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Using descriptions from popular textbooks, the authors share the importance of introducing the definition so students can make sense of reflections both on and off the coordinate grid.

Strategies range from engaging in self-education to creating inclusive classroom spaces and adopting inclusive curriculum and pedagogy.

In this paper we discuss different ways teachers can integrate science and mathematics into their curriculum. In particular, we focus on science and mathematics integration via the disciplinary practices.

Normative discourse about probability requires shared meanings for disciplinary vocabulary. Previous research indicates that students’ meanings for probability vocabulary often differ from those of mathematicians, creating a need to attend to developing students’ use of language. Current standards documents conflict in their recommendations about how this should occur. In the present study, we conducted microgenetic research to examine the vocabulary use of four students before, during, and after lessons from a cycle of design-based research attending to probability vocabulary. In characterizing students’ normative and nonnormative uses of language, we draw implications for the design of curriculum, standards, and further research. Specifically, we illustrate the importance of attending to incrementality, multidimensionality, polysemy, interrelatedness, and heterogeneity to foster students’ probability vocabulary development.

Eighth graders had a misconception regarding the concepts of impossibility and certainty as they considered the chances of obtaining a particular color when an atypical two-color spinner was in play.

The pitch of a roof, the grade of a road, and stair stringers are three applications of slope that are used extensively. The concept of slope, which is introduced fairly early in the mathematics curriculum has hands-on applications that students in a classroom can easily explore with. only the need of graph paper. Here, Wolbert discusses the tangible, real-world applications of slopes.

To learn mathematical practices, students need opportunities to engage in them. But simply providing such opportunities may not be sufficient to support all students. Simultaneously, explicitly teaching mathematical practices could be problematic if instruction becomes prescriptive. This study investigated how teachers might make mathematical practices explicit in classroom discourse. Analyses of 26 discussions from 3 mathematics classes revealed that teachers made mathematical practices explicit primarily after students had participated in them. I present a framework of 8 types of teacher moves that made mathematical practices explicit and argue that they did so without turning practices into prescriptions or reducing students' opportunities to engage in them. This suggests a need to expand conceptions of explicitness to promote access to mathematical practices.

When it comes to improving student achievement in mathematics, a core belief is that curriculum materials are of the utmost importance. However, the link between a curriculum program and student learning is affected by the ways in which teachers use resources to design learning experiences. In the first half of this commentary, I raise questions about the emphasis in research on finding the most effective curriculum programs or the most effective teaching practices as separate from one another, and I claim that research should prioritize understanding the most effective ways in which a teacher can use available materials, including expanding the definition of “effective.” In the second half of this commentary, I suggest how researchers can build on existing work to define, describe, and support teachers in learning more effective practices for using curriculum in ways that bring together research on curriculum effectiveness and teaching effectiveness.

Try these two strategies to engage students and their parents in authentic mathematical activities at home.

Composing and decomposing small sets of objects can allow young students to see the whole group as well as the subsets that created the whole.

With the proper use of KA, the teacher's role changes from a deliverer of knowledge to a facilitator or an organizer of learning experiences.

Recommendations for reforming high school mathematics curricula emphasize the importance of engaging students in mathematical investigations of societal issues (CCSSI 2010; NCTM 2000). Proponents argue that these investigations can positively influence students' social awareness and their perceptions of the use of mathematics for understanding societal problems (Frankenstein 1997; Gutstein 2006; Skovsmose 1994). In this article, I share results from my analysis of high school students' experiences as they participated in an investigation of racial bias and the death penalty. Although this classroom investigation took place about ten years ago, the lessons from students' experiences are still relevant today.

Textbooks offer some opportunities to practice modeling, but we can do more for students to experience where mathematics and science meet.

A deep understanding of continuity enables teachers to help students construct a richer concept of function and attend to precision.

Explore a new app that allows students to develop a more sophisticated understanding of fractions.

Students must be able to relate many representations of slope to form an integrated understanding of the concept.

Sixty-three teachers in a K–3 mathematics specialist certificate program conducted family projects in order to improve their skills in partnering with families around mathematics. Past studies have indicated that family involvement in children's education has many positive influences on academic achievement; however, parents' discomfort with math, and teachers' discomfort with working with parents, may be obstacles. The purpose of the present study was to examine 2 years of teachers' mathematical family projects and describe the types of projects chosen, the risks and benefits of these projects, and the quality of the parent–child interaction. It was found that the teachers implemented a variety of projects that promoted parent participation in mathematics. Teachers were also able to utilize a cycle of inquiry to examine the progress of their project. The results showed that teachers were able to create a strong connection between the math classroom and the home environment of the child, as shown, for example, by findings related to the themes of home–school connections and mathematics curriculum of the home.

A child's decision-making photo activity about pattern identification presents implications for teaching and learning patterns in the early years.

Changes to classroom rules of engagement, such as assessment, the curriculum, instruction, and the environment, can produce real results.

