When you hear the word “technology”, you probably think of computers or tablets. But calculators have become an essential electronic tool for performing more complicated arithmetic. People had been seeking simpler ways to perform calculations for centuries. Engineers, until the 1960s, used slide rules, which could perform multiplication, division, square roots, trigonometry and some other functions, but were accurate to only three digits. The answers came without giving the location of the decimal point. Also, slide rules could not be used for addition or subtraction.

**Basic Calculators
**The first electronic hand-held calculators needed to be plugged in and could only perform the basic four arithmetic operations. Initially, families and schools were reluctant to allow children to get their hands on calculators. They feared the students would not learn their facts and become so dependent on the devices they would never develop the ability to perform mental math.

As virtually everything in life, calculators provide positives and negatives, advantages and disadvantages. One real advantage is that story problems can use realistic numbers, not dumbed-down simple numbers that are easy to compute. Emphasis can properly be directed toward the problem and not the arithmetic. Another advantage is that students no longer need to learn to find square roots with paper and pencil or spend hours mastering the intricacies of multi-digit long division.

On the other hand, to prevent nonsensical results, calculator users must learn to estimate the solution before punching the buttons. Following a division, they need to know what to do with any remainder. They need to master built-in features of a constant key and memory functions. A mathematically proficient student has learned when to perform a calculation mentally and when to use a calculator. Research shows students who use calculators appropriately do the best in mathematics.

**Scientific Calculators
**Basic calculators gradually evolved by incorporating more and more features. So-called scientific calculators include pi, as well as trigonometric, exponential, and statistical functions. It also does fractions in traditional fraction form. And their screens display not only the current number, but the entire expression.

A free online version can be found at: desmos.com/scientific. Much can be gained by simply experimenting with a calculator using intuition and creativity. Anyone who can use all the buttons correctly on one of these devices is doing college-level math.

Besides the advanced mathematical operations, scientific calculators perform simple arithmetic differently in two significant ways. The first difference is the order of operations. In a basic calculator, keying in 2 + 3 × 4 will give 20; the scientific calculator will give 14. Why? While the basic calculator is adding 2 + 3 before multiplying by 4, the scientific calculator multiplies 3 × 4 before adding 2. The scientific calculator gives the mathematically correct answer since multiplication is to be performed before addition or subtraction.

The second difference is the way the calculators store the results of an operation. The basic calculator truncates, or chops off, an answer while the scientific calculator stores the actual result. For example, adding (1 ÷ 3) and (2 ÷ 3) on the basic calculator gives the answer 0.9999999 because it stores 2 ÷ 3 as 0.6666666, but the scientific calculator gives the answer 1.

It’s helpful to know that both the SAT and the ACT assessment exams allow the use of a calculator. They have a list of acceptable calculators and suggest the test taker be thoroughly familiar with its use.

**Computers
**When personal computers became available, they were accepted more enthusiastically than calculators were. Computers were often used for teaching elementary programming and keyboarding skills.

There still exists the “month myth,” the totally untrue notion that a person can learn everything they need to know to be computer literate in a month. This myth may have been true 40 years ago, but today it takes years to master finding information and navigating all the basic programs available on computers. Many state high school math standards suggest students learn computer algebra systems and dynamic geometry software.

Sometimes to practice keyboarding skills, students were instructed to write a paragraph and then type it into the computer. Unfortunately, that practice diminishes learning to compose at the keyboard, an essential skill in today’s world for students and workers. Writers also need to learn how to edit their work at the computer. Regrettably, many language arts state standards do not address teaching students the art of using the computer for composing and editing. Recent research shows, on the other hand, that note taking is often best done with paper and pencil.

**Learning Math with a Computer
**Beginning in the late 1970s, programmers were producing software that promoted learning facts through “drill and kill.” Studies show such gains made usually disappear after about a year. There are several difficulties with this approach:

What is learned by rote needs frequent review.

- A child often is not given a way to find the correct answer.
- The learning is frequently interrupted to provide a “reward,” breaking the child’s concentration.
- Deep learning is not fostered by extrinsic rewards.
- Ordinarily, children learn better when they can physically handle a manipulative.
- The child may be constrained to write the number from right to left, even though left to right may be more intuitive.

