If you are a teacher in the elementary grades or a math teacher in any grade and haven’t seen the blog Mahesh Sharma has recently started writing, do yourself a favor and browse it right now at Mathematics for All.
“Browsing” is probably the wrong word. These are long thoughtful articles on familiar topics if you have attended his workshops. But the blog is well worth the effort for anyone who teaches math.
Here’s a sampling:
Mathematics is a second language; it has its own alphabet, symbols, vocabulary, syntax, and grammar. Numeric and operational symbols are its alphabet; number and symbol combinations are its words. Equations and mathematical expressions are the sentences of this language.
Mastery of a mathematical concept is the result of an interactive process between language and quantitative and spatial experiences. Initially, concrete experiences with quantity and space form concepts and are communicated through visual representations and artifacts. Later, children learn to represent them symbolically/abstractly. Abstract symbols, formulas and equations are then applied to solving problems. This iterative and cyclic process is called mathematization.
To reverse the decline in math scores and engender in our children an interest and expertise in mathematics, educators need to pay attention to teaching methods. In countries where students do exceptionally well and in U.S. schools where students excel, there are three key attributes:
- Each grade has a focus. In other words, there are non-negotiable skills that students master in that grade;
- Everyone in the school system understands and practices the common definition of knowing;
- The key concepts and procedures are taught using efficient and elegant models—methods that are generalizable and develop mathematical ways of thinking in our students.
The Common Core State Standards in Mathematics (CCSS-M, 2010) have brought the nation’s attention to these three aspects of mathematics learning. CCSS-M stands on three legs:
- Every grade has a focus on a few key concepts to be taught and learned;
- Those topics must be taught with rigor—understanding, fluency, and applicability (in other words, with mastery);
- There should be coherence in teaching these topics—a teacher should know the trajectory of the development of the concept or procedure—where it begins and how it develops in different grades.
….as teachers, we should avoid as much as possible, giving children rules that we cannot support with mathematical reasoning. To teach mathematics ideas, we should present (a) concrete models, (b) set up patterns, (c) arrive at mathematics conjectures using concrete models and patterns, (d) use analogies, (e) use deductive and inductive reasoning, and (f) use formal proofs.