In Mathematics, The Bar Has Been Raised So We Must Reappraise
By Monica Neagoy (www.MonicaNeagoy.com)
By Monica Neagoy (www.MonicaNeagoy.com)
Whether or not your school or district has adopted the Common Core Math Standards, the fact
remains that in our technological world, the demands on citizens have been raised. It is therefore our obligation to raise the bar on the quality of mathematics instruction for all our students as well as our expectations of their mathematical know-how.
All of us are teachers of the young people who question us and learn from us, whether we are classroom teachers, home schooling parents, or adults who simply inspire the young. Children love numbers and constantly inquire about them. So let’s take the elementary math example of computation with whole numbers. Have you ever asked yourself what’s the main purpose of teaching computation—addition, subtraction, multiplication, and division—in a world where adults and children alike have iPhones, tablets, calculators, and other devices at their fingertips? Devices that compute more rapidly, more efficiently, and more correctly than we could ever do—aside from a few genius human calculators in the world, like Alexis Lemaire who can extract the 13th root of a random 100-digit number in about 14 seconds!
Deep Mathematical Thinking
The answer is not to know how to carry out the algorithm. Fifty years ago, the how-to was the main purpose. But today, if we just teach children how to add, how to subtract, or how to multiply, we will not be meeting our higher call. We would actually be failing our pedagogical mission, as the children would always lose out to calculators and computers, even cheap ones for that matter! Deep mathematical thinking or algebraic thinking is our higher purpose in teaching elementary school number and computation. But what does that mean? What does that look like in a math lesson? We don’t find much guidance in our math education history: For decades and even centuries, children have been taught whole number and fraction algorithms by rote without making sense of the successive stages. And since computation—first with whole numbers then with fractions—was the heart of elementary mathematics for most of our nation’s history, those children became today’s adults who believe, and perpetuate the belief, that mathematics is mostly about memorizing facts, learning procedures, and solving problems that have one right answer and one way to find it.
Resources to Guide Us
So we need guidance in learning how to better teach what’s really important in mathematics. For example, how to:
For example, when a child says, “If I add two odd numbers, I always get an even number because the two lonely ones find each other and then everyone has a partner,” this child is generalizing (i.e., noticing that the answer is always even) about a generalization (i.e. knowing that all even numbers have a “lonely one”). Abundant research confirms that young children, of all socioeconomic backgrounds, are capable of sophisticated mathematical thinking. We just need to nurture it, develop it, and enrich it.
To inspire teachers, teacher educators, coaches, and professional development providers with this mission of raising the bar for all students through the exploration of deep mathematics, I’ve written two books titled, Planting the Seeds of Algebra, PreK-2 and Planting the Seeds of Algebra, 3-5 (http://www.corwin.com/authors/657525)
Summer Courses to Inspire Us
My literacy colleague, Mollie Cura and I are offering five new courses next week that shed a new light on literacy and mathematics. ( (http://www.newlightonliteracyandmath.com). Mollie models how to treat students like young writers, rich with ideas, imagination, and the desire to be creative. I model the same thing but in math: how to treat children like young mathematicians, full of mathematical ideas and creativity, and how to capitalize and build on that creativity. We both believe strongly that cultivating a sense of wonder and nurturing their natural desire for learning—until it becomes a lifelong passion—are two fundamental ingredients! See our course selection here: http://www.newlightonliteracyandmath.com/#!courses/comg).
Take-Aways to Excite Us
Examples of new questions we’ll explore in our next week’s math courses include:
· What is math in the 21st century anyway?
· How can we explore addition facts algebraically?
· How do we connect the concept of area to the partial products algorithm to empower students to succeed with high school quadratic identities?
· How do we help students learn harder multiplication math facts by building on simpler facts they know?
· How do we cultivate more than the equal sharing division model starting in grade 3? Why is this so important to fractions, ratios, and proportional thinking?
· How to we make generalizing a part of each math lesson?
· How do we empower our students to thrive in math, and enjoy their lifelong math journey?
