> MOL Ybjbj >{{Q5f f 8<1|1.w"t:D1111111$3R6D'1tt'1<1x!x!x!~1x!1x!x!Q/|0@Rވr/0R101/77$070$x!'1'1x!17f o: MATHEMATICAL ETHICS: VALUES, VALENCES AND VIRTUE
Douglas Henrich
Iroquois Ridge High School, Ontario, Canada
HYPERLINK "mailto:henrichd@hdsb.ca" henrichd@hdsb.ca
ABSTRACT
In this paper I will review the themes of: gender differentiation and engagement in mathematics, ethics and separated values within mathematics, and proven teaching strategies that promote mathematical learning engagement. These themes will be unified by anecdotal success strategies that show how emphasis on the connected values of mathematics engages both male and female students.
INTRODUCTION
In a short paper by Paul Ernest (2014, University of Exeter, Questioning the Value of Mathematics), he suggests that there are negative costs to the standard way of mathematical thinking that promotes, detachment of meaning, ethical neutrality, separated values and a dehumanizing outlook when these values are applied beyond mathematics.1 Ernest suggests that the separated values of mathematics (Gilligan, 1982) promote: rules, abstraction, objectification, reason, dispassionate analysis, and impersonalilty.2 Some would argue that these are all reasonable attributes when doing mathematics, and, for the most part, that is how we teach math. Students are expected to Do the Math and there is really no room for metacognitive analysis where they are encouraged to think about what they are doing and question the procedures and the accepted correct results. Gilligan suggests that many women define themselves through relationships whereas many men define themselves through separation and active assertion through use of the I pronoun.2 It is tacitly understood that what we deem male and female characteristic behavior is present in both men and women. The use of the words many, men, women, male and female should be interpreted as some but not all.
Men readily identify with the separated values of mathematics whereas women tend to search
for connected values.2 Separated values are most pronounced within pure mathematics, while
connected values are more predominant within applied math and statistics. This might correlate
with a larger female affinity for applied math and statistics compared to pure mathematics.3
THE VALUES OF MATHEMATICS
Ernest identifies key attributes of the separated values and connected values of mathematics:4
Separated ValuesConnected ValuesRules
Abstraction
Objectification
Impersonal
Unfeeling
Atomistic
Dispassionate Reason
AnalysisRelationships
Personal Connections
Empathy
Humour
Caring
Holistic
Feelings
Intuition
Ernest suggests three reasons why separated values seem to fit mathematics so well:4
Mathematical objects results from objectification and abstraction and are naturally unfeeling and impersonal.
Mathematical structures are made up of abstract rules centred on sets of objects and their relationships.
The processes of mathematics are atomistic and object-centred based on dispassionate analysis and reason in which personal feelings play no part.
Indeed, separated values serves mathematics very well. Ernest argues that if we reject the connected values of mathematics and adopt separated values we are supporting the view that mathematics has no humour or ethical responsibility. Ernest claims that adopting this position, however, is, of itself, taking an ethical position.4 Bishop notes that there is a widespread misunderstanding that math is a value-free subject.5 Values are a question of choice. As a teacher, every lesson you choose what is actually taught through the enacted curriculum. Granted, a teacher has limited control over the intended curriculum which is usually codified through a third-party and, arguably, even less control over the attained curriculum what it is that students actually learn.5 In stronger terms, Mathematics colonizes part of our reality and reorders it contradicting the purist view of mathematics that it is a neutral sublime purity.(Gates, 2004 quoting Skovsmose, 1994)6. In essence, the way that we look at mathematical structures and mathematical objects (objectism) is a product of the enacted curriculum.4
OBJECTISM WITHIN MATHEMATICS
Ernest classifies objectism as an ontological value and uses Bishops definition of objectism in mathematics as, A world view dominated by images of material objects.5. According to Ernest, objectism has permeated mathematics since its inception in the systematic accounting in Mesopotamia and Egypt due to the need for records for trade, taxation, and scribal training.(Ernest, 2014 quoting Heyrup, 1994)4 Bishop uses cross-cultural analysis to show that the idea of counting material objects is not naturally given.5 It is a learned behavior that has both a social and a cultural context. If we were an amorphous, intelligent jelly-fish like being our counting system would be fundamentally different. We would not likely count discrete objects but would see ourselves as part of a larger continuum.
