teaching-for-transfer.dvi
Teaching for transfer
Sanjoy Mahajan
Teaching and Learning Laboratory
& Dept of Electrical Engineering and Computer Science
MIT, Room 5-122
Cambridge, MA 02139
sanjoy@mit.edu
16 April 2007
Abstract. Students, after they leave our care, are called to solve the diverse problems
offered by the world, so we should teach to increase transfer : the ability to apply
fundamental principles to new problems and contexts. This ability is rare.
The following pages are from a workshop for faculty on designing courses that pro-
mote transfer. I discuss two design principles: to name the transferable ideas and to
illustrate them with examples from diverse subjects. The discussion uses dimensional
reasoning as the example of a valuable transferable idea, illustrating it with three
diverse examples.
Copyright 2007 Sanjoy Mahajan. This document is free software. You can modify and/or redistribute it under
the terms of the GNU General Public License as published by the Free Software Foundation: either version 2 of
the License or (at your option) any later version. The source code is available at arxiv.org/abs/0704.1854v1
from the ’Other formats’ link.
Licensed under the GNU GPL v2 or later
One theme to bind them all
[I don’t give away the name of the technique yet, hence the coy section title, so that
participants can discover it by discussion.]
Pyramid volume
[I ask the participants to discuss, in small groups, the flaws in the proposed volumes.
Then we name the technique – dimensional analysis – for use in the next examples.]
Here is a pyramid with a square base. It has height h and the side of the base is b.
Comment on these proposed formulas for the volume:
b. h3 + b2 + b
c. b4 + h3
d. b4/h
[To teach for transfer, people must practice transfer to other domains. So I ask the
audience to find the worst flaw in the following argument, which comes from a domain
far from physics or mathematics.]
In many articles criticizing globalization (the kindler, gentler name for imperialism),
you can read an argument like this one [from ‘Impunity for Multinationals’,
www.globalpolicy.org/socecon/tncs/2002/0911impunity.htm, 11 Sept 2002]:
In Nigeria, a relatively economically strong country, the GDP is $99 billion.
The net worth of Exxon is $119 billion. ‘When multinationals have a net worth
higher than the GDP of the country in which they operate, what kind of power
relationship are we talking about?’ asks Laura Morosini.
Find the most egregious fault in this argument.
Tidal waves
[And a hard example of dimensional analysis, to show that the wave speed is v =
The speed of water waves in shallow water depends on the depth and on gravity, which
provides the force that drives the waves. Find a formula that connects the speed v to
the gravitational acceleration g and to the depth h.
Tidal waves on the ocean are an example of shallow-water waves (!). How fast do they
travel?
Licensed under the GNU GPL v2 or later
Why use diverse examples
Here is a line of reasoning showing why diverse examples promote transfer.
One example
Concept Example
A bare concept is difficult to grasp without examples. Examples
help learners to understand the concept and, if chosen well, to
separate the transferable concept from the illustrations. Suppose
then that you explain a concept and illustrate it with an example.
The concept and example merge in the learner’s mind, leaving
him or her uncertain about the boundaries of the concept.
Two examples
Penum
One remedy is to offer a second example. To the extent that
the second example is similar to the first, the first concept plus
example overlaps the second concept plus example. The over-
lap includes a penumbra around the concept. The penumbra is
smaller than when only one example is offered, and the two ex-
amples delimit the boundaries of the concept more clearly than
when only one example is offered. Progress!
Diverse examples
You can help the learner even more by taking the second (or
third) example from a distant field: wherefore curriculum in-
tegration. The penumbra shrinks and the concept stands out.
‘Only connect! That was the whole
of her sermon. . .Live in fragments no
longer.’ E. M. Forster, Howard’s End.

  Students, after they leave our care, are called to solve the diverse problems
of the world, so we should teach to increase transfer: the ability to apply
fundamental principles to new problems and contexts. This ability is rare. The
following pages are from a workshop for faculty on designing courses that
promote transfer. I discuss two design principles: to name the transferable
ideas and to illustrate them with examples from diverse subjects. The
discussion uses dimensional reasoning as the example of a valuable transferable
idea, illustrating it with three diverse examples.

