New Tech Network (NTN) believes all students can learn at high levels and achieve strongly in mathematics putting them on a path toward college readiness. Mathematics can often be the discipline that schools find most challenging to engage student and adult learners. Pressure from states and districts to perform well on standardized assessments, teacher attitudes towards mathematics, and inconsistent sharing of best practices can hinder student achievement in a traditional environment, let alone a ProjectBased Learning (PBL) environment.
This white paper will serve as a description and a suggested approach to mathematics in a NTN environment which blends 21st Century Skills with PBL.
Hallmarks of a Successful Math Environment
Hallmarks of a quality NTN elementary math experience include the following:
● Consistent student mathematical discourse
● Students steeped in ProblemSolving
● High levels of student achievement
● High levels of self-worth and confidence in math
● Deep conceptual understanding of the crucial standards for each grade level
Commitment to these areas in a classroom require significant attention and professional development. Some of these elements represent a shift in how teachers teach and respond to students. Too, the mindset students have about math and about themselves as mathematicians is as important as the rigorous math instruction itself (Boaler 2015).
PBL and PrBL: A TwoFold Approach
These hallmarks can be achieved through the strategic deployment of PBL and supporting ProblemBased Learning (PrBL) experiences. The shift toward PBL and PrBL as the primary instructional modes in the classroom offers opportunities to make math meaningful for students as well as challenges of which to be mindful. 2
PBL: Math authentically embedded in the project
Many PBL Units naturally offer an inroads to mathematical standards. Students are encouraged to discuss and write how the math is crucial to the final outcome of the project. Final products can often readily include equations, written explanations, tables, graphs, data, and other evidence of mathematical investigations. These products can give math meaning to students and allow for them to practice and understand mathematical modeling. Such PBL investigations also may play a crucial role in engagement by offering a handson application of math.
PrBL: Targeted, standalone math (or Math exploration time)
In addition to the math experienced within a PBL Unit, NTN also expects students to experience math in a more targeted, standalone (apart from the project) setting as well. While math in the project is often gives an impetus for application and modeling, projects are often insufficient in giving students access to math on a daily basis. While a student may calculate the cost of playground equipment, it is crucial that the student has more opportunity to practice mathematics in different contexts (both realworld and abstract) as well as test out mathematical ideas (such as, “are prime numbers always odd?”, “is an even number times an odd number always even?”).
For this reason, NTN suggests including standalone math time every day to experience the discipline of mathematics. We shall refer to the pedagogy of these activities as ProblemBased Learning (PrBL). This time may include the following, for examples:
● Number Talks
● ProblemBased lessons
● Skills Practice
● Differentiation and Enrichment
● Math Stations
Each of these activities can be rich math experiences resulting in better comprehension as well as more positive attitudes towards mathematics. Regardless of the activity, it should promote or enhance conceptual understanding, allow for and encourage student discourse, and demonstrations of learning.
Specific crucial skills
Identifying Crucial Skills and Concepts
As a school, it is critical that a conversation around standards and ideas takes place. While the Common Core State Standards provide a useful framework for districts, particular attention ought to be given to the students’ learning of the standards. This might require a schoolor districtwide conversation around math and constant monitoring of what is being taught and what students are learning. Identifying “big ideas” as a staff (i.e. Charles 2005) may help.
NTN is in the unique position of supporting both elementary and secondary schools. This allows us to synthesize certain math concepts that serve as the building blocks for Middle and High School readiness. We offer a few potential important cognitive and noncognitive skills and concepts to support here:
Collaborative problemsolving
Attending to problemsolving skills requires a longterm approach for teachers and schools and must incorporate challenging, nonroutine problems for teachers to facilitate with students (NCTM 2010). The solution to these problems may not be readily apparent to the student, but all 4
students should be able to access some (if not all) the math concepts involved. Tasks such as Jo Boaler’s “low floor, high ceiling” tasks (Boaler 2015) are excellent models of problems that are complex and necessitate flexible thinking, while being accessible for all students.
Too, students learn better when they have the opportunity to voice their own mathematical ideas and listen to mathematical ideas of others: equitable groupwork can lead to deeper conceptual understanding and sensemaking (Mulryan 1994). It is expected that students have the opportunity to work in groups consistently (if not entirely) throughout the year.
Number Sense and Decomposition
Of particular interest is students’ ability to decompose and recompose numbers towards an arithmetic problem. For instance, given the addition problem of 38 + 17, a student might calculate the answer by decomposing the numbers into parts: 30 + 10 + 8 + 7 = 40 + 15 = 55. Similarly, a student might decompose a multiplication problem. Given a the arithmetic problem of 23 x 12, a student might think of this as (20 + 3) x 12 = 240 + 36 = 276. The ability to decompose numbers into parts put them on a path towards readiness for Algebra, while allowing students creativity and agency as they design their own solutions towards arithmetic answers. These methods ought to be explicitly modeled by the instructor, investigated by students, and encouraged alongside the standard algorithm (Fuson and Beckman 2012, Kilpatrick et al 2005).
Ratio and Proportion and, relatedly, fractions
Understanding fractions as part of a whole (as well as the arithmetic manipulation of fractions) is often a difficult concept for students to fully grasp, yet it is crucial for students’ eventual treatment of algebraic expressions. Students should be able to understand, among other things,
that a fraction describes the division of a whole into equal parts, fractions are akin to percentages and/or decimals, and operations on and with fractions.
Systems for adult learning
Complicating matters at the elementary level is that teachers may be asked to instruct using a significantly different pedagogy and different methods (such as decomposition over the standard algorithm). Significant learning and professional development ought to be offered for teachers to hone their craft as it relates to mathematics instruction and promoting student discourse. NTN provides support in terms of general instructional coaching, but it might behoove a school or district to partake in generalized learning around math.
NTN believes a combination of authentically embedded math in a PBL Unit along with targeted supported time for mathematical investigations, content support, and sensemaking paves the way for the aforementioned hallmarks of successful math environment. By providing a rigorous 5
and exciting instructional experience, classrooms can be dynamic environments that will allow students to achieve in mathematics at the highest levels and putting them on the path for continued success for their secondary and postsecondary careers.
References
Boaler 2015. Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching
______ 2015. www.youcubed.org/grade/lowfloorhighceiling/
Charles, Randal (2005).
http://www.authenticeducation.org/bigideas/sample_units/math_samples/BigIdeas_NCSM_Spr0 5v7.pdf
Kilpatrick et al. (2005), Adding It Up: Helping Children Learn Mathematics. Edited by Jeremy Kilpatrick, Jane Swafford, and Bradford Findell. Washington, D.C.: National Academy Press.
Fuson and Beckman (2012) https://www.mathedleadership.org/docs/resources/journals/NCSMJournal_ST_Algorithms_Fuso n_Beckmann.pdf
Mulryan, C. M. (1994). Perceptions of Intermediate Students’ Cooperative SmallGroup Work in Mathematics. The Journal of Educational Research, 87(5), 280291
NCTM (2010).Why is Teaching With Problem Solving Important to Student Learning? http://www.nctm.org/uploadedFiles/Research_and_Advocacy/research_brief_and_clips/Researc h_brief_14__Problem_Solving.pdf?%20Target=