Traditional mathematics

Traditional mathematics (sometimes classical math education) was the predominant method 

of mathematics education in the United States in the early-to-mid 20th century. This contrasts with

 non-traditional approaches to math education.Traditional mathematics education has been challenged

 by several reform movements over the last several decades, notably new math, a now largely

 abandoned and discredited set of alternative methods, and most recently reform mathematics

 (i.e. Standards-based mathematics) based on NCTM standards, which is federally supported and

 has been widely adopted, but subject to ongoing criticism

Traditional method

The topics and methods of traditional mathematics are well documented in books 

and open source articles of many nations and languages. Major topics covered include:

• Elementary arithmetic
• Addition
• Carry
• Subtraction
• Multiplication
• Multiplication table
• Division
• Long division
• Arithmetic with fractions
• Lowest common denominator
• Arithmetic mean
• Volume

In general, traditional methods are based on direct instruction where students are shown

 one standard method of performing a task such as decimal addition, in a standard sequence. A task

 is taught in isolation rather than as only a part of a more complex project. By contrast, reform books 

may omit or even discourage standard methods of division or fractions. The formula for the

 area of a circle, for example, may be left as a discovery exercise rather than printing the formula anywhere

 in the student text where it could be quickly referenced. A page on adding fractions would ask "devise 

a method of adding that applies to these examples" rather than actually giving instructions

 to convert to a common denominator and then add the numerators. A traditional sequence early in the

 20th century would leave topics such as algebra or geometry for high school, and statistics until college,

 but newer standards require coverage of such concepts in the elementary grades. In the 21st century basic 

geometric concepts and "algebraic thinking" (number patterns) are taught every year in the United States 

beginning in kindergarten; children are taught statistical concepts of organizing data as early as first grade.

 More sophisticated concepts such as algebraic expressions with numbers and letters, geometric surface 

area and statistical means and medians occur in sixth grade in the newest standards.

Criticism of traditional math[edit]

See also: Math Wars and Reform mathematics

Criticism of traditional mathematics instruction originates with advocates of alternative methods of instruction,

 such as Reform mathematics. These critics cite studies, such as The Harmful Effects of Algorithms in Grades

 1–4, which found specific instances where traditional math instruction was less effective than alternative methods

. Advocates of alternative methods argue that traditional methods of instruction over-emphasize memorization and

 repetition, and fail to promote conceptual understanding or to present math

 as creative or exploratory. Critics also cite the fact that history of mathematics often focuses on European 

advancements and methods developed by men, thus ignoring equity issues and potentially alienating minorities

 and women.

Beyond just misplaced emphasis, critics contend that traditional mathematics instruction is ineffective, 

implying that a reformed math curriculum might be more effective. Critics note that only a small percentage of 

students achieve the highest levels of mathematics achievement such as calculus. Some argue that too few 

students master even algebra.

The general consensus of large-scale studies that compare traditional mathematics with reform mathematics 

is that students in both curricula learn basic skills to about the same level as measured by traditional 

standardized tests, but the reform mathematics students do better on tasks requiring conceptual

 understanding and problem solving.

The use of calculators became common in United States math instruction in the 1980s and 1990s. Critics 

have argued that calculator work, when not accompanied by a strong emphasis on the importance of showing 

work, allows students to get the answers to many problems without understanding the math involved. However,

 others such as Conrad Wolfram argue for a more radical use of computer-based math in a complete 

departure from traditional math.

Mathematics educators, such as Alan Schoenfeld, question whether traditional mathematics actually teach

 mathematics as understood by professional mathematicians and other experts. Instead, Schoenfeld implies, 

students come to perceive mathematics as a list of disconnected rules that must be memorized and 

parroted. Indeed, research suggests that certain approaches to traditional mathematics instruction

 impresses upon students an image of mathematics as closed to imagination and discovery, an image

 in clear opposition to how experts view the field.