teaching-for-transfer.dvi
Teaching for transfer
Sanjoy Mahajan
Teaching and Learning Laboratory
& Dept of Electrical Engineering and Computer Science
MIT, Room 5-122
Cambridge, MA 02139
sanjoy@mit.edu
16 April 2007
Abstract. Students, after they leave our care, are called to solve the diverse problems
offered by the world, so we should teach to increase transfer : the ability to apply
fundamental principles to new problems and contexts. This ability is rare.
The following pages are from a workshop for faculty on designing courses that pro-
mote transfer. I discuss two design principles: to name the transferable ideas and to
illustrate them with examples from diverse subjects. The discussion uses dimensional
reasoning as the example of a valuable transferable idea, illustrating it with three
diverse examples.
Copyright 2007 Sanjoy Mahajan. This document is free software. You can modify and/or redistribute it under
the terms of the GNU General Public License as published by the Free Software Foundation: either version 2 of
the License or (at your option) any later version. The source code is available at arxiv.org/abs/0704.1854v1
from the ’Other formats’ link.
Licensed under the GNU GPL v2 or later
One theme to bind them all
[I don’t give away the name of the technique yet, hence the coy section title, so that
participants can discover it by discussion.]
Pyramid volume
[I ask the participants to discuss, in small groups, the flaws in the proposed volumes.
Then we name the technique – dimensional analysis – for use in the next examples.]
Here is a pyramid with a square base. It has height h and the side of the base is b.
Comment on these proposed formulas for the volume:
b. h3 + b2 + b
c. b4 + h3
d. b4/h
[To teach for transfer, people must practice transfer to other domains. So I ask the
audience to find the worst flaw in the following argument, which comes from a domain
far from physics or mathematics.]
In many articles criticizing globalization (the kindler, gentler name for imperialism),
you can read an argument like this one [from ‘Impunity for Multinationals’,
www.globalpolicy.org/socecon/tncs/2002/0911impunity.htm, 11 Sept 2002]:
In Nigeria, a relatively economically strong country, the GDP is $99 billion.
The net worth of Exxon is $119 billion. ‘When multinationals have a net worth
higher than the GDP of the country in which they operate, what kind of power
relationship are we talking about?’ asks Laura Morosini.
Find the most egregious fault in this argument.
Tidal waves
[And a hard example of dimensional analysis, to show that the wave speed is v =
The speed of water waves in shallow water depends on the depth and on gravity, which
provides the force that drives the waves. Find a formula that connects the speed v to
the gravitational acceleration g and to the depth h.
Tidal waves on the ocean are an example of shallow-water waves (!). How fast do they
travel?
Licensed under the GNU GPL v2 or later
Why use diverse examples
Here is a line of reasoning showing why diverse examples promote transfer.
One example
Concept Example
A bare concept is difficult to grasp without examples. Examples
help learners to understand the concept and, if chosen well, to
separate the transferable concept from the illustrations. Suppose
then that you explain a concept and illustrate it with an example.
The concept and example merge in the learner’s mind, leaving
him or her uncertain about the boundaries of the concept.
Two examples
Penum
One remedy is to offer a second example. To the extent that
the second example is similar to the first, the first concept plus
example overlaps the second concept plus example. The over-
lap includes a penumbra around the concept. The penumbra is
smaller than when only one example is offered, and the two ex-
amples delimit the boundaries of the concept more clearly than
when only one example is offered. Progress!
Diverse examples
You can help the learner even more by taking the second (or
third) example from a distant field: wherefore curriculum in-
tegration. The penumbra shrinks and the concept stands out.
‘Only connect! That was the whole
of her sermon. . .Live in fragments no
longer.’ E. M. Forster, Howard’s End.
