Different Types of Learning Tasks and Content Delivery Styles

The following lists a number of learning tasks students may engage in by teacher instruction, and thus they are instruments of teaching a teacher may employ.  Notes regarding each are provided briefly to highlight a range of possible tools teachers may use to engage students in learning.  These are arranged generally speaking from easiest to most difficult to employ effectively.

Every single one of these types of tasks and styles has its rightful place in a course --- there is no magic bullet solution.  Exclusive use of any single style or type of task is not necessarily a good thing as there are always trade-offs to be made.  This list is not meant to be a complete or exhaustive listing of all possible learning tasks or their categorization or description, but a way of thinking about them to help create an effective mixture of learning tasks for any course of learning.

Lectures
- teacher-oriented learning task
- active instrument
- students may remain largely passive for task completion, and can passively wait it out

Theory and verbal/textual explanations are delivered by an instructor to students (focuses on "telling" over "showing").  The delivery is often best done scripted and in an organized manner, almost like a audio/visual version of a printed book delivery of theory or textual explanations.  This term describes a type of teacher-oriented activity.

New angle on math education: lessons from product design

Or a user interface for learning math

High school is when many students typically first encounter some of the greatest difficulties in learning what forms the basis for learning post-secondary level math. Those are, therefore, some very critical years in a student's math educational career. If it is important to get more students to consider entering a STEM career (and I'd argue it is [1]), then it is important to consider how math can be taught to these students so they'd be more interested in it, and be better at it later on.

The following is a long piece discussing the problems facing designing a better math educational experience, complete with the principles of design that can be used to solve such problems, closing with some guidelines for how to create a better course.  In principle, all of the following can apply equally well to other subjects, like computing science, or English, but math education is tough, so I’m picking on the hardest as an example.

Math course as a product

Just how do students experience of learning math get formed? Rather than looking at it in a traditional teacher's mindset, let's look at it in terms of a commercial product designer's mindset. That means we see students as interacting with a complete, full-fledged math education product that promises students that they'd get what they want.

Worries over losing deep conceptual knowledge: Better teaching in any subject, part 4

• This is part of the "Better teaching in any subject" series:
• Part 1: Problems with modern inquiry based methods
• Part 2: Meaning is use
• Part 3: Facilitate learning through the inner game of meaning as uses
• Part 4: Worries over losing deep conceptual knowledge

A concern some may have with facilitating learning through the inner game of meaning-as-uses is that it seems to turn everything into decomposed techniques and skills, lacking in holistic, deep, conceptual, or otherwise "meaningful" knowledge.

A moment's thought should give you comfort that that's very far from the truth.

Imagine complaining that by breaking tennis up into the various forehand and backhand techniques, players will lose sight of the holistic meaningful concepts required to understanding tennis.

I'd imagine a lot of the worry comes about from not "seeing" the holistic concepts being taught or learned when seeing only the individual techniques being learned.  Traditionally, we'd see the worksheets for practicing factoring quadratics but never see the worksheets for learning what quadratics are conceptually, or what the meaning of factoring is.  From that, we might be led to believe that there must be something wrong with worksheets or with not teaching concepts and meanings (as if we could even directly teach concepts or meanings at all).

Facilitate learning through the inner game of meaning as uses: Better teaching in any subject, part 3

• This is part of the "Better teaching in any subject" series:
• Part 1: Problems with modern inquiry based methods
• Part 2: Meaning is use
• Part 3: Facilitate learning through the inner game of meaning as uses
• Part 4: Worries over losing deep conceptual knowledge (forthcoming)

The inner game of meaning: a lesson from tennis

A lesson from the Inner Game of Tennis (Gallwey) we might draw from is that consciously and intellectually solving the problem of "what is the instructor doing" is like a fool's errand. Because the actual problem the student is trying to solve is "how do I [the student] hit that tennis ball in that situation".  Solving the former problem may help with solving the latter, but there is no guarantee of effectiveness or efficiency.

