Is there (always) a Step 0?

Examples and consideration of the unstated prerequisites of life.

(For those long-time readers of this blog, you’re familiar with my oft rant about prerequisites, but this take is about the unstated ones.)

I was thinking about part of the leadership workshop that didn’t really work for me. All the ideas made sense and the logic, justification and relevance were all fine, but it just didn’t ‘feel’ right for me. When I encounter such situations I try to go back to the basics and see if there’s some assumption about the situation (or a Step 0) that I don’t have or have missed.  I’m not sure what it is in this case, but I’m pretty confident there is something.  I’ll have to think more about it.

But then I was thinking about other situations and how that this hidden Step 0. was often a very important part of the eventual success. For example:

The practice of mise en place in cooking, where before you start the steps of a recipe, you get out all the equipment you’ll need and prep all the ingredients. I use this when I cook at home and find it makes the process much more efficient, effective, and enjoyable.  In contrast, when another member of my family cooks, they take on a more ‘just-in-time’ approach, prepping each ingredient as needed. They often seem much more frustrated with the cooking experience, and sometimes it impacts the results.

The role of assumptions in mathematics in general and in statistics in particular in answering basic questions. For example, if asked what the probability of rolling a single pip on a standard die, to get the expected answer of 1/6, there’s a long list of assumptions that have to be made about the die, the physics of rolling, the roller, etc.  This example is simple and common enough that most wouldn’t worry about these assumptions, but for more complicated situations, it is essential to clearly state all of them.

Anything based on a long-time tradition, e.g. anything within a religious tradition, where the standard interpretation or set of actions depends on a specific worldview.   I think my leadership workshop issue falls under this situation.

So then my thoughts go to wondering if most/all solutions depend on this background, a.k.a. Step 0? And, if so, then how can we effectively identify what it is?

My thought is that since there are a limited number of absolutes that all agree on, that, except in the most simple of cases, there is always a Step 0. Then the challenge becomes identifying what it is. I think you always need an expert in this case, i.e. someone who studies the situation and process and reflects on it, to provide some broader insight into the context and how the solution plays out. The key value is for the expert not to just be good at whatever it is, but to have spent time thinking about what it is (and isn’t) and how it fits in the world. The expert has to either have been or played the role of a foreigner to the world that produced the result. An expert in probability who ‘just gets it’, would never think of all the things that go into the rolling of a die, unless they deliberately stepped outside of what they understand about probability and looked at it through the eyes of another.

For our own practice, this means for those areas in which we are experts, we need to step outside and view the area as an outsider. This could be done through deliberate reflection, along with working with newcomers and those with different perspectives. Teaching is a great way to do this, as long as you pay attention to those Step 0 gaps that show up. For example, when I explain a new idea and a student doesn’t ‘get’ it, I can look at what ideas the student is building their understanding from and see if there’s a gap or if there’s some fundamental misunderstanding.

Another point of action is that once we have identified these Step 0s to remember that they exist and incorporate them appropriately in our practice. For example, if I find a specific type of gap shows up in a particular lesson, then I can either fill that gap as part of the lesson, or use part of the lesson to ‘discover’ the gap and help the students fill it on their own. The remembering part means that I’ll have to do this every time I teach the lesson.

Unstated assumptions or steps (Step 0s) exist and can greatly influence the outcome of an effort, so it worth our while to try to identify them and to regularly incorporate them into our work.


A reflection on the leadership development experience – the introductory biography

It happens with just about every new group: go around the circle and introduce yourself.In an academic setting that usually included your department, current role or position and something about your research. Often the research component can be detailed enough to include recent publications and grants.

But in the academic leadership development setting, would/should you expect something different? You might, but my limited experience, you won’t. The role part does at least include something about a leadership position to justify the seat at the table, but there’s still a focus on research achievement. Here’s some thoughts on why this is and then I’ll decide if it’s an issue:

  1. Academics define themselves based on their academic achievements. Yes, we teach and do service and other non-research activities, but ultimately we are researchers. Thus this is how we talk to each other.
  2. Academics have been primarily trained to be researchers and are good at it. They may not either have training in other areas or think of themselves as good in those areas. Thus they speak of their research as it is their ‘good thing’.
  3. Academics are experts in their field and may feel that that expertise transfers to or justifies their position in leadership. Thus, in a group, they have to establish their research credentials.
  4. Research talk is the language of the land. Thus talking about research achievements is normal, while talking about leadership achievements feels like bragging.
  5. Leadership talk is not a common language. Thus we don’t know what to talk about beyond what our current role is.

Is this an issue? If you believe in the Authentic Leadership paradigm which contains a fair amount of self-reflection, then it is an issue. At best it is shallow, as it puts the deep reveal in safe and irrelevant areas (remember, we are in a leadership development setting). At worst it sets up false identities and expectations, as participants become identified with their research rather than their leadership.

