# Journal Article

## Using Schemas to Develop Algebraic Thinking

This article describes how students generalize and formalize patterns using student developed schemas including subtracting out and building up. The article includes samples of student work including students explanations for each schema. This article could supplement readings from section II and IV.

## Using Everyday Knowledge of Decimals to Enhance Understanding

This article discusses the role of students' everyday knowledge of decimals in supporting the development of their knowledge of decimals. Students using contextual problems with decimals are able to build on their understanding of decimals. The problems used tap common misconceptions about decimal fractions so could be used to supplement section IV.

## Using Data-Collection Devices to Enhance Students' Understanding

This article examines four areas of difficulty students have with graphing and modeling. The four areas are identified as connecting graphs with physical concepts, connecting graphs with the real world, transitioning between graphs and physical events, and building graphical concepts through student discourse. The article looks at studentsí misconceptions and the role of technology in helping to form accurate graphical concepts. This article could be used in conjunction with the readings from section IV.

## Upper Elementary School Pupils' Difficulties in Modeling and Solving Nonstandard Additive Word Problems Involving Ordinal Numbers.

This article provides information about the scope and nature of studentsí difficulties with modeling and solving non-routine word problems using addition or subtraction. Highlighted misconceptions about numbers and arithmetic operations can be used to supplement section IV.

## Upper Elementary School Pupils' Difficulties in Modeling and Solving Nonstandard Additive Word Problems Involving Ordinal Numbers.

This article provides information about the scope and nature of studentsí difficulties with modeling and solving non-routine word problems using addition or subtraction. Highlighted misconceptions about numbers and arithmetic operations can be used to supplement section IV.

## Transformations and Technology: What Path to Follow?

This article describes a technology-based approach for dealing with a misconception that some students hold concerning the equivalency of geometric transformations. The article looks at students studying transformations as mathematical entities and not just as procedures to apply to shapes. Students use technology-based activities to help rectify commonly held misconceptions. This resource would be a good supplement for sections II and IV.

## The Evolution with Age of Probabilistic, Intuitively Based Misconceptions

This article summarizes the intuitively based misconceptions that many students have in regards to probability concepts. The findings highlight that many misconceptions grow stronger with age, while others grow weaker. This article would supplement the reading from section IV.

## The benefits and limits of social interaction: The case of teaching mathematical proof.

This article supplements Section IV by providing information about the need to consider social interaction within the mathematics classroom. Researchers have identified that ìstudentsí apparently bizarre mathematical behaviorsî frequently cannot be accounted for solely in terms of conceptual limitations. This article suggests moving past the ëpurely cognitiveí to the role played by social issues for students.

## Teaching Realistic Mathematics Modeling: A Teaching Experiment with Fifth Graders

This article can be used to supplement CTS Sections III and IV. The article describes the design and the results of an exploratory teaching experiment carried out to test the hypothesis that it is feasible to develop in pupils a disposition toward (more) realistic mathematical modeling. This goal is achieved by immersing them in a classroom culture in which word problems are conceived as exercises in mathematical modeling, with a focus on the assumptions and the appropriateness of the model underlying any proposed solution.

## Studying the Evolution of Students' Conceptions of Variation Using the Transformative and Conjecture-Driven Research Design

This paper summarizes a research study conducted on first year college level students taking an introductory statistics course. There are many interesting and applicable findings for high school statistics. The article focuses on studentsí thinking on variability, what students can do and transitions in their understandings. Many examples are given including, box plots, histograms, sample, center and spread. These finding could supplement sections II and IV.