Taxonomies

Oliver Knill

The Bloom Taxonomy from 1956 suggests the following steps for teaching and learning:
1. Knowledge
2. Comprehension
3. Application
4. Analysis
5. Synthesis
6. Evaluation
When building an math bot Sofia in 2004, we were not aware of the Bloom scheme and made up our own:
            Question        Concept             Objects        Bloom analogue
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1.          What            Knowledge           Terminology    Knowledge
2.          How             Skills              Algorithms     Application
3.          Why             Insight             Concepts       Comprehension
4.          Why Not         Creativity          Innovation     Synthesis
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This was ordered according to the difficulty when teaching mathematics to a machine:
Teaching definitions is no problem. More challenging is to implement getting solutions to standard tasks. Wolfram alpha can do that very well already. Teaching insight is already hard and the holy grail in education. The last step, to challenge notions and finding new ways is the hardest part and very challenging. Its difficulty is underestimated by anybody who has never done it. Its hard to be creative for a simple reason: its just hard.

In the lecture "What is mathematics" from a teaching math in a historical context course, I illustrated how arbitrary taxonomies can be. Take the classical 7 liberal arts and sciences, consisting of the trivium and quadrivium: one can see Grammar as a metaphor for "What", Rhetorics as a "How", and Logic as a "Why". The creative part is not in the Trivium. But the quadrivium includes it. Arithmetic can be seen as a metaphor for "Doing things", Geometry to "See things", Music to "Feel things" and Astronomy as "Seek new things". Slide.
Trivium:                Grammar       Rhetorics       Logic
                        What?         How?            Why?

Quadrivium   Arithmetic    Geometry        Music         Astronomy 
             Do It!        See it!         Feel it!      Seek it!
This bending of the 7 liberal arts and sciences illustrates how arbitrary taxonomies can be. As the Bloom taxonomy, it can fit the learning process. Here is a good critics of the numbered nonsense. Especially with the number 7 (seven ways to ...)

Major difference

There are two major difference between the Bloom Taxonomy and what we had done in that AI project:
A) We put "Application" before "Comprehension". 
B) We placed "Evaluation" as a separate process. 
I want to argue here that for learning processes, application should come before comprehension and that it makes sense to take out evaluation as the later is a different entity which should apply to all four steps.
The "Hitchhikers guide to the galaxy" satirically puts in Chapter 20 the "How,Why and Where phase" in that order. The joke is that "Where" should definitely come before "How" and the "Why" later. Here is the quote:
"The history of every major galactic civilization tends to pass through 
three distinct and recognizable phases, those of Survival, 
Inquiry and Sophistication, otherwise known as the How, Why, and 
Where phases. For instance, the first phase is characterized by the 
question `How can we eat?', the second by the question 
`Why do we eat?' and the third by the question, `Where shall we have lunch?'.
I really like that quote because it gets the order of "How" and "Why" right. I did not think about the "Where" yet ...

Example 1: Learning how to drive. Most learn to drive a car before gaining insight of the inner workings of a car. Even when learning to drive tanks, I first learned to drive it before being explained the concept of how steering and power is related. One can argue that it is of advantage to know the details of how a clutch rsp power stearing works in order to understand how to drive a car or tank. But only very academic or rule driven military minds do that. Yes, it is useful to know how things work and one only becomes a master if one knows it, but the understanding comes only after one has done it. On M109 tanks for example, one hits the gas pedal strongly when making a turn. This is counter intuitive but can be explained: the steering mechanism needs pressure to work. Would previous theoretical knowledge help to learn it? No, you simply experience that if you get away from the gas pedal and try to make a turn, nothing happens. Example 2: Learning a language. The fastest way to learn a language is to pick up some key phrases and play with them without being too worried at first to be grammatically correct. Then take well chosen texts and learning them. After a while, with more vocabulary and knowledge one starts to understand the connections and patterns. I learned English and French like that. For Latin, it went differently. In Latin, we first learned all the grammar and rules and then started to work with concrete texts. That was actually refreshing for a change and it helped me to understand also the grammar of other languages better. On the other hand, I never learned how to speak Latin, even after being instructed 7 years (entire middle and high school time) in it.
Example 3: Learning to cook While it is helpful to know the chemical workings of spices and ingredients of a dish, most start cooking with the cookbook. With some experience, one starts to experiment and push the boundaries. A professional cook needs to know about the physical processes going on, when cooking and what the ingredients exactly contain. But most folks just do it, use trial and error and experience from other cooks without trying to understand first, why things are done in a particular order or in a particular way. A steak tastes differently when it was cooked with an initially sizzling hot pan rather than having pan being heated up while the meat is inside. Its physics: the initially raw meet closes faster on the surface in a hot pan. Does a chef need to know this? Probably not, but the chef needs to consider a few dozen other parameters. Getting one thing wrong can ruin the task. Example 4. Learning to play an instrument While music theory is important and helps to learn to play an instrument, it is often more effective, just to play first, then later connect the dots. I myself got some basic music theory at the beginning but just enough to be able to read the music notes. Then it was playing and playing and practicing and practicing. Later in high school, when I took lessons from Sava Savof, it came a bit with a revenge that I did not learn enough music theory. Yes, our music classes in high school covered a bit of theory, also physics (like the physics of the Laplacian for drums), and other approaches to music like Stockhausen, but I never learned music theory, as it would be useful for playing piano well. Still, having put the music theory at the very beginning would have been pointless.


