State of the Blog

After a month, what can I say? Random thoughts on the state of the blog. (More for me than you)

It has been over a month since I made my first post, and I currently have 12 posted articles (counting this one) and 6 drafts.  My initial attempt was to try to dump as much of the junk running around in my head out into the blog as I could.  I now realize that there was a fair amount of partial conceived ideas hanging out there, and that once an idea is written down, it opens up what seems to be twice the opportunity for new ideas.  This is good and bad (like everything), as I have plenty of ideas for new articles (good), but I also feel like I’m spending more time thinking about these ideas (bad, only because I hoped that writing them down would release me from them). So where do I go from here:

  1. Post more often.
  2. Post on just one idea at a time and not worry how complete it is.
  3. Tighten up my voice, i.e. what do I want to be saying overall?
  4. Invite more readers and comments.

We shall see how this goes.



Canon Fodder: Beginnings

A beginning (and ending?) point for the search for the root/canon of math education.

Being a mathematician by training and profession, I’m often driven to make things outside of mathematics conform to the patterns of mathematics: abstraction, definitions, theorems, deductive proofs, etc.  I can also be pretty blind to the possibility that this won’t work for everything.  This is my attempt to make some (mathematical) sense of Education Learning Theories.

Theory = Model: A theory is just a model. A model is initially defined by the context in which is designed to work, and the question it is designed to answer.  Much can go into the context, e.g. domain, scale and resolution, and there is a natural give and take between the context and the question (and the assumptions that go into the model).

For example, in a context where we have a set of data (inputs and outputs) that are related, we wish to predict the output value at some other input.  Based on these components,  and some additional assumptions, we construct a linear function, and that linear function is our model, or our theory as to how the outputs are related to the inputs.

Of course, for a given context/question we could come up with different theories, and then we have to consider how to compare and evaluate them (if at all possible).  However, the point for now is that to even have a chance of comparing theories, we have to identify their forming context/question and consider the overlap.

Learning ≠ Physics: Considering the ‘unreasonable effectiveness’ of mathematics in describing and predicting physical phenomenon, it is natural to assume that we can use a mathematical framework for models to describe other phenomenon, like learning, with the same level of success. But, of course, learning is much more complex and there are too many variables outside the range of easy or reasonable measurement, and so, we should expect that the derived theories will focus on a limited part and will be somewhat imprecise.

Learning Here ≠ Learning There: In modeling it is common (or a common mistake) to extend the model beyond the context/questions that originally defined it. Good models are robust enough to take this abuse, but I suspect that unless stated quite broadly, Learning Theories are much more fragile and do not extend easily.

Learning Theory≠ Model: It seems, by transitivity, that I have contradicted myself here, but what I’m really saying is that Learning Theories aren’t really used in the same way that (mathematical) Models are.  Math models are used primarily (and because they can be) to predict and secondarily to understand.  They also, in a small way, form a framework/context for discussing the object being modeled.  In our example above of constructing a linear function, we could think about the general process of replacing a set of data with the parameters that define the model and then talk about different sets of data in terms of just these parameters, e.g. “Is the slope/correlation positive or negative?”. Learning theories are used with reverse emphasis.  They are primarily about creating a context, or perspective or ‘lens’ for discussing learning.  They establish a based language that those ‘here’ and those ‘there’ can use to talk about and evaluate learning.  There is some part of theories related to prediction and understanding, but it is minor.

Canon or Not?:  I don’t think I can put enough structure on learning theories to make them more ‘mathematical’, primarily due to the complexity and variety of contexts involved. I do think that there is value is pursuing more understanding and formalizing the base ideas so that we have some way to discuss, apply and teach learning theories.


LEGO and Teaching

A look at how various ways of playing with LEGO relate to teaching.

(Written on Father’s Day after thinking of how my dad started me with an Erector Set which prepared me for a lifelong love of LEGO, how much I enjoyed sharing LEGO with my children, and how much I look forward to sharing them with my future grandchildren.)

LEGO, if you don’t know, is a system (toy) for building a wide variety of things. And, as such, is a fair analogy for learning, especially constructionist learning theories. So how does LEGO teach users about ways to use LEGO?  There are three different phases: First Set, Multiple Sets, Free-form.

