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When understanding breaks down in mathematics, students feel
as though they're swimming in a sea of incomprehension.
Because this drowning sensation is so anxiety-provoking, they
avoid situations that give rise to it. This leads to more anxiety
and a vicious cycle begins. To make sure students don't fall
into this incomprehension sea, you must make sure that
they understand what they're doing at each stage.
--Brian Butterworth, author of
What Counts: How Every Brain Is Hardwired For Math
According to Jean Piaget's Period of Mental Development, we as learners, do not reach the abstract stage of thought until we are approximately 15 years old. In regard to being a mathematical learner, this is a crucial factor
(pun intended) for two reasons.
First, mathematics is a foreign language that many teach as if it is a native tongue. Without the aid of concrete representations and hands-on, minds-on engagements the desired level of understanding is unattained. To a learner
with even a moderate grasp of mathematical language, concepts and skills
oftentimes appear abstract (think algebra, geometrical planes, or even subtraction with regrouping). Even though the concept/skill may in theory seem simple, to the
"foreign-language learner" it appears complex.
Second, with the fervent call for students to become "masters of mathematical critical thinking," this mastery is
quite often not taking place. Why? Unfortunately many educators have not been adequately exposed
or trained in how to incorporate best practices that cause students to engage in quality
discussion times and justifying one's won thinking and work. During collaborative sessions, students must be taught how to verbally and symbolically verify and validate their reasoning, conjectures, and conclusions. The better students are at verbally explaining concepts, skills, and processes, the better they will be at internalizing their
learning.
These two factors are well defined and addressed within the National Council of Teachers of Mathematics (NCTM) Standards. In April 2001, the NCTM revised standards were announced. (To view the
standards, visit www.nctm.org/standards.)
The 10 standards are divided into two sections: five
content-specific standards and five process-based standards. Most
teachers have a comfort level with what they are to teach (content
skills), but not so with creating a process-oriented, inquiry-based learning environment that allows, and demands, student-driven explorations and explanations.
An interesting study revealed in USA Today, March 2003, that:
U.S. math teachers give homework more often than their Japanese counterparts, spend more time on review, and assigned more problems using basic math procedures. But the Japanese do much better in math. Go figure.
Why, of course, becomes the question. The study video recorded 638 math lessons (majority of the lessons were filmed in 1999) from classrooms in Japan, Australia, Hong Kong, Switzerland, the Netherlands, the Czech Republic, and the United States. What was interesting
in the analysis was that the two top performing countries--Hong Kong and
Japan--had very different methods for conducting their lessons. Hong Kong
consistently obtains superior results by stressing basic skills and formulas, while Japan's superiority comes from stressing how math concepts relate to one another.
The report stated in conclusion: If a classroom, school, or district incorporates
both of these best practices success is not far away. What is
thrilling
to me is that their recommended combination of both practices is
exactly what NCTM has been recommending and stressing via their standards for over a decade!
If your school or district's math scores are not acceptable,
or you would like to see them improve, there may be two other
factors to consider. One is the demand for students to
think not just regurgitate math facts. A second is the need for students to be literary
math writers. Students must now know how to explain
their thoughts in sentence form as well as in numeration form and
include the hows and whys of their thought processes.
My desire is that your school or district's mathematical abilities, classroom practices, and test scores show continuous progress and sustained gains. Proper
training, coaching, and mentoring are essential for success. Please click on the link or links below that best meet your current mathematical needs or
concerns.
Critical Thinking and Mathematics
The National Council of Teachers of Mathematics (NCTM)
include five processing standards:
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Problem Solving-Investigating, formulating, developing, and applying strategies to a wide variety of problems; verifying and interpreting results and building confidence in using mathematics in a meaningful way (real-life applications)
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Reasoning and Proof-Drawing logical conclusions through justifying using models, known facts, and relationships to explain thinking; discovering that real-life math requires critical thought, informal though, conjecturing, and validation
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Communication-Relating physical materials, pictures, diagrams, and verbal language to mathematical ideas; reflecting visually (symbols and text) and verbally (oral discussions) to mathematical environments as well as clarifying positions
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Connections-Creating links and relationships between procedural math knowledge
(Math class) and conceptual math knowledge (across the curriculum); learning to construct mental bridges between the concrete and abstract worlds of mathematical thought via engaging mathematical activities
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Representations-Creating foundational
understanding of the content/skill standards that have been
introduced, explored and mastered via illustrations and
written text that represent a variety of mathematical concepts
and theories.
