They do not provide adequate insight into the math teaching and learning processes that help students grow in math maturity and develop long lasting math knowledge, skills, and habits of
Introduction
As the Chinese proverb reminds us, the longest journey begins with the first step Maria Montessori, an Italian physician and educator and humanitarian, is best known for the Montessori philosophy of education; she taught that education is not something the teacher does, but a natural process that develops spontaneously in the human being It is not acquired by listening to words, but in virtue of experiences in which the child acts on his environment.
This book includes a number of instructional and inspirational quotations Most are drawn from two Information Age Education sites: ¥ Math Education Quotations ¥ Quotations Collected by David Moursund
Math is a foundational subject in education, and it is essential for all students to move beyond the novice level and build the knowledge and skills needed for responsible adult citizenship This math proficiency is directly relevant to many facets of everyday life and future careers, where accurate calculations, logical reasoning, and problem-solving matter Ultimately, strong math competencies prepare learners to navigate and contribute effectively in an increasingly numeric world.
This book focuses on the design and implementation of good math lessons, not on providing a collection of sample math lesson plans There are numerous math lesson plans available on the Web and from other sources, but these plans generally have three weaknesses.
1 They are not personalized to the individual strengths and weaknesses of the teacher, the teacherÕs students, and their culture
Current analyses often fall short by not providing adequate insight into the math teaching and learning processes that help students grow in math maturity and develop long-lasting math knowledge, skills, and habits of mind To raise outcomes, researchers and educators should illuminate how instructional strategies, classroom discourse, and assessment practices cultivate mathematical thinking, reasoning, and problem-solving, enabling durable understanding and confidence in mathematics By focusing on the processes behind learning—effective teaching, collaborative exploration, and formative feedback—stakeholders can foster enduring knowledge and the productive habits of mind that support lifelong success in math.
3 The person attempting to teach these lesson plans often has little personal involvement and ownership in the design and creation of the lesson plans
Take a few minutes to browse the Table of Contents and read the Preface if you haven't already, so you can discover topics that may interest you and your students These sections often highlight key themes and resources that can guide lesson planning and engage learners.
There is no need to read this book from cover to cover Find a topic that interests you, and go directly to it
Math Lesson Planning: Itếs EasyẹRight?
This section is designed to get you involved in thinking about what might constitute a good math lesson plan It is based on the written reflections of a fictitious preservice teacher
On the first day of a math education course for preservice teachers, the following assignment was given:
Throughout our years as students and through the introductory education courses we've taken, we have gained meaningful insights into how people learn and how to plan effective math instruction; this prompts us to write a private letter to our future selves about our current understanding of math lesson planning, intended not to be submitted or shared with a teacher but saved and reread at the end of the course In this self-reflection we describe how to design math lessons with clear objectives, authentic problems, structured scaffolds for diverse learners, and ongoing formative assessments that guide instruction We consider how student thinking, common misconceptions, and feedback loops should shape every lesson, and we commit to adapting plans to promote equity, engagement, and deep understanding This SEO-friendly reflection uses education-oriented keywords to reinforce professional growth in math teaching and to serve as a lasting reminder of best practices in math lesson planning.
Here is what a (hypothetical, fictitious, quite capable) preservice teacher wrote:
My goal is to teach math at the upper elementary school level, where I can help students build solid foundational skills Although math isn’t one of my favorite subjects and I haven’t always been particularly strong at it, I am proficient in arithmetic and confident that I can handle the math concepts taught in the upper elementary grades I am committed to delivering clear, engaging instruction that makes arithmetic approachable and helps students develop confidence in their own math abilities.
It seems to me that math will be one of the easier parts of my teaching assignment As I think about it, I see five components to the task
I will obtain the teacher's manual and math textbooks, and the school will provide a syllabus detailing which pages to cover, what to omit, and what to emphasize for state tests I will count the total pages to be covered over the 180-day (36-week) school year and structure the schedule to teach four days per week with one day set aside for review, short quizzes, exams, and contingencies such as snow days and fire drills We will aim to cover roughly the same number of pages on each page-coverage day, calculated by dividing the total pages by 144 (four days a week for 36 weeks).
Second, in implementing the math content to be covered I will consistently use the following plan:
When homework has been assigned, collect it and address any questions students have about it Return the in-class seatwork from the previous day and answer questions about that material This approach provides a brief review of the previous day’s content and helps reinforce learning.
Spend about 10 to 15 minutes delivering a chalk-and-talk presentation of the new material using a whiteboard and projector Don’t get bogged down answering questions; the goal is to cover the key concepts efficiently so students can then complete their assigned math seatwork.