More recently, complete math programs are available on electronic devices. While these may work for older students, especially if they teach for understanding, they have not been proven successful for younger children. The younger student needs human interaction with careful guidance, encouragement, and assessment. Ω

]]>Most people have heard of Montessori, but often they are not sure what it’s all about. The word “Montessori” refers both to a person, Dr. Maria Montessori (1870-1952), an Italian physician and educator, and the Montessori method of education she developed.

Dr. Montessori originally worked with children living in institutions and then continued with children in a housing project daycare. She introduced child-size tables and chairs and developed four categories of materials: Exercises of practical life, sensorial materials, language activities, and math materials.

**Exercises of Practical Life
**Young children love to help maintain their space, which Montessori called a Children’s House. Therefore, a Montessori preschool provides small brooms, dustpans, and sinks. To teach dressing skills, there are frames for buttoning, zippering, shoe tying, and buckling. The children also have access to polishing and pouring activities. After a child has been shown how to use a material, they are free to take it from its place on the shelf as desired, perform the activity, and return the material to the same place.

**Sensorial Materials
**To encourage the child to hone their senses, one activity requires matching Color Tablets and another, grading a set of tablets, for example, from dark blue to light blue. There are other materials, usually ten items, to be placed in order from largest to smallest in one, two, and three dimensions. Additionally, there are materials for matching weights and sounds.

**Language Activities
**Beginning language activities include the Sandpaper Letters that the child traces with their fingers in the direction they are written. The teacher uses them to teach the

Learning new vocabulary is an important part of language. Puzzle maps provide a tactile means for teaching geography vocabulary while nomenclature cards showed geometrical terms. The adult uses the three-period lesson, or name lesson, as follows:

This is. . . . (a triangle)

Show me. . . . (the triangle)

What is this?. . . . (a triangle)

**Math Materials
**The beginning math material is the Number Rods, a set of ten rods increasing in length from 10 cm to 100 cm (1 m). To make the rods countable, each 10 cm segment is painted in alternating colors of red and blue. The room has many other materials designed for counting.

Soon the child is introduced to tens, hundreds, and thousands with base-ten materials made with the ‘golden’ beads.

**My Montessori Experience
**I first heard of Montessori decades ago from reading a newspaper article. Her philosophy made sense to me and I read every book I could find about Montessori and her method. A few years later, I enrolled in the first Montessori training course offered in Minnesota and received my diploma from Mario Montessori, Maria’s son. I taught in Montessori schools for several years in the Minneapolis/St. Paul area, including one year as the special education teacher. I thoroughly enjoyed teaching those years.

While I love the way Montessori teaches reading, I slowly began to question the math progression for several reasons. Montessori used color frequently with the math materials; the ones are green, the tens are blue, and the hundreds are red, repeating in the thousands and millions. She seemed to be unaware that one out of twelve boys has some color deficiency.

Also, Montessori did not group by fives for quick quantity recognition. She wanted the children to be proficient at counting, possibly to pass the Italian basic competency test that required counting. She provided bead bars ranging from one to ten, each in a different color, a forerunner of colored blocks, which doesn’t emphasize the critical grouping of fives.

**From the Bead Frame to the AL Abacus
**I have been fascinated with abacuses for many years, even mastering adding and subtracting on the Japanese abacus. Montessori’s version of an abacus, the Bead Frame, has four horizontal rows each with ten beads; the top row beads are green followed by rows of blue, red, and green. A strip along the left side indicates, 1, 10, 100, and 1000. Some children had difficulty writing the number represented by these quantities because of this orientation.

When I presented the Bead Frame to the Montessori children, I found they had great difficulty in mastering it. To add 8 + 6, they counted out 8 beads; then to add 6, they counted 2 beads before reaching the end of the top row. At this point, they traded the row of 10 green beads for 1 blue bead, and then continued by moving 4 more green beads. Unfortunately, after the trade the children frequently forgot where they were in the count for 6. Remember that algorithms do not trade before completely combining both numbers.

I felt a simpler abacus would better serve the children. I made a few abacuses with 100 beads, ten rows each with ten beads grouped in two colors, one dark and one light. With this modified bead frame, the children didn’t need to count and they could use strategies to help them learn their facts. To enable the children to perform trades, side 2 of this abacus used two columns of beads for the ones, tens, hundreds, and thousands. With this arrangement, both addends are present before trading. The children really enjoyed the modified Bead Frame, now called AL Abacus, and made great progress mathematically.