I would argue that objectism has valenced mathematics towards a system of separate values that aligns with male-dominated assertion and separation. From the beginning, mathematics has been based on an assumed objectist conceptualization of the world.4
GENDER DIFFERENTIATION WITHIN MATHEMATICS
Based on my observations as a high school mathematics teacher, I have noted that when faced with a math problem, most (but not all) males invariably adopt a form of instrumental reasoning that will lead them directly to the solution. They become upset when their solution differs from others solutions and will quickly turn to the teacher, or textbook solutions, to resolve the conflict. They want answers, fast. To them, it is a bonus when their answer is, in fact, the correct answer but, most importantly, they must have an answer. For most males, the process is only a means to an end. In contrast most (but not all) females work best when they can attach a social context to the problem. They readily work with a partner or other groups and seem to genuinely appreciate, and enjoy, the process of working towards a solution within a social continuum. Males will talk to others as well, but only to get the answer or verify that their answer is correct.
For females, working through the math problem is a form of social play that, in itself, is its own reward.5 Even as young children, most males prefer to play games with strict rules clearly identifying winners and losers. This is especially evident in the predominately male preference for interactive video games. I am merely describing behavior that I have observed and am not making a value judgment. Bishop argues that the rule-governed criteria of mathematics has developed from the [male] pleasure and satisfaction of rule-governed behavior in games.5
Unfortunately, when we evaluate mathematics students at the high school level, the tendency is to evaluate on the basis of the students instrumental reasoning. Students are generally not allowed to engage in social dialog during tests or exams as we tend to emphasize individual evaluation of their mathematical skills and techniques. On the few occasions when I have allowed social dialog during tests, I have found that females maximize the use of social connections while males tend to be uncomfortable with the idea of talking to others during a test. Invariably, females do better than males on such tests and they also do better on that unit during exams where they are evaluated individually.
I have taught all levels of high school mathematics including calculus, algebra and data management (probability and statistics). Invariably, females tend to prefer data management as I present that course in a manner that allows for significant levels of social connections. It is not surprising that females also remain engaged with probability and statistics at post-secondary and graduate levels3 as it meets their need for connected values.
ETHICS WITHIN MATHEMATICS
Both Bishop and Ernest emphasize the need for including ethics within mathematics but do not provide a prescriptive means through which this can be done. Nor do they describe what ethics looks like in mathematics. I would suggest that the traditional ethics of philosophy is inadequate and we need to quantify a form of mathematical ethics. In support of this argument, I will first turn to Tavani (2011) who presents an interesting case why computer ethics or cyberethics are necessary for cybertechnology.7
Why Computer Ethics? We dont have: car ethics or airplane ethics. Why isnt existing ethical theory sufficient to address the concerns raised by computer technology?
ANSWER:
Computers are logically malleable and are general-purpose machines that can be shaped and molded to perform an almost endless variety of functions. These functions generate limitless possibilities for human action.
Computer technology generates policy vacuums where no explicit policies and laws exist to guide new choices made possible due to logical malleability.
There may be some confusion or conceptual muddle regarding the thing that we are trying to create new policies for.7
I have taught computer ethics to 4th Year Bachelor of Applied Information Technology students and, while they may initially argue against the need for ethics in Computer Science or Computer Technology, they do generally accept Tavanis argument that, IF ethics is desirable for cybertechnology, THEN it must be its own unique form for the reasons as detailed above.
Why mathematical ethics? At this point I can only give a partial answer and am, in fact, still actively researching this question with the hope of eventually writing a book that further explores this area. My tentative working title will be: Practical Ethics: Developing a Social Conscience. I can give anecdotal evidence as to the effect when mathematical ethics is introduced into the math curriculum as the next two problems will show.
PROMOTING MATHEMATICAL ENGAGEMENT
Ernest suggests that adopting connected values when teaching mathematics would (possibly) be all to the good.4 He clarifies, however, that mathematics itself is free from this responsibility it belongs to teachers and social institutions of mathematics not the discipline itself.4 I disagree with him on this point as I believe that he is ignoring the implied valences within mathematics this will have to be developed further as part of my research, however. He does advocate for openness, fairness and democracy both within the teaching, and doing, of mathematics. His claim is that, mathematics, like democracy, is fair because of openness and potential equal treatment of all.4
Bishop suggests that culturally responsive mathematics is possible through the development, and use, of Rich Mathematical Tasks (RMTs) within each of the five domains of mathematics (counting, locating, measuring, designing, playing and explaining) and across the six values (ideology rationalism, ideology objectism, sentimental control, sentimental progress, sentimental openness, and sentimental mystery).8 These areas are all further explored and developed both within his Mathematical Culture 2 paper in 2013 as well as within his book published in 1988.5 Space limitations preclude my discussing these domains at great lengths.
