## Finding Common Ground, Indianapolis, March 2006 Algorithms Group Report

Working Group
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:
• 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)

#### 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?

• 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 students?
• How will you guard against misinterpretation and mindless application of this document?