Traditional mathematics texts[edit]

In general, math textbooks which focus on instruction in standard arithmetic methods can be categorized as a traditional math textbook. Reform math textbooks will often focus on conceptual understanding, usually avoiding immediate instruction of the standard algorithms and frequently promoting student exploration and discovery of the relevant mathematics. The following current texts are often cited as good for those wishing for a traditional approach, often also favored by homeschoolers.

• Saxon math
• Modern Curriculum Press

Four habits of highly effective math teaching

If you were asked what were the most important principles in mathematics teaching, what would you say? I wasn't really asked, but I started thinking, and came up with these basic habits or principles that can keep your math teaching on the right track.

Habit 1: Let It Make Sense
Habit 2: Remember the Goals
Habit 3: Know Your Tools
Habit 4: Living and Loving Math

Habit 1: Let It Make Sense

Let us strive to teach for understanding of mathematical concepts and procedures, the "why"something works, and not only the "how".

This understanding, as I'm sure you realize, doesn't always come immediately. It may take even several years to grasp a concept. For example, place value is something kids understand partially at first, and then that deepens over a few years.

This is why many math curricula use spiraling: they come back to a concept the next year, and the next. And this can be very good if not done excessively (like for 5-6 years is probably excessive).

However, spiraling also has its own pitfalls: if your child doesn't get a concept, don't blindly "trust" the spiraling and think, "Well, she gets it the next year when the book comes back around to it."

The next year's schoolbook won't necessarily present the concept at the same level - the presentation might be too difficult. If a child doesn't "get it", they might need a very basic instruction for the concept again.

The "how" something works is often called procedural understanding: the child knows how to work long division, or the procedure of fraction addition or fraction division, for example. It is often possible to learn the "how" mechanically without understanding why it works. Procedures learned this way are often forgotten very easily.

The relationship between the "how" and the "why" - or between procedures and concepts - is complex. One doesn't always come totally before the other, and it also varies from child to child. And, conceptual and procedural understanding actually help each other: conceptual knowledge (understanding the "why") is important for the development of procedural fluency, while fluent procedural knowledge supports the development of further understanding and learning.

Try alternating the instruction: teach how to add fractions, and let the student practice. Then explain why it works. Go back to some practice. Back and forth. Sooner or later it should 'stick' - but it might be next year instead of this one, or after 6 months instead of in this month.

As a rule of thumb, don't totally leave a topic until the student both knows "how", and understands the "why".

Tip: you can often test a student's understanding of a topic by asking HIM to produce an example, preferably with a picture or other illustration: "Tell me an example of multiplying fractions by whole numbers, and draw a picture." Whatever gets produced can tell the teacher a lot about what has been understood.

---

Habit 2: Remember the Goals

What are the goals of your math teaching? Are they...

• to finish the book by the end of school year
• make sure the kids pass the test ...?

Or do you have goals such as:

• My student can add, simplify, and multiply fractions
• My student can divide by 10, 100, and 1000.

These are all just "subgoals". But what is the ultimate goal of learning school mathematics?

Consider these goals:

• Students need to be able to navigate their lives in this ever-so-complex modern world.
  This involves dealing with taxes, loans, credit cards, purchases, budgeting, shopping. Our youngsters need to be able to handle money wisely. All that requires good understanding of parts, proportions, and percents.
• Another very important goal of mathematics education as a whole is to enable the students to understand information aroud us. In today's world, this includes quite a bit of scientific information. Being able to read through it and make sense of it requires knowing big and small numbers, statistics, probability, percents.
• And then one more. We need to prepare our students for further studies in math and science. Not everyone ultimately needs algebra, but many do, and teens don't always know what profession they might choose or end up with.
• I'd like to add one more broad goal of math education: teaching deductive reasoning. Of course geometry is a good example of this, but when taught properly, other areas of school math can be as well.
• Then one more goal that I personally feel fairly strongly about: let students see some beauty of mathematics and learn to like it, or at the very least, make sure they do not feel negatively about mathematics.

The more you can keep these big real goals in mind, the better you can connect your subgoals to them. And the more you can keep the goals and the subgoals in mind, the better teacher you will be.

