Standard C.2. Pedagogical Knowledge and Practices for Teaching Mathematics
Well-prepared beginning teachers of mathematics have foundations of pedagogical knowledge, effective and equitable mathematics teaching practices, and positive and productive dispositions toward teaching mathematics to support students’ sense making, understanding, and reasoning.
Teaching mathematics entails not only knowing the mathematics but also knowing how to design and implement rich mathematics-learning experiences that advance students’ mathematical knowledge and proficiencies. Effective teachers are skilled in their use of high-leverage mathematics-teaching practices and use those pedagogical practices to guide both their preparation and enactment of mathematics lessons. The development of these content-focused skills and abilities form the core of work in the preparation of mathematics teachers for the upper elementary grades. In the following section, we elaborate on the knowledge and pedagogical practices specific to the teachers for the upper elementary grades.
Well-prepared beginning teachers of mathematics at the upper elementary level develop pedagogical knowledge and practices to cultivate students’ mathematical proficiency, including such components as conceptual understanding, procedural fluency, problem-solving ability, and facility with the mathematical processes essential for learning. [Elaboration of C.2.2 and C.2.3]
Central to any efforts to deliver high-quality instruction to each and every student is the development of pedagogical knowledge (Grossman, 1990; Shulman, 1986). Whereas Shulman pointed to two categorical groupings of pedagogical knowledge, general pedagogical knowledge and pedagogical content knowledge (PCK), both Ball et al. (2008) and Sowder (2007) made more specific connections to the role of PCK when one considers the unique components and elements essential to teaching mathematics. The well-prepared beginning teacher at the upper elementary level can engage students with the mathematics underlying the standard algorithms that are taught at these grades, including providing effective tools and ways to have students generate procedures themselves through multiple experiences. They help upper elementary students learn to select appropriate tools and use them to engage in mathematical practices such as seeing patterns and structure. Well-prepared beginners also are fully cognizant of the common errors and naive conceptions that emerge from not only these procedures with whole numbers, fractions, and decimals but also from the larger concepts in which they are embedded. The deeper the mathematical background of the well-prepared beginning teacher, the greater her or his potential for showing pedagogical sophistication (Holm & Kajander, 2012). Holm and Kajander suggested that when dividing a whole number by a fraction, the teacher should be able to choose an appropriate problem to clearly illustrate the repeated-subtraction or measurement-division approach, the strongest visual model, the most salient way of discussing the linkages to prior knowledge about division of whole numbers and a discussion of how to interpret the result.
The representations, notation, strategies, and language that are used in the classroom drive upper elementary grades students’ understanding of procedures and concepts. Well-prepared beginning teachers align with conventions for proper notation, (e.g., distinguishing between a multiplication symbol and the variable x) and precise language (e.g., using the verb regroup rather than borrow) so that the message to students across all three grades is consistent, providing a smooth path toward building on prior knowledge in meaningful ways that last. They recognize that rather than teaching rules or shortcuts (e.g., just append a zero at the end of a number when multiplying by 10) that are taught but are applicable for only a short time (or even not very well at all), they can use more effective instructional strategies to support students in identifying patterns and identifying constraints or boundaries of the usage of those rules when they emerge. Discussing boundaries is particularly important in upper elementary grades, where students see that rules they may have learned about whole numbers do not apply to fractions and decimals (e.g., the longer the number, the larger the number) (Karp, Bush, & Dougherty, 2014). Well-prepared beginners also understand that short cuts such as searching for key words are not effective and that, instead, word problems require attention to reading-comprehension strategies. Vignette 5.2 describes a beginning teacher's work with students who were struggling with solving word problems.
Ms. Morgan was working with a small group of third graders who were having trouble solving multiplication word problems. She asked the students to meet to discuss their strategy use.
Nela was asked how she decided to use addition to solve the problem “There are three baskets of apples on the table. Each basket contains six apples. How many apples are there in all?”
Nela responded that she saw the words “in all,” and that meant that the numbers listed in the problem should be added. She arrived at an answer of 9.
For the problem “Each student was given an equal share of stickers. If there are 25 stickers and 4 students, how many stickers will each student receive?” Rory said, “You use the word “each,” and then you know to multiply—so they each get 100 stickers.”
At this point, Ms. Morgan realized that both students were describing the use of an ineffective key-words strategy, possibly learned in previous grades. These rules or shortcuts that may have started in the primary grades were now causing serious issues. As in this case, sometimes students are mistakenly encouraged to skim through a word problem and locate the key words as a strategy to quickly choose an operation to solve the problem and then use the number(s) from the problem to carry out that operation.
Ms. Morgan had seen, in other classrooms, lists of key words that linked particular words with corresponding operations, for example, “each = multiply,” and so on. But as Nela and Rory demonstrated, these words frequently do not accurately indicate the operation that corresponds with the problem. (Also, the key-word strategy cannot be used for problems that have no key words or with multistep problems). Ms. Morgan decided to show the students three word problems with the same key words but that would be successfully solved using different operations to illustrate the pitfalls and limitations of the key-words approach. She then transitioned to an annotation approach in which one student comes to the document camera and uses the suggestions of other students to mark the word problem with highlighting and written comments to identify the important information. She next moved the group to acting out the problems using the data from the annotations with paper plates and counters. The discussion then centered on the actions and how those actions relate to the meaning of the operation selected and how problems can be sorted by their structures.
Well-prepared beginning teachers focus on sense making and reasoning when they prepare students to grasp the full meaning of a problem by comprehending the entire situation and trying to use structures, such as schema, properties of the operations, and representations to come to a reasoned solution.
