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SPÉCIALISTES DE CREBA AIDANT DES STAGIARES, 11/2017





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ORGANIC-WAY MATHEMATICS


WHO ARE WE?

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 8th-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 20th century of product of behaviorism was partly
responsible for the 21st 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 21st 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 21st 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.



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