There is an art to presenting math lessons that encompass
both the content skills and these critical-thinking process standards. When I work with educators, I like to use the analogy of a spider's web in relationship to how the standards should be taught.
The process standards
are to be woven into, through, and around the content
standards. This necessary interweaving holds true for the youngest and oldest of learners (NCTM 2001).
Fortunately, this research-based, standards-integration practice
is starting to become more commonplace in classrooms. Proof to the pudding is that many
purchasable math curricula are designed with hands-on
engagement in mind.
When process-based curriculum has been purchased, it does not mean that the best practices for using the curriculum is automatically taking place.
See the Professional Development Chart,
Teacher Training and
Coaching. Educators of all learner levels need quality training, coaching, and mentoring experiences that allow them time to explore, understand, and implement these best practices.
My desire is to help your students excel in their math abilities as
your teachers become stronger in their abilities to teach mathematics. Please feel free to contact me at anytime by E-mail or by telephone
to discuss your needs and/or goals.
Number Sense
Obtaining clear and concrete number sense concepts are the base foundation of mathematics, similar to the concrete foundation of a house. Unless the foundation is solid and in place, the house built upon it will crack and be in need of constant repair. Using this analogy, think of your students. How often do they struggle with the
"basic" mathematical concepts? Oftentimes,
educators feel they are constantly "repairing" rather than "building!"
The reason for this phenomenon is that many
educators have not been properly trained in the essential need to concentrate on the mathematical foundation of making "sense" of numbers. NCTM refers to this as Number Sense and Numeration. Much research has gone into how to best improve students' mathematical learning and has clearly shown that
Number Sense is Number One! Especially in the primary grades, following a specific pattern of introduction and utilizing specific assessment and evaluation strategies
aids educators to better serve their students needs. Many times students fall through the cracks in their foundational
learning because a sequential format of introduction and learning
is not followed. I, and a fellow colleague, Theresa Hoppingarner, have created a developmental approach to introducing the basic number and numeration concepts. When schools and districts
embrace a Number Sense First! philosophy they make marked improvements in their students' mathematical abilities in the classroom and on tests. If you would like to explore this concept in more depth, please contact me via my E-mail address or
please give me a call.
Aligning Math Standards
One of the biggest frustrations I encounter when working with educators in the area of mathematics is the inconsistency of content- and process-learning expectations both vertically (across grade levels) and horizontally (within a grade level or discipline). By this I mean that most grade levels are provided a district- or state-created scope and sequence or create one themselves. If
one refers to a scope and sequence as a planned vacation itinerary,
reality will show that when the end of the vacation arrives (the end of the calendar school year) each teacher will have had quite different trips! Likewise, when students move from grade level to grade level they quite often experience totally different planned and actual
trip itineraries! Educators must be provided with the professional time needed to create a consortium of conversation (meeting both horizontally and vertically) to establish: a)
What standards are most important from the scope and sequence or
state guidelines (Prower Standards, Ainsworth, Larry,
2003). It has been mathematically proven that if students have to learn every standard and benchmark listed they would have to attend school for 22 years!
(Marzano, 2002); b)
Once the proven skills/content/processes are determined, backwards mapping
needs to take place. This mapping technique causes discussion concerning
what prerequisites are needed before new levels of understanding
can be attempted; and c) Once backward mapping is completed and approved horizontally and vertically, strategies and techniques
can be continually shared and adopted to assure student mastery. (Much of the collaborative concept I just shared is explained in more detail in the
Curriculum Mapping and
Assessment and Evaluation links.)
If you have a desire for your school or district to work on a mathematical (or other subject area) alignment, please feel free to contact me at anytime via telephone or
E-mail.
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