It can become less of an issue if the group moves past the opening and reveals more about their leadership selves, if the research self can be used to inform the leadership self, or if the leadership identity can be brought out and celebrated in later meetings. All of these depend both on the facilitator and the buy-in from the group.


A Slight Change of Direction

Or a change for the worse, depending on your perspective.

I’ve been doing the interim administrative thing (75% time) for the last 9 months and it looks like it will go permanent and 100% this summer and so I thought I’d revisit this blog and re-purpose it accordingly.  I will continue to post on education related topics as that is still on my mind and under my authority (although I won’t be in the classroom for a while); these will tend be more at the assessment/administrative end, rather than with an in-classroom perspective. But I hope to expand and cover some topics related to administration and leadership.  Also, I’m participating in a leadership development program, and hope to post my ‘homework’ and reflections on the homework and the program here.

So, if you* were here only for the front-line teaching reports, you’ll likely be disappointed with my future posts. If you were here because you found my comments of some value (interesting or funny or sad, I don’t mind), then you’ll be okay, as I can promise more of the same.

On to the dark side!


*comments about current readers refer only to the 2 or 3 who have ever read any of my posts, assuming they are still alive.

Mosquito Netting and Open Sewers

A story that’s a metaphor for course and curriculum design.

A couple of years ago I was at a math-biology workshop and during the reception, I had a great conversation with a student from Ghana. Somehow we got on the topic of how it was popular for mathematicians (mostly westerners) to study diseases that are prevalent in Africa. He then mentioned how often the solutions that are promoted are not really practical as they don’t take into account certain elements of local society. He mentioned several examples, but the one that stuck with me was the work on malaria. At that time, it was popular for some organizations to raise money to provide mosquito netting in areas that are suffering from malaria.  My colleague said that the netting helped, but there were still mosquitoes all around, and the real problem was that there was standing water in open sewers. So it would be more helpful to fix the sewer problem rather than just supply mosquito netting.

I was left with thoughts about the appropriate of some modeling approaches I had taken, but I was more struck with wondering how often we try to solve teaching/learning problems by providing the equivalent of mosquito netting when there’s a bigger problem (open sewers) that we are ignoring.

The first example I thought of was College Algebra. I’d bet that if you talked to any math department from an open or less-selective university, you’d find that the low success rate in College Algebra was a big concern, especially with recent concerns about completion rates. I’ve been to conferences where passing rates were reported at 40-50% and ‘improvements’ to 60% were considered outstanding. In most cases, efforts to improve the pass rate focus on improving the quality or quantity of instruction (e.g. moving to open computer instruction, adding hours or study sessions, etc.).  This makes some impact, but usually doesn’t really fix the problem (and can be very costly). The ‘open sewer’ in this case comes from looking at how a typical student arrives at College Algebra.  Since the course has basically the same content as Algebra 2 in high school (which is required at my school for admittance), all students have seen the material at least once. However, something happened to bring them into College Algebra. Maybe they don’t have good math study skills, or they struggled with some prerequisite math skills (e.g. fractions). Whatever the issues may be, until they are addressed no amount of improved teaching will be able to make a significant impact.

I haven’t spent a lot of time thinking of other situations in detail, but I do use this framework when something comes up, and try to make sure that we aren’t just taking the easy/popular solution, but are considering all the issues and maybe doing the unpopular work of sewer-fixing.

Duck Theorems

Another difference between Math and the rest of STEM: the strong use of if and only if relations which lets us determine something exclusively by its properties.

A follow-up to Does Math Really Fit in STEM?

An example: Suppose you wanted to show that for an arbitrary value a, a\cdot 0 = 0, assuming you have a proper context for all the other symbols, etc. The first problem is that 0 is only really known as an (or the) additive identity, i.e. b + 0 = b. And we don’t know anything about how it works with multiplication. But there’s a way:

a + a \cdot 0 = a \cdot 1 + a \cdot 0 = a \cdot (1 + 0) = a \cdot 1 = a,

using the rest of the field axioms.  But why is this good enough? We rely on what I call a/the Duck Theorem, which in this case is the fact that a + x = a has a unique solution (0), and since a \cdot 0 also satisfies this equation, we must have a \cdot 0 = 0.

Aside: Why do I call it a Duck Theorem?  From the old saying: If it looks like a duck, walks like a duck, and sounds like a duck, it must be a duck. In this case a \cdot 0 acts like 0, and since 0 is unique, a \cdot 0 = 0.