At the start of the math period, give out the worksheets (or specify the exact problems from the text) that students will work on, and remind them of the textbook pages we have just covered, advising them to refer to those pages if they encounter difficulties Ensure some problems include answers in the back of the book or from another source so students can receive feedback on their progress As students work, circulate around the room to address individual questions and support their understanding of the assignment.
At the close of the period, wrap up by asking the class for questions about the material and addressing any difficulties using the troublesome examples from the assigned problems If most students have progressed well, collect their papers; if many students have not finished, assign the task as homework to be completed and submitted by the start of the next math period.
Every week includes a brief quiz or a longer test, so 36 class periods are devoted to quizzes and exams On days with short quizzes, the remaining math time is filled with engaging on- and off-computer math games Students who struggle with math or who have not completed required assignments receive extra instruction and deliberate practice rather than participating in the games, ensuring targeted support for every learner.
During the month before the state math exam, part of each day will be spent reviewing for the test, taking practice tests, and learning test-taking strategies
Fourth, I will make accommodations for diverse learnersẹespecially students who are particularly slow in learning math and students who are particularly fast at learning math
Students who excel in mathematics will receive additional, more challenging problems to tackle after finishing the assigned work, providing meaningful math enrichment and opportunities to deepen their understanding Alternatively, they can volunteer to help slower students in a peer-tutoring setting, reinforcing their own learning while supporting peers in need.
Students who do not complete the assignment within the allotted time must continue working on it during recess, and I will supervise them during that period, as the school is expected to have a physical education teacher to oversee recess activities.
Overview of Lesson Planning
ÒEducation is a human right with immense power to transform
On its foundation rest the cornerstones of freedom, democracy and sustainable human development.Ó (Kofi Annan; Ghanaian diplomat, seventh secretary-general of the United Nations, winner of 2001 Nobel Peace Prize; 1938-.)
Lee Iacocca argued that in a completely rational society, the best among us would be teachers, while others would take on other meaningful roles He believed that passing civilization from one generation to the next should be the greatest honor and the highest responsibility anyone could bear This perspective highlights the crucial role of education, the value of teachers, and the essential duty we all share to nurture future generations.
Human evolution endowed both proto humans and modern humans with brains optimized for learning how to tackle complex problems We can plan ahead, visualize multiple future paths, and anticipate the consequences of our actions In a world where predators outweighed our raw strength, our ancestors survived by learning from one another, cooperating, and developing tools that amplified physical abilities They passed knowledge and skills from generation to generation, enabling cultural and technological progress to accumulate over time.
Each of us is inherently both a teacher and a learner, continually learning from others and helping them learn in return Some individuals become professional teachers, steadily refining their craft to better support others’ learning As a reader of this book, you have spent years mastering foundational skills such as reading, writing, and arithmetic—abilities valued by our society—and you have repeatedly shown yourself to be a capable learner By engaging with this book, you are actively investing in your own development as a teacher, strengthening the skills you use to educate others now or in the future.
Imitation is a powerful path to learning, with many skills acquired by watching others and copying what they do In the hunter-gatherer era, children learned through imitation, guided by a show-and-do approach, personal and small-group tutoring, and trial-and-error exploration This mix of modeling, hands-on practice, and experimentation helped early humans develop essential abilities and adapt to their environment.
Human learning persisted from the start of the agricultural era about 10,000 to 11,000 years ago, with enduring learning abilities and varied paths to mastery Agriculture enabled permanent settlements and the rise of specialists such as pottery makers, basket weavers, flint knappers, and bow makers In this context, apprenticeship systems developed, where a learner studied under a master for many years to become a skilled practitioner in a narrow craft Today, apprenticeships remain a central part of our education system, continuing the long tradition of mentorship-based learning.
The governmental and business structures and the problems that emerged during the first 5,000 years of the agricultural era ultimately drove the development of reading, writing, and arithmetic A child does not learn to read by simply watching an adult read; literacy and numeracy require explicit instruction As a result, learning the 3Rs demands structured, deliberate teaching methods rather than passive observation.
During the early history of the three Rs—reading, writing, and arithmetic—education was limited to a minority Some children received one-on-one tutoring from a learned parent or a professional tutor, while others learned in classrooms under a teacher Nevertheless, only a very small portion of the population had access to these instructional forms, and the vast majority remained illiterate and innumerate.
Although illiteracy and innumeracy were once widespread globally, today literacy and numeracy are increasingly regarded as fundamental birthrights Yet vast disparities in both the quality and reach of formal education persist, leaving millions of children with unequal opportunities to learn In many regions, children grow up with little or no access to schooling, underscoring the urgent need to improve universal education for all.
Literacy and numeracy are considered a birthright of todayÕs children Good teaching helps to ensure this birthright
As a professional teacher, I designed this book to help you learn more effectively Learning is an individual and personal journey that takes place in your brain and throughout your body, shaped by how you engage with new ideas By guiding you through this book, I aim to provide practical learning strategies and support your growth, helping you take charge of your own learning process.
