers that lack a strong mathematical background are not the only problem. Some math teachers that greatly excel at the subject can be equally ineffective. If the teacher is unwilling or unable to teach the subject in a variety of ways, fails to properly assess student understanding, and does not attempt to get to know their students, the results can be as unfortunate as those from a teacher with weak math skills.

My observations were reinforced in print in the August, 2005 issue of Middle Ground: The Magazine of Middle Level Education. In that journal, University of Virginia education professor Carol Ann Tomlinson describes two classes from her middle school years that shaped not only her philosophy of education, but her overall attitude toward two subjects and more importantly, her self-esteem. Her article “Differentiating Instruction: Why Bother?” describes her seventh grade math teacher:

She was a serious math teacher. She covered math with a single-mindedness that was evident even to seventh graders. She explained the math in one way and one way only. She taught each topic one time and one time only. She used one form of assessment and one form only. She knew math, but she didn’t know about me at all.

Ms. Tomlinson goes on to describe how her suddenly found inability to “do math” damaged her self-confidence as an individual during the already difficult stage of adolescence. She also developed a profound dislike for math that has persisted into adulthood.

Thankfully, she simultaneously had an English teacher that had a completely opposite view of his students. While she acknowledges his shortcomings as a teacher, the lasting effect from her time in his class was the manner in which he got to know each student, adapting his instruction to fit their interests and abilities. Doing so kindled a great interest in literature and writing for Ms. Tomlinson, and made her realize the importance of differentiating instruction (a term that she says that she doubts anybody used at that time). By finding her talents, her English teacher showed that he wanted his students to learn and excel, while her math teacher displayed a differing attitude: “That I understood virtually nothing she was talking about was either off her radar or beyond the parameters of her interest.” This attitude was more damaging because while she felt validated in English, it could not offset the negative effects of the math class. Reading the article, it is a wonder that Ms. Tomlinson was not turned off of education completely. Instead, she discovered the reason for differentiated instruction: “Our success as teachers in helping students see themselves as competent in the subjects we teach will affect the rest of their lives.”

In addition to less than ideal instruction, much of the problem with United States math performance can be attributed to the attitudes we have toward the subject. Unlike subjects such as science, English, art and so on, answers to math problems are not open to interpretation; answers are correct or they are not. The problem with considering math in this light is that the focus has been almost completely on finding the right answer. Over the years, it seems that we have found one primary method for finding ‘the correct answer’ and that method is taught in exercises, over and over again.

American students learn how to add complex numbers in the same way, learn to multiply the same way, multiply fractions the same way and so on. Students perform these steps, in this order, get the correct answer, and repeat. The issue in teaching this way is that the focus shifts away from the thought processes needed to solve math problems. The end result becomes teachers like the woman described in Ms. Tomlinson’s article. Such a narrow view of math perpetuates attitudes like this one from a Florida school board candidate (and retired assistant principal): “one plus one still equals two.” (Solochek) Opinions like this one provide a major roadblock to the improvement of math instruction, especially when it comes from potential school board members.

The ultimate question becomes ‘what can we do to improve mathematics instruction in our schools?’ On a positive note, researchers and schools of education have the right idea. The mathematics education curriculum at Ivy Tech and the University of Southern Indiana in Evansville place a focus on answering the question ‘why?’ For example, instead of blindly multiplying fractions by multiplying the numerators and denominators, why does it work? Also of major importance is discussing the thought processes used in problem solving maths. Marking a problem wrong when the incorrect answer is not achieved does not teach a student anything; determining how the student arrived at the answer lets them know where they got off track, and allows the teacher to adjust the level of instruction that is necessary for the student to understand the concept.

As Carol Ann Tomlinson writes, differentiating instruction is also a major factor in improving school performance, and math classes are no exception. Like teachers of any subject, it is imperative that math teachers get to know their students and understand how they learn best. Creativity can exist in math classrooms just as it does in English, science, social studies or art classes. A creative maths teacher can use experiments that show how new math concepts work. Using drawings such as strip diagrams can help students visualize story problems, helping them to determine the proper method of solution. Teachers can often relate items of student interest to particular concept maths. Even small things such as using humor or inserting student names into story problems can make math more enjoyable and less of a ‘stuffy’ subject.

