Young Children and the Teaching of Algorithms: What Does Research Show?

Calculating Changes is indebted to:

Doug Clarke
Associate Professor
Mathematics Education
Australian Catholic University (St Patrick's Campus)
Email: D.Clarke@patrick.acu.edu.au
Phone: + 61 3 9953 3287

for allowing use of this material which was prepared for a presentation at the Early Numeracy Research Project Professional Development Day, Victoria, Australia, March 2001.

Although factually accurate, Doug would prefer readers to think of this article as a draft.

The article can be read and printed as a whole from this screen, or read in sections using the following links:

• Introduction
• Reasons for written algorithms
• Dangers inherent in written algorithms
• Some other research findings
• When should children meet conventional algorithms?
• Are any student-constructed algorithms okay?
• Conclusion
• References

I think a large amount of time is at present wasted on attempts to teach and to learn the standard algorithms, and that the most common results are frustration, unhappiness and a deteriorating attitude to mathematics.
(Plunkett, 1979, p.4)

A frequently-discussed issue is the role and timing of the teaching of algorithms in primary mathematics. Should algorithms be 'taught' to children or should they develop their own algorithms as necessary? What are the advantages and disadvantages of presenting algorithms to children? What can we learn from research and the advice of key scholars in the area?

What is an algorithm?

• A finite, step-by-step procedure for accomplishing a task that we wish to complete.
  (Usiskin, 1998, p. 7)
• An algorithm takes input, follows a determinate set of rules, and in a finite number of steps gives output that provides a conclusive answer.
  (Maurer, 1998, p.21)

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Adapted from Plunkett, 1979; Usiskin, 1998

• They have been traditional
• They are powerful in solving classes of problems
• They are reliable (when done correctly, they yield the correct answer every time)
• They are usually precise
• They are contracted, summarising several lines of equations
• They are fast
• They provide a written record
• They can establish a mental image
• They can be instructive
• For the teacher, they are easy to manage and mark

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Adapted from Kamii, 1998; McIntosh, 1998; Usiskin, 1998

• They do not correspond to the ways in which people tend to think about numbers. (The belief that algorithms train the mind has little basis.)
• They encourage children to give up their own thinking, leading to a loss of 'ownership of ideas'.
• They tend to "unteach" place value, thereby preventing children from developing number sense.
• They tend to make children dependent on the spatial arrangement of digits (or paper and pencil).
• They tend to lead to blind acceptance of results.
• They often result in overzealous applications.
• Adults use formal written computation for less than 25% of their calculations. (It has become increasingly unusual for standard written algorithms to be used anywhere except in the mathematics classroom.)

There has been a huge amount of evidence from our national testing that once given a calculation in vertical form, kids automatically go on to do it column by column and do not think about it.
Askew, 1999

Interestingly, according to Askew, in England, the expectation in the National Numeracy Project is that by the age of about 9, 80% of all kids should be able to mentally add or subtract two two-digit numbers, and up to that point children should not be doing any vertical computations at all.

Jamie (2nd grade girl), reported in Narode, Board, & Davenport (1993):

• [Early in the school year] Jamie added 19 and 26 mentally:
  I know I have 30 because I have a group of ten and two more tens. Then if I take 1 from the 6 and give it to the 9, I'll have another group of 10. That leaves five left, so the answer is 45.
• [After five months of school and work with conventional algorithms] Jamie attempted to add 34 and 99 by beginning to group the 9 tens and 3 tens, then stops and says:
  Oh, I have to add the ones first.
  She then grouped the units, and traded for a ten to solve the problem.
• [In the last month of the school year] Asked about the possibility of solving the problem by adding the tens first, Jamie emphatically stated:
  No, you never add the tens first.
  Instead, she suggested that another way to solve the problem might be to know the answer from memory. Finally, she was confronted with her own invented strategy as a strategy 'someone used' to add 49 + 19 (I think of 50 + 19 and then subtract one to get 68). When asked if she thought this method might work, she replied:
  If you know that way, it's okay, but it's much, much better to just add the ones first.

