Luncheon Address by Judith Sunley
On reading the CBMS document “The Mathematical Education of Teachers” (or MET), my initial reaction was, “But nobody ever taught me to think this way! How would I ever do it in a K-12 classroom, much less prepare someone else to do it? This is really hard!” It was a mathematician’s response with a bit of educator speaking. Next I wondered how anyone would ever choose to teach middle grades, with so many requirements in mathematics alone and an age group many prefer to avoid.
But I thought you probably wanted to hear more than just my personal reactions. So I went on – looking at past CBMS, MAA, AMS, NCTM, and MSEB documents, and materials from outside mathematics, thinking about how to challenge the math community to deal with the implementation issues in math and education. But I thought the working groups and plenary sessions would cover those areas pretty well.
I finally decided to focus on the extra-mathematical challenges to implementing the recommendations. I hope these more generic comments, based on the challenges of change, will stimulate some thinking “outside the box.” They build on a range of experiences at NSF, but represent my views, not those of the agency. I have heard similar ideas articulated throughout the meeting, so I know people are thinking about them, although likely not all of them.
I’ll start with a Letterman-like list of the top five challenges to change.
1. “No one wants change but a wet baby.”
I first heard this phrase from Shirley Malcom, head of education programs at AAAS. The mathematical communities have been crying for change for almost two decades – just not agreeing on things such as cloth or disposable, the softness of the diaper, its absorbency, the use of pins or sticky tape, who gets changed first, and so on. Some of us have been overwhelmed by the extent of what productive change demands of us in negotiating on such issues. This can lead us to deny the need for and inevitability of change or to argue about its details.
Although appearing radical on a first reading, the CBMS document provides mostly clarification and examples for the MAA’s 1993 recommendations in “A Call for Change.” However, it is clear that most current pre-service programs are not coming close to providing teachers with the mathematical education we envisioned in ‘93. Placing this alongside the emphasis on partnerships among groups not known for working together, we find a document that stresses change.
MET does a quite good job of avoiding what I call the tyranny of the “musts and shoulds.” The document has a lot of grassroots input in its development from a wide range of mathematical stakeholders. But as we move to the much larger community of implementers, it comes as set of directions from a group with influence, but little authority or control, over what happens. This is a recipe for resistance from those not yet convinced the diaper is wet.
2. “The greatest enemy of the better is the good, particularly if it costs less.”
If we cannot show that what is proposed is better than what we are doing now, we may be done before we start, since any change is perceived as costing more than the status quo, at least initially. And when it comes to dollars in zero-sum games, there are winners and losers. Quality doesn’t always win out over cost, even when it is hard to argue that what we have is good enough. Frank Quinn raised this with to me in the context of NSF-funded undergraduate activities.
One way to address this is to articulate the hidden costs of the status quo. The costs of our current failures are built in. For example, we don’t even see the costs associated with our failure to produce 6th graders who move with facility between integers, fractions, and decimals. But those costs are real, they are substantial, and they persist downstream. Manuel Gomez of the University of Puerto Rico told me that he made little progress in his reform work in education until he convinced the powers that be that failure was more expensive than success.
In the mathematical education of teachers, some costs are not evident in dollars. There is opportunity cost for research and graduate programs of faculty involvement in teacher education. Those currently invested in the issue have voluntarily accepted the costs relative to other activities, but we are aiming for a broader community. How do we balance potential costs with incentives to produce continuing action for the sake of better quality?
3. “There are no proofs in mathematics education.”
This has become sort of a catchphrase for mathematicians working in mathematics education. It reminds us that we have to accept a level of uncertainty in our education work that we would not accept in our mathematical work, that there is more to the work than just the mathematics. But it raises challenges for working with the broader mathematics community – and even those in science and engineering – who expect more precision than they see. It also makes the demonstration that new practices are “better” even more difficult.
We do not need proof in order to apply the analytical skills developed in our mathematics training. We do need a common set of underlying assumptions – the purposes of mathematics education, how we might create balance among purposes, and what types and quality of evidence we might expect to see when talking about change. There is an implicit set of such assumptions in the MET document that we could better articulate and share as we proceed to develop implementation strategies.
4. “When there are multiple solutions, no single answer is ‘right.’”
How many of you at some time in the past two days have said, “I like the MET recommendations, but ...” Fill in your own “but.”