First Set: The first time you encounter LEGO is usually through a boxed (or bagged) set, with an image of the completed object on the cover. When you open it, you find all the pieces (in no order) and a set of instructions. Typically one set builds one thing, although sometimes there are options. Building the set basically becomes a combination of pattern matching and picture comparison, as you try to find the pieces that are needed for each step (matching the picture to the objects) and then finding out where they are placed on the object (comparing the results of this step with the previous step). Difficulty is gauged basically by the number and size of pieces (larger number and smaller pieces make it harder). The number of pieces per step and the complexity of their assembly also influence difficulty, but due to the precision of the pieces (one of the defining aspects of LEGO: if they should go together, they can), assembly is not difficult once you have the right piece and know where it goes (dis-assembly is a different story).

Things to notice: There are a specific set of steps to take and, except for some cases with options for different final object, there are no choices to make; much like completing a puzzle. However, it is different from a puzzle in that although you have a picture of the final product, the construction is just based on the pictures step by step, and, in some cases, it is challenging to realize how what you are building ends up as the picture on the box (but it always does). The image is advertising and maybe a guide for future play.

What you learn: Building a set this way, you learn to identify and distinguish various pieces, and how to put them together. You see examples of different combinations of pieces used to build certain sub-structures, e.g. tire + wheel + axle = wheel assembly.You see examples of sub-structures being combined to build a larger structure.

Related to Teaching: This is the phase of building familiarity and some ownership with the content. You can’t construct something if you don’t have anything to work with and you don’t know how things fit together. Teaching this way is almost a ‘wax on, wax off’ Karate Kid type of experience except that (and this is important) the learned has seen and been motivated by the final result, which they will actually be able to create. Contrast this to teaching lower level math with the universal promise that ‘you’ll need this later’ (to solve some relevant? problem).

Multiple Sets: The next phase of building comes when you have multiple sets, usually from a similar theme, or you have a larger Construction or Creator set with instructions to build multiple models. It basically is one idea: you can combine sub-structures in ways different from the instructions. Sometimes the instructions will be explicit, in that at some stage you will use some sub-structure from one part in another model, but usually it just sort of implied. It can be as simple as sticking two models together (car + airplane = flying car) or just re-connecting parts differently (two dinosaurs = one with two heads + one with two tails). Or as complex as using small assemblies to create new things (Space cruiser + City ambulance = space cargo ship).

Things to notice: The instructions don’t tell you to do this. The basic building rules still hold, but now you have more options. You need different parts and some experience working with them.

What you learn: Basic identification and construction ideas are reinforced. You begin to think of some sub-structures as ‘basic’ elements and to think of results more in terms of the sub-structures. You learn to experiment (some) with failures in engineering or aesthetics helping you form some general guidelines.

Related to Teaching: It would be easy to push this stage like it was done in the First Set by providing specific examples or experiences where they had to (re-)combine existing material into new ideas. However, if you make it happen, it didn’t really happen. This stage is really driven by the learner’s own natural curiosity stirred by some basics techniques, enough stuff, and maybe a small prompt. It takes time and space, and the need for such is different for each student.  It takes careful prompting through examples and questioning to open a learner up to this stage.

Free-form: This phase is an extension of the last, as once you decide that you don’t have to follow the instructions exactly, then you can ask if you need instructions at all. Models are now built based on imagination, experimental construction, and other high level skills. Single objects and sub-structures take on new identities (2×4 brick = house = foot = rock = ….).

Things to notice: The goals are now what the builder imagines, and the builder also gets to (re)define things along the way to fit his or her goals.

What you learn: There’s a lot of rule-breaking in this phase, but it is built on some foundational concepts. The meta-take-away is this idea of keeping true to the basic/necessary rules and objects, but being free in all other ways is okay/good. Although once you have this idea, you then may start questioning the basic rules (and objects). For me, I’m okay with connecting a plate vertically to a brick, taking the construction off into a new direction, but I’m not okay with using non-LEGO parts (or even some of the super-specialty LEGO parts).

Related to Teaching: This is probably the ultimate goal: a student has enough skill with the basics and he or she can creatively use those skills to build something new. But like with Multiple Sets, it is not easy to achieve and comes more from opportunity (supplies and time) and opportunistic teacher input (questioning and prompting).

A Warning:  To entice more users, LEGO has been partnering with modern media (see e.g. Superman), and even creating their own (e.g. LEGO video games), and it is tempting to adopt this in our own teaching. This does work at the first level, but one potential weakness is that to mimic the media, LEGO often adds specialty pieces that have limited general usage. This is not bad as one can always choose not to use those pieces in later phases. However, the problem comes when the user expects that anything that he or she builds will have the same level of reality, and future building is limited by this expectation. In teaching this comes when we motivate some work with a real world application and sell the idea that this is why we teach math at all, then when we come to some more mundane or just less interesting material, students have no interest in it.