(Find some downloadable worksheets on my website for these explorationshttp://www.monicaneagoy.com/media/downloads/)
Blog Posts to Inform Us
See our blog posts on Corwin Connect that will give you lots of new teaching ideas:
http://bit.ly/New_Light_on_Literacy_and_Math
http://bit.ly/5_Ideas_to_Give_Meaning_to_Fraction_Division
All of us are teachers of the young people who question us and learn from us, whether we are classroom teachers, home schooling parents, or adults who simply inspire the young. Children love numbers and constantly inquire about them. So let’s take the elementary math example of computation with whole numbers. Have you ever asked yourself what’s the main purpose of teaching computation—addition, subtraction, multiplication, and division—in a world where adults and children alike have iPhones, tablets, calculators, and other devices at their fingertips? Devices that compute more rapidly, more efficiently, and more correctly than we could ever do—aside from a few genius human calculators in the world, like Alexis Lemaire who can extract the 13th root of a random 100-digit number in about 14 seconds!
The answer is not to know how to carry out the algorithm. Fifty years ago, the how-to was the main purpose. But today, if we just teach children how to add, how to subtract, or how to multiply, we will not be meeting our higher call. We would actually be failing our pedagogical mission, as the children would always lose out to calculators and computers, even cheap ones for that matter! Deep mathematical thinking or algebraic thinking is our higher purpose in teaching elementary school number and computation. But what does that mean? What does that look like in a math lesson? We don’t find much guidance in our math education history: For decades and even centuries, children have been taught whole number and fraction algorithms by rote without making sense of the successive stages. And since computation—first with whole numbers then with fractions—was the heart of elementary mathematics for most of our nation’s history, those children became today’s adults who believe, and perpetuate the belief, that mathematics is mostly about memorizing facts, learning procedures, and solving problems that have one right answer and one way to find it.
Resources to Guide Us
So we need guidance in learning how to better teach what’s really important in mathematics. For example, how to:
- Foster ways of thinking, doing, and communicating about mathematics with understanding
- Guide students to make connections and analyze relationships
- Notice structure of number and properties of operations
- Observe and study change
- Conjecture, justify, symbolize, and mathematize … and most importantly, generalize
- And beyond that, generalize about generalizations!
For example, when a child says, “If I add two odd numbers, I always get an even number because the two lonely ones find each other and then everyone has a partner,” this child is generalizing (i.e., noticing that the answer is always even) about a generalization (i.e. knowing that all even numbers have a “lonely one”). Abundant research confirms that young children, of all socioeconomic backgrounds, are capable of sophisticated mathematical thinking. We just need to nurture it, develop it, and enrich it.
To inspire teachers, teacher educators, coaches, and professional development providers with this mission of raising the bar for all students through the exploration of deep mathematics, I’ve written two books titled, Planting the Seeds of Algebra, PreK-2 and Planting the Seeds of Algebra, 3-5 (http://www.corwin.com/authors/657525)
Summer Courses to Inspire Us
My literacy colleague, Mollie Cura and I are offering five new courses next week that shed a new light on literacy and mathematics. ( (http://www.newlightonliteracyandmath.com). Mollie models how to treat students like young writers, rich with ideas, imagination, and the desire to be creative. I model the same thing but in math: how to treat children like young mathematicians, full of mathematical ideas and creativity, and how to capitalize and build on that creativity. We both believe strongly that cultivating a sense of wonder and nurturing their natural desire for learning—until it becomes a lifelong passion—are two fundamental ingredients! See our course selection here: http://www.newlightonliteracyandmath.com/#!courses/comg).
Take-Aways to Excite Us
Examples of new questions we’ll explore in our next week’s math courses include:
· What is math in the 21st century anyway?
· How can we explore addition facts algebraically?
· How do we connect the concept of area to the partial products algorithm to empower students to succeed with high school quadratic identities?
· How do we help students learn harder multiplication math facts by building on simpler facts they know?
· How do we cultivate more than the equal sharing division model starting in grade 3? Why is this so important to fractions, ratios, and proportional thinking?
· How to we make generalizing a part of each math lesson?
· How do we empower our students to thrive in math, and enjoy their lifelong math journey?
(Find some downloadable worksheets on my website for these explorationshttp://www.monicaneagoy.com/media/downloads/)
Blog Posts to Inform Us
See our blog posts on Corwin Connect that will give you lots of new teaching ideas:
http://bit.ly/New_Light_on_Literacy_and_Math
http://bit.ly/5_Ideas_to_Give_Meaning_to_Fraction_Division
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