Reforming education is a difficult task. Much of this difficulty stems from getting people to agree on the most effective methods of reform. If the fact that American students are struggling in their mathematical abilities is believed to be true, it must be concluded that our typical methods of instruction – following specific steps repeatedly through daily homework practice – are ineffective. If we want to improve the ability of our students to solve problems and perform well in math, we must change the way that we teach the subject. Understanding the needs of each student - like we often do in reading instruction - can be done in math as well. Teaching concepts in multiple ways can accommodate the wider variety of learners that enter our classrooms. Teachers today have access to a wealth of resources; using these resources along with some creativity can go a long way in achieving the kind of math improvement that all of us can agree upon.

WORKS CITED
Solochek, Jeffrey S. “Pasco County schools switch to new math program.” St. Petersburg Times, August 22, 2010. Retrieved from http://www.tampabay.com/news/education/k12/pasco-county-schools-switch-to-new-math-program/1116669
Tomlinson, Carol Ann. “Differentiating Instruction: Why Bother?” Middle Ground: The Magazine of Middle Level Education, August, 2005.

ESRC Mathematical Relationships: identities and participation

By Jenny Shaw

I know very little about maths education per se, but have some indirect experience of it having spent the best part of a weekend sitting at my kitchen table beside my son struggling with the maths tuition class of his electronic engineering degree. His girlfriend was there too, and both us were feeling both his and our anxiety. When, eventually, he finished the task it felt like a tremendous achievement, though he said he was sure he had got it all wrong and would fail. Whether or not our sitting there had helped or not, it was clear that feelings and unconscious states of mind were flowing around the room; and on reflection I have wondered how far this might be compared to what psychoanalyst Wilfrid Bion called ‘reverie’ or more specifically ‘maternal reverie’[1]. There was some discussion, I gather, at the last seminar about Bion’s theory of thinking, and the terms K and -K referring to types if learning and knowledge, and the ‘K link’ referring to the relationship of mutual dependency between infant and mother were raised. This last flows into the concept of reverie which was the term Bion used for the way mothers metabolise, process and transform the painful experiences their infants are busy expelling. Many of us know that just having someone there while you struggle can be comforting, and some children, or adults, seem especially to need that companionship- but it is not only comfort.

What I want to talk about today is obviously speculative, but internal states and dilemmas probably have something to do with liking or not liking a subject, and whether someone feels they are good at it or not. And both, together, account for a good part of the disaffection or turning away from maths which is the theme of this series. Because the  psychodynamic approach has been chosen as one of the theoretical perspectives to be carried through the series I take it that there is some acceptance that unconscious processes and associations have an  impact, that early life experience is especially formative, that feelings come before cognition, and that relationships are the fundamental source or driver of many of life’s patterns. Generally speaking, I am taking an ‘object relations’ approach, the development of which marked a departure from Freud’s stress on instincts, but nevertheless want to mention Melanie Klein’s notion of the epistemophilic instinct, the desire to learn about the world, but also the need to understand what inhibits it. The child psychotherapist Hamish Canham notes that at the heart of all learning is a tension between this desire to learn and a different desire not to know the truth and in writing of this he quoted Ronald Britton on how ‘new knowledge arouses our hostility, threatens our security, challenges our claims to ominisicence, reveals our ignorance and sense of helplessness and releases our latent hatred of all things new or foreign’[2]. It is easy to find examples of this, at the level of both the individual and society, we need only think of Thomas Kuhn’s discussion of paradigms and paradigm shifts. But let me stay with Canham who cites Roger Money-Kyrle’s claim that there are three key facts of life which, if not accepted, impede learning, one of which is recognizing the centrality of the relationship between parents to mental life. Evading knowing about this fact (and again there is plenty of evidence that such evasion occurs as, for example, in the ‘family romance’), can make for difficulties in learning and especially, Canham suggests, biology, but also maths. And he then gives an example: of a 6 year patient whose parents had separated, but were thinking of getting back together again. Based on some drawings done in the session, of faces and lines connecting them (but not mother and father) and of the girl then deciding to do some maths – he interprets the difficulty she has at doing a sum she had set herself (1+46 = 46) as being due to her difficulty in seeing her parents as a couple and that, as a result, the girl could only see half the sum. This is only one example, but many of us have our own ‘funny’ ways of doing certain calculations which are perhaps signs of an inner shaping of mental operations and of resistance to doing them in the way we are taught/told.

