The MSLC offers a training program on Core Concepts in Mathematics Teaching and Learning Through the lens of Tutoring
The Core Concepts in Mathematics Teaching and Learning curriculum is research-based, drawing on mathematics education literature. The training uses undergraduate tutoring as the context in which we explore these concepts. NCTM’s (2014) Principles to Action Mathematical Teaching Practices (MTP) are used as a framing for the training. Participants are required to attend eight training sessions, each an hour long. As a whole, training sessions are designed to draw on mathematics education research to inform not only the content of the sessions but also the delivery methods. In addition, NCTM’s (2014) MTP are heavily embedded into the training as an underlying framework of productive practices. Further, as part of the training requirements, tutors are asked to engage with or submit materials through a learning management system. These materials consist of discussion posts, recordings of tutoring sessions and reflections on those sessions, as well as some additional training assignments.
The first training session focuses on the goals of mathematics education. The training focuses on the broad goals of mathematics teaching and learning, as well as the narrower goals of tasks students bring to the tutor room. The broad perspective focused on goals of mathematics education from the perspective of the National Research Council (2001), Schoenfeld (1992), and De Corte et al. (2000). The types of knowledge, skills, and dispositions students need to be mathematically competent are discussed. The narrower goals of the task cover aspects of MTP 1, Establish Mathematics Goals of Learning (NCTM, 2014). While tutors might not have curricular control over goals or be working with students long-term, they can attend to the goals of a particular task. Awareness of the task goals are important in order to communicate those goals to the student, monitor the student’s progress toward the goals, and make in-the-moment decisions. As part of this training, ideas surrounding where the students have been (what they know) and where they are going are discussed. This mathematical knowledge for teaching (Hill, et al., 2004) is dependent on the courses the tutors work with, however, the goal is to give participants resources to better understand the course curriculum as well as develop habits of considering this type of knowledge.
The second training session covers how students learn and types of learning experiences and processes which support development of mathematical competence. This training focuses on De Corte (1995) and NCTM’s (2014) descriptions of the types of learning experiences students should have as well as NCTM’s (2000) mathematical process standards. The descriptions of productive learning experiences and processes drew on several theoretical perspectives, however, the goal of this training is not to delve deep into theory. Rather the goal is to give the tutors a sense of how these theories inform practice. This training draws on the participants’ own experiences as learners and helped them reflect on how the types of experiences and learning environments described by De Corte and NCTM match their own productive learning experiences. In addition, because MTP 6, Build Procedural Fluency from Conceptual Understanding (NCTM, 2014), closely aligns with the goals of mathematics instruction and productive learning experiences, tutor actions which support this practice are discussed.
The third training session focuses on supporting the motivational, affective, and epistemic aspects of learning mathematics as well as supporting students’ productive struggle, MTP 7, Supporting Productive Struggle in Learning Mathematics (NCTM, 2014). Supporting students as they struggle with mathematics is an important aspect of tutoring. Often students lack confidence or may have stopped short of their full effort because they believe they cannot be successful. Tutors need to help students push back against the idea that they are “not a math person”.
The fourth training session extends the ideas of the prior trainings to discuss the role of beliefs and social norms in tutoring. As part of this training, tutors review case studies to examine the roles of the tutor and student. The questions used to frame this discussion draw on video-club professional development activities reported in the mathematics teacher education literature (e.g., Huffer-Ackles, Fuson, & Sherin, 2014; Stockero, 1998; Van Zoest & Stockero, 2008; Van Zoest, Stockero, & Taylor, 2011). In addition, as a post-training activity, participants are asked to reflect on their own tutoring sessions in a similar manner. Also included in this training are aspects of MTP 2, Implement Tasks that Promote Reasoning and Problem Solving (NCTM, 2014). Much of this practice is not applicable to tutoring because the tutors do not select the tasks in tutoring. However, as aspects of the norms of the tutoring session, tutors should support students without taking over their thinking or lowering the cognitive demand of the task while encouraging use of varied strategies (NCTM, 2014, p.24). In addition, MTP 4, Facilitate Meaningful Mathematical Discourse, applies to the norms of the session. While the discourse in a tutoring session may be different from a classroom since there is only one student, the general idea of a dialog and communication of mathematics still applies. After this training session, tutors are asked to listen to recordings of their tutoring sessions and write written reflections comparing their performance at the beginning of the semester to their current performance (about halfway through the semester).
The fifth session continues to build on productive tutoring norms and discourse by discussing MTP 5, Pose Purposeful Questions (NCTM, 2014). The purpose of this training is to give participants the tools to draw out student thinking and elicit student communication. Eliciting student thinking is necessary in order to build on and confirm students’ understandings. In addition, tutors need to develop questioning skills that do more than gather information but also draw attention to connections and encourage students to explain their thinking.
