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A great read in how mathematics is prevalent in all areas of ones life in ways that they might not even imagine! Su does an amazing job at capturing human desires and how mathematics takes a role in every single one of them. Not only that, he ends each chapter (that is titled after a human desire) with related math problems that you can explore on your own!

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Winner of the Mathematics Association of America's 2021 Euler Book Prize, this is an inclusive vision of mathematics-its beauty, its humanity, and its power to build virtues that help us all flourish "This is perhaps the most important mathematics book of our time. Francis Su shows mathematics is an experience of the mind and, most important, of the heart."-James Tanton, Global Math Project "A good book is an entertaining read. A great book holds up a mirror that allows us to more clearly see ourselves and the world we live in. Francis Su's Mathematics for Human Flourishing is both a good book and a great book."- MAA Reviews

For mathematician Francis Su, a society without mathematical affection is like a city without concerts, parks, or museums. To miss out on mathematics is to live without experiencing some of humanity's most beautiful ideas.

In this profound book, written for a wide audience but especially for those disenchanted by their past experiences, an award‑winning mathematician and educator weaves parables, puzzles, and personal reflections to show how mathematics meets basic human desires-such as for play, beauty, freedom, justice, and love-and cultivates virtues essential for human flourishing. These desires and virtues, and the stories told here, reveal how mathematics is intimately tied to being human. Some lessons emerge from those who have struggled, including philosopher Simone Weil, whose own mathematical contributions were overshadowed by her brother's, and Christopher Jackson, who discovered mathematics as an inmate in a federal prison. Christopher's letters to the author appear throughout the book and show how this intellectual pursuit can-and must-be open to all.

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Amazon.com: Mathematics for Human Flourishing: 9780300237139: Su, Francis, Jackson, Christopher: Books

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Khan Academy includes many helpful tools to students such as video explanations, practice problems, and solutions for a variety of different subjects. These videos can help strengthen a student's knowledge on a topic and provide more examples to the student as needed.

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The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a function; Exponential models; and Quadratic equations, functions, and graphs. Khan Academy's Algebra 1 course is built to deliver a comprehensive, illuminating, engaging, and Common Core aligned experience!

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Paul's Online Math Notes provides notes, examples, and practice problems for a variety of different math subjects from Algebra I to Calculus III. Students may find this helpful if they feel as if they're in need of more structured notes and more practice problems.

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Welcome to my online math tutorials and notes. The intent of this site is to provide a complete set of free online (and downloadable) notes and/or tutorials for classes that I teach at Lamar University. I've tried to write the notes/tutorials in such a way that they should be accessible to anyone wanting to learn the subject regardless of whether you are in my classes or not. In other words, they do not assume you've got any prior knowledge other than the standard set of prerequisite material needed for that class. In other words, it is assumed that you know Algebra and Trig prior to reading the Calculus I notes, know Calculus I prior to reading the Calculus II notes, etc. The assumptions about your background that I've made are given with each description below.

I'd like to thank Shane F, Fred J., Mike K. and David A. for all the typos that they've found and sent my way! I've tried to proof read these pages and catch as many typos as I could, however it just isn't possible to catch all of them when you are also the person who wrote the material. Fred, Mike and David have caught quite a few typos that I'd missed and been nice enough to send them my way. Thanks again Fred, Mike and David!

If you are one of my current students and are here looking for homework assignments I've got a set of links that will get you to the right pages listed here.

At present I've gotten the notes/tutorials for my Algebra (Math 1314), Calculus I (Math 2413), Calculus II (Math 2414), Calculus III (Math 2415) and Differential Equations (Math 3301) class online. I've also got a couple of Review/Extras available as well. Among the reviews/extras that I've got are an Algebra/Trig review for my Calculus Students, a Complex Number primer, a set of Common Math Errors, and some tips on How to Study Math.

I've made most of the pages on this site available for download as well. These downloadable versions are in pdf format. Each subject on this site is available as a complete download and in the case of very large documents I've also split them up into smaller portions that mostly correspond to each of the individual topics. To get the downloadable version of any topic navigate to that topic and then under the Download menu you will be presented an option to download the topic.

Here is a complete listing of all the subjects that are currently available on this site as well as brief descriptions of each.

Algebra Cheat Sheets - This is as many common algebra facts, properties, formulas, and functions that I could think of. There is also a page of common algebra errors included. There are two versions of the cheat sheet available. One is full sized and is currently four pages. The other version is a reduced version that contains exactly the same information as the full version except it has just been shrunk down so two pages print of the front and two pages print on the back of a single piece of paper.

Trig Cheat Sheets - Here is a set of common trig facts, properties and formulas. A unit circle (completely filled out) is also included. There are two versions of the cheat sheet available. One is full sized and is currently four pages. The other version is a reduced version that contains exactly the same information as the full version except it has just been shrunk down so two pages print of the front and two pages print on the back of a single piece of paper.

Calculus Cheat Sheets - These are a series of Calculus Cheat Sheets that covers most of a standard Calculus I course and a few topics from a Calculus II course. There are four different cheat sheets here. One contains all the information, one has just Limits information, one has just Derivatives information and the final one has just Integrals information. Each cheat sheets comes in two versions. One that is full sized and another that has been reduced, with exactly the same information as the full sized version, that prints two pages on the front and/or back of each page of paper.

Common Derivatives and Integrals - Here is a set of common derivatives and integrals that are used somewhat regularly in a Calculus I or Calculus II class. Also included are reminders on several integration techniques. here are two versions of the cheat sheet available. One is full sized and is currently four pages. The other version is a reduced version that contains exactly the same information as the full version except it has just been shrunk down so two pages print of the front and two pages print on the back of a single piece of paper.

Table of Laplace Transforms - Here is a list of Laplace transforms for a differential equations class. This table gives many of the commonly used Laplace transforms and formulas. It is currently two pages long with the first page being the Laplace transforms and the second being some information/facts about some of the entries.

All of the classes, with the exception of Differential Equations, have practice problems (with solutions) you can use for practice as well as a set of assignment problems (without solutions/answers) for instructors to use if they wish.

Algebra (Math 1314) [ Notes] [ Practice Problems] [ Assignment Problems] - Topics included in this set of notes/tutorial are :

Preliminaries - Exponent Properties, Rational Exponents, Negative Exponents, Radicals, Polynomials, Factoring, Rational Expressions, Complex Numbers

Solving Equations and Inequalities - Linear Equations, Quadratic Equations, Completing the Square, Quadratic Formula, Applications of Linear and Quadratic Equations, Reducible to Quadratic Form, Equations with Radicals, Linear Inequalities, Polynomial & Rational Inequalities, Absolute Value Equations & Inequalities.

