The Relationship Between High-School Mathematics Teachers\u27 Beliefs and Their Practices in Regards to Intellectual Quality

This study examines the relationship between mathematics teachers’ beliefs and instructional practices related to learning, pedagogy, and mathematics in regards to components of intellectual quality for eight high-school mathematics teachers. Research has demonstrated that the higher the degree of intellectual quality for instruction is rated the higher student achievement is on standardized assessments. The findings in this study demonstrate a consistent pattern between teachers espoused beliefs and their instructional practices. Even though teachers’ practices changed as they wrote curricular units to be more in line with intellectual quality characteristics, their beliefs stayed consistent over an 18 month period and were correlated to their instructional practices at the beginning and end of the project

The Importance of Spatial Reasoning in Early Childhood Mathematics

It is important to recognize the critical role spatial reasoning, relational thinking, and mathematical modeling play in the overall development of students’ central understanding of mathematics. Spatial reasoning predicts students’ later success in higher levels of mathematics, such as proportional thinking and algebraic reasoning. The National Research Council report implores educators to recognize the importance of developing spatial reasoning skills with students across all areas of mathematics. This chapter describes a study that used the Primary Math Assessment—Screener and Diagnostic to assess students’ spatial reasoning and relational thinking. The results highlighted curricular resources to improve students’ understanding of mathematics. Students’ mathematical spatial reasoning improved significantly

Building Place Value Understanding Through Modeling and Structure

Place value is a concept in which students in elementary school struggle and instruction and curricular materials continue to introduce and teach place value in a disconnected fashion. This study introduced place value through a modeling perspective, focusing specifically on using the bar model to represent units and quantity. The investigation piloted a place value module highlighting the use of the bar model in four first grade classrooms with high percentages of diverse learners, many from low-income families and with limited English language proficiency. The results indicated students successfully described the differences between units of 1 and 10 and could build and describe numbers in their teens and twenties. Students’ vocabulary and understanding of place value improved over a three-week period, suggesting visual models can be used as an effective model to promote place value understanding

A Glimpse into Secondary Students’ Understanding of Functions

In this article we examine how secondary school students think about functional relationships. More specifically, we examined seven students’ intuitive knowledge in regards to representing two real-world situations with functions. We found students do not tend to represent functional relationships with coordinate graphs even though they are able to do so. Instead, these students tend to represent the physical characteristics of the situation. In addition, we discovered that middleschool students had sophisticated ideas of dependency and covariance. All the students were able to use their models of the situation to generalize and make predictions. These findings suggest that secondary students have the ability to describe covariant and dependent relations and that their models of functions tend to be more intuitive than mathematical-even for the students in algebra II and calculus. Our work suggests a possible framework that begins describing a way of analyzing students’ understanding of functions

Expect the Unexpected When Teaching Probability

Probability has recently made its way into many textbook series and standards documents (NCTM, 2000; NGA, 2010). When students engage in probability problem solving many unexpected situations can arise due to the counterintuitive nature of probability concepts. These situations can be difficult for students and challenging for teachers to analyse during teaching. Recently, as facilitators of a Mathematics Science Partnership grant workshop on probability, we had the opportunity to engage middle school teachers in professional development workshops as well as in their classrooms. In this article, we discuss a rich probability task used with these teachers along with two scenarios that represent challenging aspects of probability for students, and challenging teaching. In these two scenarios, we explain the underlying probabilistic concepts that proved difficult for students. For each probability challenge, we discuss how the process of analysing student thinking can inform teaching strategies that may guide student conceptual development

Five Key Ideas to Teach Fractions and Decimals with Understanding

The teaching of fractions and decimals is a significant challenge for many teachers due to the inherent difficulty of the topic for students as well as the lack of high-quality, modernized curricular materials. This article examines the key ideas of teaching fractions and decimals for understanding that are evident in the current research literature and the curricular materials and teaching strategies from high-achieving nations

Using Solution Strategies to Examine and Promote High School Students’ Understanding of Exponential Functions: One Teacher’s Attempt

Much research has been conducted on how elementary students develop mathematical understanding and subsequently how teachers might use this information. This article builds on this type of work by investigating how one high-school algebra teacher designs and conducts a lesson on exponential functions. Through a lesson study format she studies with her colleagues how other algebra students have mathematically modeled a bacteria growth problem with no prior formal instruction. Analysis revealed that the teacher was able to use students’ algebraic thinking to structure her class and begin promoting mathematical understanding. The implications for building on students’ conceptions of algebra are explored throughout the paper

Developing Mathematical Thinking: Changing Teachers’ Knowledge and Instruction

In the present research, we evaluated the effectiveness of a multi-year professional development program in mathematics for elementary teachers. Each year the program focused on a different domain of mathematics. We found the program increased teachers’ knowledge of (a) number and operations, (b) measurement and geometry, and (c) probability and statistics. We also examined the relation between mathematical knowledge and teaching practices. Across the three domains neither pretest nor posttest mathematical knowledge were related to classroom teaching practices. However, change in knowledge was positively related to six different dimensions of teaching practice for number and operations, and for measurement and geometry; and was positively related to four or six dimensions for probability and statistics. That is, those teachers with greater changes in knowledge demonstrated more effective instruction

Instructional Learning Teams: A Case Study

Changing teacher practices to improve student learning is a challenge. For teachers’ practices to change, faculties within schools must build communities of practice. However, supporting teachers’ collaborative learning within a Professional Learning Team can be an elusive challenge. We found through the Instructional Learning Team (ILT) model of professional development that teachers have a focused model to make effective changes to their practice. ILTs promote school improvement by providing a process through which teachers collaboratively focus on sustained reflection about student learning tasks, instruction, and student work using the Japanese Lesson Study and critiquing their work using Newmann’s (1996) Intellectual Quality framework. We followed two teams of teachers over a semester and qualitatively examined changes in four elements of professional learning: shared ideas and values, focus on student learning, reflective dialogue, and deprivatization of practice. Through the ILT process all four elements of professional learning communities increased. This process of changing practice through examining instructional tasks, practices and student work has a direct impact on helping teachers move toward implementing the Common Core State Standards (CCSS)

Statewide Mathematics Professional Development: Teacher Knowledge, Self-Efficacy, and Beliefs

We examined the impact of a state mandated K-12 mathematics professional development course on knowledge, self-efficacy and beliefs of nearly 4,000 teachers and administrators. Participants completed the Mathematical Thinking for Instruction course, emphasizing student thinking, problem-solving, and content knowledge specific to mathematics instruction. Inventories utilizing items fromthe Learning Mathematics for Teaching project (2005) measured changes in participants’ Mathematical Knowledge for Teaching (MKT) and an end-of-course self-evaluation enabled analysis of changes in MKT, self-efficacy and beliefs. Statistically significant changes were found in all three variables. This study adds to our understanding of the potential usefulness of mandating professional development as a policy vehicle for influencing educators’ mathematics knowledge and beliefs