There are many things in the background that make this work, all of which rely upon the exceptionality gained from using if and only if and the corresponding deductive arguments. The definitions involved exclude any other options. The Duck Theorem provides a distinctive uniqueness (which could be rewritten, in our case, as x + a = a if and only if x = 0.) I’m not an expert on the other STEM fields, but I’m not aware of many (or any) cases where some physical result can be used to 100% identify some particular object or property. Most definitions are descriptive and, unless they’ve been abstracted into mathematics, are not given in an if an only if form. (A mammal is warm-blooded (as well as other things), but being warm-blooded doesn’t make you a mammal). Of course I’m only talking about how math relates to math, this doesn’t always extend to the results when math is applied (e.g. statistics is as precise within its own context, but it doesn’t extend the if and only if nature to the results of a statistical test as understood in the application area, i.e. statistical results are not if and only if/causal).

This all is not to say math is better (I’d argue that this feature makes math worthy of the many jokes made about math); it is just different, and maybe different enough that method proven to be good for teaching physics, biology, engineering, programming, etc., might not translate over to being good methods for teaching math.

The Hand You’re Dealt

Some thoughts on what it means in teaching to work with the hand you’re dealt

A month ago or so I was talking to a colleague about a course that we both have taught in the past and we got around to the issue that the students often entered the course without as complete of a background in a particular area of mathematics (the details don’t matter, but what comes next does). We had different opinions as to what one should do about this. And my colleague commented, “You have to face reality and deal with the hand you’ve been dealt.” I nodded, but then realized that what I thought that meant was completely different than what he meant.

I took it to mean that we had to work with the knowledge that our students came with, possibly avoiding or deemphasizing parts of the course that required a knowledge deeper than they could handle, or maybe we should rethink about how those elements of the course are presented. My colleague thought it meant we should take time out of the course to (re)teach the relevant material, possibly taking a good part out of the course to do so. Personally, I don’t choose to reteach prerequisite material.  I am willing to remind students of this knowledge and provide resources or references for them to learn it on their own, but I don’t want to go backwards. But now I wonder if this makes me weaken the course or avoid the tougher topics. Or if I’m challenging them enough.

Another twist to the interpretation is what it means when we say a student doesn’t have the necessary background. There’s a big difference between the student actually not ever seeing the relevant material, between having seen it but not being currently that fluent in it. I tend to assume that the latter situation is true, which justifies not doing an extensive review or reteach, but also doing some short, just-in-time reminders of the relevant material in the current context.  Good or bad, this approach avoids going too in depth into the background material and forces me and the students to figure out the parts that are useful for the course material.

I’m still not sure how these choices impact the course; it is good to be aware and maybe I can pay more attention to why I’m making the decisions I make.

Assessment Basics

Assessment has been on my mind and so I capture here what I consider to be the basics.

Having just finished the development of a big (think state-wide) assessment, I took a pause to think about all the conversations I’ve had about assessment and decided I’d try to organize those thoughts here, for what I’d say to someone new to thinking about assessment.

The most important thing to know about assessment is that assessment doesn’t just mean test. There are lots of different ways to assess student learning (and teacher performance, and …), which leads us to the second point that not every assessment leads to a grade  because at it’s root, assessment is about feedback and that doesn’t always have to be a numerical or letter grade. The short conversation with a student during office hours where you hear what she thinks about a topic and can confirm and correct, is a type of assessment. The interesting part about the feedback focus of assessment is that assessment is for the student and for the teacher.  Every bit of information reflects on what the student has learned or not learned, and thus reflects on the effectiveness of the instructional technique. In some cases the assessment is primarily about providing feedback to the teacher. You will hear that there are two types of assessments: formative and summativewith formative being mostly about feedback and summative being mostly about evaluating the amount of learning (i.e. grades). However, even the most pure summative assessment (think of a final exam or professional entrance exam), does end up have some formative components at least to the students on how they prepared for the exam, or how their exam taking strategy worked, and for the teacher as to his or her effectiveness (especially when he or she can tie some element of the assessment directly to some element of the class).  Unfortunately, most of us still have to give grades and thus we often have to use assessments to form those grades, but there’s a distinction between low-stakes assessments and higher-stakes assessments. For example, homework should be more formative and can (and should?) use less formal evaluation schemes (e.g completion grades with feedback), while exams and the like should be evaluated more carefully. This brings up the last point, assessments are best when they are carefully tied to specific outcomes. This didn’t come first, because if you limit yourself to gradeable test-like assessments, it is difficult to match up all the learning outcomes with those assessments, but if you can see that office-hour conversation, or that brief give-and-take during class, or even that puzzled glance from your students as possible assessments, then it is possible and pleasurable to look at how and what you do in class and the connection to what outcomes you expect. And as a bonus point, rubrics are a useful tool for both the students and the teacher, especially for non-traditional assessments. As students are not always used to various assessments, a rubric can help them understand what to expect, while for teachers, a well thought-out rubric can help with giving effective feedback and in grading.