Switching "How" and "Why"

How quickly "concepts" should be introduced into the learning process is a hotly debated point and there will never be agreement: what should come first, the "How" or the "Why", when learning. I hope the examples before have made a point that for most learning processes the "How" comes before "Why". I know however that especially in academic circles, many would put first the "Why" and then the "How". But this often has a crippling effect on learning, especially in practical matters like using software. The reason is simple: for software of many other human made things, the "How" often does not have a good answer or is too complex even for an engineer. But it has a funny effect that intelligent people can often be rather helpless in practical matters when using a new tool. Kids are usually very good at exploring new things because the "How" is more like a game type approach in contrast to the more analytic approach which adults chose. Playing around first is faster and more efficient in learning things. And once, the "How" is mastered, the "Why" starts to become accessible. Since there is so much disagreement in this matter, lets argue again why having things the other way can have a paralyzing effect: trying to decipher a manual of an older watch or camera or learn a programming language by first learning the syntax rules or theoretical properties of a language or first learning music theory is harder than just playing around with that thing, just programming some examples or playing around with the music instrument. Most folks do what kids do naturally: they play first with it before analyzing it. When learning a language, asking first to learn the structure and workings of grammar makes sense but it can lead to frustration and therefore loss of interest. Its better first to learn to talk, repeating sentences even blindly before understanding them and then fill the gaps. The examples themselves then explain the grammar. Similarly when playing a musical instrument, it is better first to learn how to play and only later learn the music theory behind it. In math, the meaning and interpretation of things usually come only much after mastering "how to do things". In my opinion, the Bloom mixup of "How" and "Why" has serious consequences: students do no more know the basics if the basics are questioned very early on. It is expected now that the student comes up with a clever or new way to compute things even in basic arithmetic. Having to be creative all the time produces stress and uncertainty. But creativity only flourishes, once there is enough firm knowledge built in. Creativity to a large degree also just comes from ``going the wrong way". But going the wrong way all the time is not helpful and the lack of success hinders learning. Its like stuffing too much paper and wood onto a freshly lit fire. The fire first needs to be developed a bit before it can handle the challenge.

"Evaluation" as a different entity

"Testing and evaluation" is an important step as it allows to pinpoint misconceptions. But it should not be part of the taxonomy as the evaluation part appears everywhere. The assessment of "understanding" is a complicated process and it is so important that it merits its own taxonomy: A way to assess the understanding is to check whether one is
being able read it
being able to write it
being able to teach it
being able to program it
Each is a multiple times harder than the previous one. Reading or hearing it is passive but it can be challenging already. Take a book about a subject you are unfamiliar with and you know that this stage can be challenging. Writing requires to reformulate it or solve problems. Teaching requires to be able to answer unexpected questions about it and programming it requires to understand every detail, every special case, every limiting case. Teaching it also could mean to give a talk about it or explain it to somebody else. Programming it can be overwhelming and is the ultimate test whether one understands the material. A machine is unforgiving. The smallest misconception prevents things from working. And if things don't work, there is nobody else to blame. We simply don't understand it well enough yet.

