ORGANIC-WAY MATHEMATICS

Organic-way Mathematics: The Essence of Problem Solving (OMEPS) is the flagship of Organic-way Mathematics Education. Underpinned by an 8-year doctoral study (Joseph 2020) under the aegis of Concordia University Chicago, OMEPS has captivated teachers and students through several years of informal piloting at public schools in Brooklyn, New York as of 2007 and in the Artibonite Department of Haiti, as of 2017, particularly in Saint-Marc and in Gonaïves. Organic-way Mathematics: The Essence of Problem Solving is based on the developmental constructivism of Jean Piaget blended with the socio-cultural theory of Lev Vygotsky. The confluence of the two constructivist philosophies frames the tenets of our Organic-way education project, which includes Organic-way Mathematics: Profound Conceptual Understanding (MAPCU). In opposition to prominent behaviorists such as John Broadus (J.B.) Watson, Edward Thorndike, and Burrhus Frederic (B.F.) Skinner, constructivists believe in the students’ agency. That is, the students must play an active part in the learning process, as evidenced by these two quotes: “To understand is to discover, or to reconstruct by discovery, and such conditions must be complied with, if in the future, individuals are formed, who are capable of production, and not repetition” (Piaget 1973). “Children construct their knowledge. Development cannot be separated from its social context. Learning can lead to development. Language plays a central role in mental development. Students learn best in the company of others, or the more able persons” (Vygotsky 1978). In practice, the constructivist epistemology engenders constructed-response questions, requiring justifications of answers in contrast to multiple-choice questions. In fact, constructed-response questions compel students to compose their own responses from scratch. On the other hand, despite its limited benefits and popularity among students, multiple-choice questions (MCQ) are proven less effective at leveraging deeper conceptual understanding. For the MCQs leave too much room for students to guess the correct answer, prompting the creation of OMEPS. Organic-way Mathematics: The Essence of Problem Solving (OMEPS) has come to serve as a supplement to any math program. Based on Joseph (2020), the traditional math programs are insufficient to leverage student profound understanding of essential concepts. To fill in the gaps, OMEPs have come to suggest a way for the teacher to create space for a community of learners, including herself, to “do” mathematics, negotiate meanings, and arrive at the most efficient solution. It is worth noting that OMEPS does not introduce mathematical concepts. Rather, it presumes that the concepts embedded in the OMEPS problem have already been covered. In sum, on purpose OMEPS Grades 5-6-7 focuses on a troublesome group of foundational and fundamental concepts (e.g., fraction, percentage, decimal, ratio, and proportion). These concepts are arguably the most troubling to upper-grade students. The lack of mastery of these topics created some profound gaps that are difficult to be filled in later grades. For example, whereas a high-school student may tackle with ease the function f(x) = 5x + 6, she may be mystified by the function f(x) =1/2 x + 5x + 6 due to the presence of the fraction ½ as the slope. That’s why the main intent of OMEPS is to demystify the nature of fractions, among the others, through real-life application, continuous reinforcement, and standards-based assessment. Below are underscored the key characteristics of a typical 90-minute OMEPS lesson.

Characteristics of OMEPS Grades 5, 6, and 7
 Project-based and student-centered learning.
 Real-life application, reinforcement, and assessment of the most foundational and fundamental concepts
—The Gang of Five—Fraction, Decimal, Percentage, Ratio, & Proportion.
 Prototype problems or tasks with differing solution strategies.
 Emphasis on annotated reading of mathematical story problems.
 Logical reasoning, rigor, and flexibility in alignment with any state curriculum and standard.
 Hands-on approach with realia (household items and environmental materials).
 Usage of multiple intelligences (visual, auditory, kinesthetic, etc.)
 Emphasis on small-group cooperative learning and whole-class discussions, and reflection.
 One-to-one support for low-performing students

Below is a striking testimonial highlighting the potency of OMEPS. “Organic Math Organic-way Math is an instructional resource that closely follows Piaget’s constructivist theory. At the beginning of each problem, students find themselves in a state of disequilibrium, thus forcing them to activate their prior knowledge. Organic Math encourages the exploration of problem solving. Each student actively participates in trying to solve the task in his or her unique way. It encompasses cooperative group work, spiral review of previous math skills while still introducing new concepts. The premise is to teach / re-teach multiple math skills that are embedded in the word problem; skills that are not taught in isolation. Organic Math is a flexible tool that can be utilized in the classroom at the teacher’s discretion. It aids in the preparation of State Assessments because it is aligned with the State Standards. Following are some key points of Organic Mathematics:

 Flexibility in using the program in class.
 Spiral Review: All skills are taught in conjunction with one another.
 Kids are allowed to be in disequilibrium, and it is acceptable.
 Teachers are truly only facilitators. Through accountable discourse, students find the
solutions or approximate them.
 Word problems are interesting and connected with ELA (English Language Arts, i.e.,
reading and writing)
 Flexibility and Test Prep.

I conclude by saying that I would choose Organic Math over many other programs, some of which we are currently using officially. Irma Mendez, 6C Classroom Teacher The Bilingual School/PS 189K, Brooklyn, New York.

LEVERAGING STUDENT MATH PROFICIENCY, NOTABLY PROBLEM SOLVING

More students fail in math than in any other subject (Schmoker, 2011). Yet in K-12 America’s classrooms, mathematics is second most valued discipline after English. All the reasons for which students flunk mathematics are not known. One possible cause, though, is that learners cannot find satisfying answers to their perennial question: “Why do I need to know this…?” (Bonnesen & al., May 2021). Unfortunately, based on teachers’ misperceptions of the nature of mathematics, students remain mystified and incompetent, instead of becoming engaged and productive, fueling teachers’ exasperations and parents’ worries.

So, Organic-way Mathematics, a research-based pedagogy crystalizing the author’s ideas over two decades of teaching experience and research combined has come to make a positive difference with a focus on problem solving and profound conceptual understanding by addressing two essential questions:

1. Why do we have to elevate student math problem-solving proficiency?

2. How can we leverage student math achievement?

1. WHY MUST WE ELEVATE STUDENT MATH PROBLEM-SOLVING PROFICIENCY?

Leveraging student math proficiency, principally at problem solving, is of utmost significance. Singham (2005) remarks that student continuous failure in mathematics has a horrific and disproportionate impact on high school graduation rates and college prospects.  The National Council of Teachers of Mathematics (NCTM 2000) sheds lights on why we must elevate student math problem-solving proficiency, arguing that:

“Problem solving is a hallmark of mathematical activity and a major means of developing mathematical knowledge…Problem solving is natural to young children because the world is new to them, and they exhibit curiosity, intelligence, and flexibility as they face new situations…Problem solving is the cornerstone of school mathematics.”

Congenially to NCTM, Leinwand (2000) showcases the preeminence of problem-solving classes by contrasting them to traditional mathematics classes:

If one believes that mathematics is appropriate and necessary for all students, problem solving and applications are the primary goals of the mathematics program; …then, more group work during class would be evident as students collaborate, teachers will rely less on lectures, …and performance task, scored holistically, would take the place of short- answer, percentage-correct test…” 

Other terms that characterize problem solving include constructed-response question, open-ended problem, and inquiry-based problem solving. In adhesion, Bayazit (2013) posits that open-ended problem solving is the most significant cognitive activity in professional and everyday life. Moreover, the inquiry acknowledges that problems are at the heart of cognition for both individual learning and development.

In agreement, Douglas et al. (2012) affirms that open-ended problem solving is a central skill to engineering practice. It is imperative for engineering students to develop expertise in solving these types of problems. Problem solving is so important that transcends most societal and disciplinary boundaries. Additionally, in today’s global job market, mathematical knowledge with critical reasoning and decision making is highly valued. The National Council of Mathematics supports this proposition, stating that quality math instruction is fundamental for a strong economy.

Another reason for leveraging student math proficiency is the long-standing and troubling student underachievement in mathematics. Despite the increased interest in math problem solving by researchers and practitioners, students in general continue to struggle (Krawec & Huang, 2012). For example, there was no significant growth on the New York State math examinations from 2017 to 2019. In fact, 40.2%, 44.5%, and 46.7% of students scored at or above grade level respectively in 2017, 2018, and 2019. Krawec & Huang echoes Bryant & Hammill (2000) in that success in overall math achievement is highly correlated with math problem solving. These views have been triangulated by a growing body of research, including Davis, Maeda, & Kahan (2008).