Teachers in an urban high school design a learning environment for at-risk mathematics students.

Incorporate simple and complex figures, distortion and proportion, and visual reasoning into your discussion of similarity.

Teaching a visual subject to a visually challenged student inspires strategies that benefit all students in a class.

Teachers learn to appreciate varied student representations of the Flower Bed problem after working through it on their own.

Students expand their perspective when they consider counting with large numbers.

Budgeting credit card debt and calculating college costs provide real-world mathematical contexts.

One secondary school's mathematics department decides to change its geometry curriculum.

More than just accommodation, this framework is designed to help teachers reach all students—no matter their level of ability.

A team teaching activity allowed students to answer these driving questions, “How far did your car climb?” and “What gear ratio did you use?”

The concrete, pictorial, and abstract methods of this lesson give students access to investigate, isolate, define, and use prime numbers.

aWvranese students consistently rank ■W|jl|Ei ,1Щ1 on international exams SiflHHpmatics and science. In 2007, ШШ^п recorded the highest TIMSS BBi score for eighth grade (Mullis, aBtin, and Foy 2008). The central pHfcation agency in Taiwan publishes ШШШеа mathematics curriculum 9HHMH|s, which textbooks and nMBHPexams follow closely. In May each year, all ninth graders are expected to take the Basic Competency Test. It is a seventy-minute timed exam with thirty-four multiplechoice problems that measure students' proficiency in the following areas

This study employed a cluster randomized trial design to evaluate the effectiveness of a research-based intervention for improving the mathematics education of very young children. This intervention includes the Building Blocks mathematics curriculum, which is structured in research-based learning trajectories, and congruous professional development emphasizing teaching for understanding via learning trajectories and technology. A total of 42 schools serving low-resource communities were randomly selected and randomly assigned to 3 treatment groups using a randomized block design involving 1,375 preschoolers in 106 classrooms. Teachers implemented the intervention with adequate fidelity. Pre- to posttest scores revealed that the children in the Building Blocks group learned more mathematics than the children in the control group (effect size, g = 0.72). Specific components of a measure of the quantity and quality of classroom mathematics environments and teaching partially mediated the treatment effect.

Articles in this department showcase students' in-depth thinking and work on problems previously published in Teaching Children Mathematics. This month's scenario challenges students to consider elapsed time and requires them to convert between different units of time measure.

This retrospective study examined the impact of prior mathematics achievement on the relationship between high school mathematics curricula and student postsecondary mathematics performance. The sample (N = 4,144 from 266 high schools) was partitioned into 3 strata by ACT mathematics scores. Students completing 3 or more years of a commercially developed curriculum, the University of Chicago School Mathematics Project curriculum, or National Science Foundation-funded curriculum comprised the sample. Of interest were comparisons of the difficulty level and grade in their initial and subsequent college mathematics courses, and the number of mathematics courses completed over 8 semesters of college work. In general, high school curriculum was not differentially related to the pattern of mathematics grades that students earned over time or to the difficulty levels of the students' mathematics course-taking patterns. There also was no relationship between high school curricula and the number of college mathematics courses completed.

Recent research on mathematics reforms in the United States indicates that the reforms are not yet widely implemented. Generally, this claim results from looking at the extent to which teachers use curricular materials or engage in particular classroom practices. This article moves beyond disparate questions of use and practice to examine interactions between teachers and curricula as evidenced by their enactments of whole-number lessons from a Standards-based curriculum. Specifically, we analyze videorecorded 1st- and 2nd-grade classroom lessons in terms of students' opportunities to reason and communicate about mathematics. This analysis indicates that the level of fidelity to the written curriculum differs from the level of fidelity to the authors' intended curriculum during lesson enactments. Drawing on this analysis, this article explores how curricula support and hinder teachers as they engage students in opportunities to learn mathematics and how teachers' instructional moves and choices impact the enactment of curricula.

How this teacher develops composition of ten with second graders was dramatically reshaped by the 2006 release of NCTM's Curriculum Focal Points.

Secondary school geometry is perhaps most succinctly described as the study of shape. Many aspects of shape are stud ied, such as properties of and relation ships among shapes, location of shapes, transformations of shapes, and reasoning about shape. Consider an important counterpoint to this shape story or perhaps chapter zero in the story?the study of vertex-edge graphs, which are geometric objects for which shape is not an essential characteristic. A vertex-edge graph is a diagram consisting of points (vertices) and line segments or arcs (edges) that connect some of the points. In contrast to what is taught in most geometry courses, shape, size, and location are not essential characteristics of a vertex edge graph. What really matters is how the vertices are connected by the edges. Figure 1 shows an example of a vertex-edge graph. The term vertex edge graph is used to distinguish this type of graph from other graphs, such as graphs of functions or data plots. Sometimes a vertex-edge graph is called a network. The most common and formal term is simply graph. In this article, we will use whichever term is most appropriate in the context.