Because of the unique cognitive and physical characteristics of each student, the solution to the actual problem is always unique anyway.  It always require each student to solve it anew.  The instructor can only point in a general direction, but the student has to go the final distance to arriving at a personalized solution.

If a student's energy is devoted to solving the problem of "what is the instructor doing", then the student will have little energy left for what is more important: solving the actual problem anew for themselves in a way that fits their own unique cognitive and physical characteristics.

How to facilitate learning proper meanings from proper uses

"Meaning is use" means that meaning comes from a variety of particular uses, and students need to look and see while teachers show the varieties of uses properly in order to learn the proper meanings from their proper uses.  But because every student brings with them a different set of prior learning and experiences, a way of conceptual thinking (and physical doing) that works for one student may not work for another.

Meaning is use: Better teaching in any subject, part 2

• This is part of the "Better teaching in any subject" series:
• Part 1: Problems with modern inquiry based methods
• Part 2: Meaning is use
• Part 3: Facilitate learning through the inner game of meaning as uses
  (forthcoming)
• Part 4: Worries over losing deep conceptual knowledge (forthcoming)

Meaning is use

Meaning as use is a philosophical concept of meaning from Wittgenstein:

"For a large class of cases of the employment of the word 'meaning' --- though not for all --- this way can be explained in this way: the meaning of a word is its use in the language" (Philosophical Investigations).

Many traditional and folk understanding of meaning explains meaning in terms of mental representations, or idealized objects in some (sometimes mathematical) objective space, etc. --- i.e. stuff in people's heads or in some Platonic ideal space that has no practical significance for teachers in the classroom.

So while we may not necessarily agree that meaning is philosophically just its use in the language, it's certainly practical to see it that way for teaching!  Because we can, as Wittgenstein urges, look and see the variety of cases in which a word is used, but we cannot look into the heads and minds of students --- and more importantly, nor can students look into the minds of teachers in learning what the teacher meant.

"So different is this new perspective that Wittgenstein repeats: 'Don't think, but look!' (PI 66); and such looking is done vis a vis particular cases, not generalizations." [1]

Meaning as discussed usually refers to meaning of words in a language, but math is no different.  Math is itself a natural language, with a grammar and semantics that's evolved in the mathematical community, used to talk about things and their relationships.  We need not look further than many science research papers wherein authors write mathematical notations and formulas, interweaved with English prose, to see how math is very much a language we can talk about things with.

If we accept that meaning (of words) comes from their use, then the meaning of a thing like a math formula is also built up from the use of it.  The proper meaning of a math formula doesn't come from the instructor explaining it, and it doesn't even come from students discussing and talking about it.  The meaning of a math formula comes from the proper use of it.

Problems with modern inquiry based methods: Better teaching in any subject, part 1

• This is part of the "Better teaching in any subject" series:
• Part 1: Problems with modern inquiry based methods
• Part 2: Meaning is use (forthcoming)
• Part 3: Facilitate learning through the inner game of meaning as uses
  (forthcoming)
• Part 4: Worries over losing deep conceptual knowledge (forthcoming)

Problems with experiential, discovery, inquiry, and constructivist learning and teaching

In education, teachers nowadays are often taught constructivism and other modern inquiry based teaching and learning methods.  Those teaching methods purport to help educators teach children in a way that helps the kids construct their own meaning of what they are to learn.  One of the central claims is that meaning is constructed through experiencing, and reflecting on those experiences, on the basis of concepts and meanings learned previously.

By "modern inquiry based" teaching and learning methods, I mean the constellation of academic philosophies and folk understandings of experiential, discovery, inquiry, and constructivist learning and teaching methods.

What's frustrating is that the core understandings in modern inquiry based teaching and learning methods are not so much as wrong, but are just not very helpful to teachers.  Not helpful because the core pedagogical ideas basically only tell teachers that kids must learn from experiences and reflection.  Since we're not privy to see or control all the stuff that happens in the kids' heads anyway, therefore all the philosophically interesting parts of constructivism have no practical significance in the classroom.

