Brownell (2004/1947) argued for the teaching of meaningful mathematics. Mathematics that is taught relationally and with understanding underlies this type mathematical instruction (Brownell, 2004/1947). Battista (2002/1999) provided evidence that meaningful mathematics instruction can take place in an inquiry based classroom and that counter to skeptics from Brownell’s time, teaching conceptually may even require less time for the same or better student achievement results. Jacobson and Lehrer (2002/2000) add to these results by relating geometry teachers’ content and pedagogical knowledge to student achievement in an inquiry based classroom. In fact, Pesek and Kirshner (2002/2000) described how the teaching of concepts instrumentally before relationally can lead to many interferences with learning. Although many of the ideas from Brownell’s (2004/1947) time on how learning should take place in a mathematical classroom have not changed, these results are becoming experimentally verified in recent research.

Many studies of the day look much deeper than test results through random sampling of students. Studies such as Watson and Moritz (2002/2000) assess student understanding of a concept and categorize these results in ways that can foster better instruction. Watson and Moritz (2002/2000) looked at a particular concept, statistical sampling, to categorize student understanding into levels that were then subcategorized. Research assessing students’ understanding of a concept is important for proper curriculum alignment and standards trajectory (Watson & Moritz, 2002/2000). These categories of understanding also help teachers and curriculum developers create meaningful tasks that reach students in their zones of proximal development and move them to new areas of enlightenment.

With strong emphasis in recent years on standards and assessment, many teachers have been pulled towards teaching instrumentally rather than conceptually (Pesek & Kirshner, 2002/2000); however, this discussion was even highlighted before this time (Brownell, 2004/1947). All students, young and old, and even non-students look for shortcuts and easy methods to solve problems (Pesek & Kirshner, 2002/2000). Instrumental teaching fosters these shortcut methods that students and parents enjoy, but interfere with understanding (Pesek & Kirshner, 2002/2000). Many students who “enjoy” this “luxury” of mathematical instruction perform algorithms and apply formulas haphazardly, without thought, and many times outside applicable context (Pesek & Kirshner, 2002/2000). Students receiving instrumental instruction are also much less likely to understand concepts such as area outside the sphere of the mathematical classroom to real life applications (Pesek & Kirshner, 2002/2000).

Jacobs (2002/1998) discussed possible outcomes for teaching for understanding through invented arithmetic algorithms rather than traditional algorithms. Jacob’s (2002/1998) suggest that students gain understanding of place value and become better problem solvers when given the opportunity to build their own algorithms for operations with whole numbers. These findings do not suggest that these student invented algorithms are more efficient routes for computation (Jacob, 2002/1998), but as Pesek and Kirshner (2002/2000) suggest that the implications for allowing students time to gain relational understanding increases retention and ability to interconnect concepts (Brownell, 2004/1947).

Lampert (2004/1990) offered an environment for educating students very similar to the preceding studies; however, her environment focused on the creation of a mathematical community similar to mathematicians. This community fostered collaboration and questioning, but at the same time required confidence and courage to share solutions (Lampert, 2004/1990). Lampert (2004/1990) had to purposefully create this environment by entertaining actions by students and herself that give students wording that would allow for this community to succeed. Students were allowed to give solutions and explain reasoning while at the same time allowing modification of thought processes based on other students’ responses. Lampert’s (2004/1990) study focused on this environment and the creation of an activity to build laws of exponents when multiplying like bases with findings suggesting that it is possible for students, particularly in elementary grades, to engage in reasoning and sense making with argumentation.

The culture of a classroom is largely created by the teacher; however, there are external forces that do influence how an environment is embraced and in essence is effective. Some of these may be administration policies, school/system pedagogical emphasis, and students past classroom experiences. There are also times when a deeper look into student misconceptions is required outside regular intervention and pedagogical techniques because these misconceptions may be very unique to a student’s language proficiency or auditory ability (Montis, 2002/2000). A large part of all of these research articles however hinge in teachers’ ability to create meaningful and engaging tasks. Moving the emphasis from a teacher being the giver and assessor of all knowledge to an environment where students make “mathematics” meaningful to themselves and question the ideas of others will increase retention and achievement of students. These research articles highlight the mathematical practice standards within the Common Core State Standards for Mathematics.

Battista, M. (2002). Learning in an inquiry-based classroom: Fifth graders’ enumeration of cubes in 3d arrays. In J. Sowder & B. Schappelle (Eds.), *Lessons learned from research* (pp. 75-83). Reston, VA: National Council of Teachers of Mathematics. (Reprinted from Journal for Research in Mathematics Education, 30, pp. 417-448, by M. Battista, 1999)

Brownell, W. A. (2004). The place of meaning in the teaching of arithmetic. In Carpenter, J. A. Dossey & J. L. Koehler (Eds.). *Classics in mathematics education research *(pp. 9-14). Reston, VA: National Council of Teachers of Mathematics. (Reprinted from *Elementary School Journal*, 47, pp. 256-265, by W. Brownell, 1947)

Jacobson, C. & Lehrer, R. (2002). Teacher appropriation and student learning of geometry through design. In J. Sowder & B. Schappelle (Eds.), *Lessons learned from research* (pp. 85-91). Reston, VA: National Council of Teachers of Mathematics. (Reprinted from *Journal for Research in Mathematics Education*, 31, pp. 71-88, by C. Jacobson & R. Lehrer, 2000)

Jacobs, V. (2002). A longitudinal study of invention and understanding: Children’s multidigit addition and subtraction. In J. Sowder & B. Schappelle (Eds.), *Lessons learned from research* (pp. 93-100). Reston, VA: National Council of Teachers of Mathematics. (Reprinted from Journal for Research in Mathematics Education, 29, pp. 3-20, by V. Jacobs, 1998)

Lampert, M. (2004). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. In Carpenter, J. A. Dossey & J. L. Koehler (Eds.). *Classics in mathematics education research *(pp. 153-171). Reston, VA: National Council of Teachers of Mathematics. (Reprinted from *American Educational Research Journal*, 27, pp. 29-63, by M. Lampert, 1990)

Montis, K. (2002). When a student perpetually struggles. In J. Sowder & B. Schappelle (Eds.), *Lessons learned from research* (pp. 125-128). Reston, VA: National Council of Teachers of Mathematics. (Reprinted from *Journal for Research in Mathematics Education, *31, pp. 541-556, by K. Montis, 2000)

Pesek, D. & Kirshner, D. (2002). Interference of instrumental instruction in subsequent relational learning. In J. Sowder & B. Schappelle (Eds.), *Lessons learned from research* (pp. 101-107). Reston, VA: National Council of Teachers of Mathematics. (Reprinted from *Journal for Research in Mathematics Education*, 31, pp. 524-540, by D. Pesek & D. Kirshner, 2000)

Watson, J. & Moritz, J. (2002). Developing concepts of sampling for statistical literacy. In J. Sowder & B. Schappelle (Eds.), *Lessons learned from research* (pp. 117-124). Reston, VA: National Council of Teachers of Mathematics. (Reprinted from *Journal for Research in Mathematics Education*, 31, pp. 44-70, by J. Watson & J. Moritz, 2000)