Graphing and Functions - Graphing Lines, Circles, and Piecewise Functions, Function Definition, Function Notation, Function Composition, Inverse Functions.

Common Graphs - Parabolas, Ellipses, Hyperbolas, Absolute Value, Square Root, Constant Function, Rational Functions, Shifts, Reflections, Symmetry.

Polynomial Functions - Dividing Polynomials, Zeroes/Roots of Polynomials, Finding Zeroes of Polynomials, Graphing Polynomials, Partial Fractions.

Exponential and Logarithm Functions - Exponential Functions, Logarithm Functions, Solving Exponential Functions, Solving Logarithm Functions, Applications.

Systems of Equations - Substitution Method, Elimination Method, Augmented Matrix, Nonlinear Systems.

The Algebra notes/tutorial assume that you've had some exposure to the basics of Algebra. In particular it is assumed that the exponents and factoring sections will be more of a review for you. Also, it is assumed that you've seen the basics of graphing equations. Graphing particular types of equations is covered extensively in the notes, however, it is assumed that you understand the basic coordinate system and how to plot points.

Calculus I (Math 2413) [ Notes] [ Practice Problems] [ Assignment Problems] - Topics included in this set of notes/tutorial are :

Algebra/Trig Review - Trig Functions and Equations, Exponential Functions and Equations, Logarithm Functions and Equations.

Limits - Concepts, Definition, Computing, One-Sided Limits, Continuity, Limits Involving Infinity, L'Hospitals Rule

Derivatives - Definition, Interpretations, Derivative Formulas, Power Rule, Product Rule, Quotient Rule, Chain Rule, Higher Order Derivatives, Implicit Differentiation, Logarithmic Differentiation, Derivatives of Trig Functions, Exponential Functions, Logarithm Functions, Inverse Trig Functions, and Hyperbolic Trig Functions.

Applications of Derivatives - Related Rates, Critical Points, Minimum and Maximum Values, Increasing/Decreasing Functions, Inflection Points, Concavity, Optimization

Integration - Definition, Indefinite Integrals, Definite Integrals, Substitution Rule, Evaluating Definite Integrals, Fundamental Theorem of Calculus

Applications of Integrals - Average Function Value, Area Between Curves, Solids of Revolution, Work.

The Calculus I notes/tutorial assume that you've got a working knowledge of Algebra and Trig. There is some review of a couple of Algebra and Trig topics, but for the most part it is assumed that you do have a decent background in Algebra and Trig. These notes assume no prior knowledge of Calculus.

Calculus II (Math 2414) [ Notes] [ Practice Problems] [ Assignment Problems] - Topics included in this set of notes/tutorial are :

Integration Techniques - Integration by Parts, Integrals Involving Trig Functions, Trig Substitutions, Integration using Partial Fractions, Integrals Involving Roots, Integrals Involving Quadratics, Integration Strategy, Improper Integrals, Comparison Test for Improper Integrals, and Approximating Definite Integrals.

Applications of Integrals - Arc Length, Surface Area, Center of Mass/Centroid, Hydrostatic Pressure and Force, Probability.

Parametric Equations and Polar Coordinates - Parametric Equations & Curves, Calculus with Parametric Equations (Tangents, Areas, Arc Length and Surface Area), Polar Coordinates, Calculus with Polar Coordinates (Tangents, Areas, Arc Length and Surface Area).

Sequences and Series - Sequences, Series, Convergence/Divergence of Series, Absolute Series, Integral Test, Comparison Test, Limit Comparison Test, Alternating Series Test, Ratio Test, Root Test, Estimating the Value of a Series, Power Series, Taylor Series, Binomial Series

Vectors - Basics, Magnitude, Unit Vector, Arithmetic, Dot Product, Cross Product, Projection

Three Dimensional Coordinate System - Equations of Lines, Equations of Planes, Quadratic Surfaces, Functions of Multiple Variables, Vector Functions, Limits, Derivatives, and Integrals of Vector Functions, Tangent Vectors, Normal Vectors, Binormal Vectors, Curvature, Cylindrical Coordinates, Spherical Coordinates

The Calculus II notes/tutorial assume that you've got a working knowledge Calculus I, including Limits, Derivatives, and Integration (up to basic substitution). It is also assumed that you have a fairly good knowledge of Trig. Several topics rely heavily on trig and knowledge of trig functions.

Calculus III (Math 2415) [ Notes] [ Practice Problems] [ Assignment Problems] - Topics included in this set of notes/tutorial are :

Three Dimensional Coordinate System - Equations of Lines, Equations of Planes, Quadratic Surfaces, Functions of Multiple Variables, Vector Functions, Limits, Derivatives, and Integrals of Vector Functions, Tangent Vectors, Normal Vectors, Binormal Vectors, Curvature, Cylindrical Coordinates, Spherical Coordinates

Partial Derivatives - Limits, Partial Derivatives, Higher Order Partial Derivatives, Differentials, Chain Rule, Directional Derivatives, Gradient.

Applications of Partial Derivatives - Tangent Plane, Normal Line, Relative Extrema, Absolute Extrema, Optimization, Lagrange Multipliers.

Multiple Integrals - Iterated Integrals, Double Integrals, Double Integrals in Polar Coordinates, Triple Integrals, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Change of Variables, Surface Area.

Line Integrals - Vector Fields, Line Integrals With Respect to Arc Length, Line Integrals With Respect to x and y, Line Integrals of Vector Fields, Fundamental Theorem of Line Integrals, Conservative Vector Fields, Potential Functions, Green's Theorem, Curl, Divergence.

Surface Integrals - Parametric Surfaces, Surface Integrals, Surface Integrals of Vector Fields, Stokes' Theorem, Divergence Theorem.

The Calculus III notes/tutorial assume that you've got a working knowledge Calculus I, including limits, derivatives and integration. It also assumes that the reader has a good knowledge of several Calculus II topics including some integration techniques, parametric equations, vectors, and knowledge of three dimensional space.

Differential Equations (Math 3301) [ Notes] - Topics included in this set of notes/tutorial are :

First Order Differential Equations - Linear Equations, Separable Equations, Exact Equations, Equilibrium Solutions, Modeling Problems.

Second Order Differential Equations - Homogeneous and Nonhomogeneous Second Order Differential Equations, Fundamental Set of Solutions, Undetermined Coefficients, Variation of Parameters, Mechanical Vibrations

Laplace Transforms - Definition, Inverse Transforms, Step Functions, Heaviside Functions, Dirac-Delta Function, Solving IVP's, Nonhomogeneous IVP, Nonconstant Coefficient IVP, Convolution Integral.