I would suggest that developing mathematical problems in accordance with mathematical ethics strictures will meet both the requirements of Ernest and of Bishop. Although I have not yet fully developed the concept of mathematical ethics, I have prepared a paper on Mathematical Ethics that I have presented at a number of venues.
For the most part, it has been well received and is available online at the following link:
HYPERLINK "http://www.fields.utoronto.ca/programs/mathed/meetings/minutes/12-13/HenrichSept2012.pdf" http://www.fields.utoronto.ca/programs/mathed/meetings/minutes/12-13/HenrichSept2012.pdf
As part of this paper, I have developed the following mathematical ethics problem.
1.0 Standard Unethical Grade 12 Calculus Optimization Problem
Problem Description: XYZ Corporation produces a commercial product that is in great demand by consumers on a national basis. Unfortunately, near the plant where it is produced there is a large population of dove tailed turtles who are adversely affected by contaminants from the plant. XYZ has a filtering process that is expensive and any increase in filtering effectiveness reduces their profit. Dove tailed turtles are not a protected species hence there are no environmental rules regarding XYZs level of contaminant. Clearly, no filtering at all would maximize XYZs profitability but would destroy the dove tailed turtle population.
A local environmental group monitors XYZs contaminant level and maintains a website showing the percentage mortality rate of the dove tailed turtles due to XYZs contaminants. XYZ has noticed that the higher the mortality percentage, the less items are bought and the lower their profitability. Their Marketing Department and Research Group has established the following Revenue Function, R(x), as a function of Dove Tail Turtle Mortality expressed as a decimal between 0 to 1 representing mortality percentage:
EMBED Equation.3
Y=R(x) is expressed in billions of dollars and represents the revenue generated. Since XYZ has fixed operating costs of one billion dollars, the profit function, P(x) is given by: P(x)=R(x) 1.
Problem Statement:
Sketch the graph of EMBED Equation.3
Find the optimal dove tail turtle mortality rate percentage that will maximize revenue.
State what the maximum profit will be.
Solution:
Using standard calculus optimization techniques we find that a dove tail turtle mortality rate of 13.5% will generate a maximum profit of $541,341,133.00 for XYZ Corporation.
2.0 An Ethical Grade 12 Calculus Optimization Problem
You have been hired as the mathematical consultant for XYZ Corporation. There problem is as stated in 1.0 above. They have asked you to help them optimize their profit and request that you:
Problem Statement:
Sketch the graph of EMBED Equation.3
Find the optimal dove tail turtle mortality rate percentage that will maximize revenue.
State what the maximum profit will be.
As a successful student of Grade 12 calculus, you easily do the math and arrive at the solution as stated in 1.0. But dont collect your money just yet. You also understand that mathematics is an intentional human activity and carries with it a certain social responsibility. Are there any ethical considerations that you would bring to your clients attention?
Solution:
For a complete solution click on the link below which will take you to my Mathematical Ethics paper that I presented at the Fields Institute, University of Toronto in September, 2012. The solution is on Page 19 of the paper.9
HYPERLINK "http://www.fields.utoronto.ca/programs/mathed/meetings/minutes/12-13/HenrichSept2012.pdf" http://www.fields.utoronto.ca/programs/mathed/meetings/minutes/12-13/HenrichSept2012.pdf
Alternatively, type in mathematical ethics in any search engine and this paper should be one of the first to appear.
Approach:
Present Problem 1.0 to the class in its exact form as above without any further statement or qualifiers. Invariably, the high performing males in the class will race towards the solution and will quickly present the solution as above. After a few minutes, someone, usually a female will quietly raise their hand and say, But this problem is just wrong. When asked to explain they will. At the point, the rest of the class will generally jump in and agree with them including the males who raced to the solution. Explain the context of the problem in terms of mathematical ethics and the implied social responsibility that everyone has and then present Problem 2.0 to them. Follow up with the full solution as detailed below.