The sixth training session focuses on building on and understanding student thinking. This training covers MTP 8, Elicit and Use Evidence of Student Thinking (NCTM, 2014). Teuscher, Moore, and Carlson (2016) discussed the importance of decentering, or trying to take the student’s perspective to understand the thinking behind their mathematics. It is not enough just to attempt to understand the student’s thinking when they’re incorrect, but it’s also important to verify the student’s understanding when they give a correct answer. Further, the purpose of eliciting student thinking is to work with and build on the student’s understandings in order to focus their thinking rather than funnel it down a pre-determined path (Herbel-Eisenmann & Breyfogle, 2007; Wood, 1998).
The seventh session focuses on use of representations, MTP 3, Use and Connect Mathematical Representations (NCTM, 2014), as a means of mathematical sense-making. In particular, use of representations both with real-world problem contexts and non-context tasks are discussed. As part of this session, case study analysis similar to training session four were employed as a training method. Several cases were presented with each demonstrating various degrees of effective use of representations by the tutor. Tutors were asked to reflect on their own use of representations in their mathematics learning as well as discuss the quality of use of representations in the case studies.
The eighth and final training session covers cognitive apprenticeship. This training occurrs last because while it combined some of the general ideas of the Mathematics Teaching Practices (NCTM, 2014), it did not directly reflect the practices. However, the ideas of cognitive apprenticeship are particularly relevant to the tutoring situation. First, in a tutoring session, there are no other students to contribute mathematical ideas to move the mathematics forward. While tutors should encourage productive struggle, there will be times when students reach a point of unproductive struggle. In addition, cognitive apprenticeship (Collins et al., 1989) stems from research in which a more capable other is assisting a learner, often a single learner, similar to the tutoring context. Further, cognitive apprenticeship is closely connected to Schoenfeld’s (1983; 1985) work in problem solving. The tutoring situation is essentially problem solving as students have reached an impasse on a task. Ideas from cognitive apprenticeship can give tutors additional strategies for building on student understandings in tutoring sessions. This final training session also utilizes case studies for participants to examine how cognitive apprenticeship practices look in action.
As a culminating activity, participants are asked to again listen to recordings of their tutoring sessions and reflect on their performance. In particular, they are asked to examine one of their most recent sessions, answering several prompts. They then compared their initial and final recordings and reflect on how their behaviors have changed.
Collins, A., Brown, J. S., & Newman, S. E. (1989). Cognitive apprenticeship: Teaching the crafts of reading, writing, and mathematics. In L. B. Resnick (Ed.), Knowing, learning and instruction: Essays in honor of Robert Glaser (pp. 453-494). Lawrence Erlbaum Associates.
De Corte, E. (1995). Fostering cognitive growth A perspective from research on mathematics learning and instruction. Educational Psychologist, 30(1), 37-46.
De Corte, E., Verschaffel, L., & Op't Eynde, P. (2000). Self-regulation: A characteristics and a goal of mathematics education. In M. Boekaerts, P. R. Pintrich, & M. Zeidner, Handbook of self-regulation (pp. 687-729). San Diego, CA: Academic Press.
Herbel-Eisenmann, B. A., & Breyfogle, M. L. (2005). Questioning our patterns of questioning. Mathematics Teaching in the Middle School, 10(9), 484-489.
Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers' mathematics knowledge for teaching. The Elementary School Journal, 105(1), 11-30.
Hufferd-Ackles, K., Fuson, K. C., & Sherin, M. G. (2004). Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education, 35(2), 81-116.
National Council of Teachers of Mathematics (NCTM). (2014). Principles to action. Reston, VA: NCTM.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM: National Council of Teachers of Mathematics.
National Research Council. (2001). Adding it up: Helping children learn mathematics. (J. Kilpatrick, J. Swafford, & B. Findell, Eds.) Washington, DC: National Academy Press.
Schoenfeld, A. H. (1983). Beyond purely cognitive: Belief systems, social cognitions and metacognitions as driving forces in intellectual performance. Cognitive Science, 7, 329-363.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press, INC.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan.
Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (2011). Situating the study of teacher noticing. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics Teacher Noticing (pp. 3-13). New York: Routledge.
Stockero, S. L. (2008). Using a video-based curriculum to develop a reflective stance in prospective mathematics teachers. Journal of Mathematics Teacher Education, 11, 373-394.
Teuscher, D., Moore, K. C., & Carlson, M. P. (2016). Decentering: A construct to analyze and explain teacher actions as they relate to student thinking. Journal of Mathematics Teacher Education, 19, 433-456.
Van Zoest, L. R., & Stockero, S. L. (2008). Synergistic scaffolds as a means to support preservice teacher learning. Teaching and Teacher Education, 24(8), 2038-2048.
Van Zoest, L. R., Stockero, S. L., & Taylor, C. E. (2012). The durability of professional and sociomathematical norms intentionally fostered in an early pedagogy course. Journal of Mathematics Teacher Education, 15, 293-315.
Wood, T. (1998). Alternative patterns of communication in mathematics classes: Funneling or focusing? In H. Steinbring, M. G. Bartolini Bussi, & A. Sierpinska (Eds.), Language and Communication in the Mathematics Classroom (pp. 167-178). Reston, VA: National Council of Teacher of Mathematics.