Systems of Differential Equations - Matrix Form, Eigenvalues/Eigenvectors, Phase Plane, Nonhomogeneous Systems, Laplace Transforms.

Series Solutions - Series Solutions, Euler Differential Equations.

Higher Order Differential Equations - n th order differential equations, Undetermined Coefficients, Variation of Parameters, 3 x 3 Systems of Differential Equations.

Boundary Value Problems & Fourier Series - Boundary Value Problems, Eigenvalues and Eigenfunctions, Orthogonal Functions, Fourier Sine Series, Fourier Cosine Series, Fourier Series.

Partial Differential Equations - Heat Equation, Wave Equation, Laplace's Equation, Separation of Variables.

These notes assume no prior knowledge of differential equations. A good grasp of Calculus is required however. This includes a working knowledge of differentiation and integration.

Algebra/Trig Review - This is an Algebra Review and Trig Review that was originally written for my Calculus I students. It it still geared mostly towards Calculus students with occasional comments on how a topic will be used in a Calculus class. However, anyone needing a review of some of the basic algebra, trig, exponential functions and logarithms should find the information of use.

Not all the topics covered in an Algebra or Trig class are covered in this review. I've mostly covered topics that are of particular importance to students in a Calculus class. I have included a couple of topics that are not that important to a Calculus class, but students do seem to have trouble with on occasion. As time permits I will be adding more sections as well.

The review is in the form of a problem set with the first solution containing detailed information on how to work that type of problem. Later solutions are usually not as detailed, but may contain more/new information as required.

Complex Number Primer - This is a brief introduction to some of the basic ideas involved with Complex Numbers. The topics covered are a brief review of arithmetic with complex numbers, the complex conjugate, modulus, polar and exponential form and computing powers and roots of complex numbers.

Note that this primer does assume that you've at least seen some complex numbers prior to reading. The purpose of this document is go a little beyond what most people see when the first are introduced to complex numbers in say a College Algebra class. Also, this document is in no way intended to be a complete picture of complex numbers nor do I cover all the concepts involved (that's a whole class in and of itself).

Common Math Errors - As with the Algebra/Trig review this was originally written for my Calculus I class. However, only one of the five sections that I've given here directly addresses the topic of Calculus. The other four sections are more general errors or cover Algebra and Trig errors. There are a couple of calculus examples in the first four sections, but in all of these cases I've also tried to provide non Calculus examples as well.

This portion of the site should be of interest to anyone looking for common math errors. If you aren't in a Calculus class or haven't taken Calculus you should just ignore the last section.

How To Study Math - This is a short section with some advice on how to best study mathematics.

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Welcome to my math notes site. Contained in this site are the notes (free and downloadable) that I use to teach Algebra, Calculus (I, II and III) as well as Differential Equations at Lamar University. The notes contain the usual topics that are taught in those courses as well as a few extra topics that I decided to include just because I wanted to.

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Desmos is a quick, free graphing calculator available to students at any given time! It allows for students to visualize equations and graphs in the xy-plane.

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Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.

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Free online 3D grapher from GeoGebra: graph 3D functions, plot surfaces, construct solids and much more!

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Uh oh! Wolfram|Alpha doesn't run without JavaScript.

Please enable JavaScript. If you don't know how, you can find instructions . Once you've done that, refresh this page to start using Wolfram|Alpha.

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Wolfram|Alpha: Making the world's knowledge computable

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Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music...

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Symbolab Math Calculator - Step by Step calculator

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Symbolab: equation search and math solver - solves algebra, trigonometry and calculus problems step by step

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Free math problem solver answers your calculus homework questions with step-by-step explanations.

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Limitless Mind: Learn, Lead, and Live Without Barriers

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Limitless Mind: Learn, Lead, and Live Without Barriers

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I really enjoyed this read. While it is not specifically marketed as a math education book, the author, Jo Boaler, is a education professor at Stanford and was a math education student herself and much of the material within this book relate back to math ed as a result of those consequences. A great read!

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Neuroplasticity - the reason our brains are not 'fixed'.

When we let go of the idea that our brains are fixed, we stop believing that our genetics determine our lives' pathways, and instead learn that our brains are incredibly adaptable. The knowledge that every time we learn something new our brains change comes from perhaps the most important research of this decade 'brain plasticity' also known as ' neuroplasticity'.

If you want your child to adopt growth mindset, avoid labeling them as 'smart'.

Starting at the age of three, children are prone to adopting a damaging fixed mindset from a seemingly innocuous word that's used everywhere - the word 'smart'. Parents regularly praise their children by telling how smart they're in order to build up their confidence. Unfortunately, children start to think 'Oh good, I'm smart', but then later when they struggle or fail, as everyone does at some point, they start to change their thinking 'Oh, I'm not so smart after all'. And so, they end up constantly evaluating themselves against the fixed idea.

Don't get this wrong. It's fine to praise people, but always praise for what they did and not them as people.

Instead of praising "You solved that tricky problem like that? That is so smart! say "I loved your solution to the problem; It's so creative."

Carol Dweck is a pioneering researcher in the field of motivation, why people succeed (or don't) and how to foster success. One of Carol Dweck's most famous studies gave two groups of students a challenging task. On completion, one group was praised for being 'really smart', while the other was praised for working hard. Both groups were then offered a choice between two follow-up tasks, one that's easy and one that's challenging. 90% of students praised for working hard chose the challenging task whereas majority of those praised for being smart chose the easy task so they can continue to look smart.

If your student or team become disillusioned because others are ahead of them, champion one word - "Yet".

You are not terrible at drawing. You just haven't learned to draw well yet. This may seem like a small linguistic change but it's important one. It moves the focus from the perceived personal lack to progress of learning.

Your student or team might ultimately fail on a problem but praise the correct thinking/effort.

'False growth mindset' thinking is telling students or your team to simply try harder and praising them for effort even if they fail. This backfire because people know praise is a consolation prize. Instead you should praise the learning - the effort that led to some result that could be used to build on moving forward.

Key Message #1 Every time we learn, our brains form, strengthen or connect neural pathways. We need to replace the idea that learning ability is fixed, with recognition that we're all on a growth journey.

Researchers found brains are literally growing new connections and pathways as the adults study, learn and when the pathways are no longer needed, they faded away (this is why continuous learning and applying what you learn is important).

Researches now know when we learn something, we grow the brain in 3 ways:

1. New pathway is formed.

2. Pathway that's already there is strengthened.

3. A connection is formed between two previously unconnected pathways.

In another study, 'learning disabled' students developed their brains to an extent that allowed them to function in almost the same way as 'regular performers'.