I have tried this several times with my calculus classes and it has always played out as I have described above. I often have female students approach me after class and tell me that they find the ethical version of the problem to be one of the most interesting, and engaging, problems that they have worked on. I also get similar feedback from teachers who have attended my presentations on mathematical ethics and have tried the above problems with their class.
Drawbacks:
Ethics is a Waste of Time
Some may argue that Problem 2.0 detracts from the real math of the problem and that it is a waste of the students time to worry about the ethics of a math problem once they have solved it. To those individuals, I merely advise them to present such problems to their classes and see what the reaction will be. I can almost guarantee that the level of engagement and understanding will go up. In pedagogical terms, the enacted curriculum will be more in line with the attained curriculum. These problems will not be in line with the intended curriculum, at least in North America, as I am not aware of any math curriculum in North America that specifically includes mathematical ethics within the intended curriculum. Note: I dont consider character education to be the same as mathematical ethics. Worldwide, Australia would be the exception. Australia has specifically mandated that ethics be included within the mathematics curriculum at both the secondary and post-secondary level. A lot of the research on ethics in mathematics comes from Australia.
Preparing Ethical Math Questions
The time required to prepare ethical math questions such as 2.0 can be significant. Each has to be individually prepared by the teacher as you cant just go online, type in mathematical ethics problems and except for anything to come up. It takes time to prepare ethical math questions that are engaging, relevant to the curriculum and realistic. My hope is that others will decide to prepare similar questions and will make them available online.
CONCLUSION
Does this mean that you should change the way you teach mathematics? Not at all. Atweh suggests that students should be engaged in meaningful and authentic real world problems that develops both mathematical capability and also develops an understanding of the social world and how to contribute to its transformation.10 Reformatting mathematics problems based on mathematical ethics should meet this objective.
REFERENCES
1Ernest, Paul. Is Mathematics Harmful (as well as Beneficial)?
HYPERLINK "http://www.youtube.com/watch?v=RCngE2hZyMg" http://www.youtube.com/watch?v=RCngE2hZyMg, Mathematics Culture 3, April 15, 2014 (accessed August 4, 2014)2Gilligan, Carol. (1982). In a Different Voice, Cambridge, Massachusetts:
Harvard University Press.3Whiteley, Walter. Differential Interests of Women among areas of Mathematics and
Statistics. In Philosophy of Mathematics Education Journal No. 28, October, 2014
(P. Ernest, Ed)4Ernest, Paul. The Values and Mathematics: Overt and Covert
Part 1: HYPERLINK "http://www.youtube.com/watch?v=Jak61rsez5g" http://www.youtube.com/watch?v=Jak61rsez5g
Part 2: HYPERLINK "http://www.youtube.com/watch?v=ZyvRr2gCfVQ" http://www.youtube.com/watch?v=ZyvRr2gCfVQ,
Mathematics Culture 2, September 24, 2013 (accessed August 4, 2014)5Bishop, Alan. J. (1988). Mathematical enculturation: A cultural perspective on
mathematics education. Dordrecht: D. Reidel Publishing Company.6Gates, Peter. (2004). Lives, Learning and Liberty: The Impact and
Responsibilities of Mathematics Education. Vol I 71 80. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education. 7Tavani, Herman T. (2011). Ethics and Technology, 3rd Edition.
New Jersey: John Wiley & Sons, Inc. 8Bishop, Alan J. "What would the mathematics curriculum look like if instead of
techniques, mathematical values were the focus?
Part 1: HYPERLINK "http://www.youtube.com/watch?v=vaOkqpcEUWA\\" http://www.youtube.com/watch?v=vaOkqpcEUWA\
Part 2: HYPERLINK "http://www.youtube.com/watch?v=vaOkqpcEUWA" http://www.youtube.com/watch?v=vaOkqpcEUWA
Mathematics Culture 2, September 24, 2013 (accessed August 4, 2014)9Henrich, Douglas. Mathematical Ethics: A Problem-based Approach. Presented at The Fields Institute, Math Ed Forum, University of Toronto, September, 2012. 10Arweh, Bill and Brady, Kate (2009). Socially Response-able Mathematics
Education: Implications of an Ethical Approach. Eurasia Journal of
Mathematics, Science and Technology Education, 5(3), 267-276.
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