For example, adding, simplifying, and multiplying fractions all connects with a broader goal of understanding parts or part and whole. It will soon lead to ratios, proportions, and percent. Also, all fraction operations are a needed basis for solving rational equations and doing the operations with rational expressions (during algebra studies).

Tying in with the goals, remember that the BOOK or CURRICULUM is just a tool to achieve the goals -- not a goal in itself. Don't ever be a slave to any math book.

---

Habit 3: Know Your Tools

Math teacher's tools are quite numerous nowadays.

First of all of course comes a black or white board, or paper - something to write on, pencil, compass, protractor, ruler, eraser.
And the book you're using.
Then we also have computer software, animations and activities online, animated lessons and such.
There are workbooks, fun books, worktexts, online texts.
Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on.
And then there are games, games, games.

The choices are so numerous it's daunting. What's a teacher to do?

Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.

There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity to spice up your math lessons.

Basic tools

1. The board and/or paper to write on. Essential. Easy to use.


2. The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. Check my curriculum pages for some help. Two things to keep in mind:
   

   i) Now matter what book you're using, YOU as the teacher have the control. Don't be a slave to the curriculum. You can skip pages, rearrange the order in which to teach the material, supplement it, and so on.
   
   ii) Don't despair if the book you're using doesn't seem to be the perfect choice for your student. You can quite likely sell it on homeschool swap boards, and buy some other one.

3. Manipulatives. I once saw a question asked by a homeschooling parent, on the lines, "What manipulatives must I use and when?" The person was under the impression that manipulatives are a 'must' thing.
   Manipulatives are definitely emphasized in these days. They are usually very good, but they're not the end goal of math education, and there is no need to go hog wild over them.
   Manipulatives are something the student manipulates with his hands to get a better grasp of something. But the goal is to learn to do math without them.
   Some very helpful manipulatives are
   
   • A 100-bead basic abacus
   • something to illustrate hundreds/tens/ones place value. I made my daughter ten-bags by putting marbles into little plastic bags.
   • some sort of fraction manipulatives. You can just make pie models out of cardboard, even.
   Often, drawing pictures can take place of manipulatives, especially after the first few grades and on.
   Check out also some virtual manipulatives.
4. Geometry and measuring tools. These are pretty essential, I'd say. For geometry however,dynamic software can these days replace compass and ruler and easily be far better.

The extras

These are, obviously, too many to even start listing.

• Some game or games are good for drilling basic facts. Games are nice for about any topic. Here's one that I played with playing cards with my dd; and now she seems to have learned the sums that add to 10. And here's a game that's worth 1000 worksheets. Of course the internet is full of online math games.


• I would definitely use some math software if teaching graphing, algebra, or calculus. CheckMathProf for example, or Math Mechanixs. I've listed a few more here.


• If you're ready to add something new to your toolbox from the online world, try The Math Forum's MathTools - a library of technology tools, lessons, activities, and support materials. Check also my pages listing math games and activities online.

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Habit 4: Living and Loving Math

You are the teacher. You show the way - also with your attitudes, your way of life.

Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day?

And then: do you like math? Love it? Are you happy to teach it? Enthusiastic?

Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your kids a way of life, and if math is a natural part of it or not.

Math is not a drudgery, nor something just confined to math lessons.

Some ideas:

• Let it make sense. This alone can usually make math quite a difference and kids will stay interested.


• Read through some fun math books, such as Theoni Pappas books, or puzzle-type books. Get to know some interesting math topics besides just schoolbook arithmetic. And, there are lots of story books (math readers) to teach math concepts - see a lists by concepts here.


• Try including a bit about math history. This might work best in a homeschooling environment where there is no horrible rush to get through the thick book before the year is over. At LivingMath.net you can find a math history course designed for homeschoolers.


• When you use math in your daily life, explain how you're doing it, and include the children if possible. Figure it out together.

Miscellaneous Math Teaching tips

• The child needs to know the basic addition and multiplication facts very well, or she will have difficulties with fractions, decimals, etc.  These basic facts need to be known by heart.
  