Many of us have grown feeling judged for everything, often feeling not good enough and worrying about being found out. When people let go of fixed mindsets, they become liberated. When we face a challenging uncertain situation rather than turning away because of fear of not being good enough, we should dive in, knowing that the situation presents opportunities for brain growth.

"If I grew up in a world where no one was labeled as gifted, I would have asked a lot more questions."

Key Message #2 The times when we are struggling and making mistakes are the best times for bran growth.

Brain researches show increased brain activity when people struggle and make mistakes and decreased activity when they get work correct. Unfortunately, most learners still think they should always be getting right the first time, and many feel if they struggle, they're not good learners when this is the very best thing they can be doing.

Understand that world-class experts - pianists, chess players, novelists, athletes - practiced for around ten thousand hours over twenty years. Success isn't related to tests of intelligence but to amount of deliberate practice they undertook.

When you're in the valley, you're in the deep dark trenches of change. Honor that time, work through it and someday you'll be on the mountain looking back and be grateful.

Key Message #3 When we change our beliefs, our bodies and brains physically change as well.

Researches found those people who thought they were doing more exercise were actually healthier than those who thought they were doing less, even when the amount of exercise was the same. The difference between negative and positive thinkers is incredible. Negative thinkers are 71% more likely to die in the follow-up period than those who thought positively about their exercise.

The mind has remarkable power over the body and its muscles. Researchers again found strength can be improved by mental training without moving your arms and legs. You might think 'Cool then we no longer need to go to the gym', and you are in part correct. When our brain imagines in a focused way the development of muscles, they actually strengthen through the development of enhanced signaling in the brain.

Two siblings lined up their toys but in a moment of excitement, the little girl fell of the bed. Her brother heard a crash and peered over to see his sister landed on her hands and knees. Worrying she might be hurt and that she was about to erupt in a loud wail that would wake up their parents, he said "Amy did you see how you landed? No human lands on all fours like that. You... you're a unicorn!" We don't always have to be an older brother, instead we have knowledge of ways to handle failure to develop a positive mindset and creativity in solving problems.

Key Message #4 Neural pathways and learning are optimized when considering ideas with a multidimensional approach.

Effort is key but it's not the only thing. You need to try new strategies and seek input from others when you're stuck. Despite Einstein's incredible achievements, he was quick to point they didn't come from a gift or special talent but dedication and hard work. In his words:

"A person who never made a mistake never tried something new. It's not that I'm smart. It's just that I stay with problems longer. I have no special talent. I'm only passionately curious. In the middle of difficulty lies opportunity."

Einstein's brain which sits on microscope slides in a museum in Philadelphia has been examined for special qualities. Many visitors have stared at his brain and not noticed anything remarkable, but researchers found something interesting. What's different in the brains of people who are outliers in their fields is they have more active connections between different brain areas, and more flexibility in their thinking. It's not something they're born with. It's something they developed through learning.

Fold a diamond paper this way:

Put the problem in the center of the diamond and use the four quadrants to invite different forms of thinking.

Key Message #5 Speed of thinking is not a measure of aptitude. Learning is optimized when we approach ideas and life with creativity and flexibility.

Adam Grant is a renowned American organizational psychologist. In one of his studies, Grant concludes:

"Although we rely on them (gifted and geniuses) to keep the world running smoothly, they keep us running on a treadmill". Those who go on the change the world are creative and flexible thinkers, people who think outside the box.

Key Message #6 Connecting with people and ideas enhances neural pathways and learning.

If you aren't getting push backs, you're probably not being disruptive enough. Whoever made a difference by being like everyone else? New ideas will never be easy for some people to accept yet they're very important. Ideas that are hardest for people to accept are those that go against the status quo and they may be the most important of all.

One of the teachers found two kids in his class disagreeing with each other. In their words: "I think I know what you were thinking, but it's really this...". "Oh yeah, that is what I was thinking". And those are first graders!

Thinking is inherently social. Even when we read alone, we're interacting with the author's thoughts. It's very important, perhaps the essence of learning when we develop the capacity to connect with another person's ideas to build them into our thinking.

Don't fall prey to the thinking that you do less as you grow older.

As people age, many start to believe that they're capable of less and this in turn influences important decisions they make late in their lives. Researchers have found elderly people who pursued more leisure activities have a 38% lower risk of developing dementia. They are also happier and live longer in general than those who fall prey to limiting beliefs. So instead of retiring to a life of minimal activity, we really should fill our retirement years with new challenges and learning opportunities.

Above all, remember that when you fall from the metaphorical bunk bed, which we all do from time to time, don't despair. Say out loud "I'm a unicorn!" and get back to work.

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Neuroplasticity - the reason our brains are not 'fixed'. When we let go of the idea that our brains are fixed, we stop believing that our genetics determine our lives' pathways, and instead learn that our brains are incredibly adaptable.

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The Math Myth: And Other STEM Delusions [Hacker, Andrew] on Amazon.com. *FREE* shipping on qualifying offers. The Math Myth: And Other STEM Delusions

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Reading the World with Maths: Goals for a Criticalmathematical Literacy Curriculum

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Reading the World with Maths: Goals for a Criticalmathematical Literacy Curriculum

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Reading the World with Maths: Goals for a Criticalmathematical Literacy Curriculum

Marilyn Frankenstein

University of Massachusetts, USA
In this paper I suggest ways that teachers can introduce mathematics as a tool to interpret and challenge inequities in our society. The teaching methods suggested also make maths more accessible and applicable because it is learned in the context of real-life, meaningful experiences. This suggestion is particularly useful for teachers who are creating an interdisciplinary math and social studies curriculum. The examples are based on my work teaching at the College of Public and Community Service, Umass/Boston. My students are primarily working-class adults who did not receive adequate mathematics instruction when they were in high school. Many of them were tracked out of college preparation. Therefore, the ideas presented in this article can be applied to the secondary classroom.
When my students examine data and questions such as the one shown in the box below they are introduced to the four goals of the criticalmathematical literacy curriculum.
1 Understanding the mathematics.
2 Understanding the mathematics of political knowledge.

3 Understanding the politics of mathematical knowledge.

4 Understanding the politics of knowledge.
Clearly, calculating the various percentages for the unemployment rate requires goal number one, an understanding of mathematics. Criticalmathematical literacy goes beyond this to include the other three goals mentioned above. The mathematics of political knowledge is illustrated here by reflecting on how the unemployment data deepens our understanding of the situation of working people in the United States. The politics of mathematical knowledge involves the choice of who counts as unemployed. In class, I emphasize that once we decide which categories make up the numerator (number of unemployed) and the denominator (total labor force), changing that fraction to a decimal fraction and then to a percent does not involve political struggle - that involves understanding the mathematics. But, the decision of who counts where does involve political struggle - so the unemployment rate is not a neutral description of the situation of working people in the United States. And, this discussion generalizes to a consideration of the politics of all knowledge.

