# 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.

# Biography

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.

So my first list, penned one morning at a McDonald’s was this:

1. Ronald Reagan (first election and president during college years; ‘won’ the cold war, seemed to see beyond party lines)
2. Jesus Christ (a classic, meaningful and influential; not comfortable with following the “Jesus as a Business Leader’ path, but for relationships, a good choice)
3. Bill Gates (I thought I needed an actual business leader and I like tech; unique in some sense by managing to stay in charge from dorm-room startup to international multi-billion dollar company)
4. Cathy Family (Chick Fil A, probably Don, current CEO) (Another business choice; preservation of their family values in running the business in face of opposition (and lost profit), read some about the discussions related to the controversy and their efforts to learn and adjust)
5. Martin Luther King, Jr (another classic, clear mission, unique solutions)

I then started thinking more about people I’ve served with or under and came up with a few more. I did limit myself to not mention people that I currently work with, because that could be weird and I think identifying strengths and weaknesses takes time. I think I’ll save them for another time.

I’ve also thought (and worried) about what this list says about me. One thought is that as a mathematician I connect more to ideas than the person generating the idea, so maybe it makes sense that I don’t really have a strongly identified set of admired leaders.  I think in the next phase where we look at characteristics of these leaders, I’ll hopefully see an admired list of values.

# Taking a Left Turn

A contemplation of non-linear improvement and its connection to (micro)management.

A little exercise:

Think of one thing that characterizes poor teaching (by whatever measure).  Now, think of one thing that characterizes good teaching. Is there a straight path from one to the other?

In many examples I can think of, you can get from one to the other, but not by a straight path.  For example, poor teaching is often disorganized, while good teaching is engaging, but I don’t think that just becoming more organized (the straight path from disorganized) will necessarily make you engaging. As one goes from poor to good, from disorganized to engaging, they have to take a turn somewhere. But where should this happen?

Now we get to talk some math.  There’s an optimization algorithm called Steepest Descent (SD) in which you go off in a straight path searching for the best spot along the path. The key is picking the direction and for SD, you choose the one that gives you the best initial improvement. So, from our example, if you are disorganized, your initial direction is one that increases your organizing skills.

In nice cases it is easy to see how SD eventually finds the optimal solution, however even in simple cases it can be shown that the optimal solution is only reached in the limit, i.e. after an infinite number of steps. You move towards the optimal solution, but taking a zig-zag path.

We can bring this back to our question of improvement, as there are two ways to ‘fix’ SD relevant to ‘taking a turn’. The first in Conjugate Gradient (CG) and in simplest terms, you put the ‘turn’ in the choice of the initial direction by turning it so that it doesn’t ‘go back’ in the previous directions. In our example, that would mean after you optimize all you can by becoming more organized, you figure out your next area of improvement, but make sure that you minimize any aspects that have to do with organization.

CG works when SD works, and in nice cases CG can find the solution in a finite number of steps. Even in less than nice cases, it works better than SD.

The other fix uses some sort of Dogleg (DL) process, and is typically for non-linear problems. In this case you start your step in the SD direction, but then at some time you change your path towards another point. The key is that the path choices are built not on the original problem, but on an idealized approximation which makes the base calculations easier. The final answer is based on the original problem. In our example, this would mean that you would start becoming more organized, but based on education literature or other’s experiences, you’d at some point turn towards some theoretical ideal. You’d be using your real experience to measure the success and stop at the best point along that path.

So why do I bring all this up? I’ve been involved in various efforts to ‘improve’ things, and when those efforts have been successful it has been when the overall goal is clear and the action taker is also a key decider on the thing. I’d say this works because the improvement uses a CG or DL type approach. For example, if someone is trying to improve their teaching, when they know what sort of broad results they want, and are involved in identifying the issues and the solution, then there is a higher likelihood for real improvement. In the case where the action taker is responding to someone else’s choices or someone else has the broad vision, there is usually much more frustration and little improvement. I’d argue that outside management (micromanagement) takes a more SD type approach, as they work from only a snapshot of the situation and usually focus on places that give the quickest response.

There are no real surprises here. It seems most modern leadership/management models tend toward a collaborative/shared-governance perspective which captures the broader or holistic view inherent in the CG and DL type approaches. But it is good to be on the look-out for the less effective, zig-zag approach inherent in a SD type approach.

# 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.