Finding Common Ground, Indianapolis, March 2006

Algorithms Group Report



Working Group
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

  1. What is the mathematical nature of algorithms?
  2. What algorithms do we want to teach, and what are the mathematical ideas behind them?
  3. Why do we want to teach algorithms, and what is the value of those ideas?
  4. How do we teach them in such a way that the students appreciate that value?


What is an Algorithm?

Key characteristics of an algorithm for the purposes of our discussion are:

Two additional properties of algorithms are:

Question 4

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 common algorithm.

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 algorithms.

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 algorithm.

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?