Taxonomy for solving problems

An other taxonomy for creativity can be seen on the front of the Boston science museum which are all related to inquiry
discover
imagine
search
innovate
It does not claim to be a taxonomy as the later aims to be a classification. Assume however it would: how does one make discoveries, where does one search and explore to innovate? The words themselves are quite useless since they do not provide any constructive path to research: if it were then the order and equip it with context:
imagine       What is the goal or vision?
search        What has been done? Look around. 
discover      Recognize that something is new.
innovate      Improve and realize the project
would be better. The most famous problem solving advise is the "How to solve it" guidelines by George Polya
what is the question     Understand the problem
outline a path           Make a plan
carry out the plan       Realize the plan
review and check         Revise and improve
It was an extremely important book for me as it helped me to improve my problem solving skills. The most difficult part is certainly the creative step to find the solution: also here there are models. An example is the "snowflake model" of creativity of David Perkins:
Allow complexity and disorganization. Enjoy challenge and chaos.
Solve problems. Find creative solutions. Feel for good questions. 
Think in opposites, metaphors or analogies. Challenge assumptions.
Take risks. Accept failure. Learn from mistakes. Enjoy incompetence.
Scrutinize own ideas. Seek criticism.Put aside own ego. Test ideas.
Catalysis is enjoyment, satisfaction, and challenge for the work itself. 

Learning taxonomies

Also teaching guidelines exist. There are 7 principles for smart teaching which appears in the book "How learning works".
Preknowledge influences
Organize knowledge
Motivation determines
Get component skills 
Practice and feedback
Climate and culture 
Self monitor learning
This is not a taxonomy and also not ordered. It is a list which contains probably anything a group would come up in an hour when being asked "What matters when you teach?" Maybe it is because the authors are mostly from non STEM areas (there is only one statistician, the rest are psychologists) that they forgot to mention the most important point for "How learning works": learning only works if the one who teaches knows the stuff.
The teacher has to know the subject well.
Not only is it important to understand the material which is taught, but way beyond. And this is my main critics of teaching models of pure online or flipped classroom teaching: there are models where the teacher can hide behind ignorance: for Moocs, the videos are simply "talking heads" like the news anchors in TV or actors in film. They don't have to understand anything. Similarly, in a flipped classroom, where the teacher does no more have to reorganize learn and present the material, but simply guide through some pre-canned worksheets (usually where solutions are given). Its the fast food of teaching and destroys teaching culture. Doing work in groups or alone or doing presentations is of course extremely important, but it should come together with good instruction. But the later should not just consist of pre-canned videos. Not that flipped classrooms or video instruction should not happen. They are ancient flavors of teaching (I took a course in electronics 35 years ago in a Telekolleg which was essentially a MOOCs, there was just no web but TV, but it is pretty much the same model it came with a book and worksheets and tests. New with MOOC's is the peer review and discussion. Most of my teachers since first grade used flipped classroom components and it was nice, for a change). My critics is the propaganda behind "replacing all instruction by flipped classrooms" and hype about massive online courses is that the proponents often sound as if it is universal. But there are pitfalls: for example, online discussions and peer review are tricky: most of them on the web are terrible if not moderated. Slashdot, wikipedia and stack overflow and stack exchange are well moderated. It needs a critical mass of actively involved experts so that nonsense is weeded out.

Our group made once a brainstorm experiment on March 9, 2015 to see "what teaching should accomplish besides the content" and here it what we came up within 10 minutes: (this is paraphrased only and merge the points and order them differently as they had found:)
Ability to build models
Acquire problem solving skills
Hone communication skills
Ability to think independently
Ability for conceptional thinking
Develop sanity check habits
Practice perseverance and work habits
Appreciation for application
Appreciation for beauty
Appreciate connections with the world
Appreciate connections to history
One problem with trying to accomplish all this at once is "overload". I myself have seen the problem "overload" in my own teaching: for many years I had used technology almost in every class and made every lecture connected with a historical fact. Applications can be tricky since applications by definition deal with other subjects which requires expertise and knowledge way beyond the situation taught. Teaching appreciation for beauty and connection with the world can be especially challenging. For all this, one has to keep in mind that
Transferring knowledge and insight
is still by far the most important point. If that does not work, all the others will fail. The success of dry online lectures like "Khan academy" comes exactly from the fact that they are "no nonsense" approaches: no coating in stories, cryptic motivation, mumbo jumbo, pedagogical tricks or politics: knowledge pure.

Advise

While working on mathematics myself or when teaching, I also like to contemplate on the process of how things are found or how a lesson is planned. This is almost as interesting than the actual mathematics or the teaching itself. So what is would be my list of advise for research and teaching? Having scolded taxonomies above, don't take this too seriously and make your own.
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