We believe that by encouraging students to use only one method (algorithmic) to solve problems, they lose some of their capacity for flexible and creative thought. They become less willing to attempt problems in alternative ways, and they become afraid to take risks. Furthermore, there is a high probability that the students will lost conceptual knowledge in the process of gaining procedural knowledge.
Narode, Board, & Davenport, 1993, p.260

Hope (1989, pp.13-14) emphasised five understandings that children should develop in relation to calculation and number sense:

1. Calculation is done for a purpose (we should avoid problems that have little interest or purpose for students).
2. The choice of calculation procedure depends on the context.
3. Calculators can often be simplified (children should look for ways to simplify a calculation before carrying it out).
4. The context can help in evaluating the reasonableness of a calculated answer.
5. Calculated answers must be interpreted.

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Kamii & Dominick (1998)

Twice as many third graders who had not been taught written algorithms as those who had, successfully answered 6 + 53 + 185 (50% compared with 25%). Interestingly, the answers of the 'no-algorithms group' were all in the range 221-284, while the others ranged from 29 to 838.

Hope & Sherrill (1987)

Unskilled mental calculators among U.S. secondary school students almost exclusively favoured the mental analog of the pencil-and-paper algorithm. . . . Skilled mental calculators, on the other hand, used a variety of strategies, involving primarily different forms of distributivity and factoring.

Jones (1973)

Eighty British 11-year olds were asked to calculate 67+38, 83-26, 17x6 and 116÷4, and were free to use written or mental methods. Over half of the 320 calculations were successfully completed by non-standard methods.

Narode, Board, & Davenport (1993)

A year-long study of first, second and third grade students indicated that conceptual knowledge may be extinguished through an emphasis on procedural knowledge. The students' prior understandings of place value in double-digit addition and subtraction became subordinate to and subverted by teacher taught algorithms which the children accorded higher status than their own, successful invented strategies.

Groves & Stacey (1998)

Victorian children with long-term experience with calculators performed better overall than children without such experience; both on questions for which they could use any tool of their choice and on mental computation.

Lampert (1989)

A four-year teaching experiment was conducted with fourth- and fifth-grade students in Michigan, in which children were encouraged to develop their own methods for problems like the multiplication of large numbers. There was a particular emphasis on relating a given operation to a story or drawing with which it corresponded. Students of diverse abilities were able to do both multiplication of large numbers and make sense of what they were doing.

Ross (1989)

Sixty students in grades two to five were interviewed regarding their understanding of two-digit numbers. She concluded that:
pupils need to engage in problem-solving tasks that challenge them to think about useful ways to partition and compose numbers. The addition and subtraction of numbers through ninety-nine are appropriate topics for second graders, but we should encourage pupils to find sums and differences in their own ways. (p.51)

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Some scholars (eg: Alistair McIntosh) believe that primary children should never be taught written algorithms, arguing that they get in the way of the most important computation method: mental computation.

Others argue for delaying the presentation of conventional algorithms until children have had considerable experience at creating their own:
If we challenge students with more opportunities for making estimates and mental computations and show them the conventional algorithms only after they have experienced a fertile period of inventing their own efficient procedures for solving problems like 32 + 59, we can expect more young students to demonstrate the numeric partitioning flexibility that is characteristic of those with good number sense. If students then learn a conventional algorithm, they will view it as one of many ways to find a sum or difference; they'll be able to choose from and use, as appropriate, estimation, mental computation, and calculators, as well as invented and conventional algorithms.
Ross, 1989, p.51

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Early in school, the answer is probably 'yes', but over time, we want to encourage children to consider whether the procedures are:

1. Efficient enough to be used regularly without considerable loss of time.
2. Mathematically valid.
3. Generalisable (can the algorithm be applied to the full range of problems of the type being solved?)
   Campbell, Rowan, & Suarez, 1998, p.51

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What does the CSF say about algorithms?
[The Curriculum & Standards Framework is the official policy document of Victoria, Australia.]