  One of the best ways to start children with math is to have them skip-count up and down from a very young age.  Use a number line to show what the 'skips' or steps mean.  if your child can master the skip counting by twos, threes, fours, etc., she has learned a lot about addition and later on multiplication tables will be an easy fare!  See also this article How to drill multiplication tables.
• Use manipulatives and pictures in your teaching.  Almost all mathematical concepts can be illustrated with pictures, which can even take a place of concrete manipulatives.  For example, if you can condition your child to draw lots of pie pictures when studying fractions, he/she can learn to visualize fractions as 'pies'.  Then he/she won't make the addition mistake 1/2 + 1/4 = 2/6.  Also certain kind of software can take place of the manipulatives.
• In geometry have your child or children DRAW a lot.  See examples in the Math Mammoth Geometry ebook.
• When studying time, money, measuring, homeschoolers have an advantage since they can study those subjects in their natural settings.  Involve your child when you measure, count money, check the time.
• In middle school years, it's important to get familiar with functions, relations, and patterns - these develop algebraic thinking. Check this article about algebraic reasoning from MathCounts.org.
• If you need to know the whys and wherefores of some particular math topic deeper than the textbook tells you, check Dr. Math's archives.  The answers provided there are mathematically "sound doctrine", whereas math textbooks can contain all kinds of errors.

The Advantages of the Use of Computers in Education

Computers are useful tools in different field of applications. Computers are very much useful in the field of engineering and architecture specially in the aspects of design and calculations. These machines are also beneficial in their application in the field of business and economics as it allows users to store and retrieve large amount of data in a less amount of time.

Computers are now use in different fields of education. In the field of teaching in particular, the wide use of computers are evident in different school in elementary, high-schools, colleges and universities.

The use of computers in learning provide both advantages and disadvantages in the learning processes. Computers are advantageous in the sense that these machines teach more effectively in technical sense, they can reach and teach more students and kept students more focus with the subject.

The use of computer technology in learning allows the teacher to individualized the learning instructions as well as the technology grants the students the autonomy and making them to learn with their own.

With the access of the internet, it is also possible to teach those students or learners that are located in the remote or far places provided these places have internet signals. These technological capability make the teacher to reach as well as to teach more learners.

The internet access will also allow the students to get different educational resources from all over the world, thus making the world like a classroom of learning. It enables him or her to access different ideas as well as learnings and knowledge coming from abroad.

In teaching kids or young learners, the use of computers in the teaching processes is very much advantageous. The use of computer machines could catch the attention of the kids, making them attentive as well as participative in teaching and learning activities. Making use with these technology will allow the teachers to enhance his/her teaching style that could possibly enhance the learnings of the students.

The use of computer technology in education makes us prepared for the future. Computers as used in education allows the learners to learn modern tools and knowledge that will make him or her ready for the possible technological changes in the future.

Even though the computer technology and the internet provides advantages in the teaching and learning process, we cannot deny the fact that still these technology also provides disadvantages in the educational processes and these things should also be considered.