The power of LEGO is the power of good teaching and that is empowering a student to discover his or her own creativity and goals, and to give them the basic skills, time and space, to achieve those goals.

Best of Math Mole Volume 2

More puzzles, mostly weird ones!

Following the pattern in Best of Math Mole Volume 1, here’s some more problems with comments and solutions (at the end).

2.1: Write down 5 odd numbers that add up to 14.

Just because someone likes math it doesn’t mean he or she doesn’t have other interests.  The problem is one of many where the puzzle writer (not me) showed off some word play to go with the number play. It can be a good problem to use in a set where the problems include heavy and/or tricky calculations.

2.2: A simple calculation. Let [a] be the symbol for a^a so for example [2] = 2^2 = 4. Then let <b> be the symbol for b with b [‘s around it, so that <2> = [[2]] = [4] = 256. Finally, let (c) be the symbol for c with c <‘s around it. Calculate (2).

This is from The Mathematical Experience by Davis and Hersch, which is a great resource for interesting math ideas and a great book to share with the mathematically inclined students. It is hard to classify this problem and I’m not sure the answer is terribly satisfying, but it does open up some interesting conversations about what an answer is, how numbers can be represented, etc.

2.3: tamreF’s Last Theorem: Prove that n^x + n^y = n^z has no solution (x,y,z) in the positive integers for n>2.

This might not be as interesting, but at the time Fermat’s Last Theorem was popular even outside of mathematics and so being able to give students a chance to work with something that at is sort of similar to the FLT was a good thing.


Soln2.1: First you should complain that it is impossible as the sum of 5 odd numbers (by traditional definition) must be odd, but once I stand firm of the claim that there is an answer, you should open up to different ideas for ‘odd.  Then there are various options, like 3, 3, 3, 3, 2, where using 2 as an odd number or being different from 3 is an odd choice.  Let the groans begin.

Soln2.2: We have (2) = <<2>> = <256> = [[...[[256]]...]]] where there should be 256 [‘s in the last expression. That part is easy, but now trying to evaluate it is challenging: [something weird is happening with the formatting]

[256] = 256^{256} \approx 3 \times 10^{316}.

Now raise it to itself! And again and again 252 more times.  I have no idea how big a number this would be or really any way to represent it.  You can try logs, but you really need some kind of repeated log like

\log^n(x) = \log(\log(...(\log(x))...))

with n logs involved. But then we’ve just replaced one precise, but meaningless number with another one (suppose \log^{256} x = 1, what is x?

Soln2.3: First it does hold for n=2 with 2^k + 2^k = 2^{k+1} for any k. So for n>2 what can we do? The clever proof is to look at it in base n and then n^x looks like 10…000 and so the sum of such numbers can only be another such number (in base n) when n=2.  Another approach is to consider w.l.o.g. x \le y < z and then after dividing through by n^x we have 1 + n^s = n^t with 0 \le s < t and then look at the equation mod n and you end up with either 1+0=0 (0<s) or 1+1=0 (0=s) and neither is true for n>2.

Common Core Arguments

Common Core (for math) is a good thing?

I suspect this is happening in other places, but I’ve been enjoying various letters-to-the-editor in our local paper making arguments for and against the Common Core State Standards (CCSS).  I’m not here to argue one side or the other (I’m basically positive, and, for transparency, I serve on a statewide committee related to CCSS connection to higher ed).  What I’m finding interesting in the arguments is that both sides are actually right, but they are arguing for different things.

Those supporting CCSS focus their arguments on the higher standards and the focus on college/career readiness.  In my state, it was not long ago that the high school standards for college-readiness in math was a math ACT score of 191! And, sadly, the percentage of students that achieved this (low) standard, was not very high.  Efforts to raise the standards had begun, and one could view CCSS as a continuation (and finalization) of these higher standards.  Also the focusing of the topics and the increase on rigor and reasoning is a good thing at least for those going into math in higher education.  The intent of CCSS is relatively clear, but the results will depend on how well it is implemented.  One reasonable concern is the amount and type of computer based testing that is planned.