Statistics is the key to decision making in the information age. The importance of statistical thinking for life and work is reflected in state and provincial mathematics frameworks and in national standards.

Elapsed-time problems are notoriously difficult for children (Monroe, Orme, and Erickson 2002). Instruction on techniques for teaching and learning elapsed time is not emphasized in current mathematics education literature. Nor is it addressed in Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (NCTM 2006). This absence of instruction may be due to a position held by some mathematics educators that telling time and determining elapsed time are life skills rather than skills delegated to mathematics instruction. (See Editor's note.) However, time is addressed in the Principles and Standards for School Mathematics (NCTM 2000) Measurement Standards for both the pre-K–2 and 3–5 grade bands. And regardless of one's perspective on the delegation of time as a discrete content area, determining elapsed time is encountered during mathematics instruction by most children at some point and is often met with frustration. This is especially true when the start or end time falls between the hour and half hour. Children find it challenging to keep track of unit changes between hours and minutes. On a national assessment, only 58 percent of eighth-grade students were able to correctly identify that 150 minutes equals 2 1/2 hours (Jones and Arbaugh 2004). Elapsed-time instruction often focuses on converting units and keeping track of those conversions rather than on counting up or back from one time to another. This article examines how students are able to make sense of elapsed-time problems when instruction is connected to open number-line strategies. Adapting this technique–typically used for recording addition and subtraction counting strategies–provides a method for supporting students' thinking about elapsed-time problems.

Our children live in a highly sophisticated world composed of interlocking complex systems. An appreciation and understanding of such systems is critical for making effective decisions about our lives as individuals and as community members. The activities presented in this article provide rich opportunities for children to experience an introduction to complex systems during which they think mathematically about relevant relationships, patterns, and regularities in dealing with authentic problems (English 2006; Lesh and Doerr 2003).

Joe is a wonderful artist. He can visualize pictures in his mind and then draw them with great detail. He reads at grade level but seems frustrated by mathematics and is still working to develop fluency with whole-number computation. Chue loves mathematics. Numbers make sense to him, and the language barrier does not seem quite so overwhelming when mathematics class begins. Maria always follows directions and does her work quickly and quietly. She reads above grade level and easily understands patterns and relationships with numbers when new concepts are introduced in mathematics class.

This article includes a fictional story featuring Etty Wanda, a girl who likes to point out mathematical errors, and Mrs. Grey, an evil sorceress who poses a paradoxical math problem that asks about comparing area with perimeter.

In a traditional mathematics curriculum, middle school students' experiences with functions in algebra are usually limited to linear relationships. However, a number of realworld experiences (e.g., compound interest, population growth, and the amount of a drug remaining in a subject's bloodstream) involve exponential functions. In this article, I describe two problems involving exponential functions and some basic probability concepts that were used in an eighthgrade class, which led to greater student understanding about the general form of an exponential function. I also describe how students used graphing calculators to help them make sense of the values that are used in the general form and how they relate to the context of the problem.

Inherent in every middle school curriculum is the need for students to measure—the mass of iron filings for a chemical reaction, the appropriate volume when mixing paint, or the length of a board for a shelf. The list is never ending. As endless as the need to measure is the list of questions often heard in the classroom: “What side of the ruler are centimeters on?” “Is this in ounces or milliliters?” “This thermometer—is it in Celsius or Fahrenheit?” The underlying question that we as teachers ask is this: “Why are our students having so much difficulty measuring?”

PRINCIPLES AND STANDARDS FOR SCHOOL Mathematics (NCTM 2000) places proportionality among the major concepts connecting different topics in the mathematics curriculum at the middle school level (p. 217). What concerns us about many of the problems presented to students, however, is that they are often posed purely as a ratio or proportion from the start. Often the statement of a problem is a giveaway that a proportion is involved. For example, the question “If 15 students out of 20 get a problem correct, how many students in a class of 28 would we expect to get the problem correct?” does not tap the depth of proportional reasoning that is required for meaningful problem solving.

Zachary's grandmother was walking him out of preschool. He looked at the tiled walk-way and yelled, “Look, Grandma! Hexagons! Hexagons all over the walk. You can put them together with no spaces!”

What is proportional reasoning? many word problems emphasize realworld situations, with the focus on a single variable. Word problems involving proportional reasoning situations are more complex. Solving these types of problems requires students to identify the two variables involved, then identify the relationship between these two variables. Helping students develop a true understanding of proportional reasoning cannot be accomplished with one isolated topic or situation. For students to use proportional reasoning skills, especially in situations involving rational numbers or unfamiliar story settings, several prerequisite topics must be mastered, including equivalence, fraction, ratio, and scale (Levin 1999; Weinberg 2002). Until these foundational topics are seen as interconnected concepts rather than as isolated skills, students may learn how to solve a single proportion item but fail to recognize the many proportional instances found in and out of the classroom.

When we talk about our prekindergarten curriculum development project

This op-ed piece gives a historical perspective on curriculum change.