*Teaching and Learning about Whole Numbers in Primary School*, a topical study from ICME-13, focuses on how children are taught and learn about whole numbers by examining two meanings of whole numbers. First is the analytical meaning, which is defined by the standard number system students are exposed to in elementary school. Second is the representational meaning, the use of whole numbers as quantities. The book presents many different approaches to making whole numbers meaningful in the classroom and compares the outcomes of diverse methods of teaching students about whole numbers.

The interest in whole numbers in mathematics education has existed for over 100 years, since the founding of the journal *L’Enseignment Mathématique *in 1899 by Henri Fehr and Charles-Ange Laisant. This survey covers research since the 1970s. In addition, the book pinpoints a shift from teaching written algorithms to teaching arithmetic that results in calculations.

The survey makes clear that whole numbers and quantities are *not* the same thing. Thompson argued in 1993 that one could think of quantities without representing them by a number. On page 3, Thompson gives us this example: “Percy has more books than Susan and Susan has more books than Deborah; therefore, Percy has more books than Deborah.” Thompson also gives us an example of quantifying relations between quantities by using numbers but without representing the quantities themselves as numbers. “Percy has 5 more books than Susan and Susan has 10 more books than Deborah; therefore, Percy has 15 more books than Deborah.” If one knows the relations among the whole numbers, the answer would be 5 + 10 = 15 regardless of what the numbers mean.

In section 2.3.2, “Teaching and Learning Mental Arithmetic,” the survey talks about six main categories of mental strategies for addition and subtraction of whole numbers. While these categories are helpful in examining students’ solutions, they do not provide a deep insight into individual processes of doing a calculation. On page 9 of the book, the various dimensions of the process of mental calculation are presented: methods of calculations, cognitive elements, and tools for solution. On page 10, an example is given of a student, Andrew, a second grader who performs the calculation 46 – 19 = 27. Here, Andrew’s solution appears that he borrowed one from the 4 and made 6 into 16. However, what is not obvious is *how* he figured out the difference. As pointed out, this was carried out using mental math and is not part of the observable behavior.

Section 2.5.3, “Instruction Based on Pre-designed Diagrams,” presents a diagram that has received a lot of interest in the study of problem solving, which is known as the Singapore Model Method. Page 27 contains an example where the bars represent quantities and arrows represent relations between quantities. What is interesting about this diagram is the common difference between the expected and the observed parts of the problem. Students will tend to guess what the whole numbers of prizes will be and then adjust the solution by trial and error, or make other types of adjustment. The diagram works not only as a tool for the reasoning part of the problem(s) but also gives the opportunity for students to work in their own zone of proximal development and with the teacher.

The last part of the survey, “Knowledge of Numbers and Arithmetic Teaching,” considers the teacher’s knowledge of how to effectively teach whole numbers. Here, several points of view are cited. The distinction between expository and discovery methods led to the study of teachers’ attitudes towards mathematics by mainly asking the question: Do teachers see mathematics as a set of ready-made concepts or as an activity?

One of the most interesting study results is outlined in section 3.2, “Describing Teachers’ Pedagogical Content Knowledge.” Here division and fraction arithmetic are considered on pages 32–33. Two of the arithmetic tasks presented to teachers were ¼ ÷ 4 and 320 ÷ 1/3. Eighty-three percent answered these questions correctly; the errors observed in the study indicated the incorrect use of the division algorithm or that 320 ÷ 1/3 = 320/3 = 106.666. One of the word problems presented to the teachers was: A 5 m long stick is divided into 15 equal sticks, what is the length of each stick? All but one of the teachers (96.7%) answered the problem correctly. 90% of the teachers noticed points where the students could make common mistakes. But only four (13.3%) predicted the students might swap the dividend and the divisor in the problem due to the incorrect belief that dividends must be larger than divisors.

This study also surveyed the pedagogical connect knowledge of teachers and asked questions such as “why is 2/3 ÷ 1/3 = 2?” This was a very eye-opening study, as it not only increased the knowledge of mathematics for the teachers but it also gave them ideas that they could take away and use in the classroom.

The final section of the book, “Summary and Challenges for the Future,” provides recent developments on the study and teaching of whole numbers and arithmetic and addresses the major themes to be studied in the future. One of the items I found to be interesting was the idea that mathematical education research could include teachers as learners and as teachers.

There is an increasing push for more training, after-school programs, and more professional development than in years past. The main idea is that learning mathematics is necessary but not sufficient for learning how to teach mathematics. Throughout the book, differences between content knowledge and pedagogical content knowledge were emphasized. These two kinds of knowledge must be considered separately. If a teacher were to know, say, the algorithms for a calculation, but not have any pedagogical knowledge, they will have difficulty explaining the concepts to the students and will subsequently not look for the common errors and misunderstanding students make. This is precisely the reason why these workshops, seminars, and research studies are so important for the development of teachers. The only downside is that mathematics education is constantly evolving. With this constant change, however, research will always be new, which in turn will make for fresh ideas for teaching our students.

Peter Olszewski is a Mathematics Lecturer at The Pennsylvania State University, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. His reach fields are in mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks. He can be reached at pto2@psu.edu. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar and bass, reading, gardening, traveling, and painting landscapes.