Example: Unemployment Rate

In the United States, the unemployment rate is defined as the number of people unemployed, divided by the number of people in the labor force. Here are some figures from December 1994. (All numbers in thousands, rounded off to the nearest hundred thousand).

. In your opinion, which of these groups should be considered unemployed? Why?

. Which should be considered part of the labor force? Why?

. Given your selections, calculate the unemployment rate in 1994.

1. 101,400: Employed full-time

2. 19,000: Employed part-time, want part-time work

3. 4,000: Employed part-time, want full-time work

4. 5,600: Not employed, looked for work in last month, not on temporary layoff

5. 1,100: Not employed, on temporary layoff

6. 400: Not employed, want a job now, looked for work in last year, stopped looking because discouraged about prospects of finding work

7. 1,400: Not employed, want a job now, looked for work in last year, stopped looking for other reasons

8. 60,700: Not employed, don't want a job now (adults)

For discussion: The US official definition counts 4 and 5 as unemployed and 1 through 5 as part of the labor force, giving an unemployment rate of 5.1%. If we count 4 through 8 plus half of 3 as unemployed, the rate would be 9.3%. Further, in 1994 the Bureau of Labor Statistics stopped issuing its U-7 rate, a measure which included categories 2 and 3 and 6 through 8, so now researchers will not be able to determine "alternative" unemployment rates (Saunders, 1994).

In this article, I will develop the meaning of each of these goals, focusing on illustrations of how to realize them in their interconnected complexity. Underlying all these ideas is my belief that the development of self-confidence is a prerequisite for all learning, and that self-confidence develops from grappling with complex material and from understanding the politics of knowledge.
Almost all my students know how to do basic addition, subtraction, multiplication and division, although many would have trouble multiplying decimal fractions, adding fractions or doing long division. All can pronounce the words, but many have trouble succinctly expressing the main idea of a reading. Almost all have trouble with basic math word problems. Most have internalized negative self-images about their knowledge and ability in mathematics. In my beginning lessons I have students read excerpts where the main idea is supported by numerical details and where the politics of mathematical knowledge is brought to the fore. Then the curriculum moves on to the development of the Hindu-Arabic place-value numeral system, the meaning of numbers, and the meaning of the operations.
I start lessons with a graph, chart, or short reading which requires knowledge of the math skill scheduled for that day. When the discussion runs into a question about a math skill, I stop and teach that skill. This is a non-linear way of learning basic numeracy because questions often arise that involve future math topes. I handle this by previewing. The scheduled topic is formally taught. Other topics are also discussed so that students' immediate questions are answered and so that when the formal time comes for them in the syllabus, students will already have some familiarity with them. For example, if we are studying the meaning of fractions and find that in 1985, 2/100 of the Senate were women, we usually preview how to change this fraction to a percent. We also discuss how no learning is linear and how all of us are continually reviewing, recreating, as well as previewing in the ongoing process of making meaning. Further, there are other aspects about learning which greatly strengthen students' understanding of mathematics:
(A) breaking down the dichotomy between learning and teaching mathematics;
(B) considering the interactions of culture and the development of mathematical knowledge; and

(C) studying even the simplest of mathematical topics through deep and complicated questions.

These are explained in more detail below.
When students teach, rather than explain, they learn more mathematics, and they also learn about teaching. They are then empowered to proceed to learn more mathematics. As humanistic, politically concerned educators, we often talk about what we learn from our students when we teach. Peggy McIntosh (1990) goes so far as to define teaching as "the development of self through the development of others." Certainly when we teach we learn about learning. I also introduce research on math education so that students can analyze for themselves why they did not previously learn mathematics. I argue that learning develops through teaching and through reflecting on teaching and learning. So, students' mathematical understandings are deepened when they learn about mathematics teaching as they learn mathematics. Underlying this argument if Paulo Freire's concept that learning and teaching are part of the same process, and are different moments in the cycle of gaining existing knowledge, re-creating that knowledge and producing new knowledge (Freire, 1982).
Students gain greater control over mathematics problem-solving when, in addition to evaluating their own work, they can create their own problems. When students can understand what questions it makes sense to ask from given numerical information, and can identify decisions that are involved in creating different kinds of problems, they can more easily solve problems others create. Further, critical mathematical literacy involves both interpreting and critically analyzing other people's use of numbers in arguments. To do the latter you need practice in determining what kinds of questions can be asked and answered from the available numerical data, and what kinds of situations can be clarified through numerical data. Freire's concept of problem-posing education emphasizes that problems with neat, pared down data and clear-cut solutions give a false picture of how mathematics can help us "read the world". Real life is messy, with many problems intersecting and interacting. Real life poses problems whose solutions require dialogue and collective action. Traditional problem-solving curricula isolate and simplify particular aspects of reality in order to give students practice in techniques. Freirian problem-posing is intended to reveal the inter-connections and complexities of real-life situations where "often, problems are not solved, only a better understanding of their nature may be possible" (Connolly, 1981). A classroom application of this idea is to have students create their own reviews and tests. In this way they learn to grapple with mathematics pedagogy issues such as: what are the key concepts and topics to include on a review of a particular curriculum unit? What are clear, fair and challenging questions to ask in order to evaluate understanding of those concepts and topics?
This aspect is best described with the following example.
Example: When we are learning the algorithm for comparing the size of numbers, I ask students to think about how culture interacts with mathematical knowledge in the following situation:
At first students try to explain the boy's answer by hypothesizing that each of the girl's links was smaller than each of the boy's or that she built a wider, shorter tower. But after reading the information, they see that this could not be the case, since the girl's answer was correct. We speculate about how the culture of sexism - that boys always do better or have more than girls - blocked the knowledge of comparing sizes that the boy clearly understood in a different situation.
Most educational materials and learning environments in the United States, especially those labelled as "developmental" or "remedial", consist of very superficial, easy work. They involve rote or formulaic problem-solving experiences. Students get trained to think about successful learning as getting high marks on school or standardized tests. I argue that this is a major reason that what is learned is not retained and not used. Further, making the curriculum more complicated, where each problem contains a variety of learning experiences, teaches in the non-linear, holistic way in which knowledge is developed in context. This way of teaching leads to a more clear understanding of the subject matter.
Example: In the text below, Sklar and Sleicher demonstrate how numbers presented out of context can be very misleading. I ask students to read the text and discuss the calculations Sklar and Sleicher performed to get their calculation of the US expenditure on the 1990 Nicaraguan election. ($17.5 million - population of Nicaragua = $5 per person.) This reviews their understanding of the meaning of the operations. Then I ask the students to consider the complexities of understanding the $17.5 million expenditure. This deepens their understanding of how different numerical descriptions illuminate or obscure the context of US policy in Nicaragua, and how in real-life just comparing the size of the numbers, out of context, obscures understanding.
On the basis of relative population, Holly Sklar has calculated that the $17.5 million US expenditure on the Nicaraguan election is $5 per person and is equivalent to an expenditure of $1.2 billion in the United States. That's one comparison all right, but it may be more relevant to base the comparison on the effect of the expenditure on the economy or on the election, i.e. to account for the difference in per capita income, which is at least 30/1 or an equivalent election expenditure in the United States of a staggering $30 billion! Is there any doubt that such an expenditure would decisively affect a US election? (Sleicher, 1990). I argue , along with Freire (1970) and Freire and Macedo (1987), that the underlying context for critical adult education, and criticalmathematical literacy, is "to read the world". To accomplish this goal, students learn how mathematics skills and concepts can be used to understand the institutional structures of our society. This happens through: a. understanding the different kinds of numerical descriptions of the world (such as fractions, percents, graphs) and the meaning of the sizes of numbers, and
b. using calculations to follow and verify the logic of someone's argument, to restate information, and to understand how raw data are collected and transformed into numerical descriptions of the world. The purpose underlying all the calculations is to understand better the information and the arguments and to be able to question the decisions that were involved in choosing the numbers and the operations.