**From the AL Abacus to RightStart Math
**When the time came for me to decide on a topic for my doctoral dissertation, I chose to replicate features of primary mathematics as taught in other countries, especially in East Asia, combined with some Montessori principles. The lessons I wrote became RightStart Grade 1 and Level B, first edition. The school where I conducted my research was amazed at the children’s progress and asked me to continue writing lessons for more grades, and thus was born RightStart Mathematics. Ω

Miss Koyabe has a French mother and Kenyan father. She was born in France and came to Arizona as a baby. The homeschooler enrolled in a theatre workshop at age 9 and she was smitten with musical theatre. She made her debut at the Hale Theatre in Gilbert, Arizona.

Homeschooling allowed Koyabe to really work on dance – ballet, hip-hop, jazz, contemporary – and voice and acting. She states, “It was a big asset for me because it gave me more of the freedom an adult might have while still being in school. I think the biggest hurdle to overcome was time. Having flexibility of my schedule made a huge difference.” Be sure to watch for Phoebe’s name in upcoming musicals, plays and T.V. shows, even.

]]>• Ages 4-7, *Color It, Say It, Play It and Create It!* A Piano Method To Teach Little Fingers To

Play! 51 pgs.

• Ages 5-adult, *Drill It and Kill It – Read Music Like a Pro* – a Complete Piano Method for the

Beginner

• Beginner skill level, *How To Play Chords and Improv On the Piano*

• Beginner-Intermediate skill level, *Fake It ‘Til You Make It* – A guide to playing a Fake Book

on the piano

All written and ^{©} Copyright, by Kathi Kerr, www.melodymusicstudios.com

====================================================================

Assessed by Michael Leppert

Kathi Kerr, owner of Melody Music Studios and author of a complete line of instruction books, brings her wealth of knowledge to anyone interested in learning and developing keyboard skills in a fun and relaxed way. These books are excellent resources for developing any desired level of competence from a living room Lizt to a full-fledged professional.

- The first volume shown above, is the launching pad for little students to become acclimated to the world of the piano and experience the pleasure of accomplishing skill development. As the title states, Kathi incorporates coloring the keys of drawings of white piano keyboards in the book, into the process of becoming familiar with the keyboard. Part of the coloring process is learning the names of the keys being colored.
**(This is a brilliant application of the scientific fact that information taught through play only requires 10-12 repetitions to imprint the brain vs. 400 repetitions by rote.)**Ms. Kerr continues to present more musical information such as rhythm, note and rest values, pitch values and applying these values to playing familiar simple songs. - The second volume addresses, in great detail, development of the all-important skill called sight-reading. This is the ability to sit down to a never-before-seen piece of music and within 5-8 minutes of preparation, being able to play it virtually correctly. Professional recording musicians and orchestral players have to possess as strong a sight-reading ability as text reading to an editor or writer. In my opinion, these two volumes should follow sequentially.
- Third in Ms. Kerr’s line of excellent instruction books, addresses playing chords and improvising on the piano. These two skills follow those developed in the above two books. Chords are like the scaffolding that melodies are hung upon and learning to play them is built upon the initial note-reading skills and expands one’s sight-reading ability. Songwriters and composers know that our Western ears hear chords progress in a certain pleasing or (intentionally) displeasing manner, called “chord progressions”. After some initial discussions of music theory (the facts of music), Kathi teaches the concept of progressions and provides actual examples that are immediately familiar from hearing decades of popular songs use them. Finally, she teaches chord inversions. Space does not allow for a discussion of this part of theory, but a deep knowledge of using inversions is absolutely necessary for the professional or semi-pro musician and to a purely amateur player who wishes to enjoy playing comfortably without anxiety.
- The last volume in Melody Music’s line teaches the ability to use Fake Books, which are typically, very large volumes of just the melodies of popular songs and a chord shorthand allowing pro musicians, such as piano bar pianists, to take requests at large. This edition is invaluable for an aspiring performer who envisions playing weddings, bar mitzvahs and other events that require a fluid knowledge of hundreds of songs, appealing to multiple generations of listeners and dancers.

If your child – or yourself – desires to develop keyboard confidence enough to play for others with joy and ease, possibly even to make a living, please visit Melody Music Studios website and see Kathi Kerr’s excellent line of instruction books. MjL

]]>Many readers of this website are no strangers to the fact that high school science curricula have been in need of a new voice for many years, especially one that is both Christian and upholds a rigorous academic model. Though they not the majority in America, many believing scientists and engineers who homeschool their kids, as well as educators at Christian schools with a strong STEM focus, have lamented various insalubrious features they see among the variety of curriculum choices.