Numbers, tables, geometric shapes such as cones are all, as are words, capable of eliciting associations- not all them negative. But some are, and negativity runs through the language of mathematics which is full of destruction - cancelling, killing, subtracting, taking away, dividing and, of course, negative numbers. And though there is adding and multiplication too, the death-dealing words outnumber the more benign or creative ones and, I am suggesting, resonate at the unconscious level. The power of maths is often described and thought of as ‘awesome’, and something with an ‘awesome power’ is potentially very threatening and easily associated with punishment. Words like ‘mean’ are ambiguous or have several meanings, and though they have a specific meaning used in maths, they carry the other meanings too. Gianna Williams, for example, writes of an adolescent patient who was very particular about time, and never let her forget the mean-ness of Greenwich mean time[3]. The binariness of some maths, of one and nought, does not allow for any in-betweeness, and may encourage a sense of ‘sudden death’. For paediatrician and psychoanalyst Donald Winnicott indeterminacy and imprecision was especially important in fostering growth and creativity[4], but maths at the levels studied in school is very prescriptive, rule bound, and breeds the feeling that there is nothing between being very good at maths or a complete failure. However creative maths can be at the higher level it is not at the lower ones and our concern today is about how so few get to enjoy maths in the most creative way. And one possibility, which I am suggesting today, is that there is a binary quality about the way maths is learned in school which inhibits the form of understanding which is based on identification and that unlike some other subjects, say history, it is harder or even impossible to understand what it takes to be good at maths. And just as someone who isn’t good in this way can’t see how those who are can be so, reciprocally, those who are good at maths cannot imagine how those who aren’t fail to see the light. The precision of maths may, at times, be a comfort, and in the context of gender differences in subject choice it has often been mooted that because of the ‘male wound’ (the social demand’ for boys to dis-identify with their mothers) boys lean towards subjects which give some relief from the messiness of feelings which they have been prevented from expressing[5]. But precision can also represent heartlessness. It is a terrible tension and Wilfrid Bion, a psychoanalyst mentioned at the Manchester seminar, drew heavily on mathematical imagery in his attempt to explain the nature of  psychoanalysis was and, indeed, concludes his book Attention and Interpretation by prefacing the index as follows: ‘This indeed, like the rest of the book, is the outcome of an attempt at precision. The failure of the attempt will be clear; what may not be clear is the following dilemma; ‘precision’ is too often a distortion of the reality, ‘imprecision’ too often indistinguishable from confusion’. A no-win situation.

My main point so far is that maths has connotations with death, and many of the anxieties which come, for many, from doing maths, hark back to the primitive anxieties of early life when all was chaos, unpredictability and often full of fear. The existential fears of infancy are grounded in whether mother is there, in which case all is well, or whether she is not, when everything falls apart. This is the ur experience, life or death, or ‘one or none’. With the growth of a capacity for symbolisation, things change, and things, material things, can come to stand for the mother, and she can be ‘a little bit there’. This is a the point of the ‘transitional object’ as described by Donald Winnicott and the series of tricks or illusions which help an infant build the internal apparatus for dealing with separation and deprivation. It is indeterminacy, not precision, which Winnicott sees as the critical feature of the ‘transitional object’, and in his account of what goes on between a mother and her infant Winnicott stresses that what is given is experienced as found or created. And this has led me to wonder, from a position of ignorance, whether there is something of this underpinning the terms ‘constructivist’ or ‘fuzzy maths’? Both Winnicott and Bion attempted to grasp and explain the flow or passage of feelings between individuals and the psychological techniques, mainly projective identification and introjection, which powered those flows. Which brings me back to my kitchen table.

I don’t know what happened next, what marks my son got, only that he has does not appreciate the ‘digital marking’ which gives a tick or a cross, but not feedback which would help him learn from the wrong answers. Earlier I suggested that the situation of the shared feelings around my table might be comparable to Bion’s notion of maternal reverie. Neither my son’s girlfriend nor I could think for my son, we weren’t making sense of the problem he was struggling with, but we were both intently thinking about him, we had him in mind and he was not left quite alone with the struggle.  As with a small child a mother may not explain something, but because she is emotionally engaged with and tolerates it, this may enable the child to make the shift. The reverie concept involves an exchange of experience between mother and infant where a painful feeling, like an infant feeling it is dying, is projected into the mother and after some time there is re-introjected in a way which make the feeling tolerable because it has been ‘shared’ -which means the infant does not feel not alone with the terror. But if the ‘mother’ does not accept the projection, and bats it away, the infant’s feeling is not returned in a more tolerable understood form and  loses all meaning – it becomes a ‘nameless dread’. In Bion’s description the infant still projectively identifies with ‘the mother’, but instead of having a sense of her as a receptive and understanding person the baby is presented with a ‘wilfully misunderstanding object’; but, nonetheless, it is the one with which the infant identifies. I have put ‘mother’ in inverted commas because I think a subject or discipline, like maths, can be experienced in a similar way as a person and have argued this elsewhere[6], (and one reason the term ‘object’ is routinely used in the psychoanalytic literature, is because things or parts of people can be our ‘object’ too). And even if this example is a bit close to home, the obligation to use academic jargon may capture some of the awkwardness of identification with a wilfully complex and misunderstanding object.