Whether it's the squiggles of printed text, the raised dots of Braille, listening to music or audio books, watching television, or interacting with a computer game, information flows into the brain The brain processes this information, interpreting and understanding it based on years of education and experience.
Constructivism is a learning theory that describes how the brain actively acquires and constructs new knowledge, linking it to existing understanding and storing it for future use It emphasizes that learning arises from constructing meaning rather than passively receiving information, as learners interpret new ideas through their prior knowledge and experiences In mathematics education, constructivism highlights the importance of connecting new mathematical concepts to what students already know, fostering deeper understanding through exploration, problem solving, and reflection Consequently, constructivism remains a foundational framework for teaching mathematics, guiding instructional strategies that help learners build, organize, and retain mathematical knowledge.
Both teachers and learners face a common challenge: we don’t know exactly what the other understands, and the learner must grasp new concepts from the teacher’s perspective This is especially daunting in math and other vertically structured disciplines, where each grade level assumes mastery of prerequisites—the content of earlier courses and earlier parts of the current course Unfortunately, those assumptions are often wrong, and many students fail to learn or forget much of what they are taught in school Effective math instruction requires regularly assessing prior knowledge, identifying learning gaps, and providing deliberate scaffolding to bridge past concepts with new ones, so students build a coherent, long-lasting understanding.
Many have heard the saying "use it or lose it," and this principle also applies to the brain and learning When students acquire specific math knowledge and skills, they tend to lose them over time through disuse, making the learning difficult to retrieve As a result, what they learned may fail to transfer to new situations they encounter in the future, undermining both retention and the ability to solve problems.
Students vary greatly in how much math they encounter and use at school—both in math classes and throughout the school day—and in their lives outside of school Consequently, every student you teach comes with a unique preparation for constructing new mathematical knowledge and skills, which highlights the need for instructional approaches that support diverse math experiences and readiness.
It's interesting to compare life outside of school with a student's life during a course In schools, learning content is divided into disciplines, and a typical school day is organized into courses or units that focus on different subject areas such as the fine and performing arts, physical education, math, language arts, social science, and science In middle school and high school, these courses are usually taught by teachers who specialize in their subject, even though they may not be familiar with the day-to-day details of instruction students receive in other subjects.
What is Mathematics?
"A mathematician, like a painter or poet, is a maker of patterns
Patterns endure because they are built from ideas, a notion echoed by G H Hardy when he suggests that ideas give permanence to patterns Algebra is the language of mathematics—and, in turn, the language of the information age—providing the essential framework to understand nature As the Rosetta Stone of nature and a passport to advanced mathematics, the language of algebra translates complex patterns into universal concepts It is the logical structure of algebra, not merely the solutions to its equations, that has made algebra a central pillar of classical education.
As an adult, you know a lot about mathematics and apply it every day in real life You use math to manage money, plan time, measure distance, compute area and weight, and analyze statistical data—turning numbers into practical solutions This everyday mastery shapes your view of mathematics as a powerful tool for solving real-world problems and gives you a reliable reality check you can share as you help children learn math through both informal experiences and formal education Your mathematical literacy empowers you to navigate daily challenges and support the next generation in building a strong, confident foundation in math.
Your answer to the question "what is math?" is likely different from many others, and when we educate our children, it's helpful to reach a shared understanding of what math is and which concepts are most important for kids to learn Establishing that consensus guides instruction, clarifies learning goals, and helps ensure children build a solid foundation in mathematics.
This chapter investigates the question “What is math?” and provides essential background for designing and implementing effective math lesson plans It offers insights to help you compare different perspectives on math with your own understanding and with your approach to teaching As you read others’ insights, reflect on how their ideas fit with your beliefs about what math is and how it should be taught, so you can develop more coherent, student-centered instruction.
Algebra is the language of mathematics, which itself is the language of the information age
A Short and Useful Answer to the What is Math Question
Here is one of my favorite quotes: ÒGod created the natural numbers All the rest is the work of man.Ó (Leopold Kronecker; German mathematician and logician; 1823Ð1891.)
Humans appear to have an innate sense of small quantities linked to the natural numbers 1, 2, and 3 Very young infants demonstrate an inborn understanding of these small quantities, reflecting early numeracy Cross-cultural studies of aboriginal tribes show varied quantity vocabularies: some languages only have terms for one and two, others include one, two, and three, and some communities possess more extensive numerical vocabularies for quantity.