Level 2: students use materials and diagrams to model addition and subtraction problems, and also use informal written methods based on place value to solve addition and subtraction problems with two-digit numbers. (p.39)

Level 3: students refine known written methods used to add and subtract three-digit numbers and decimal numbers involving tenths, and to multiply and divide by single-digits and multiples of 10. (p.62)

So where do we go?
I believe that there is no place for introducing conventional algorithms to children in the first three years of school. By giving arithmetic a problem solving focus, and by providing a whole range of problems for children to solve (preferably in story contexts of interest to children), we redefine the role of students in arithmetic, in the words of Lampert (1989), from the task of
remembering what to do and in what order to do it, to a problem of figuring out why arithmetic rules make sense in the first place (p. 34).
My suggestion would be to use a variety of such problem types, with increasingly large numbers, challenging children to solve them, by any method that makes sense to them. Through sharing their methods, children can make a start on the process of evaluating the various methods for their mathematically validity, their efficiency, and their generalisability, though not in these terms!

Concluding thought (from 1830)

The learner should never be told directly how to perform any operation in arithmetic. . . . Nothing gives scholars so much confidence in their own powers and stimulates them so much to use their own efforts as to allow them to pursue their own methods and to encourage them in them.
Colburn, 1912, p.463

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Askew, M. (1999). Woe betide the kids who has measles. Australian Primary Classroom, 4(2), 27-31.
Board of Studies. (2000). Mathematics curriculum and standards framework II. Carlton, Victoria: Author.
Campbell, P. F., Rowan, T. E., & Suarez, A. (1998). What criteria for student-invented algorithms? In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (Yearbook of the National Council of Teachers of Mathematics, pp. 49-55). Reston, VA: NCTM.
Colburn, W. (1970). Teaching of arithmetic. Elementary School Journal, 12, 463-480.
Groves, S., & Stacey, K. (1998). Calculators in primary mathematics: Exploring number before teaching algorithms. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (Yearbook of the National Council of Teachers of Mathematics, pp. 120-129). Reston, VA: NCTM.
Hope, J. (1989). Promoting number sense in school. Arithmetic Teacher, 36(6), 12-16.
Hope, J. A., & Sherrill, J. M. (1987). Characteristics of unskilled and skilled mental calculators. Journal for Research in Mathematics Education, 18, 98-111.
Jones, D. A. (1973). An investigation of the differences between boys and girls during the formative years in the methods used to solve mathematical problems. Unpublished M.Phil thesis, University of London.
Kamii, C., & Dominick, A. (1998). The harmful effects of algorithms in grades 1-4. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (Yearbook of the National Council of Teachers of Mathematics, pp. 130-140). Reston, VA: NCTM.
Lampert, M. (1989). Arithmetic as problem solving (Research into Practice series). Arithmetic Teacher, 36(7), 34-36.
McIntosh, A. (1998). Teaching mental algorithms constructively. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (Yearbook of the National Council of Teachers of Mathematics, pp. 44-48). Reston, VA: NCTM.
Maurer, S. B. (1998). What is an algorithm? What is an answer? In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (Yearbook of the National Council of Teachers of Mathematics, pp. 21-31). Reston, VA: NCTM.
Narode, R., Board, J., & Davenport, L. (1993). Algorithms supplant understanding: Case studies of primary students' strategies for double-digit addition and subtraction. In J. R. Becker & B. J. Preece (Eds.), Proceedings of the fifteenth annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (Vol 1, pp. 254-260). San Jose, CA: Center for Mathematics and Computer Science Education, San Jose State University.
Plunkett, S. (1979). Decomposition and all that rot. Mathematics in School, 8, 2-5.
Ross, S. H. (1989). Parts, wholes, and place value: A developmental view. Arithmetic Teacher, 136(6), 46-51.
Usiskin, Z. (1998). Paper-and-pencil algorithms in a calculator and computer age. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (Yearbook of the National Council of Teachers of Mathematics, pp. 7-20). Reston, VA: NCTM.

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