Teacher education is a complex system with diverse goals, objectives, and pathways involving students, mathematics and education programs, academic institutions, and K-12 school systems with many points of individual and collective decision making. CBMS recognizes that its recommendations must be implemented as appropriate locally and that all implementations will not be the same. Part of the value of a summit like this is that we can get some examples of practices that are effective in their local situations.
But we aim toward a level of consistency in approach that may be difficult to attain. With no universal solution on which all can agree, any local solution is more vulnerable to criticism.
For example, I am concerned that smaller institutions may not be able to afford gradations in their mathematics offerings to allow for depth in fundamental ideas for teachers at the same time they prepare majors at expected levels of knowledge and skill. Larger institutions may achieve such gradations at the risk of having the mathematics educators they produce perceived as somehow “second class” relative to the traditional major. Either outcome could call even successful programs into question.
5. “There is more to education than mathematics.”
The biggest challenge of all is that this change (which may be expensive and for which there is no certainty of positive outcomes given the many elements of diversity) depends critically, but only a little on mathematics-oriented implementers. CBMS recognizes this, addressing the document primarily to mathematics faculty and mathematics educators, but including language for other education-oriented groups and recommendations for partnerships.
I see the set of those with interests in K-12 mathematics education and capable of influencing teacher education programs as even wider than suggested in MET. In addition to those mentioned, I would include the faculty and deans of schools of education, college and university administrators, school superintendents and personnel officers, school principals, faculty in the natural and social sciences, and so on.
To me, the challenge lies not so much in identifying and contacting such groups, but in communicating with them effectively. Mathematics education and the mathematical education of teachers represent only a small part of the spectrum of issues such groups face on a daily basis. In some instances there are few people with mathematical expertise. Their perspectives on mathematics education are very different from ours.
When we who are steeped in mathematics speak to those with many issues and concerns beyond mathematics education, we need to find a way to get across a message that is coherent, understandable, and urgent.
So how do we face up to these challenges as a spur to the process of change and keep ourselves honest as mathematicians and mathematics educators? It won’t be easy. But we have grown over the past 20 years, and now have some advantages that can help us convert potential obstacles into reinforcement for change.
First, we are now experienced change agents – in mathematics and in education. While our past results have not come easily or completely, we can see changing practice all around us. We are beginning to understand how to influence change.
No one can argue that where we are in the mathematical education of teachers is “good enough.” We can build on that to shape the future.
Moreover, the number of people engaged in the effort is growing. All you need do to see that is look around at this summit. We are building networks of interested colleagues, both inside and outside mathematics. Both higher education and K-12 education are undergoing significant change, and we are able to use our networks to tap into a broader spectrum of partners.
We also have a growing research base that provides credibility for what we are proposing. Moreover, the research base speaks directly to the fact that there is no single solution to issues in mathematics education, that balanced approaches are critical to success. The research shows us the importance of experimentation to get bounds on our areas of uncertainty, and the experimental approach permits us to take advantage of diversity and devise strategies that are optimal in particular circumstances. Involving prospective teachers in research, along with the faculty that teach them, may help everyone understand how much more there is to know.
Finally, there is nothing mathematicians like better than grappling with a really hard problem. They might prefer to do it theoretically, without involving the messy real world, but they will rise to the challenges required once engaged in working something out. We are bringing a lot of firepower to bear on addressing the mathematical education of teachers, perhaps for the first time ever.
Finally, I wanted to take time to plant some seeds for further thought and action so you do not leave without something a bit unexpected. No answers here, just questions.
I’ll start with some explicitly mathematical thoughts.
The MET document contains several statements describing the mathematical knowledge needed by prospective teachers as “quite different from that required by students pursuing other mathematics-related professions.” Usually the difference had to do with deep understanding of fundamental principles. I can’t help but ask whether this is not important for all those interested in mathematics. It certainly couldn’t hurt mathematics majors, or even those expecting to be professional users of mathematics, to have such knowledge, but are we ready for the significant change in undergraduate programs it implies? How would our colleagues in other fields who rely upon us in training their students react?
This observation is perhaps most relevant at the high school level, where the MET examples are few and the recommendations for change minimal. As we move from high school to middle and elementary school, we get more detailed examples and recommendations that appear more substantive. Yet, if we look at the results of the Third International Mathematics and Science Study (TIMSS), we see US students doing fairly well at grade 4, about average at grade 8, and well below average at grade 12. On the surface this looks inconsistent. I can guess at reasons for the disconnect, but have we really thought about the implications of TIMSS in preparation for high school teaching?
