It is a joy to see how far math teaching quality has improved since I was a girl. Prejudices have lifted and it is expected that all students will be taught math equitability. It is likely that you were not taught math equitably or correctly in school so learning math anew and learning how to teach math is essential.

Elementary and Middle School Mathematics, Teaching Developmentally by Van de Walle, Karp and Bay-Williams summarizes the evidence based (proven) practices and the research on which they are based. Here are brief but vital notes taken while studying the text of Chapter six.

The National Research Council in 2001 concluded that all students are best served when attention is given to the following principles:

1. Learning with understanding is based on connecting and organizing knowledge around the conceptual ideas.

2. Learning builds on what students already know.

3. Instruction in school should take advantage of students informational knowledge of mathematics.

Many children will be able to ascertain the meanings of words through general conversation, or so it appears. However, defining the meaning of specific words with explicit instruction, and defining procedures with explicit instruction, can benefit most students except those who are wholly unchallenged. For the mathematically gifted additional special attention and work can be provided, and there are a wealth of resources from the national Council for teaching mathematics, online, to do so.

Inquiry-based instruction where students are asking questions trying to figure things out and coming up with ways to figure things out should not preclude explicit instruction. Although they are two different ways of teaching they can be incorporated. Providing students with explicit instruction for vocabulary, and specific method while ensuring that students are given ample opportunity to inquire, discover, think the pair and share in an inquiry-based exercise, it’s vital to develop thinking-ability.

It is important to remember that experiential-based learning centers on active problem-solving and the construction of knowledge so it will produce deeper understanding of mathematics. Active problem solving enhances student ability to retain, generalize, and apply information skills long-term.

“The best explicit instruction is scaffolding, meaning it moves from a highly structured, single-strategy approach to multiple models, including examples and non-examples. It also includes immediate area correction with the fading of prompts to help students move to independence.”

“Experiential-based learning that centers on active problem-solving and the construction of knowledge that produces deeper understanding of mathematics and enhances student ability to retain, generalize, and apply information and all skills that are vital to long term success in mathematics. Explicit extraction, to be effective, must include making mathematical relationships explicit so students don’t just learn how to do that day’s mathematics, but make connection to other mathematical ideas.” This is one of the major findings in how students learn. Is

“The instructional focus moves from concrete representations or manipulatives and models to semi concrete representations such as drawings or pictures and images finally to abstraction using only numerals or mentally solving problems.” This is a well researched and proven, evidence based practice for teaching math.

it is fascinating to learn that conceptual knowledge, for example what is multiplication, is universally understood and is part of the universal language of mathematics. However, procedures such as how to multiply and more importantly, symbols used to multiply are culturally determined and are not universal. This is extremely important because many students are tripped up by this; trying to please teachers and do exactly the way they are told. When teachers enforce a specific method on students they can confuse their minds by not allowing them to understand different ways of doing it and especially culturally diverse ways of doing mathematics procedures.

Vocabulary is such an important subject in mathematics and there are many words that have meanings in mathematics and also have a different meaning in English. Some of these words are: on, even, product, mean, some, factor, acute, but, division, difference, similar, angle, factor, tangent, obtusus, acute, circular, adjacent, variable, radical, proportion, matrix, and irrational. This is of utmost importance because in the elementary grades students are just beginning to learn that words have more than one meaning. It is of even greater importance for students who are English language learners.

The National Library for virtual manipulatives provides a wealth of manipulative resources for teaching math. From kindergarten to 12th grade.: Numbers and operations, algebra, geometry, measurements as well as data analysis and probability, are all included with manipulatives for learning. All of these are valuable in teaching lessons using an overhead projector and for children to practice on their own at a computer using the applets