Cognition as a Collaboration with Nature

• This is part of a series, along with the prior Group Preference Discovery: a parable of choosing movies, and Trust and Other Elements of Collaboration.

In the previous posts' parable and examination on trust and collaboration, the group was always two or more intelligent agents.  An interesting angle to think about is to recast those person-vs-person games into a person-vs-nature story, from which we might learn that intelligence cannot figure out the world without assuming by charity that nature must be out to help it (rather than out to harm it).

This post is extremely speculative, but is an interesting thought experiment, one might say.

Consider, if an animal acts in the world only to get feedback in the form of a veto vs no-veto, as in the form of die or not-die, well that at first seems like how evolution works but it ignores a lot of information the animal receives from nature beyond die-or-not.

After all, there's no a priori reason why an animal should think that walking off a ledge should kill it, and yet it learns not to along with lots of other stuff to do or not do.  Not all of that is encoded genetically via evolution: certainly the appearance of every ledge or hilltop or predator can't all be encoded into DNA. Some basics are, of course, but much is learned once born.  How does it learn it if the only feedback it can ever get is die-or-not?

Trust and Other Elements of Collaboration

• This is part of a series, along with the prior Group Preference Discovery: a parable of choosing movies, and the forthcoming Cognition as a Collaboration with Nature, to be ready next week.

In the previous post, I examined through a parable comparing two different kinds of group decision making strategies: (1) "freely" throwing out suggestions, vs (2) veto-or-not the other group member's choice.

From that, we talked about communication in terms of information content and how it affects whether we can really get to know a person better (as a friend, let's say) in what is a repeated game (as in game theory, or micro-economics).

Now, we can also talk about what it means to collaborate, not just cooperate, but arrive at decisions via collaboration.  That's what we'll talk about presently.

See, both strategies above can be cooperative.  In fact, the veto-or-not strategy could be seen as an extreme form of cooperation via deferring totally to the other party unless the choice picked is just too far outside the bounds of one's bare minimum acceptability.

Though it's cooperative, it really is another classic example of what non-collaboration looks like.  The party who engages in veto-or-not behaviour is purposefully, even if unintentionally due to ignorance, withholding information from the other parties.  That was clear in the previous post's parable where we saw how veto-or-not communicates exactly one bit of information, a true vs false, or 1 vs 0, etc.

Group Preference Discovery: a parable of choosing movies

• This is part of a series, along with the forthcoming Trust and Other Elements of Collaboration, and Cognition as a Collaboration with Nature, to be ready in the next two weeks.

This is a parable from which we'll see elements of game theory and information content to forward an understanding about communication in a group.

A group of friends is deciding what movie to go watch.  To keep things simple, let's say the group is just of two people, Patrick and Sally.

Well, every time they go see a movie, Patrick asks Sally what movie she'd like to see.  Almost inevitably, Sally always defers back to Patrick to go see whatever he wants instead.  No matter how much Patrick tries to get her to pick or even make some of her own suggestions, she pretty well never does.  Once in a while, she may veto a movie Patrick chose or say that, after seeing the movie, that next time they should see something else for whatever reason (bad actor, terrible director, etc.).

Patrick is mostly fine with choosing, but does Sally realize that her not revealing her preferences for what she wants to see, it makes it next to impossible for Patrick to get to know her via getting to know what she likes, desires, and prefers absent other people's choices.

Patrick doesn't get to know her because by only ever vetoing or not-vetoing choices, she provides just one bit of information regarding any movie he decides on:  i.e. veto vs no-veto, true vs false, 1 vs 0. It's literally one bit of data.  Worse, Patrick can't even tell if a no-veto is due to her really liking a choice he made or if it's just barely passing by her very lowest minimum standards of movie watching.