DR: Many teachers are going toward the Core
Curriculum State Standards in Math (CCSSM) a little bit skeptical- wondering how
is this going to be any different from the standards we had in the past?
Mahesh: The main difference between these standards
and most of the previous state standards and frameworks is that the CCSSM are
mathematically sound- they have focus, rigor, and coherence. They are aimed at helping develop student’s
mathematical way of thinking. They
should serve as the foundation for creating proper school math textbooks,
better teacher preparation, effective lessons and meaningful student engagement
and higher achievement and interest in mathematics. When teachers understand the intent and
content of these standards, align their curriculum with CCSSM expectations and
convert them into daily lesson plans coupled with appropriate progress
monitoring, then children’s mathematics achievement will rise.
DR: Why do you
feel that some teachers are resistant to the changes?
Mahesh: The standards call for examining how teachers
teach and how and what students, at each grade level, learn, know and
communicate the mathematics they know, not what is covered. CCSSM are demanding and is not business as
usual. For students to learn the
standards well, we need to teach them with understanding, fluency and
applications. We need to convey the
mathematician’s art; asking simple questions.
This will take a lot of planning and introspection to our pedagogy. The
standards call for our own understanding of mathematics and its structure.
DR: How does CCSSM differ at the elementary level
from middle and high school?
Mahesh: The focus of
math content in the CCSSM in the elementary grades is on mastering key arithmetic
concepts and procedures with deeper understanding so that the students are
prepared for more demanding mathematics in middle and high school and
beyond. Such a focus means mastering
certain non-negotiable skills at each grade level.
DR: Non-negotiable skills? Such as….
Mahesh: Students
know number concept at the end of Kindergarten, addition facts by the end of
first grade, additive reasoning by the end of second grade, multiplication by
the end of third grade, and division by the end of fourth grade. This means students should master number
concept, number sense and numeracy. Numeracy
means a student’s ability to execute whole number operations correctly, consistently,
fluency in standard form and understanding by the end of fourth grade. Then, students can learn mathematics
better. Another example, fractions- the
concept is introduced, in CCSSM, earlier in the third or fourth grade, first by
using the same denominator, but by fifth grade, children have mastered the
operations on fractions with deeper understanding. Similarly, multiplication is introduced in
the second grade as repeated addition, equal groups of objects, and
arrays. By the end of third grade,
however, children should have mastered the concept as repeated addition, groups
of arrays, the area of a rectangle so they understand the distributive property
of multiplication over addition and subtraction, have automatized
multiplication facts, and are prepared to multiply fractions in fifth grade
using the area model of multiplication then apply multiplication of fractions
to master the operations of addition and subtraction on fractions with
efficiency and understanding. By the end
of sixth grade they have mastered operations on whole numbers, fractions
(parts-to-whole, comparison of quantities, decimals, percent, ratio,
proportion, scale factor, etc.) and integers.
DR: What about students who understand how to
multiply earlier than the others- is it fair to hold them in the same classroom
than those who are just learning the concept?
Mahesh: A student may be able to do a procedure, but
not fully understand the concept behind it so when it comes time to use this
concept in a different way, they are no longer that ‘gifted’ student they were
in earlier grades. With a focus of
mastery of non-negotiable skills, the teacher can pay attention to children’s
errors, misconceptions, or lack of mastery in proper time. At present, we devote a great deal of time on
preview, pre-testing, and our classrooms have become so diverse in mastery and
preparation. Teachers find meeting
individual needs to be very difficult.
They have multiple preparations for the same class with only coverage as
a focus, rather than mastery. While many parents would differ in this notion,
there are very few ‘gifted’ students who are just that; where they understand
the procedure earlier, as well as the concept.
Procedure without concept is a recipe that cannot work later on in their
math learning. There are some gifted
children in every classroom, but their
needs must be met, first by deepening their understanding, broadening their
conceptual schemas, and increasing their fluency in all aspects of mathematics-
linguistic, conceptual, procedural, and skills.
Narrow acceleration of procedures for these children takes them to a
situation that they “burn out”- they plateau in their reach, or simply memorize
and apply mindless procedures on sheets after sheets. They need to engage in mathematics
conversation, have a chance to present their thinking about a problem, critique
others’ reasoning, make connections and apply their knowledge in meaningful
problem solving alone and with others. A
truly gifted child (who has master of linguistic, conceptual, procedural and
efficient skills) needs to be with his peers of similar thinking- in an
accelerated (honors) program with a competent teacher who knows
mathematics. This means vertical
acceleration.
DR: How do you feel about the concept of a
‘flipped’ classroom where students watch a lesson on-line and then do their
practice in school under the guidance of a teacher?
Mahesh: First off, the quality of the videos
available to students is poor. The
largest outlet, Khan Academy, teaches only procedural methods without the
development of any conceptual schemas.
He teaches how to solve a problem, but not how to think about it, why
does that procedure work, what are the relations of that particular procedure
to other math concepts. They are very
good when a competent teacher has introduced the topic and related the concept,
developed the linguistic and conceptual schemas for the mathematical idea. Teaching is about having a discussion;
discussing all the possibilities.
Students have no interest in the concept if they have been introduced to
the procedure first. Changing one’s pedagogy to a flipped classroom
is simply ‘rearranging the chairs on the Titanic.’ A teaching lesson has four components: linguistic, conceptual, procedural, and
skill. The first two require
conversation rather than lecture. A
video simply cannot provide what a teacher can deliver. A teacher has three roles: didactic,
Socratic, and coaching. The videos,
books, collecting information from the internet, etc. are didactic components
of teaching. They provide
information. This information needs to
be converted into knowledge—language of mathematics, conceptual schemas, and
skills. This conversion takes place by
discussion, by exchange of ideas, messing around with methods and information,
guided discovery, etc. The role of the
teacher is to pose problems, ask questions, scaffold their thinking, provide
examples and counter examples, refute their findings, provide encouragement as
they wrestle with ideas. It is not just
practice exercises after watching a video.
It is developing mathematical way of thinking—discerning patterns,
extending patterns, creating patterns and applying patterns. Teacher’s role is not just to supervise
practice. Moreover, CCSSM suggests
certain mathematics instructional practices (they are 8 of them) videos do not
involve even two of them. They are wonderful for some students, but not for all
students. If you want to use them (judiciously selected), use them after you
have introduced the language, conepts and the procedural idea in the
class. They will be wonderful as
practice tools and reinforcement. Your
classroom should be augmented by going to them not before your introduction of
the idea. We need to use these available resources, but not usurp your role
(didactic, Socratic, and coaching) as a teacher. They should be in the mix of all of the
resources you as an effective teacher should use.
Mahesh Sharma is the former President and Professor of Mathematics Education at Cambridge College where he taught mathematics and mathematics education for more than thirty-five years to undergraduate and graduate students. He is internationally known for his groundbreaking work in mathematics learning problems and education, particularly dyscalculia and other specific learning disabilities in mathematics. Mahesh is an author, teacher, teacher-trainer, researcher, consultant to public and private schools, and a public lecturer. He was the Chief Editor and Publisher of Focus on Learning Problems in Mathematics, an international, interdisciplinary research mathematics journal with readership in more than 90 countries, and the Editor of The Math Notebook, a practical source of information for parents and teachers devoted to improving teaching and learning for all children.
Do you have questions you would like Mahesh to answer regarding techniques that would enhance your teaching using the Common Core? Post them below, and he will respond on this blog.
Do you have questions you would like Mahesh to answer regarding techniques that would enhance your teaching using the Common Core? Post them below, and he will respond on this blog.
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