We’ve all seen them … lovely compilations of ideas ‘that work’, often published by an esteemed leader or organization with the goal of moving our profession forward.
A recent post by Justin Baeder (see http://blogs.edweek.org/edweek/on_performance/2011/03/doing_what_works_doesnt_really_work.html) got me thinking about my response to these ‘what works’ lists. Most of the entries on the lists are discrete items, and their inclusion might have some evidence-based rationale. However, they are seldom presented with a conceptual framework to understand what is productive about them. Justin Baeder also points out that the ‘what works’ mentality often misses the importance of system thinking … life (and teaching) is much more complex than using the right set of tools.
It’s not that I do not want to see change happen. Obviously (or maybe not so) I have invested considerable time and energy to bring about change. My concern: Change without understanding context or concepts is just like our students doing 100 problems practicing ‘solving equations’. The result may (on occasion) look pretty, and it appears that we are doing the right things.
Our work represents our understanding and wisdom; our work represents professional effort. Replicating ‘what works’ in the field normally avoids the understanding and wisdom … after all, “X” was listed as a ‘best practice’ by an expert, so it must be a good thing to do.
Learning is a complex process, involving elements of a multitude of sciences (psychology, biology, sociology, and anthropology to name a few). Each ‘practice’ exists in a complex web of practices and their meanings. Transferring a best practice to a new context assumes a fundamental equivalence of the contexts.
Getting there will involve developing our understanding and wisdom. We will find that there are concepts and theories with a track record of predicting positive outcomes. At a sufficiently high level, we can all use the same concepts and theories to produce good results. Seek the understanding and wisdom. As Albert Einstein was reported to have said:
The significant problems we have cannot be solved at the same level of thinking with which we created them.
Sometimes, we are so accustomed to seeing the world in the same ways that we can not completely process new information.
When people look at the curriculum described in the New Life model (http://dm-live.wikispaces.com) there is no label that looks like a beginning algebra course. Does this mean that algebra is not a valid domain for developmental mathematics?
The issue here is “What are the powerful ideas of mathematics” … ideas that will help students prepare for a variety of opportunities in college and in life. If we focus on the good stuff — ideas with power — we will find ourselves renewed and our students rewarded.
In the case of ‘beginning algebra’, we normally sort through a topic list that has phrases like ‘expressions’, ‘linear equations’, ‘graphing equations in two variables’, ‘systems of equations’, ‘exponents and polynomials’. These phrases are so entrenched in developmental mathematics that our textbooks are almost required to use this list to create their table of contents. If these phrases are not used, we wonder if there is any algebra involved.
Instead, think about two fundamental ideas of algebra (just to start) — proportional quantities and variables. Proportionality is fundamental to many concepts that students either need to use or will encounter in future courses. This includes unit conversions and dimension analysis — work that exceeds the simple mechanical conversions we often limit ourselves to. Proportionality can be studied in various representations; the numeric level is the applied level, but we should abstract this information to the graphical level and the symbolic level.
When we look at proportionality from a symbolic viewpoint, there is an opportunity to develop a deeper understanding of variables and constants. Students leave our existing courses convinced that any letter is something to be solved for; the power of ‘variable’ focuses on the ability to represent information and relationships in a way that communicates well.
Rather than the demise of beginning algebra, we are actually looking at the revival of algebra. Algebra is a set of powerful ideas; the reformulated curriculum focuses on these ideas … and avoids the artificial ‘procedural’ content that has been performing an impersonation of ‘algebra’ in our curriculum.
This site is meant to help everybody gain enthusiasm for the re-definition of developmental mathematics. History has saddled us with a curriculum mis-matched with needs, and we have an opportunity to create something much better.
We can go beyond a ‘redesign’ that only makes the instructional delivery more efficient. You can examine the mathematics that your diverse students really need in their academic life and in society, and we can create courses that show the vitality of our discipline.