Also, well-prepared beginners support the learning of each and every student. This approach is particularly important in Multi Tiered Systems of Support (MTSS) such as Response to Intervention (RtI), because students are usually identified for formal special education services starting in the third grade. This process requires the careful assessment of students to pinpoint their strengths and gaps so that instruction and interventions can be targeted, whether for students with disabilities or for students who may be identified as gifted with a high interest in or a talent for mathematics. With reference to emerging multilingual learners, the well-prepared beginning teacher incorporates the appropriate linguistic practices and strategies needed, including home-language connections and relevant academic language and discourse practices to support students when they move to more complex mathematics vocabulary. Instruction builds on relevant contexts and the need to build on students’ lived experiences in and out of the school setting. All this knowledge about, and emphasis on, teaching individual learners precludes the use of curriculum interventions via generic computer programs, basic worksheets, or Internet searches for merely attractive or fun ideas that do not support the development of significant mathematical thinking.
Although all well-prepared beginning teachers strive to align mathematical concepts across the grades, this alignment is particularly crucial for teachers of upper elementary grades who bridge work in primary grades and later work in such courses as Algebra I. A pressing challenge is teaching in ways that support the development of mathematical ideas over time while resisting the practice of teaching only the mathematics that appears in the standards for one’s own grade level. For example, well-prepared beginning teachers of Grade 5 invest in knowing middle level content so that they are positioned to support students’ readiness even when some of those ideas are not well represented in the fifth-grade standards. The idea of continuity of development certainly applies to teaching across upper elementary grades. For example, the responsibility for the use of number lines is represented most strongly in third-grade standards. Well-prepared beginning teachers in Grades 4 and 5 build on the development and use of the number line even though it is not specifically articulated in the standards for their grade levels. In sum, well-prepared beginners have strategic understandings of the trajectory of the representations used and take responsibility for meeting grade-level standards and reinforcing what came before (e.g., the meaning of the equal sign) and what is still to come in later grades (e.g., fifth-graders' more sophisticated use of vertical and horizontal number lines for locating points on a coordinate graph). Finally, teachers of the upper elementary grades likely have some students who need help on early childhood content and some who are ready to learn middle level content.
Well-prepared beginning teachers of mathematics at the upper elementary level effectively use technology tools, physical models, and mathematical representations to build student understanding of the topics at these grade levels. [Elaboration of C.1.6 and C.2.3]
Well-prepared beginners know when to use different manipulatives and various technologies to support students in developing understanding of mathematical concepts and to create opportunities for collective consideration of mathematical ideas such as multiplication, fractions, area, volume, and coordinate geometry. They judiciously select particular representations on the basis of mathematical considerations, knowledge of their students, and other relevant factors. For example, to develop deep understandings of fractions, students must flexibly use three representations: area, linear measurement, and set models. The set model is the most complex of the three representations, so well-prepared beginners begin fractions modeling using area models and linear measurement models that connect to the number line before using the set model. Furthermore, they flexibly and resourcefully think about what representations are available in their current classrooms, schools, and wider communities; they advocate for resources to enhance their abilities to convey mathematical ideas for students to explore and discuss. This consideration of resources might include helping students’ utilize calculators responsibly, giving them access to operations with large numbers and decimals that would be extraordinarily cumbersome to calculate by hand.
Well-prepared beginners also understand that meaning is not inherent in a tool or representation but that it needs to be developed through a combination of exploration, carefully orchestrated experiences, and explicit dialog focused on meaning-making (Ball, 1992). As a result they support students’ developing connections between these representations, attending to links between and among equations, situations, manipulatives, tables, and graphs, using various tools including technology.
Well-prepared beginning teachers of mathematics at the upper elementary level learn to use both formal and informal assessment tools and strategies to gather evidence of students’ mathematical thinking in ways appropriate for young learners, such as the use of observations, interviews, questioning, paper-and-pencil and computer-based tasks, and digital records, including audio and video. [Elaboration of C.3.1]
Well-prepared beginners recognize the many valued mathematical-learning outcomes that need to be assessed. They do not focus on particular outcomes to the detriment of gaining insights into others. For instance, in a unit on geometric measurement, they assess more than students’ application of learned formulas; they also examine outcomes such as students’ understanding of the concept of area, ability to use mathematical tools such as protractors, and attention to precision when they measure the volume of a prism. They seek to assess valued learning outcomes such as engagement in mathematical practices and mathematical dispositions, even when routes to assessing them may not be straightforward.
Well-prepared beginners utilize multiple ways to assess learning outcomes. For instance, when focusing on students’ fluency with multiplication facts, they know that timed tests are not the only, first, or necessarily best approach. They recognize that fluency has several components and that timed tests do not support ability to assess strategy use, efficiency, or flexibility. They recognize that they may get a sense of students’ accuracy, but primarily accuracy under the time pressure of speed. Well-prepared beginning teachers are fully aware of the negative outcomes of timed tests, which include movement away from number sense and mental computation and toward planting a seed for a negative attitude toward the study of mathematics.
Instead of relying on timed tests, the well-prepared beginner appreciates the value of looking at individual performance on assessments to pinpoint the strengths of students who are struggling (e.g., two or more grades below their peers). Using diagnostic interviews and other individualized assessments of students’ thinking, they can find the gaps in foundational knowledge from previous grades as well as position instruction near the point at which students are strong in their understanding. In this way the movement forward is not in fits and leaps (as would be would be with a more gross measure of student performance in a large-scale assessment) but targeted to specific needs and built on sound footing from the learner’s perspective.