There are essentially two options. Secular publishers may contain more or less mainstream scientific content but often fail to inspire enthusiasm in the Christian classroom. For Christian educators, the primary deficiency is the lack of integration with or even acknowledgment of their faith at even the most benign and ecumenical level. As science teachers with a faith perspective, we never tire of rejoicing over the marvels of the cosmos and wonder at the benevolent Creator who made them. Educators who want to integrate their faith convictions into the study of science are required to supplement if they want students to reflect on the intersection of faith and science.

What’s worse, some secular science texts can sound downright hostile to religious views or arrogant about the unquestionable certainty of the scientific knowledge, which itself is an unscientific posture.

On the other hand, Christian faith-based texts either tiptoe around controversial issues (evolution in biology texts, old earth creationism in earth science, climate change in environmental science) or they dispense with mainstream views in a few lopsided polemical paragraphs.

Where integration is attempted, Christian texts often overwhelm the science with artificial and disconnected faith content: too-frequent reminders that God made everything, devotional insets, forced biblical tie-ins, etc. Or, they present a tame, mollified view of the faith, for example smoothing over global problems and manmade catastrophes with exonerative refuge in divine omnipotence and foreknowledge.

Christian texts can also speak disparagingly about non-Christian scientists, insinuating that they are deceived, suggesting that they are duped by the devil to lead children astray, incredulous that supposedly educated people could fall for theories so obviously full of holes. Non-Christian scientists are categorically “arrogant” and motivated by a desire to undermine the Christian faith. This is unfortunate. Not only should Christian interaction with those outside the faith should avoid becoming adversarial, it should strive to promote charity and goodwill. It should rejoice in the common ground we share with them. It should give the benefit of the doubt. “Love…hopes all things, believes all things.”

At the visual level, secular and faith-based textbooks alike can suffer from stultifying page layout. In the worst cases, you find silly graphics, overblown fonts, gaudy imagery or sometimes campy clipart pictures—textbooks that seem to be imitating popular media in an attempt to keep students' attention, shouting at the student from the page.

What we set in front of students says a lot about what we think of them. Would we use Disney princesses, Lego figures, or anime to illustrate our science texts? Of course not. So let’s press that a little further—what kind of graphical design would most respect, most dignify, most ennoble students?

What visual presentation would make them feel like they are being summoned and welcomed to an adult enterprise? Why would we fail to treat them like the kind of mature human beings we are trying to lead them to become?

Do teenagers know when they are being spoken down to? You better believe it. And they resent But too often current textbook production thinking has taken a subject with natural, built-in fascination and wonder, and has made it either intimidating or vitiating.

I believe students, especially high school-aged students, will respond to being treated like young adults. They will recognize when, at last, they are not being sold to, psychologically coddled, or propagandized. They will know when teachers and materials are engaging them with respect and dignity.

I also believe that faith-based education that is honest and blunt about the great problems of the world gives them that respect, treats them as real agents and stewards in the world, and prepares them to enter Christian womanhood and manhood with stronger minds. Finally, when we bring students fully into the scientific controversies of the day, they feel the adults are trusting them to think about issues for themselves, and that is the essence of education.

Full disclosure: I work for a science curriculum publisher, Novare Science & Math. And of course I am interested in company success. But at Novare, we were all schoolteachers and parents before we were publishers.

But for those interested or involved in secondary education, it is my pleasure to offer Novare Science & Math for consideration to those. We are currently producing the first new General Biology to come from a Christian publisher in a while. This book, written by Dr. Heather Ayala of Belmont Abbey College and educator Katie Rogstad, will be an answer to many of the wishes of Christian biology teachers. Here is an excerpt:

“Consider how God provides everything we need for life, and how wonderful it is that all His creatures support one another in their needs for matter and energy. Not only that, but He designed these processes in a sustainable way, such that the cycle continues on and on without ever being exhausted (at least so long as the sun continues to shine!). Let us thank Him for His goodness to us and resolve to care for the earth He designed for us so that we don’t compromise this perfectly designed environment through carelessness, waste, and pollution.”