It may be lonely at the top, but it can't be boring - at least not with all that money. Last week the federal government released figures showing that the richest 1 percent of American households was worth more than the bottom 90 percent combined. And while these numbers were widely reported, we found them so shocking that we thought they were worth repeating. So here goes: In 1989 the top 1 percent of Americans (about 934,000 households) combined for a net worth of $5.7 trillion; the bottom 90 percent (about 84 million households) could only scrape together $4.8 trillion in net worth.

Example: Students are asked to discuss how numbers support Helen Keller's main point and to reflect on why she sometimes uses fractions and other times uses whole numbers. Information about the politics of knowledge is included as a context in which to set her views.
Samples from time-and-motion studies, conducted by General Electric. Published in a 1960 handbook to provide office managers with standards by which clerical labor should be organized (Braverman, 1974).
Open and close Minutes
Open side drawer of standard desk 0.014

Open center drawer 0.026

Close side drawer 0.015

Get up from chair 0.039
Sit down in chair 0.033
Turn in swivel chair 0.009
Example: I ask students to create and solve some mathematics problems using the information in the following article (In These Times, April 29 - May 5, 1992). Doing the division problems implicit in this article deepens understanding of the economic data, and shows how powerfully numerical data reveal the structure of our institutions. Perhaps the most dramatic example of the politics involved in seemingly neutral mathematical descriptions of our world is the choice of a map to visualize that world. Any two-dimensional map of our three-dimensional Earth will, of course, contain mathematical distortions. The political struggle/choice centers around which of these distortions are acceptable to us and what other understandings of ours are distorted by these false pictures. For example, the map with which most people are familiar, the Mercator map, greatly enlarges the size of "Europe"(3) and shrinks the size of Africa. Most people do not realize that the area of what is commonly referred to as "Europe" is smaller than 20% of the area of Africa. Created in 1569, the Mercator map highly distorts land areas, but preserves compass direction, making it very helpful to navigators who sailed from Europe in the sixteenth century.Example: Students practice reading a complicated graph and solving multiplication and division problems in order to understand how particular payment structures transfer money from the poor to the rich.(2)The Rate Watcher's Guide (Morgan, 1980) details why under declining block rate structures, low-income citizens who use electricity only for basic necessities pay the highest rates, and large users with luxuries like trash compactors, heated swimming pools or central air-conditioning pay the lowest rates. A 1972 study conducted in Michigan, for example, found that residents of a poor urban area in Detroit paid 66% moreper unit of electricity than did wealthy residents of nearby Bloomfield Hills. Researchers concluded that "approximately $10,000,000 every year leave the city of Detroit to support the quantity discounts of suburban residents".Example: Students are asked to discuss what numerical understandings they need in order to decipher the following chart. They see that a recognition of how very small these decimal fractions are, so small that watches cannot even measure the units of time, illuminates the viciousness of time-motion studies in capitalist management strategies.
When used in textbooks and other media, combined with the general (mis)perception that size relates to various measures of so-called "significance", the Mercator map distorts popular perceptions of the relative importance of various areas of the world. For example, when a US university professor asked his students to rank certain countries by size they "rated the Soviet Union larger than the continent of Africa, though in fact it is much smaller" (Kaiser, 1991), associating "power" with size.

Political struggles to change to the Peter's projection, a more accurate map in terms of land area, have been successful with the United Nations Development Program, the World Council of Churches, and some educational institutions (Kaiser, 1991). However, anecdotal evidence from many talks I've given around the world suggest that the Mercator is still widely perceived as the way the world really looks.

As Wood (1992) emphasizes:
In a variety of situations, statistical descriptions don't simply or neutrally record what's out there. There are political struggles/choices involved in: which data are collected; which numbers represent the most accurate data; which definitions should guide how the data are counted; which methods should guide how the data are collected; which ways the data should be disaggregated; and which are the most truthful ways to describe the data to the public.
Example: Political struggle/choice over which definitions should guide how data are counted. In 1988, the US Census Bureau introduced an "alternative poverty line", changing the figure for a family of three from $9453 to $8580, thereby preventing 3.6 million people whose family income fell between those figures from receiving food stamps, free school meals and other welfare benefits. At the same time, the Joint Economic Committee of Congress argued that "updating the assessments of household consumption needs ... would almost double the poverty rate, to 24 percent" (Cockburn, 1989). Note that the US poverty line is startlingly low. Various assessments of the smallest amount needed by a family of four to purchase basic necessities in 1991 was 155% of the official poverty line. "Since the [census] bureau defines the [working poor] out of poverty, the dominant image of the poor that remains is of people who are unemployed or on the welfare rolls. The real poverty line reveals the opposite: a majority of the poor among able-bodied, non-elderly heads of households normally work full-time. The total number of adults who remain poor despite normally working full-time is nearly 10 million more than double the number of adults on welfare. Two-thirds of them are high school or college-educated and half are over 33. Poverty in the US is a problem of low-wage jobs far more than it is of welfare dependency, lack of education or work inexperience. Defining families who earn less than 155% of the official poverty line as poor would result in about one person in ever four being considered poor in the United States" (Schwartz and Volgy, 1993).

