Learning to use math in science is a non-trivial task. It involves many different skills (not usually taught in a math class) that help blend physical knowledge with mathematical symbology. One of these is the idea of quantification—that physical quantities can be assigned specific numbers (with a unit). A second is to develop an intuition for scale. One way to help students develop these skills is to teach estimation: the ability to consider a physical situation and put reasonable approximate numbers to it.

1.
This paper is part of a series of papers analyzing thinking about physics with math and providing methods for teaching it as part of our instruction on specific content topics
.
E.
Redish
, “
Using math in physics—Overview
,”
Phys. Teach.
59
,
314
(May
2021
).
2.
E.
Redish
and
D.
Hammer
, “
Reinventing college physics for biologists: Explicating an epistemological curriculum
,”
Am. J. Phys.
77
,
629
642
(July
2009
).
3.
E. F.
Redish
et al, “
NEXUS/Physics: An interdisciplinary repurposing of physics for biologists
,”
Am. J. Phys.
82
,
368
377
(May
2014
).
4.
J.
Tuminaro
and
E.
Redish
, “
Elements of a cognitive model of physics problem solving: Epistemic games
,”
Phys. Rev. ST Phys. Educ. Res.
3
,
020101
(
2007
).
5.
D.
Hammer
, “
Student resources for learning introductory physics
,”
Am. J. Phys.
68
,
S52
S59
(June
2000
);
E.
Redish
, “
Oersted Lecture 2013: How should we think about how our students think?
Am. J. Phys.
82
,
537
551
(June
2014
).
6.
E.
Redish
, “
Using math in physics: 1. Dimensional analysis
,”
Phys. Teach.
59
,
397
(Sept.
2021
).
7.
E.
Redish
, “
Using math in physics: 4. Toy models
,”
Phys. Teach.
(accepted for publication).
8.
Looking at a log-graph paper will show you why. The error by being off by one at one sig fig can be ∼100% if the first digit is 1, but only 10% if it’s 9. So a “one-and-a-half sig fig” rule—two sig figs if the first digit is about 1 or 2, and one sig fig if it’s about 8 or 9
.
9.
I always subtract a point each on estimation problems for keeping more than three significant figures or not using powers of 10 notation. Of course, in calculational problems and in lab, students are always supposed to use the appropriate number of sig figs for the situation at hand
.
10.
S.
Mahajan
, “
Don’t demean the geometric mean
,”
Am. J. Phys.
87
,
75
(Jan.
2019
).
11.
Readers can view the supplementary material
at TPT Online, https://doi.org/10.1119/5.0021823, under the Supplemental tab.
12.
This number is not an accident but arises from the original definition of the meter as setting the distance from the North Pole to the equator (on the line running through Paris) as 10,000 km. So this sets the circumference of Earth as 4 x 107 m. This number is accurate with the current definition of the meter to better than 1%
.
This also illustrates the idea that if you don’t remember a number, sometimes a story helps you reconstruct it.
13.
There is a lovely discussion of this point in E. Mazur
, “
The problem with problems
,”
Opt. Photonics News
7
(
6
),
59
60
(
1996
).
14.
E.
Redish
, “
Using math in science: 5. Functional dependence and scaling
,”
Phys. Teach.
(accepted for publication).
15.
E.
Mazur
,
Peer Instruction
(
Prentice Hall
,
1997
).
16.
B.
Dreyfus
,
personal communication
, Oct. 8,
2014
.
17.
The Living Physics Portal
, https://www.livingphysicsportal.org.
18.
S.
Mahajan
, “
Welcome to the back of the envelope
,”
Am. J. Phys.
85
,
959
(Dec.
2017
);
S.
Mahajan
, “
Making Fermi estimates by tapping the vast knowledge in our gut
,”
Am. J. Phys.
86
,
536
(July
2018
); and many more!
19.
S.
Mahajan
,
Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving
(
MIT Press
,
2010
).

Supplementary Material

AAPT members receive access to The Physics Teacher and the American Journal of Physics as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.