Those against CCSS focus their arguments on the source and commercial interests involved in the creation of the CCSS.  It is clear that the motivation and early (or even most of the) participants that developed CCSS came from business and politics and unfortunate (but not necessarily evil) that more educators weren’t involved.  It is also clear that some commercial companies will be making big money on the implementation and testing involved in CCSS. It is a problem, and might be the ultimate downfall of CCSS, that the content specialists weren’t brought into the process early (or enough).

Either way you look at it, CCSS is a big deal. Maybe it’s best contribution to education will be all the discussions that it has spawned. I look forward to the next round of letters, especially after the first year of tests and more than the usual number of students fail.


1For reference, ACT uses 22 for ‘College Readiness’ and many/most area colleges would probably send a student with a score below 22 into remediation; even a 22 only gets you entry into College Algebra which does earn college credit, but one might argue that it isn’t really indicative of college-readiness.

C4C: Open

What an open classroom, learning experience and university might look like.

Continuing the prior post, Changes for Change, I focus on the characteristic of being open.

As described, being open comes in two forms: open to many people (think Open Enrollment or Open Door, or even better, Open House) and open to input and change from others (think Open Source).  Let’s break it down on how this looks at various levels.

At the colleague-to-colleague level the goal is to connect my course and its elements to my colleagues for the purpose of feedback, leveraged development, and renewed innovation.  The feedback part is obvious: if I have a literal and figurative Open Door Policy for my class and my materials and I seek out ‘visits’ from my colleagues and honest conversation, then I’ll get feedback.  I can add structure to the process by organizing and focusing the input toward specific topics or issues.  I would hope that my colleagues would reciprocate and invite me into their courses.  Coordinating with my colleagues and their courses opens up the idea of shared resources and shared development.  And then, as I see and understand what others are doing in their classes, I get new ideas and new encouragement to try to ideas.

At the class level, openness takes on a somewhat different meaning in that it’s not just being open, but developing the attitude that learning should be open. That means pushing the boundaries of FERPA and other policies and creating space and mechanisms for sharing.  It starts with the initial contact with the material; through discussions in-class and online I need the students to take ownership of the material and, more importantly, some responsibility for others facility with the material.   It then moves to feedback; by using peer evaluation, formal and informal groups, class presentations, etc. they learn to trust each other and themselves as guides and experts. And finally we have assessment; grades must still be individual but contributions to the class must play an important role in evaluation learning.

At the university level, there are several policy changes that can increase openness. Current evaluation of teaching is primarily summative and based on meeting traditional standards, so we’d have to create formal (and informal) opportunities for evaluation that is focused on improvement not punishment. Also, to encourage sharing of innovation, we need to allow for a richer set of evaluations by other innovators that allow for experimentation.  A simple change involves sharing of resources; currently we use a LMS that requires permissions to see what resources someone is using in his/her course.  There must be some way to allow easier sharing or at least mirror resources on an open site. Related to sharing we need to open avenues for co-teaching situations.  Finally, we need to add more campus-wide sharing and celebration of teaching.  There should be ‘teacher of the week/month’, regular open seminars, and an annual festivals/conferences.

Changes for Change

An overview of what personal and institutional changes would help change the way we teach.

I’ll admit it up front: I’m a policy wonk. I like having policies and I like thinking about policies.  I was once on a committee and we were considering adding a new graduation requirement and I came up with a list of 22 rules without breaking a sweat. Except for such committee assignments, I try not to inflict this interest/ability on others, but it doesn’t keep me from thinking about how changes I might make personally might end up as institution-wide policy.  Thus we have this post.

It isn’t a question of “flip or don’t flip”, but a question of what position or attitude should I (we) have to support beneficial, productive, and necessary changes in my (our) teaching.  Here’s my list (with longer posts on each to follow):

Open: I need to break down the barriers between my class and the rest of the world (think Open Enrollment) and I need to provide ways to share my teaching resources and look for more community input (think Open Source).

Agile: I need to have a flexible framework and be able to quickly adjust to changes that occur in my classroom.

Forgiving: I need ways to evaluate my teaching both locally and globally, not for punishment, but for growth, and I need ways to allow for things to not go well.

Fault Tolerant: I need to design and implement my course both expecting the best of everyone and everything involved, but also being able to allow for and adjust to things that don’t fit my plan, and still achieve my goals.

Entrepreneurial: I need to find ways to discover and explore new ideas, and means to bring them to my classroom.

Note: As I write these characteristics, I can’t help but see that they are also some of the big things I want for my students. So, as I explore ideas about creating a personal and campus culture based on these ideas, I should also think about how to add my students and classroom into the mix.