The current widely used human languages have words as well as math notation for 1, 2, 3, 4,
5, and so on In this sense, the term natural numbers has come to mean the numbers 1, 2, 3, 4, 5,
Natural numbers exhibit clear patterns: starting at 1, every second number—1, 3, 5, 7, —is odd, while starting at 2, every second number—2, 4, 6, 8, —is even When you add two consecutive natural numbers, the sum is always odd, and when you multiply two consecutive natural numbers, the product is always even Some natural numbers have exactly two natural-number divisors, and these are known as prime numbers.
With this background, we can summarize what mathematicians do Here is my 3-step summary:
Within mathematics, a mathematician detects a potential pattern, articulates it clearly, and formulates a conjecture asserting that the pattern holds for a precisely defined set of cases For example, one might conjecture that the sum of two consecutive natural numbers is always an odd natural number, and that this statement is true for every pair of consecutive naturals.
2 The mathematician tests the conjecture in a number of cases, looking for counter examples and developing a ÒfeelingÓ for why the conjecture might be correct Examples:
1 + 2 = 3, an odd natural number; 2 + 3 = 5, an odd natural number; 3 + 4 = 7, an odd natural number But, what about using larger natural numbers? Example: 17 + 18 = 35, an odd natural number; 1,644 + 1,645 = 3,289, an odd natural number
Starting with the pairs (1, 2), (2, 3), and (3, 4), we observe that the sums form a sequence that increases by 2 at each step This yields the sequence 3, 5, 7, 9, 11, 13, 15, all of which are odd numbers A mathematician exploring a pattern looks for additional relationships beyond the obvious You may notice that the last digit of every odd number generated in this sequence is one of 1, 3, 5, 7, or 9 The question remains: are there any odd natural numbers whose last digit is something else?
Mathematicians seek to develop a proof—a convincing argument that a conjecture is true for both themselves and the broader audience The journey can yield a counterexample that disproves the conjecture, or it may reduce to exhaustively checking a finite set of possibilities, where a computer can dramatically speed up the verification of every case When the situation involves infinitely many possibilities, specialized mathematical techniques are used to handle infinity; historically, infinity posed great difficulties, and topics such as Zeno’s paradox remain engaging examples for students Ultimately, progress depends on having the right knowledge, skills, and persistence, because some conjectures remain unsolved for centuries despite rigorous effort.
Patterns exist in every discipline, and exploring patterns in non-mathematical fields often reveals how mathematics can be useful Some patterns may be unfamiliar to mathematics, and when they captivate mathematicians, their study can become part of the mathematical toolkit Through this ongoing process, math becomes applicable to more disciplines and continues to broaden its relevance across diverse fields of study.
In math, a proof is an argument that is very convincing to both self and others It stands the scrutiny of others over time
Long before we had written language and schools, people learned to count and make other simple uses of math
Very young children have an innate ability to distinguish the quantities one, two, and three—a basic form of number sense that supports early math learning Alongside this numerical intuition, they also possess an inborn capacity to learn natural languages, such as English and Spanish, and this language aptitude helps fuel cognitive development that lays the groundwork for math education.
From early childhood, caregivers help children learn to count and determine how many objects are in a small set As children develop stronger math understanding and skills, formal math education content begins to be introduced This progression includes exposure to various numeral systems, such as Roman numerals (I, II, III, IV, V) and Hindu-Arabic numerals (1, 2, 3, 4, 5, …), which provide different ways to represent quantities and support mathematical literacy.
Mathematics encompasses diverse numeral systems, including Chinese numerals (一, 二, 三, 四, 五) and Arabic numerals, and it defines symbols for addition, subtraction, multiplication, and division, as well as fractions and decimals The field is organized into subdisciplines such as algebra, geometry, statistics, probability, and calculus, each addressing different mathematical concepts and problems There are also various systems for measuring distance, area, and quantity, with the metric system and the English (imperial) system serving as prominent examples Together, these elements form the core of mathematical reasoning and its wide range of real-world applications.
We have developed effective aids for math problem solving, designed around the everyday tools you already use You likely own or have easy access to a calculator, computer, clock or wristwatch, and a ruler, and you routinely rely on these aids to perform arithmetic calculations, determine the time of day, and measure length By leveraging these familiar devices as problem-solving tools, you can tackle math challenges more efficiently, apply mathematical concepts to real-world tasks, and boost accuracy in your results.
A car contains a speedometer and an odometer Nowadays, it may contain a Global
Positioning System (GPS) that can display a map and give you oral instructions as to when and where to turn to get to a specified destination
Humans have developed many aids to doing and using math, ranging from calculators to computer systems When a readily available aid can solve or substantially aid in solving a type of school math problem, we must ask what we want students to learn about that problem type This remains a challenging and open question in math education, guiding decisions about the balance between understanding concepts, mastering procedures, and leveraging tools to reason and model.