Mathematics is a soft subject

Reflections on three years in a flipped classroom

Hans Georg Schaathun and Welie Annett Schaathun

NTNU, Noregs Teknisk-Naturvitskaplege Universitet

June 2016

12For whosoever hath, to him shall be given, and he shall have more abundance: but whosoever hath not, from him shall be taken away even that he hath.

13Therefore speak I to them in parables: because they seeing see not; and hearing they hear not, neither do they understand.

Matthew 13.12-13

Accurate description of my classes
Serious: we teach abstract thinking
- students who think abstractly learn a lot
- students who do not get confused

I want to start this talk with the words of the most famous educator, or pedagogue in our part of the world. «Those who have, shall be given, and those who do not shall loose the little they have.» You can see King James' version on the slide.

The words, the Matthew principle, is often quoted in the context of economics. However, the original context was that of teaching and learning. It explains the use of parables.

Anyway, it seems to me to be a pretty accurate description of many of the classes I have taugh. «Seeing, they do not see. Listening, they do not hear.» Those who know a lot when they start the module, learn a lot during it. Those who knew little, tend to forget the little they knew.

Seriously, it is a challenge in our mathematical and abstract subjects. Many students have a good grasp of mathematical concepts as they appear in a concrete and practical situations, but they ability make generic abstractions. When we try to teach abstraction, we risk confusing them so that they loose the intuitive grasp that they had, without gaining any new insight.

Parables, of course, is a well-known idea. We call them examples.

Pure mathematicians frown upon them. «Why on Earth do you want an example? We have proved it!»

Farewell, Lecture!

2013

New module: Discrete Mathematics

iAJC ajcann.wordpress.com [CC BY-SA 2.0]

2013 in hindsight

Positive feedback.
Old-fashioned.
- Lectures (on video)
- Individual exercises
Most students lag behind.
Simple problems are difficult for students.
They don't ask for help.

Njål Foldnes

Лев Семёнович Вы́готский (1896--1934)

Lev Semyonovich Vygotsky

2015

Online lectures and exercise compendium
$2\times2$h teacher-led sessions
$1\times2$h student-led tutorial (compulsory)
Two optional textbooks
Written exam

Contribution to https://giti-math-class.wikispaces.com/, licensed under a Creative Commons Attribution Share-Alike 3.0 License.

Jean Piaget (1896--1980)

		How do you think flipped classroom works?
		Very well	Well	Neither	Badly	Very badly
How much do you learn from flipped classroom compared to traditional methods?	More	1	7	1	--	--
	Same	--	5	2	--	--
	Less	--	1	2	3	3
	Don't know	--	1	--	--	--

How do you rate the amount of each activity in the classroom?

	Too much	Appropriate	Too Little
Presentation of theory	--	15 (2)	10 (3)
Plenary discussions	5 (3)	21 (3)	--
Group discussions	6 (3)	17 (1)	3 (2)
Individual exercises	2 (1)	14 (2)	9 (2)
Student presentations	11 (4)	13 (1)	--

How much do you learn from each activity?

	Very much (3)	Much (2)	Something (1)	Nothing (0)	Score
Solutions by teacher	8 (3)	9 (2)	8 (1)	---	2.00
Individual exercises	4 (1)	10 (1)	11 (4)	---	1.72
Theory on blackboard	3 (0)	12 (3)	9 (3)	1 (0)	1.68
Solutions on video	3 (1)	12 (2)	8 (2)	2 (1)	1.64
Group discussions	2 (0)	12 (2)	11 (4)	---	1.64
Theory on video	3 (0)	8 (1)	12 (4)	2 (1)	1.48
Plenary discussions	1 (0)	8 (1)	15 (4)	1 (1)	1.36
Solutions by students	1 (0)	4 (1)	16 (3)	4 (2)	1.08
Own presentation	---	6 (1)	14 (1)	5 (4)	1.04
Read textbook	1 (1)	2 (0)	13 (1)	5 (2)	0.72

$$\hat\mu=1.46 \quad \hat\sigma=0.15$$

The Magic Number $7\pm2$ - George Armitage Miller (1956)

Klinger -- Blatny, 1988

Same pieces - new position

20-30 seconds
Discussion
- Feedback loop
- Address student schemata
- Fill gaps in student scha

Memories in working memory last about 20-30 seconds. Information which is coded into schemata, usually by amending or augmenting old ones, will survive in long-term memories. The rest is lost.

We need to give the students time to encode the schemata before within the 20-30 seconds span.

We know that prompt feedback is important for learning. If it takes more than 30 seconds to get an answer, the question will be gone from working memory.

Discussion establish a feedback loop, and the iterations can be down to 20 seconds or less.

Flipped classroom can potentially solve some problems with the conventional 2h lecture, but not the most important one. Two-hour chunks are not compatible with the human brain. We need to iterate presentation, activity, contemplation, and feedback in much shorter cycles.