Among other things, the newer approaches to mathematics teaching attempt to have students gain an appreciation of the subject by means of seeing patterns and generalizations which emerge from a study of certain mathematical instances. Hopefully, the students get a feeling for the beauty and power of mathematics from their realization of the existence of the structure, regularity, and order which permeate every branch of the subject. Yet there is another way in which students can sense the strength of mathematics. This is in the recognition of the fact that, at times, seemingly different situations actually have elements in common which, when seen from a certain point of view, produce strikingly analogous results. It is the purpose of this article to give an example of such an analogue. It has to do with the graphs of equations of two variables involving absolute values.

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### James M. Moser

Because the statement of the problem has a very appealing simplicity—it is very succinct, lacks nuances and tricky conditions, and is easy to understand—one might assume that the solution would be equally forthright and clear.

### James M. Moser,

*The Psychology of Mathematics for Instruction* is a text that should be in the professional library of every mathematics educator and researcher. The text does accomplish the purpose of its authors, that is, “to give shape and direction to an emerging branch of study concerned with how expert thought in mathematics proceeds, how that expertise develops, and how instruction can enhance the process of mathematics learning” (p. v). Throughout the development of the text, the authors present mathematics instruction as being dependent on the structure of mathematics and the human being's capacities to pursue intellectual activity.

### James M. Moser

The Commission on Mathematics, in its well-known report written in 1959, called for the “judicious usc of unifying ideas” in mathematical instruction. While it is true that the report was written for the main purpose of improving secondary mathematics instruction, this particular recommendation has just as much importance for elementary education as it does for secondary.

### James M. Moser

This article does not address the particular issue of whether manipulatives should be used in an elementary school mathematics program that includes arithmetic, geometry, statistics, elements of algebra, measurement, and the solution of routine and nonroutine problems in each of these areas. That issue has been addressed elsewhere in this special issue. Rather, the assumption is made that the reader is at least favorably inclined toward the use of manipulative materials. The issues that are considered here are posed as questions that teachers should resolve for themselves. Even though teachers may be convinced that manipulatives should be a part of instruction, some theoretical basis for their use should be developed. The writer is indebted to Richard Skemp (1981) for the quotation. “There is nothing quite so practical as a good theory.” Skemp does not take credit for it, claiming he heard it from someone else, who may even have read it somewhere. No matter, it is sound advice.

### James M. Moser

This is a true story. I do have a daughter named Kathy who is in sixth grade attending a local school here in my hometown of Madison. The school is using one of the recent textbook series that emphasizes the “basics.” Kathy first learned division in fourth grade, spent a great deal of time last year in trying to master the skill of long division, and is at it again this year. The following letter to Kathy's teacher is essentially what I wrote a few evenings ago. After thinking about it a bit, I have added a bit to this more formal rendition. But basically it went something like this.

### James M. Moser

While going through my files a hort time ago, I found the article “Does Mathematic Have to Be So Awful?” (Bereiter 1971). As one might expect from the title, Bereiter overstated his case, just a readers of this column might think I have over tated mine. Yet the title ha enough truth in it for both of us to go on. The key notion is *awful*, a word with several meanings that are not necessarily disjoint. According to *Webster's, awful* can mean (1) inspiring, or filled with, awe, as in afraid, terrified, deeply respectful, or reverential; (2) extremely disagreeable or objectionable: or (3) exceedingly great.

### James M. Moser

### Edited by Howard F. Fehr

### Thomas P. Carpenter and James M. Moser

Children's solutions to simple addition and subtraction word problems were studied in a 3-year longitudinal study that followed 88 children from Grades 1 through 3. The children were able to solve the problems using a variety of modeling and counting strategies even before they received formal instruction in arithmetic. The invented strategies continued to be used after several years of formal instruction. Four levels of problem-solving ability were found. At the first level, children could solve problems only by externally modeling them with physical objects. Modeling strategies were gradually replaced with more sophisticated counting strategies. The results of the study are at variance with important aspects of models of children's performance proposed by Briars and Larkin and by Riley, Greeno, and Heller.

### James M. Moser and Thomas P. Carpenter

Do you think many first-grade children could solve this comparison problem? Joe won 6 prizes at the fair. His sister Connie won 9 prizes. How many more prizes did Connie win than Joe? Most curricular programs apparently assume that word problems are difficult for children of all ages, and that children must master symbolic addition and subtraction operations before they will be able to solve even simple word problems. Recent National Assessment of Educational Progress (NAEP) results (Carpenter et al. 1980) also give some credence to the belief that children are poor at problem solving. We believe. however, that young children are *good* problem solvers.