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Teaching a lesson in Countability

Course goal

First I would think about the stories I want to tell because that’s how it sits in my mind. It doesn’t sit in my mind as a collection of learning objectives.

Mental schema

Hilbert’s hotel and bijections

Stories that need to be told

What is the purpose of bijections? They give you the basic underpinnings of counting things. So that has to come up in a learning objective because that is really fundamental.

Explanation

It would be something about understanding that the student that when you count something you are making a bijection and that’s how we’re going to generalize counting to infinity. You can’t count to infinity, that’s the problem, but every time that you count you’re taking a set of numbers and you are matching up each number with one object. So 1 goes to this thing, 2 goes to that thing. I am labeling each object with a number. And that is making a bijection

Explanation

I would say “The student will be able to…” what do they need to understand about this concept. What do they need to do with it? I would say, they need to be able to tell me what a bijection is. So “They’ll need to be able to state the definition of a bijection”, so that’s one learning objective

Formulating learning objective

They need to be able to explain how counting is creating a bijection with natural numbers.

Formulating learning objective

They need to be able to explain how this then extends to comparing the size of two infinite sets.

Formulating learning objective

Explain what it means for one infinite set to be the same size of another infinite set using bijections

Formulating learning objective

All that stuff about the Hilbert hotel and all the weird and interesting examples… maybe they could provide some examples of where this applies

Secondary objective

The Hilbert hotel, I don’t care if they remember anything about that… it’s completely irrelevant… no problems at all related to the hotel… that’s just for color. Even the examples are for color.

Relevant/irrelevant material

It would be nice if they could do that (provide interesting examples)

Teacher decides what is important

I would not use the ABCD method - the main point of the learning objectives from my perspective. I don’t like the the ABCD method it’s too formulaic - it forgets the purpose of the learning objective - which is to communicate something to the student and to communicate something to the teacher. What do you want the student to get out of the learning objective. You both should get something out of it. It’s a rule of thumb someone put together.

What method to use.

What do you want to communicate to the students. It helps them focus their energy and attention during the lecture because they know what they need to get out of it. If you don’t tell them upfront, they might want to memorize all about the Hilbert hotel - which is irrelevant.

Importance of learning objectives

Creating the assessment for a learning objective is going to help you realize whether you’ve gotten your learning objectives right. You just asked me the question just now, I realized that I am missing some learning objectives. A student hearing the learning objective I gave, what are they going to do? The student is going to go home and memorize the definition of a bijection. They are going to go over those things and be like, I’m done. But I would be very unhappy if they said they’d be done. I’m missing something important.

Purpose of assessment

I’m missing that the student will be able to use a bijection to prove new theorems or statements about countability. That’s super important.

Missing learning objective

What would happen is that when I go to create assessments, and then, wait a minute, these have nothing to do with these learning objectives. So why am I writing these assessments?

Cycle of learning objective and assessment

It’s iterative. It’s not like you write down your learning objectives and then you write your assessments.

Cycle of learning objective and assessment

For me I start thinking about what I want to say first and then I think about why do I want to say that. What’s important about that? What do I want them to get out of that? Oh that’s because I need them to know how to do this. So that gives me a learning objective. Then I start thinking about what do I want the students to do. I want them to do this problem, that problem. Why? Because of this learning objective that I forgot to include. Or now I realize that this learning objective is slightly wrong and I have to change it.

Cycle of learning objective and assessment

It’s not like one week or one day. It’s like one minute. You are going to go back and forth.

Cycle of learning objective and assessment

It’s not like I sit down and write all my learning objectives, then all my assessments, then all my content. You are not going to end up with something good. It’s not linear at all.

Cycle of learning objective and assessment