This leads to another of Bion’s contributions, the distinction between, and pairing of, the container/contained[7]. The point of the container (mother usually) is to contain the child’s experiences and fears, and Bion writes that it permits ‘an emotional realization of a learning experience which becomes progressively more complex as it constantly recurs in different forms throughout mental development, finally encompassing the possibility of whole hierarchies of hypotheses, and scientific deductive systems’. However it seems that for Bion maths was, perhaps, the final test, and that if psychoanalysis could be represented in terms of mathematics it would become the container. What may be interesting for this audience is that Bion used mathematics as the idiom for expressing his ideas about the nature of thinking. It is abtuse stuff and there is a schema or grid for plotting how thought develops from a ‘preconception’ which meets/mates with a ‘realization’ to make a ‘conception’ and produces satisfaction, (a harvest of self-sensations). Bion emphasises that the ‘crux’ in the development of thought lies in the ‘decision between modification or evasion of frustration’ and that the development of mathematical elements is ‘analogous to the development of conceptions’. But he also writes that ‘Mathematical elements, namely straight lines, points, circles and something corresponding to what later becomes known by the names of numbers, derive from realisations of two-ness as in breast and infant, two eyes, two feet and so on….’

I fear I have laboured this point enough, and that what we need to focus on is what breaks the link and the circuit of mutual understanding/thinking and leads to attacks. The link in the first place is that between infant and the breast, or patient and analyst, and the mechanism of projective identification-and the attacks on mathematics or the teacher of it, is a result of non-meaning dominating and unmet needs being responded to with envy and hostility – which were beautifully illustrated by some of Steven Blake and Tamara Bibby’s transcripts from the last seminar. The relationship with the person, the teacher, becomes the relationship with the discipline and the ‘object’, now maths, goes bad, and ‘pleasure’ comes only from reviling/attacking it and the person (teacher) associated with it. When communication between the one ‘who can do maths’, (teacher) and ‘the one who can’t’ (child or student), is ruptured, a different process is set up, misunderstanding arises. Negative and toxic feelings are projected, and introjected, the ‘container’ doesn’t understand what is going on, and with non-meaning dominant the infant has hostile, envious feelings (thinks the mother/teacher does know, and has what is wanted, but won’t give it- the good things in the ‘breast’ are withheld).

I am not sure what practical lessons we can draw from all this, but I want to end by citing Margot Waddell noting that the kind of thinking which will be going on in any learning-situation is based in processes for which the mother/infant relationship is the prototype’, and that there is inevitably a very ‘complex relationship between emotional and cognitive learning and their underlying mental states’. Waddell then outlines three broad ways of relating (‘primary identifying with’) another person: adhesive, projective and introjective and gives an example of three ways a mother might help her child solve a jigsaw puzzle[8]. Teachers, as I have argued elsewhere, are emotionally ‘in loco parentis’, and much of what goes on schools and colleges is piggy-backed on to the parent/child relationship- with the consequence that it is not just the pupil/teacher relationship which is anchored in early life experience, but the pupil/discipline relationship too[9]. These traces and links, though they become more attenuated and abstract as children move from primary to secondary school, and have more and different teachers, as well as different rooms to frame their learning, remain the template or paradigm for all later learning. The challenge for maths educators is how to be aware of unconscious and primitive understanding of relationships in ways which foster the manipulation of their more abstract representations. I know policy makers like solutions from the moment the problem is identified rather than understood, and that they like ‘trigger points’ too, as these give a clue to points of intervention. I am loathe to either give suggestions or simply evade the need for them so I would end on suggesting that there may be scope for re-thinking the order in which maths is taught – the latency period for example with its characteristic joy in amassing facts, and joy in being tested stands out but also further experiments such as those teaching topology to primary school children such as reported by the Sauvys [10].

P.S. I had planned to start this talk with Howard Becker’s reverse engineering point – of  how,  if we wanted to turn off students from maths, might we go about it?[11] And one of the most obvious answers might be by giving lots of tests and impersonal marking (digital) systems. And though not all testing is counterproductive, it generally evokes anxiety and it may be no accident that other subjects suffering ‘flight’, like languages, are also ones which rely heavily on testing as part of their pedagogy. I was very shocked to learn at the first seminar in this series, on testing and assessment, that any item which most students could do was removed- a sign that ranking or stretching was prioritised over learning, and that almost all mathematicians felt that they were failures. Is it any wonder?