Computer Assisted Instruction (CAI)
Computer Assisted InstructionTerminology
Use of computer in education is referred by many names such as
• Computer Assisted Instruction (CAI)
• Computer Aided Instruction (CAI)
• Computer Assisted Learning (CAL)
• Computer Based Education (CBE)
• Computer Based Instruction (CBI)
• Computer Enriched Instruction (CEI)
• Computer Managed Instruction (CMI)
New Terminology• Web Based Training
• Web Based Learning
• Web Based Instruction
Computer-based education (CBE) and computer-based instruction (CBI) are the broadest terms
 and can refer to virtually any kind of computer use in educational settings. Computer-assisted
 instruction (CAI) Computer Aided Instruction (CAI) is a narrower term and most often refers to
drill-and-practice, tutorial, or simulation activities. Computer-managed instruction (CMI)
 Computer-managed instruction is an instructional strategy whereby the computer is used to provide
 learning objectives, learning resources, record keeping, progress tracking, and assessment of learner
 performance. Computer based tools and applications are used to assist the teacher or school
administrator in the management of the learner and instructional process.
Computer Assisted Instruction (CAI)A self-learning technique, usually offline/online, involving interaction of the student with programmed
 instructional materials.
Computer-assisted instruction (CAI) is an interactive instructional technique whereby a computer
 is used to present the instructional material and monitor the learning that takes place.
CAI uses a combination of text, graphics, sound and video in enhancing the learning process.
The computer has many purposes in the classroom, and it can be utilized to help a student
 in all areas of the curriculum.
CAI refers to the use of the computer as a tool to facilitate and improve instruction.
CAI programs use tutorials, drill and practice, simulation, and problem solving
approaches to present topics, and they test the student's understanding.
Typical CAI provides1. text or multimedia content
2. multiple-choice questions
3. problems
4. immediate feedback
5. notes on incorrect responses
6. summarizes students' performance
7. exercises for practice
8. Worksheets and tests.
Types of Computer Assisted Instruction1. Drill-and-practice Drill and practice provide opportunities or students to repeatedly practice
 the skills that have previously been presented and that further practice is necessary for mastery.
2. Tutorial Tutorial activity includes both the presentation of information and its extension into different
 forms of work, including drill and practice, games and simulation.
3. Games Game software often creates a contest to achieve the highest score and either
 beat others or beat the computer.
4. Simulation Simulation software can provide an approximation of reality that does
 not require the expense of real life or its risks.
5. Discovery Discovery approach provides a large database of information specific
 to a course or content area and challenges the learner to analyze, compare, infer and
 evaluate based on their explorations of the data.
6. Problem Solving This approach helps children develop specific problem solving skills and strategies.
Advantages of CAI• one-to-one interaction
• great motivator
• freedom to experiment with different options
• instantaneous response/immediate feedback to the answers elicited
• Self pacing - allow students to proceed at their own pace
• Helps teacher can devote more time to individual students
• Privacy helps the shy and slow learner to learns
• Individual attention
• learn more and more rapidly
• multimedia helps to understand difficult concepts through multi sensory approach
• self directed learning – students can decide when, where, and what to learn
Limitations of CAI
• may feel overwhelmed by the information and resources available
• over use of multimedia may divert the attention from the content
• learning becomes too mechanical
• non availability of good CAI packages
• lack of infrastructure

