Finding Common Ground, Indianapolis, March 2006
Algorithms Group Report
Richard Askey (University of Wisconsin)
Doug Clements (SUNY Buﬀalo)
Guershon Harel (facilitator, UC San Diego)
Audrey Jackson (Saint Louis Public Schools)
William McCallum (University of Arizona)
Joseph G. Rosenstein (Rutgers University)
Susan Jo Russell (Education Research Collaborative, TERC)
April 6, 2006
The four questions
- What is the mathematical nature of algorithms?
- What algorithms do we want to teach, and what are the
mathematical ideas behind them?
- Why do we want to teach algorithms, and what is the value of
- How do we teach them in such a way that the students appreciate
What is an Algorithm?
Key characteristics of an algorithm for the purposes of our discussion
- recurrence of a procedure within the overall process that occurs
until the process is exhausted
- generalizability to a large class of problems.
Two additional properties of algorithms are:
- Efficiency (running time relative to size of input)
- Transparency (visibility of the underlying properties being used)
Common algorithms for addition, subtraction, multiplication, and
division should be studied by students once they have developed a good
understanding of the meaning of those operations and have developed
ﬂuency with some more transparent algorithms and some other methods
that are not algorithms. While common algorithms for arithmetical
operations have the important qualities of eﬃciency and
generalizability, those qualities compete with the transparency that is
beneﬁcial for students who are at the stage of developing understanding
of the underlying concepts.
An example of what we mean by a transparent algorithm is the algorithm
below, which can help students become ﬂuent 48 with computation and
regrouping. The transparency of this 16 algorithm results from the
close link between the steps taken 14 and the explanations of those
steps. (Thus, when you add 4 50 tens and 1 ten, you get 5 tens, or 50;
you don’t get 5.)
The common algorithm below lacks this transparency: The compression of
this algorithm can obscure the mathematical structure of what is taking
place. Many students 148 never understand that when they “carry 1,” the
“1” refers to a 10. Moreover, the 1 that is “carried” magically appears
as a superscript, a notational complication that is problematic ifone
doesn’t understand why it is being introduced.
However, once students understand algorithms such as the ﬁrst algorithm
above, and have rehearsed both the method and its explanation, so that
they know it well (i.e., have reached ﬂuency in both the computational
and conceptual aspects of addition), they then can proceed to the
Students also use methods that are not algorithms but embody important
mathematical ideas. An example of what we mean by a method that is not
an algorithm is the regrouping that occurs in the addition problem 74 +
28 = 72 + 30 = 102. This is an adaptation of a method which should be
taught earlier when students learn addition facts up to 20 by
decomposing into 10s.
Once the common algorithm is introduced, students should compare how
diﬀerent algorithms and methods work in a variety of cases. They can
compare eﬃcency and transparency of algorithms, decide which methods
are algorithms and which are not, and judge the correctness of
For example, one student calculates 169/13 using long division, another
uses the fact that 10 × 13 = 130, and that 3 more 13s make 169.
They can compare where the second student’s 130 appears in the long
division algorithm and thus come to a better understanding of that
In all of this, there is disagreement about the relative amount of time
that should be spent on diﬀerent components. For example, one position
is that most students need to spend a great deal of time progressing
from transparent to common algorithms, and that the invention and
analysis of algorithms is a vibrant valid mathematical activity in
itself. Another position is that an early treatment of common
algorithms is important because it forms a good basis for a rich study
of word problems. Some of us think that both approaches could be
workable, others don’t.
Where do YOU go from here?
- How will this be used so that it actually has an eﬀect?
- How is this understanding of common ground to be used to help
others ﬁnd common ground?
- How will this document be integrated with the previous common
ground document and other documents from this meeting?
- How will this be used in a way that is beneﬁcial to teachers and
- How will you guard against misinterpretation and mindless
application of this document?