I should add that this is a rather unusual comment for me to be making as I usually feel we speak too much to teachers at the high school level. We don’t usually regard those and the middle and elementary grades to be “mathematics teachers,” so we tend to overlook them.
The next items are not explicitly mathematical, but address aspects of teacher education.
While there is never any explicit statement that the recommendations refer to those programs preparing teachers as part of their traditional undergraduate program, this assumption seems to run implicitly through the recommendations. How do we handle situations in which students might be non-traditional in some way – part-time students, students doing a year beyond the undergraduate degree for certification, adults who wish to transition to teaching from some other career, etc.? How do these recommendations fit into alternate modes for obtaining teacher certification? Can we blend these recommendations with some in-service activity so that those involved have some sense of the challenges they will face in the classroom?
Over the past two years, several people have tried to convince me that pre-service teacher education and in-service professional development were distinct activities that should not be addressed simultaneously. I have increasingly come to believe the opposite, namely, that they should form a continuum of activity, all parts of which are suitable for involvement of faculty in higher education. Rather than being “two sides of a coin,” we should regard them as parts of the surface of a Moebius strip. This is explicitly recommended in the CBMS document, but we are not provided much detail on how to integrate the two. This is an issue that deserves additional attention. As usual, change demands more change.
And finally, here are a couple of ideas that may seem off the wall to some, if not all of you. I noted earlier that there are many communities that must be involved in change efforts in mathematics education. Changing the mathematical education of teachers is a social process, not – or not fully – a mathematical process. There are many interested parties with peripheral interests in the area. When appropriately engaged, they can be helpful. (For example, Gordon Ambach helped connect math and science education programs with the Council of Chief State School Officers, something very valuable for NSF activities.) When ignored, these communities can have a devastating impact on our ability to introduce high quality mathematics to prospective teachers.
I suggest we need to find out something about facilitating social change processes and work to adapt what we learn to this situation. This is not something that will come easily to most mathematicians, but it could make a difference between success and one more lost opportunity. I’m suggesting three areas for particular attention.
The first is assessment, a major issue in both K-12 and higher education right now. I’m using assessment in a couple of senses. First is the traditional view of student assessment where the math community is already involved in developing new approaches. Question: How will we assess teacher knowledge in the new paradigms we are putting forth? Second is assessment of progress toward stated objectives in change efforts. Assessment of knowledge is usually part of assessments of progress, which in turn depend on the existence of theories and models for actions that include intermediate objectives and plans for measuring progress toward them.
We have a tendency to want to act in mathematics education without conscious attention to things like theories and intermediate outcomes. When we fail to think about how we link actions and objectives and how we measure progress, we deprive ourselves of some major tools that we can use both to convince people we are making progress and to modify our approaches as needed. Think about the endpoint for the CBMS recommendations. How do we get from here to there? How will we know we are successfully on the way? How might local and global pathways differ?
Finally, our efforts in the mathematical education of teachers involve linking with groups having very different goals and objectives -– some related to mathematics and some not. There are social theories regarding diffusion of ideas within and across networks, interactions in multidisciplinary groups, and so on, that might inform our change efforts. I would suggest we start looking at how we can put them into action in this very complex, dynamic social process of change. We will need to translate the ideas into language mathematicians and mathematics educators understand, much as if we were translating physical problems into the language of mathematics for purposes of application. But I think it’s time to stretch our minds, once again, and make ourselves into a "learning community” that takes advantage of every idea it can find to reach its goals.
As I was looking through a variety of material over the past two months, I came across the following.
It is told that Albert Einstein once handed his secretary an exam to be distributed to his graduate students. The secretary scanned the paper and objected, “But Professor Einstein, these are the same questions you used last year. Won’t the students already know the answers?” “It’s all right, you see,” replied Einstein, “the questions are the same, but the answers are different.”
If the answers can change so rapidly in physics, how much more so in the math education of teachers. We need constantly to be searching for today’s answers and laying the groundwork for tomorrow’s.
If anyone is interested in discussing any of these ideas further, I would be happy to do so, either in questions in the time remaining, or in follow-up discussions after the summit. Thank you for the opportunity to put these ideas on the table.