In California the longitudinal study by Davis et al. (2008) finds that underachieving students routinely skip constructed-response questions while answering the multiple-choice items. According to the quantitative inquiry, 16% of Blacks students and 10% of Hispanics ignore the constructed-response questions. In contrast, typically more achieving, 8% Caucasians and 5% Asians provided “blank responses” to CRQs. Considering this revelation, Inoue & Buczynski (2011) theorizes that teachers’ pedagogical weaknesses were responsible for student struggle with problems that require justifications of answers.

Additionally, the quadrennial Trend International for Mathematics & Science Study (TIMSS 2015) reports that American students are routinely outperformed by international counterparts. Similarly, the triennial Programme Internationale for Student Assessment (PISA 2015) finds that American students held 39th place on the competition centered on open-ended problem solving. Closer to us, the biennial National Assessment of Educational Progress (NAEP 2017) reports no significant progress among 8^th-grade students scoring below the 300-point proficiency threshold.

The conundrum with mathematics has been noted by multiple national watchdogs. In Everybody Counts, the National Research Council (NRC 1989) outlines the challenges facing math educators and suggests that high-quality education in mathematics is essential. “Our national goal must be to make mathematics education the best in the world” (NRC 1989, p. 88). Likewise, in Before It’s Too Late, the National Commission on Mathematics and Science Teaching for the 21st Century (The Glenn Commission 2000), among others, argues that math education is a major problem for America’s ability to compete on the global stage. Glenn was convinced that “the future well-being of our nation and people depends not just on how we educate our children, but on how well we educate them in math and science” (p. 4).

Acknowledging the significance of problem solving is one step in the right direction. But knowing how to apply the power of problem solving to leverage student potential would be a higher level. This lingering and vexing dilemma continues to frustrate math teachers and unnerve school leaders, reeling for best practices.  In the last three decades, to improve mathematics education in New York, multiple curricular reforms have been undertaken. Most memorable were the 1980’s Seven Key Ideas, the 1990’s New Standards, and the 2010’s Common Core Learning Standards (CCLS).

The Common Core Learning Standards, a federally approved alteration of the nationwide Common Core State Standards, was a bit more oriented towards constructed questions responses. Still, CCLS drew the ire of parents and teachers as soon as it was released for having failed to involve those key stakeholders during the conception. Another reason for CCLs rapid demise was a hasty roll-out, leaving teachers unprepared, worried, and baffled.  As a result, in 2020 the New York Next Generation Learning Standards came into being.

The antiquated paradigm of the 20^th century of product of behaviorism was partly responsible for the 21^st century students’ woes at profound understanding. O’Brien & Moss (2004) admits that rote memorization of arithmetic facts, a staple of traditional math, isn’t as important as making sense of math concepts and applying them to the everyday world. But despite the increased interest in math problem solving by researchers and practitioners, students in general continue to struggle (Krawec & Huang, 2012). As a result, business leaders, among other entities, have grown frustrated at the lack of college readiness and career readiness among high-school graduates in this 21^st Century. As Bryant, Bryant, & Hammill (2000) note, success in math achievement is highly correlated with math problem solving.

2. HOW CAN WE LEVERAGE STUDENT MATHEMATICAL PROFICIENCY?

Raising nation-wide student achievement in mathematics appeared an uphill and recurring battle. Numerous researchers hypothesize on how to leverage student mathematical proficiency. Roberts (2016) posits that the concentration on problem solving in the classroom will not only influence students’ thinking and problem-solving skills, but also improve students’ analysis skills and standardized test scores.

Moreover, Bayazit (2013) notes that students who follow problem-based mathematics curricula tend to outperform their counterparts both in mathematical achievement and in problem solving. Additionally, Gullie (2011) finds that instruction around constructed-response questions in earlier grades can predict achievement on standardized exams around open-ended problem solving in subsequent grades. Furthermore, the potency of problem solving is further explained by Douglas et al. (2012), stating that problem solvers usually generate and manage a complex array of possible solution paths.