mathematics is everywhere

Mathematics: Everywhere & Everyday English   华语   Bahasa Melayu   日本語  Mathematics expresses itself everywhere, in almost every facet of life - in nature all around us, and in the technologies in our hands. Mathematics is the language of science and engineering - describing our understanding of all that we observe. The Mathematics Everywhere & Everyday Exhibition explores the many wonders and uses of mathematics in our lives. This exhibition is divided into nine areas focusing on different aspects of mathematics. Gallery Pathways worksheets prepared for primary and secondary school classes are available on request and booking. Please view our Enrichment Programmes – Gallery Pathways for more information on booking, etc.   Introduction & Concepts Mathematics has been around since the beginnings of time and it most probably began with counting. Learn the history of mathematics, and get to know some of the greatest mathematical minds and their contributions. In this section, you are also introduced to some basic mathematical concepts and counting tools – from giant electronic calculators to the abaci of different cultures. Highlight The Ames Room shows the effects of scale and illusion. In Counting Counting various quantities is one of the activities that people engage in from young. However, sometimes we wonder, just how big is one million? See how much space one million saga seeds occupy and be awed by numbers that you can relate to in your daily lives like the number of heartbeats in a typical lifetime. Sometimes, numbers can be used to fool people. Some exhibits in this section provide a few examples of well-known mathematical tricks.     In Nature 1, 1, 2, 3, 5, 8, 13... This is the Fibonacci Sequence, where each number is derived from adding the previous two numbers. This sequence of numbers can be found in many natural patterns like in pineapples, sunflowers, nautilus and pine cones. Our eyes are usually drawn to objects that are symmetrical. Leonardo Da Vinci’s Vitruvian Man is often used as a representation of symmetry in the human body. The Soothing Symmetries exhibit anchors this section and explains what symmetry means to science and in mathematics, as well as in everyday life.   In Shapes, Curves & Patterns Circles, squares and triangles are just a few of the shapes that are familiar to us through our daily lives. Discover the usefulness, beauty and 'cleanness' of the round shape, and learn more about the other shapes that can be found around us. This section also features the Möbius strip that has several curious properties.       In Games & Puzzles Many, if not all puzzles and games require mathematical logic and deduction. This section uses the fun and excitement of various popular games and puzzles, and the exhilaration of solving them, to attract and engage the visitors to realise the mathematics in fun and games. Highlight The Odds Are Stacked Against You! explores the odds of various casino games like roulette, blackjack and slot machines. This will set many thinking about their chances of winning in gambling.     In Time & The Heavens Mathematics was developed to understand the cycles of nature as observed in the seasons. Ancient people understood the need to define time in relation to celestial movements for agricultural, astronomical, astrological and navigational reasons. This section looks at the mathematics of astronomy, its relationship to the inventions of various cultural-historical calendars, and the division of time into units of hours, minutes and seconds.       Challenges & Controversies When we talk about mathematics, equations and formulas will pop into our mind. There are hundreds of equations in mathematics, but which is the Most ‘Beautiful’ Equation of all? You are invited to vote for your favourite. What do modern mathematicians actually do? What problems are of interest today? These are some of the issues explored in this section. Some outstanding challenges that remain unresolved are still intriguing many mathematicians. Discover what some of these challenges are.   In Real Applications Mathematic is used in our everyday lives; from figuring out the amount needed to buy your lunch to calculating the bank’s interest. This section explores some of the real life applications of mathematics. For example, internet banking is getting more and more common these days, and we depend on cryptology – the study of protecting information using codes – to keep our transactions safe. Learn more about how it is done in this section.         Awesome, Fearsome Calculus Calculus is the study of change and it is one of the most important fields in mathematics. Isaac Newton and Gottfried Leibniz are usually credited with the invention of calculus. Newton used calculus in his laws of motion and gravitational attraction. Explore an introductory calculus lesson by taking something to the limit in Know Your Limits, and learn more about slopes, tangents, integration and differentiation in this section's other exhibits. Back to Top

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instructional materials in mathematics

tips and advice for math teachers

As a math teacher, you need to find the best instructional materials for mathematics education. It's important for students to get a proper math education, but many students find it difficult to learn.

In many cases, mathematics education is one of the most difficult disciplines to teach. Math classes, particularly at higher levels, are usually stale and boring, consisting of the teacher giving a lecture at the chalkboard. But supplementary instructional materials in mathematics can change the way you teach lesson.

If you want to be a great math teacher:

1. Use a variety of math class educational materials to cater to different learning styles. One of the biggest problems with understanding math is that it's hard for some students to grasp the concepts in a traditional setting. It's up to you to find other ways of teaching.

2. Get students the help they need early on. Math ideas tend to build upon each other. If you don't grasp one concept, then it's hard to move on to the next. If a student is struggling, make sure that he has access to math tutoring.

3. Use games in your class to make it fun. Mathematics education doesn't have to be boring.

Find the best math class books to teach your students - A good math book should explain the concepts and have plenty of practice problems that increase in difficulty. For Jr. High and above, you may want to consider a math textbook that has answers to every other question in the back. This can help students know that they are on the right track. 

Use instructional materials for math that help students pass exams - There are a number of standardized tests that require mathematics knowledge, from state proficiency exams to the SAT. If you are tutoring in math, it's especially important for you to help your students understand all of the concepts they will need to pass the exam with flying colors. 

Look for additional mathematics instructional materials to make class more fun

There's no need to stand in front of the class and lecture every day. Do your best to make sure that math class is fun for your students and their success will come naturally. ...
Help your students learn how to use instructional materials for mathematics education. It's always best if students know where they can go to find answers.