Angela T. Barlow

Problem Contexts for Thinking about Equality: An Additional Resource.

Stop and think for a moment. If these students were given the equation 5 + 6 = ___ + 4, would they put a 7 in the blank? Probably not. Th eir comments demonstrate what is referred to as an operational view of the equal sign (Carpenter, Franke, & Levi, 2003). Students who view the equal sign operationally believe that the symbol indicates that they should be performing an operation. Th us, these students would most likely fi ll in the blank with an 11 (the sum of 5 and 6) or a 15 (the sum of all numbers in the equation). In contrast, students with a relational view of the equal sign recognize the relationship that the symbol represents between the two sides of the equation (Carpenter et al., 2003). Students with a relational understanding will state that the equal sign indicates that one side of the equation is the same as the other side. A student with a relational view would know to put a 7 in the blank in order to achieve the equal relationship, resulting in the same amount on each side of the symbol. It has been well-documented that many students do not understand the meaning of the equal sign (Carpenter et al., 2003; Faulkner, Levi, & Carpenter, 1999; Knuth, Alibali, Hattikudur, McNeil, & Stephens, 2008; Knuth, Stephens, McNeil, & Alibali, 2006; Molina & Ambrose, 2006). Th us, researchers have called for instruction that specifi cally addresses such misconceptions (Carpenter et al., 2003; Faulkner et al., 1999), and have indicated that such work must start at the elementary level (Knuth, Alibali, et al., 2008). In response to these recommendations, some state curriculums require that students gain a full understanding of the equal sign. For example, the state of Mississippi has the following objective included in its 3rd-grade curriculum: “Create models for the concept of equality, recognizing that the equal sign (=) denotes equivalent terms such that 4 + 3 = 7, 4 + 3 = 6 + 1 or 7 = 5 + 2” (Mississippi Department of Education, 2007, p. 26). Similarly, Connecticut’s 2nd-grade curriculum states, “Demonstrate an understanding of equivalence or balance of sets using objects, models, diagrams, numbers whole number [sic] relationships (operations) and the equals sign, e.g. 2 + 3 = 5 is the same as 5 = 2 + 3 and the same as 4 + 1 = 5” (Connecticut State Department of Education, 2007, p. 20). In addition, the Common Core Standards explicitly state that students in 1st grade need to gain an understanding of the equal sign (Common Core State Standards Initiative, 2010). In response to curriculum requirements and the research recommendations, mathematics educators have sought to design tasks or experiences that will aid students in developing an accurate understanding of the equal sign. As we were planning instruction for our classroom, a review of the available resources revealed tasks that fell into two An Additional Resource

Exploring Pedagogical Content Knowledge of Biology Graduate Teaching Assistants Through Their Participation in Lesson Study

ABSTRACT Graduate teaching assistants (GTAs) are responsible for teaching the majority of biology undergraduate laboratory sections, although many feel underprepared to do so. This study explored the impact of biology GTA participation in a professional development model known as lesson study. Using a case study methodology with multiple qualitative data sources, this study found that lesson study was beneficial for this group of GTAs in that it modified critical aspects of their beliefs about biology instruction. Each participant felt that lesson study helped revise their teaching and changes were seen in some aspects of the participants’ Pedagogical Content Knowledge (PCK). Despite this, there was an observed disconnect between participants’ vocalized intent and classroom practice. This disconnect could be attributed to the difficulty of implementing new strategies, the short duration of the lesson study, and the instructional inexperience of the participants in the study.

But what if the goal is to model with mathematics

The Getting to School task demonstrates students’ need to engage in three decision-making processes as means for authentic engagement in modeling.

Inspection-worthy mistakes: Which? And why?

Carefully select and leverage student errors for whole-class discussions to benefit the learning of all.

Transitioning from practicing teacher to teacher leader: a case study

ABSTRACT Given US students’ lack of international competitiveness in mathematics and science, many states have adopted the Common Core State Standards for Mathematics (CCSSM). To help ensure that the Standards for Mathematical Practice within the CCSSM and effective teaching practices are implemented appropriately, many teachers are emerging as teacher leaders with the intent of supporting teachers with this implementation. Unfortunately, not all of these teachers are necessarily prepared in the field of teacher leadership. Recognizing the importance of having well-prepared teacher leaders, this article describes the case of Ms. Hodges, as she transitioned from mathematics teacher to teacher leader. Data revealed that several factors affected Ms. Hodges’ transition, some positively and others negatively. These results as well as implications for teacher leader development are provided.

Examining Mistakes to Shift Student Thinking.

teachers and students can have different perspectives on the value of discussing students’ mathematical mistakes, despite various classroom evidence that such discussions can help foster strong conceptual understanding (Boaler 2016). Some teachers consider student mistakes to be an opportunity to correct errors in individual student thinking. Others view the public inspection of mistakes as an opportunity for all students in the classroom to learn. Examining Mistakes to Shift Student Thinking

Developing Algebraic Reasoning through Variation in the U.S.

Historically, algebra in the U.S. has been viewed “as a gatekeeper to a college education and the careers such education affords” (Kilpatrick & Izsak, 2008, p. 11). As such, current curriculum documents emphasize the need to support all students in learning algebra (Common Core State Standards Initiative [CCSSI], 2010; National Council of Teachers of Mathematics [NCTM], 1989, 2000).

Matches or Discrepancies

Efforts to pursue high-quality mathematics teaching have led to ever-increasing research interests in exploring the practices of mathematics classrooms in highachieving countries, including China. There have been studies on how Chinese students learn mathematics (Fan, Wong, Cai, & Li, 2004), how Chinese teachers teach mathematics (Li & Huang, 2012), and the characteristics of effective mathematics teaching valued by Chinese teachers (Cai & Wang, 2010; Huang, Li, & He, 2010; Li, 2011).