Written by Dr. James Carlovsky
Important moments of learning do occur in a classroom. For many classroom teachers, these moments can be difficult to describe. “Skilled teachers often recognize when important mathematical moments occur during a lesson” (Leatham et al., 2015, p. 88). This article intends to shed light on these important moments of learning on behalf of the classroom students and their teachers.
Pedagogical Moves
So, what does a teacher do? These classroom student moments of learning often can lead to a classroom teacher action, also known as a pedagogical move (Jacobs et al., 2010). These moments of learning may lead a classroom teacher to ask a specific question, to revise a portion of a lesson, to refine what a classroom student has expressed, or even to test a classroom student in a formal or informal way (Lesh et al., 2003) So, what do teachers do in the process of student learning? Sherin suggested that “emphasis on understanding the ideas that students offer is one of the hallmarks of mathematics education”(2001, p. 84). A teacher should keep a focus on the student and their thought process.
Learn from History!
So, what do teachers usually do with their math lessons? A common perspective on learning in mathematics in the 20th century was that learning is the accumulation of knowledge, that practice solidifies mastery, and that knowledge is demonstrated by the ability to solve particular classes of problems (Schoenfeld, 2006).
Get Students to Talk!
Walshaw (2008) demonstrated that effective instructional practices demand students’ mathematical talk. Teaching focused solely on the acceptance of all answers and solutions does not strike at the core of what mathematics discourse truly entails. Mathematics teaching cannot be solely algorithmic or a fixation on the number of correct or incorrect answers that a student completes on a given assignment or assessment. Mathematics instruction and mathematics learning are so much more. When a teacher “presses a student to elaborate on an idea, attempts to encourage students to make their reasoning explicit, or follows up on a student’s answer or question with encouragement to think more deeply” (Morrone, et al., 2004, p. 29), the teacher is learning what students actually know and is providing an incentive for them to enrich that knowledge.
Notice What Students Say!
Davies and Walker (2005) focused on how teachers learned to notice significant instances of their students’ knowledge probing. Knowledge probing could be described as grappling with content in a lesson. It is important to note that these teachers no longer focused their attention on lesson delivery, but rather on student interactions within a lesson. Other research is now beginning to probe how teachers question, revise and refine, test their content knowledge, and extend their knowledge to more powerful forms of classroom teaching (Lesh et al., 2003; Lesh & Doerr, 2003) for moments of learning on behalf of the students.
Critical Moments
An article by Thames and Ball (2013, p. 31) mentioned moments of learning as “critical hinge moments. In sizing up a student’s answer, a teacher also has to coordinate hearing a student’s thinking with a sense of whether or not the mathematical point is crucial at the moment.” Some critical moments are opportunities to open up a topic to a student for learning as well as to help the students achieve the objective of a lesson.
What’s Your Goal?
A classroom teacher can have a problem, can get all kinds of interesting things to come up, and can raise an interesting side trip during the lesson. However, more importantly, the teacher must keep an eye on the goals for the lesson and must coordinate decisions with whatever students are doing to get to that goal. This careful eye is the definition of skillful teaching, according to Ball (2005; 2010). Ball (1993) paid attention to the ideas of a student, not dismissing them as incorrect simply because they do not fit into the mathematical curriculum. The National Council of Teachers of Mathematics noted: “Effective teaching involves observing students [and] listening carefully to their ideas and explanations” (2000, p. 19).
In Summary
Researchers have found that teachers’ use of student thinking during mathematics instruction supports student learning of mathematics (Carpenter et al., 2006; Franke et al., 2007; National Council of Teachers of Mathematics, 2014; Van Zoest et al., 2016). Moments of learning do not have much value in the classroom for a teacher unless that classroom teacher notices that student learning or thinking and makes a pedagogical move. In summary, teachers should . . .
- Talk less and get students to talk more
- Notice what students say
- Adapt the lesson (make a pedagogical move)
- Allow student discussion to meet the lesson’s goal
Dr. James Carlovsky (’02, MS Ed ’10) serves as a professor of mathematics and instructional technology at Martin Luther College.
References
Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93(4), 373–397.
Ball, D. L., & Forzani, F. M. (2010). Teaching skillful teaching. Education Leadership, 68(4), 40–45. https://doi.org/10.1136/bmj.2.6037.698-a
Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching. American Educator, (Fall), 14–22. https://doi.org/10.1016/j.cedpsych.2006.02.001
Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (2006). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3–20. https://doi.org/10.2307/749715
Davies, N., & Walker, K. (2005). Learning to notice: One aspect of teachers’ content knowledge in the numeracy classroom. In 28th Conference of Mathematics Education Research Group of Australasia, New Zealand (pp. 273–280). Retrieved from http://www.merga.net.au/documents/RP272005.pdf
Franke, M. L., Kazemi, E., & Battey, D. (2007). Mathematics teaching and classroom practice. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 225–256). Charlotte, NC: Information Age Publishing.
Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202. https://doi.org/10.2307/20720130
Leatham, K. R., Peterson, B. E., Stockero, S. L., & Van Zoest, L. R. (2015). Conceptualizing mathematically significant pedagogical opportunities to build on student thinking. Journal for Research in Mathematics Education, 46(1), 88–124. https://doi.org/10.5951/jresematheduc.46.1.0088
Lesh, R., Cramer, K., Doerr, H. M., Post, T., & Zawojewski, J. (2003). Using a translation model for curriculum development and classroom instruction. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism. Models and modeling perspectives on mathematics problem solving, learning, and teaching. (pp. 449–463). Mahwah, NJ: Lawrence Erlbaum Associates.
Lesh, R., & Doerr, H. M. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 3–33). Mahwah, NJ: Lawrence Erlbaum Associates.
Morrone, A. S., Harkness, S. S., Ambrosio, B. D., & Caulfield, R. (2004). Patterns of instructional discourse that promote the perception of mastery goals in a social constructive mathematics course. Educational Studies in Mathematics, 56(1), 19–38.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics.
Schoenfeld, A. H. (2006). What doesn’t work: The challenge and failure of the what works clearinghouse to conduct meaningful reviews of studies of mathematics curricula. Educational Researcher, 35(2), 13–21.
Sherin, M. G. (2001). Developing a vision of classroom events. In T. Wood, B. S. Nelson, & J. Warfield (Eds.), Beyond classical pedagogy: Teaching elementary school mathematics (pp. 75–93). Hillsdale, NJ: Erlbaum.
Thames, M. H., & Ball, D. L. (2013). Making progress in U.S. mathematics education: Lessons learned—past, present, and future. In K. R. Leatham (Ed.), Vital directions for mathematics education research (pp. 15–44). New York, NY: Springer.
Van Zoest, L. R., Leatham, K. R., Peterson, B. E., & Stockero, S. L. (2016). Conceptualizing the teaching practice of building on student mathematical thinking. In Proceedings of the 38th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1281–1288).
Walshaw, M., & Anthony, G. (2008). The teacher’s role in classroom discourse: A review of recent research into mathematics classrooms. Review of Educational Research, 78(3), 516–551. https://doi.org/10.3102/0034654308320292