The production team is writing the textbook they wished for when they were young. Novare hopes that the book blesses and serves students and teachers as they investigate the wonders that modern biology has discovered about God’s amazing creation. More information at http://novarescienceandmath.com

]]>*The list of educators, psychologists and brain researchers who have dissected and diagnosed the failure of America’s schools is endless.*

by Carolyn Forte

The modern homeschool movement was pioneered in the 1960s and 70s by brave souls who refused to accept the lock-step rigidity of our public-school system. They were studied by a few academics like John Holt, Dr. and Mrs. Raymond Moore and Dr. Ruth Beechick, who saw great value in this new and revolutionary way of educating children. All of them had spent decades inside America’s schools and knew their faults. They were intrigued by the unorthodox style many homeschoolers adopted, studied the results and produced more than a score of books to encourage and help us do the same. Within a few years, many others, including Dr. Mary Hood, John Taylor Gatto and David and Micki Colfax contributed volumes to the homeschool bookshelf.

These authors were all very different, but the common thread in all their writings is the idea that ordinary parents can do much better than the schools and should not look to the education establishment as the ultimate authority on learning. Since all of us were products of school, public or private, these ideas took time to sink in. However, since in the early 1980s it wasn’t easy to get standard texts for our children, most of us were forced to teach our children outside the box. Many followed closely the educational philosophy of Dr. Raymond and Dorothy Moore who developed *The Moore Formula* from their observations of homeschooling families. They advocated delaying formal schooling until at least age eight and incorporating real work and service projects into the “school” day.

These ideas sounded radical at the time and they probably still sound radical today. I encounter very few homeschool families who follow this advice, even though the fruit of that out-of-the box homeschool tree has been spectacular. I suspect that fear is the main reason so few homeschool parents today will step off the well-beaten but booby-trapped path of conventional schooling. Though there is still opposition in some quarters, homeschooling has gone mainstream. Most of the public recognizes it as a legitimate alternative and thousands of companies have targeted homeschoolers as an emerging market. Where we had to improvise out of the local library and teacher supply store, homeschoolers today are bombarded right in their homes with e-marketers hawking every kind of educational material and program known to man. Type *homeschool* into any search engine and you will be presented with a vast array of charter schools and products, all of which are “guaranteed” to provide easy homeschool success. Beware.

The average homeschool family lasts for two years. Many others persist by repeatedly changing curriculums, programs, co-op classes, charters or Private School Programs. They are looking for the perfect homeschool path, but it is often illusive. There can be many reasons for this, but after observing successful, as well as struggling, homeschool families for several decades, I believe that most problems stem from the mistaken assumption that the public schools provide the proper model for curriculum. The truth is that they provide a near-perfect model for educational failure. You don’t have to take my word for it. Read award-winning teacher, John Taylor Gatto’s *Weapons of Mass Instruction*, Dr. Raymond and Dorothy Moore’s *Better Late Than Early *or Dr. Jane Healy’s *Endangered Minds.* The list of educators, psychologists and brain researchers who have dissected and diagnosed the failure of America’s schools is endless.

As a former public-school teacher, I never wanted to copy anything the schools did and it has always astonished me that so may parents will reject the public schools only to strive mightily to “keep up” with them. As John Taylor Gatto has said, “The schools are anti-human.” Children cannot be put on a conveyor belt, rigidly scheduled, force-fed thousands of bits of information to be regurgitated on command and then be expected to think and use that information intelligently and creatively. Our schools actively seek to repress creativity and independent thought. They endanger children’s minds by presenting abstract concepts long before the developing brain is ready for them, often creating neural roadblocks. They force inactivity and close visual work on very young children who should be developing visual skills, large and small muscle coordination and balance, with active and creative play. Each child is a unique human being with a unique learning style, talents and interests, as well as developmental schedule. We don’t expect all children to walk at 11 months of age. Why do we expect all children to read at 5 and multiply at 7? A half century ago, children were not expected to read until age 6 or 7 and multiplication was introduced at age 9.

There are other pitfalls with copying the public schools. They major in the minors. School today is all “academics” with little time or value given to anything else. Art, music, drama and all things creative are shoved to the sidelines — if they are provided at all. Thousands of hours are devoted to busy work, which has little, if any, real intellectual value. Dr. Moore once said that it takes about two-and-one-half years to prepare for high school. I didn’t believe him at the time, but I saw with my own eyes how right he was. You see, once you can read, write and cypher, there is nothing you cannot learn. Once the student is ready to learn, it takes only a few hours of instruction to learn to read (about 20 hrs.) and learn all of arithmetic (about 30 hrs.). Since we start way too early today, it takes much longer to learn these basics.

