## List of Contents

**January 2008**

**News from Around the World**

**Number Sense, Numeracy and Mathematical Literacy**

*Broekman, H.*

**Teaching Non-Traditional Applications in Engineering Mathematics: A Case Study**

*Gruenwald, N., Sauerbier, G., Zverkova, T., Klymchuk, S.*

**Teaching- Research, NYCity Model**

**Procedural Knowledge and Written Thought In Pre-Algebraic Mathematics**

*Baker, W., Czarnocha, B.*

**Limits of Sequences – Foundations of Understanding in Calculus**

*Czarnocha, B., Giraldo, J., Prabhu, V.*

**Investigations of Students’ Logical Thinking in Calculus**

*Czarnocha, B., Prabhu, V.*

**The Effectiveness of the “Do Math” Approaches –the Bridge to Close the Cognitive Gap Between Arithmetic and Algebra**

*Menil, V., Dias, O.*

**Integration of research and practice in the TR-NYC methodology of teaching-research**

*Czarnocha, B., Prabhu, V.*

**Voices from the Field**

**Learning Trajectory – Positioning a fraction on the number line**

*Czarnocha, B.*

**Evolution of the Didactic Contract**

*Prabhu, V.*

**Quantitative Literacy at BCC**

*QL Group at BCC*

**Editorial**

The first issue of the Mathematics Teaching-Research Journal online in 2008 brings information from mathematics classrooms around the World:

**Teaching Non-Traditional Applications in Engineering Mathematics: A Case Study,** whose idea originated through German, Ukrainian and New Zealand collaboration. It focuses on the development of students’ competency in different steps of the mathematical modeling process, an essential knowledge for anyone wishing to improve students’ achievements on the International Test PISA. This new exam assessing 15-years old teenagers’ knowledge of mathematics at the moment they are graduating from their middle schools. Mathematics modeling is one of the several categories, which comprise the rubric of the International Test and its decomposition into 6 gradation levels is roughly in accordance with the steps in the development of this competency.

**Using a Continuum of Classroom Assessment Tools to Make Good Decisions for Mathematics Teaching and Learning** asks how to make good decisions in the classroom. What are good decisions in the mathematical classrooms? Good decisions, both from the point of view of teachers and students, are those, which increase the effectiveness of mathematics understanding and mastery. Effectiveness, on the other hand is measured by assessment instruments, which are powerful tools in the hands of teachers, who are refining their instruction in accordance with the student needs. The author leads us into the intricacy of the relationship between student needs and design of instruction.

**Number Sense, Numeracy and Mathematical Literacy** – analyzes the difference between the three ways to understand Quantitative Literacy, a subject, whose significance is steadily increasing at CUNY. In characteristic, concrete terms of a classroom teacher examples of each are provided and their discussion leads to the classification of types of thinking which may take place in the classroom.

The first issue of the Mathematics Teaching-Research Journal on Line in 2008 continues with the review of the work of the Bronx Teaching-Research group which gave rise to the Teaching-Research, NYC Model. The elements of the model have been described in previous issues of this journal. We provide a review of the efforts introducing research methods into classroom and discuss different theories of learning which have been used by teacher-researchers of the group. **The work Procedural Knowledge and Conceptual Thought in Pre-algebraic Mathematics** describes the teaching experiment on the relation between mathematics and language conducted during the years of 1999-2002 – one of the first efforts of the teaching-research group spurred by the desire to improve learning of mathematics with the help of natural language. We continue with the discussion of the **Limits of Sequences – Foundations of Understanding in Calculus** focused on student understanding of the concept of the limit as seen through the excerpts from mathematical essays of students written during the teaching-experiment Introducing Indivisibles into Calculus Instruction (NSF-ROLE #0126141). The concern about the mastery of elements of logic necessary for that understanding is developed in the next paper **Investigations of Students’ Logical Thinking in Calculus** where the possibility of understanding certain student logical errors with the help of the theory of the schema is outlined. The paper contains also elements of the methodology of the TR-NYC Model. A more recent work in the paper: **The Effectiveness of the “Do Math” Approaches-the Bridge to Close the Cognitive Gap Between Arithmetic and Algebra** describes the Teaching-Experiment conducted last year at Hostos CC investigating the effectiveness of the remedial instruction based on Math XL platform. The paper addresses the achievement rates on the ACT Compass proficiency exam of CUNY and shows that the instruction designed by authors dramatically increased the passing rates at Hostos CC. **Integration of Research and Practice in the TR NYC Model of Teaching-Research** summarizes the experience of the TR group in applying different theories of learning to work in the mathematics classroom – one of the concerns of the educational establishment. It discusses the subtle process of coordination between the theory and classroom reality showing where lies the usefulness of theoretical approaches for the classroom teacher. This theme continues in the short reports of the new section Voices from the Field, where **Trajectories Of Learning – Positioning fractions on the real line** introduces the concept of trajectory of teaching and/or learning formulated by Martin Simon, while the **Evolution of the Didactic Contract** brings in the concept of the didactic contract formulated by Guy Brousseau.