Example: Political struggle/choice over which ways data should be disaggregated. The US Government rarely collects health data broken down by social class. In 1986, when it did this for heart and cerebrovascular disease, it found enormous gaps:
"The death rate from heart disease, for example, was 2.3 times higher among unskilled blue-collar operators than among managers and professionals. By contrast, the morality rate from heart disease in 1986 for blacks was 1.3 times higher than for whites ... the way in which statistics are kept does not help to make white and black workers aware of the commonality of their predicament" (Navarro, 1991).
There are many aspects of the politics of knowledge that are integrated into this curriculum. Some involve reconsidering what counts as mathematical knowledge and re-presenting an accurate picture of the contributions of all the world's peoples to the development of mathematical knowledge. Others involve how mathematical knowledge is learned in schools. Winter (1991), for example, theorizes that the problems so many encounter in understanding mathematics are not due to the discipline's "difficult abstractions", but due to the cultural form in which mathematics is presented. Sklar (1993), for a different aspect, cites a US study that recorded the differential treatment of Black and White students in math classes.

And, of course, Freire (1970) theorizes about the politics of "banking education", when teachers deposit knowledge in students' empty minds.

Underlying all these issues are more general concerns I argue should form the foundation of all learning, concerns about what counts as knowledge and why. I think that one of the most significant contributions of Paulo Freire (1982) to the development of a critical literacy is the idea that:

In a non-trivial way we can learn a great deal from intellectual diversity. Most of the burning social, political, economic and ethical questions of our time remain unanswered. In the United States we live in a society of enormous wealth and we have significant hunger and homelessness; although we have engaged in medical and scientific research for scores of years, we are not any closer to changing the prognosis for most cancers. Certainly we can learn from the perspectives and philosophies of people whose knowledge has developed in a variety of intellectual and experiential conditions. Currently "the intellectual activity of those without power is always labelled non-intellectual" (Freire and Macedo, 1987). When we see this as a political situation, as part of our "regime of truth", we can realize that all people have knowledge, all people are continually creating knowledge, doing intellectual work, and all of us have a lot to learn.
(1) Thanks to my friend, Umass/Lowell economist Chris Tilly, for this problem. Understanding the Mathematics.
(2) This situation has changed in Massachusetts, which now has a flat rate structure, and my reference did not contain real data for Michigan. So although the context-setting data are real, the numbers used to understand the concept of declining block rates are realistic, but not real.

(3) Grossman (1994) argues that "Europe has always been a political and cultural definition. Geographically, Europe does not exist, since it is only a peninsula on the vast Eurasian continent." He goes on to discuss the history and various contributions of geographer's attempts to "draw the eastern limits of 'western civilization' and the white race" (p.39).

Braverman, H. (1974) Labour and monopoly capital, New York, Monthly Review Press. it does not follow that they cannot express very complex analyses of social, political, economic, ethical and other issues. And many people with an excellent grasp of reading, writing and mathematics skills need to learn about the world, about philosophy, about psychology, about justice and many other areas in order to deepen their understandings.Frankenstein, M. (1995) Equity in mathematics education: Class in the world outside the class, in Secada, W. G., Fennema, E. and Adajian, L. B. (Eds.) New directions for equity in mathematics education, pp165-190, Cambridge, UK, Cambridge University Press.
Frankenstein, M. (1997) Breaking down the dichotomy between teaching and learning mathematics, in Freire, P. J., Fraser, D., Macedo, et al (Eds.) Mentoring the mentor: a critical dialogue with Paulo Freire, pp59-88, New York, Peter Lang.

Freire, P. (1982) Education for critical consciousness. Unpublished Boston College course notes taken from Frankenstein, M. July 5-15.

Lerman, S. (1993) Personal communication with the author. March 26.

McIntosh, P. (1990) November 14. Interaction phases of personal and pedagogical change. Talk at Rutgers University/Newark, NJ.
Saunders, B. (1994) September 26. Number game, The Nation, 259.9:295-296.

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Reading the World with Maths: Goals for a Criticalmathematical Literacy Curriculum

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Reading the World with Maths: Goals for a Criticalmathematical Literacy Curriculum Marilyn Frankenstein University of Massachusetts, USA In this paper I suggest ways that teachers can introduce mathematics as a tool to interpret and challenge inequities in our society. The teaching methods suggested also make maths more accessible and applicable because it is learned in the context of real-life, meaningful experiences.

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Nctm Past President Cathy L. Seeley shares her messages on today's most relevant topics and issues in education. Based on Cathy L. Seeley's award-winning nctm President's Messages, and including dozens of new messages, this must-have k-12 resource offers straight talk and common sense about some of today's most important, thought-provoking issues in education. With topics ranging from the impact of rising expectations and the trap of timed tests to the role of technology and the phenomenon of jumping on bandwagons, this book provides a base for lively discussion among elementary, middle, and high school teachers; leaders; policy makers; and families. This book contains 41 messages included in three sections: (1) School Mathematics for the 21st Century: Elementary and Secondary Mathematics in America; (2) Great Ideas Whose Time Has Come (and Gone?): Mathematics Issues Facing Schools and Districts; and (3) Real Students and Real Teachers: Mathematics in Today's Classroom. This book also contains the following: (1) Foreword by Marilyn Burns; (2) Introduction; (3) How to Use This Book; (4) Afterword: The Sum of the Parts Is Greater than Some of the Parts; (5) Acknowledgments; (6) Readings and References; (7) Index; and (8) About the Author.

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Nctm Past President Cathy L. Seeley shares her messages on today's most relevant topics and issues in education. Based on Cathy L. Seeley's award-winning nctm President's Messages, and including dozens of new messages, this must-have k-12 resource offers straight talk and common sense about some of today's most important, thought-provoking issues in education.

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Creating a magazine gives students a chance to learn about real-world applications of math.

"When will I ever use this?" If you're a math teacher, this is a question you're very familiar with. As the math curriculum becomes ever more abstract in middle and high school, it's difficult for students to see when and where the material they're learning will be applicable.

Just telling them that the material will be useful for their following math courses or future academic career doesn't cut it-and it shouldn't. This is when the " unreasonable effectiveness of mathematics," as explored through reading and writing, is the ace up the sleeves of teachers.

Reading and writing are complex, fundamental, integrative learning skills that should be used to their potential in math class. Reading is a two-part process that is similar to mathematics in that it requires first the transfer of encoded information to the reader and then the comprehension of that information. And writing is one of the most powerful modes of learning because it engages both hemispheres of the brain -one side generates ideas, the other structures them.