Our schools, and that includes Charter Schools, are also obsessed with standardized testing. The conveyer belt includes shackles to be certain that no one escapes. Since the new tests are Common Core aligned, one must adhere to the most absurd curriculum ever devised to torture children in order to feel marginally comfortable with the tests. The fact that these tests have NO bearing on any future opportunities and have NO academic value, is not well understood by the average teacher, let alone the vast majority of parents. Thus, these tests, written by psychologists, not teachers, drive the curriculum for all traditional schools, public, private and charter. Instead of learning in the real world, too many homeschool families follow this model and push too much too soon, often with inappropriate methods from mind- numbing textbooks and/or online courses.

Sadly, because this developmentally and educationally inappropriate content takes so much time, students have little time left to learn from family life, play, travel, work and creative activities. Dr. Moore said that every child should create and/or work in some sort of business, depending on the child’s age and abilities. He recommended starting small businesses or working within a family business. This teaches many things ranging from economics to personal relationships, not to mention reading and math. A child who wants to earn a profit will be willing to learn the math necessary to avoid a loss.

Dr. Moore also recommended spending a third of one’s time on service to others. This could mean helping an elderly neighbor with a chore like yard work, or carrying in groceries. It could be volunteering at a local charity, church, community center, community theater, nature center, or nursing home. There are thousands of ways to help others in need and each provides a valuable learning experience.

Our society is obsessed with paperwork and test scores, but too much paperwork might even lower test scores by limiting the volume of reading and stifling opportunities for thought and creativity. Where children used to write three short sentences for a third grade assignment, they are now told to write a “research paper.” At the same time, they are not taught penmanship, so the act of writing becomes difficult and tedious, especially for little boys, whose small muscle coordination lags behind girls’ by many months.

Starting academics late as Dr. Moore advised or slowing down the pace of studies to a more developmentally appropriate schedule will do a lot of good for young children, but formal academics shouldn’t be the exclusive focus if you want a truly educated and competent child. In his *Moore Formula, *Dr. Moore recommended spending no more than a third of your time in formal academics. He also warned against packaged curriculums and advocated reading a great number of real books instead. When you allow much more time for work, service, exploration and the development of talents and interests, students have a chance to acquire useful skills and see how their academic studies are useful in real life situations.

This will sound to many, like an abandonment of “rigorous standards” but as I learned over many years of watching homeschoolers create their own educational pathways, the *Moore Formula* is the most rigorous standard of all. Students who devote significant time pursuing a passionate interest, have real direction and develop excellent work habits. They do not finish college or even grad school, only to discover that they hate the career they spent so much time and money to prepare for. They know how to function independently in the world among a diverse population. They have already experienced setbacks and roadblocks and have learned to pivot to a new course of action when necessary. These are vital skills that no textbook can teach. In short, don’t major in the minors. Academic skills are only the beginning and they don’t take as long to acquire as a standard time-wasting school curriculum will demand. Never confuse wisdom with knowledge. C.F.

In mathematics we can’t do much without using language. Even though the mathematician may write a mass of equations, she needs words to explain their meaning. Good math instruction requires careful attention to the terms used. We don’t want students to have to relearn the mathematical meaning of words.

**Number Words**

The first math words we want a child to learn are the names for quantities. For quantities one to ten, the words are arbitrary. Rather than continuing to memorize a growing number of unrelated words, people hundreds of years ago began to think of numbers grouped into manageable chunks.

The Romans grouped into fives, V, as well as tens, X, when recording numbers. They had no symbol for 2 or 3, so they doubled or tripled the symbol for 1. Thus, 2 is written II and 3 is written III. With the tens, 2 tens is written XX and 30 tens XXX. Also 5 tens is L and 6 tens is LX. Although only four different symbols are needed to record numbers from 1 to 99, larger numbers require more symbols, C for 100, D for 500, and M for 1000.

Note that the early Roman numerals represented 4 as IIII and 9 as VIIII. Only later did 4 became IV, meaning one less than five, and 9 became IX, one less than ten. However, the economy gained in writing those numerals turned out to be a major obstacle to performing calculations with the numerals. It is interesting to note that most clocks with Roman numerals use the early four, IIII, but the later nine, IX.