Teachers’ beliefs dictate what is going to transpire in the math classroom. There is a strong correlation between teachers who believe in in the outdated paradigm (computations, rote memorization, lectures, practice sheets, and “teaching to the test”) and student underachievement.  On the contrary, if teachers view mathematics as a dynamic, problem solving, inquiry-based, and reflective endeavor, among other 21^st century characteristics, students will deepen understanding and long-lasting memory. Consequently, Organic-way Mathematics: The Essence of Problem Solving (OMEPS) has come to be the first book of the Organic-way movement, a pedagogical tool to assist in establishing a culture of mathematical problem solving within a community of learners, regardless of language and grade level.

Conceived in the mid 2000’s and supported by the eight-year case-study research (Joseph 2020), Organic-way Mathematics: The Essence of Problem Solving (OMEPS) has been field-tested informally by the author (Math Coach) in support of teachers at Public School 189 in Brooklyn with remarkable success. This is the feedback provided by a participating teacher upon receiving her students’ scores on the 2007 New York State mathematics examinations:

“Organic Math [ now Organic-way Math] is an instructional resource that closely follows Piaget’s constructivist theory. At the beginning of each problem, students find themselves in a state of disequilibrium, thus forcing them to activate their prior knowledge. Organic Math encourages the exploration of problem solving. Each student actively participates in trying to solve the task in his or her unique way. It encompasses cooperative group work, spiral review of previous math skills while still introducing new concepts. The premise is to teach / re-teach multiple math skills that are embedded in the word problem; skills that are not taught in isolation. Organic Math is a flexible tool that can be utilized in the classroom at the teacher’s discretion. It aids in the preparation of State Assessments because it is aligned with the State Standards. Following are some key points of Organic Mathematics:

• Flexibility in using the program in class.
• Spiral Review: All skills are taught in conjunction with one another.
• Kids are allowed to be in disequilibrium, and it is acceptable.
• Teachers are truly only facilitators. Through accountable discourse, students find the solutions or approximate them.
• Word problems are interesting and connected with ELA (English Language Arts, i.e., reading and writing)
• Flexibility and Test Prep.

I conclude by saying that I would choose Organic Math over many other programs, some of which we are currently using officially.

             Irma Mendez

6C Classroom Teacher

The Bilingual School/PS 189K  

Brooklyn, New York                                                                                              

                                                                                        June 17, 2007

Despite local accolades from teachers of different grades, Ms. Mendez’s testimony could not be scientifically validated. The author was compelled to be enrolled in a doctoral program (2012-2020) culminating in dissertation titled “A Case Study of Mathematics Teachers’ Perceptions of Haitian Students Solving Constructed-Response Questions in New York Middle School.”  This 242-page document pertains to all communities of mathematics learners. To access the full ProQuest peer-reviewed study Joseph (2020), use the link below: https://drive.google.com/file/d/1SWoTjZUIJnGl8npz_y5s04B1apd3DCtL/view

Underpinning Organic-way Mathematics, the qualitative inquiry Joseph (2020) finds multiple barriers and enablers to students, particularly English language learners, challenged by constructed-response questions. For example, the research participants overwhelmingly agree that the lack of reading comprehension was a daunting hindrance, compelling substantial revisions to the original stages of Organic-way Mathematics.

Joseph (2020) was supported by a confluence of constructivist philosophies advanced by such luminaries as Robert Stake, Robert Yin, Sharan Merriam, Jean Piaget, Lev Vygotsky, and Jerome Bruner. Also discussed in the theoretical framework were contrasting epistemological stances of prominent positivists and behaviorists, i.e., Auguste Comte, John Watson, J.B. Skinner, and Edward Thorndike.

The softer ontological positions of Karl Popper and Alfred Bandura appear to astonishingly coalesce with the pragmatism of John Dewey, the critical pedagogy of Paulo Freire, and the Socratic Method for Teaching and Learning (SMTL), among other frameworks.

Most mathematics curricula of competitive countries (e.g., Chinese Taipei, Japan, Singapore, Hong-Kong, South Korea, and Australia) are problem-based, which inspired the development of Organic-way Mathematics: The Essence of Problem Solving (OMEPS). The flagship document of Organic-way Mathematics has, moreover, been influenced by other frameworks comprising Charlotte Danielson (2007) Enhancing Professional Practice: A Framework for Teaching.