Effective writing also clarifies and organizes a student's thoughts, and the slow pace of writing is conducive to student learning because it allows them to reason carefully to make sure they're correct before they state their thoughts. Studies have shown that writing is valuable specifically for the math classroom-for example, it seems that a student's ability to explain concepts in writing is related to the ability to comprehend and apply them.

And of course written explanations also allow the teacher to understand and assess the student's thinking in a way that computational steps alone may not provide.

Writing in Math Class

Although writing prompts for the elementary-and less frequently for the middle school-math classroom have been available for some time, they generally focus on attitudes and dispositions, likes and dislikes, and thoughts, concerns, and feelings about math or math class. They include journals, math autobiographies, letters to the teacher, and freewriting activities. They involve writing about a concept or a process the students are studying, or center on metacognitive skills such as effort, goals, expectations, study habits, and so on.

These prompts can be adapted for high school students, but I've found it more useful to use reading and writing to illustrate how mathematics is embedded in every aspect of life. Reading and writing are particularly effective in developing a quantitative understanding of the world around us because they can be used to lead students to reflect on everyday experiences. One assignment that I've found effective is having students create a magazine of excerpts of articles about various applications of mathematics in real life.

The magazine-I've included an issue here-is composed of summaries of published articles, so to get students started, I keep an updated list of articles that highlight the ways math is present in different facets of the real world-from the probability that a soccer team ends up in a particular World Cup group to gerrymandering, fractals in Jackson Pollock's paintings, black holes, the fairness of social media, invisibility cloaks, and more.

I select articles from a variety of newspapers, journals, and magazines, such as The Washington Post, The New York Times (especially the Upshot section), The Economist, New Scientist, Scientific American, and The Atlantic, as well as from blogs, websites, podcasts, videos, and other online resources. The app MathFeed ( downloadable for free in the Apple App Store), curated by mathematician Francis Su, is my most valued and trusted source-it allows me to find the most important news and views about math in the media in a single place.

After picking an article that has piqued their interest-from the list I provide or other sources of their choice-the students read it carefully, summarize it, and post it on the online magazine. (If they want to work with an article they have found, they have to ask me first so we don't end up with repetitions in the magazine.) They know they're required to understand the article content properly, edit their summary thoroughly, provide a visual to aid their readers' comprehension, and cite their source.

I give them access to the draft of the magazine, and they also have to upload their article and take care of any technical issues that may arise.

For assessment, I've created a simple rubric that looks at content understanding, clarity of communication, editing, critical thinking, initiative, and creativity.

This has been one of my most popular projects-students are amazed to discover some of the myriad applications of math. They also enjoy being able to choose articles according to their interests, and appreciate that the end product, the magazine, can be easily shared with family and friends and stored in their own e-portfolios.

This is a valuable activity that encourages students who enjoy reading and writing more than the computational side of math and gives all a deeper understanding and stronger appreciation of the usefulness and effectiveness of mathematics.

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Creating a magazine gives students a chance to learn about real-world applications of math.

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Teacher Desmos is a free site available to students to make interactive activities for their students! With Teacher Desmos, you are able to have students connect to an activity (as they would a Kahoot) and carry through various activities such as graphing, drawing, etc. different math concepts and topics.

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With this source, teachers are able to preview mathematics curriculum in a variety of high school math topics. They are able to access prep, lessons, and practice for topics and can also receive student lessons for these topics as well.

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IM 9-12 Math™ certifiedby Illustrative Mathematics® Meets all expectations in all three EdReports Gateways Ready to get started with IM 9-12 Math™ ? IM 9-12 MATH DESIGN FEATURES A coherent progression of learning IM K-12 Math™ is fully aligned to the focus, rigor, and coherence of the standards, and has been given the stamp of approval by EdReports, earning perfect scores in the areas of Focus & Coherence and Rigor & Mathematical Practices.

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Aplus Math is a interactive source free to use that helps teachers create worksheets and includes flashcards and games. Teachers can find this source useful for their math worksheet section.

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Can Figure It helps students through geometric proofs simply by making a free account. It helps students visualize many geometric concepts and provides more geometry material for that student to access. (Teachers are also able to make classrooms in this site for their students to connect to)

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"CanFigureIt Geometry was exactly what I was looking for! It is an interactive way to complete proofs online, which is especially useful for remote learning and/or differentiated learning. One of the frustrating parts of teaching proofs is the lack of students having the opportunity for immediate feedback.

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AAA Math Provides free lessons for a variety of different math levels (available in Spanish as well!). They promote lessons that help all types of learners and include various interactive lessons.

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AAA Math features a comprehensive set of thousands of interactive arithmetic lessons.

There is no cost or registration required to practice your math on the AAAMath.com web site.

Unlimited practice is available on each topic which allows thorough mastery of the concepts.

A wide range of lessons (Kindergarten through Eighth grade level) enables learning or review to occur at each individual's current level.

Immediate feedback prevents practicing and learning incorrect methods, which is a common result of traditional homework and worksheets. Practice can continue as long as desired in a non-threatening format which helps build self-esteem and confidence.

Please feel free to try the lessons by clicking on one of the grades at the top or a topic area on the left side of the page.

Remember to add the site to your Favorite Places and please tell others about the site. It is an excellent way to learn or review math.

Note: Due to heavy traffic a page may not load properly. The same content is available at: AAAKnow.com, AAAStudy.com, 321Know.com and AAAMath.com

© Copyright 2011 J. Banfill. All Rights Reserved.Legal Notice

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AAA Math features a comprehensive set of interactive arithmetic lessons. Unlimited practice is available on each topic which allows thorough mastery of the concepts.

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This site includes a wide variety of resources for our students! Ranging from math lessons to calculators to games, there are a lot of resources for our students to take advantage of!

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Math explained in easy language, plus puzzles, games, worksheets and an illustrated dictionary. For K-12 kids, teachers and parents.

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Math Planet is practically a free textbook on high school math topics ranging from pre-algebra to geometry. There is a compilation of many useful topics as well as a linked video lesson to the topics for students if they desire.

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Math planet is an online resource where one can study math for free. Take our high school math courses in Pre-algebra, Algebra 1, Algebra 2 and Geometry. We have also prepared practice tests for the SAT and ACT. We have recently launched a basic programming course in Python language for beginners.

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Mathplanet.com is an online resource where one can study math for free. Take our high school math courses in Pre-algebra, Algebra 1, Algebra 2 and Geometry. We have also prepared practice tests for the SAT and ACT.

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Math Central is a cumulative math resource provided in three different languages! This site includes a "resource room", a "teacher bulletin board", and a database with numerous problems available for free to anyone who wishes to access it.

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Recursos de enseñanza para maestros de matemáticas, un servicio de pregunta y respuesta para maestros y alumnos, un problema del mes, y mucho más