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[1] See Wilfrid Bion (1967) Second Thoughts. Selected Papers on Psychoanalysis, London, William Heinemann Medical Books

[2] See Canham (2002) ‘Where do babies come from?’ reprinted as ch 1 ‘What makes children want to learn?’ in B.Youell (2005) The Learning Relationship

[3] Gianna Williams (1997) Internal Landscapes and Foreign Bodies. Eating Disorders and Other Pathologies, London, Duckworth. Tavistock Clinic Series.

[4] Donald Winnicott (1971) Playing and Reality¸London, Tavistock.

[5] See Jan Harding and Michael Sutoris (1987) ‘An Object-Relations Account of the Differential Involvement of Boys and Girls in Science and Technology’ in Alison Kelly (ed) Science for Girls Milton Keynes, Open University Press and Liam Hudson and Bernadine Jacot (1991) The Way Men Think: Intellect, Intimacy and the Erotic Imagination  New Haven, Yale University Press

[6] Jenny Shaw (1995) Education, Gender and Anxiety, London, Taylor and Francis.

[7] Wilfrid Bion (1970) ‘The Container and Contained’ in Attention and Interpretation: a scientific approach to insight in psychoanalysis and groups,London, Tavistock Publications

[8] Margot Waddell (1998) Inside Lives.Psychoanalysis and The Growth of the Personality. London, Tavistock Clinic Series. The adhesive mode is a form of mimicking or parroting where a child seeks to be someone else, a teacher or parent maybe: the projective is where a child acts ‘as if’ they are another, towards, say, a sibling; and the introjective is where by seeking understanding and engaging in the task the child is enabled to build a secure sense of self derived from introjecting the thoughtful qualities of mind nurtured by the mother/teacher or, in Bion’s world, the ‘thinking breast’.

[9] See Shaw op cit.

[10] Jean and Simone Sauvy (1974) The Child’s Discovery of Space, Harmondsworth, Penguin Education.

[11] Howard Becker (1998) Tricks of the Trade. How to think about your research while you are doing it Chicago, Chicago University Press

Use Flash Cards to Teach Preschool and Kindergarten Maths

What is a flashcard?

A flashcard is a piece of paper or card with a letter, a number, a word or a picture written on it. Sometimes there is something written on the other side (e.g. translation, answer, name, etc.)

Why are flashcards useful?

Flashcards are useful because we can use them many times. They provide variety and interest for children. We can use them for literacy practice, both English and other langauges (letters at P1 and words at P2-P3, etc.), for maths literacy (learning numbers and symbols +, x, =, etc.), for memorising vocabulary and for encouraging children to be creative. Each teacher can keep his / her own set, and can add to them all the time. You can keep your flash cards in a box, divided into different categories (letters, pictures, nouns, verbs, etc.)

How do I make flashcards?

Basic Flashcards

You just need paper and a marker pen. If you need a marker pen, ask your head teacher. Even if you don’t have a marker pen, you can make them with a bic (see picture below), or with coloured chalk; blue and purple work best.

Hanging Flashcards

These are very good when you want to combine flashcards in an order (e.g. maths sums, or sentences in English). You need larger pieces of paper / card, which are folded. You also need a piece of string that you can hang beneath or across the chalk board (see picture below). You can write on both sides of the card and flip them over.

      

What activities can I do with flashcards?

Basic Literacy

1) Show & Elicit (P1-4) Show the flashcard and elicit the word / number / name / translation

2) Point at (P1-3) Flashcards are held up by students in different parts of the room. Tell the other students to point at one card.

3) Give me (P1-4) Put many flashcards on the floor. Tell one student to bring you one card.

Sequence Formation

1) Make a sentence (P2-6) Stand several students in a line at the front of the class, each with one flashcard. The others must tell them to stand in the correct order.

2) Creative maths (P1-3) Display number and symbol flashcards. Ask students: ‘Who can make a sum?’

Vocabulary learning

1) Translate (P1-6) Show English, elicit Kinyarwanda (on the back) or vice versa.

2) Vocabulary Box (P1-6) Keep the class flashcards in a box. Each day, one student can come to the front to test the others on the pronunciation / meaning / translation of the cards.

These are simple ideas. There are many more ways to use flashcards.

What will you use flashcards for?

More creative maths for kindergarten and preschool, visit www.eimaths.com