A monumental improvement occurred in recording numbers with the introduction of the familiar Hindu numerals. Each quantity from zero to nine is written with a distinct digit. For numbers larger than ten, the digits are reused, but the place of the digit in the number determines its value, hence the term, place value.

Unfortunately, the words for numbers in the Indo-European languages predated the Hindu numerals and lacked the simplicity and clarity of the written numerals. Children today struggle matching the irregular number words to the corresponding symbols.

Remarkably, the East Asian languages were changed to make their number words consistent with the Hindu numerals. For example, eleven became ten-1; twelve became ten-2; twenty-three, 2-ten 3; and forty-seven, 4-ten 7. Many of the children speaking these languages understand place value before they even start school, giving them a great advantage in learning arithmetic. Happily, English-speaking children can gain the same benefit by using these transparent number words for a short period of time.

**Simply Incorrect Words**

Probably the term that aggravates me the most is “number sentence.” A sentence is a group of words that make a complete thought. How does the **equation** 2 + 3 = 5 fit that definition? Using the term number sentence confuses the learner in both math and language. One third grader when asked to write a number sentence wrote: Two plus three equals five. The term equation means to make equal. This equality is a fundamental principle of mathematics. Fortunately, this ill-advised term of number sentence is disappearing from textbooks and tests.

A close second in annoyance is “take away.” First of all, it is bad English to say “seven take away five.” If this is a declarative sentence, shouldn’t there be an s after take: Seven takes away five. That’s kind of bold of seven. Or, if it’s an imperative sentence, shouldn’t there be a comma after seven: Seven, take away five. Now, seven is kind of brazen. Secondly, in England take way is fast food. Seriously, using take away limits a child’s understanding of subtraction. Often, subtraction is not about taking something away, but comparing, find a missing part, or adding up. Again, I’m happy to report this phrase is disappearing from texts and tests. Oh, what should we say? How about the correct term, *minus*?

The next word to censure is “timesing,” which will never make it into a math dictionary. Timesing, referring to multiplication, is a babyish nonword and yes, nonword is a word. On

the other hand, multiply and multiple are authentic mathematical words. The expression,

3 × 2, is best read as “three multiplied by two” or even “three taken two times.” Saying “three times two” doesn’t really describe the situation. This wording started in the 20th century and may be confusing to some children because time is associated with clocks.

Another word, fairly new to elementary arithmetic vocabulary, is *regroup*. According to the dictionary, regrouping is what a military unit does after a defeat. While adults think of this word as re-group, children learn it as a word to describe a process and not as equality. After all, have you ever witnessed a child regroup their toys and then talk about regrouping them? The old-fashioned words carry and borrow work just fine; they are mathematical words and programmers still use them. The old argument that to borrow implied something needing to be returned isn’t valid: languages borrow from each all the time. However, an even better word is *trade*, which children do understand and it does imply equality.

**Geometrical Words**

Preschool children are often taught non-mathematical words for geometrical shapes. Instead of ellipse, they learn *oval*, but an oval can also be egg-shaped or shaped like a running track. And the mathematical name for a diamond is *rhombus*. They also are taught that rectangles are “long and low,” disallowing squares.

When discussing the area of a rectangle, textbooks usually name the sides as *l *for length and *w* for width. Yet, when discussing triangles, the sides are named *b* for base and *h* for height. If the sides of rectangles and triangles had corresponding names, it would greatly help students see the relationship between the areas of a triangle and a rectangle. I think the best terms are *width*, for the distance from side to side, and *height*, for the perpendicular distance from the width. One day, I casually mentioned to Kim, an honor student in her senior year of high school, that squares are rectangles. She replied, “They are? They have different formulas!” Her textbook used *s* for the sides of a square.

**Some Oddities**

*Diagonal* and *similar* are two words having a mathematical meaning at odds with everyday usage. The common meaning of diagonal is a line that is neither horizontal or vertical, or a road that does not travel north and south or east and west. Contrast that with the mathematical definition: a line in a polygon (a closed figure with straight lines) drawn between any two non-adjacent vertices. Such a line could also be horizontal or vertical. Simply rotate the polygon with its diagonal until the diagonal is horizontal or vertical.

Amazingly, the usual meaning of *similar* is contrary to its mathematical meaning. In everyday use, similar means not exact, but almost the same; in mathematics similar means identical, but either shrunk or enlarged proportionately.

Did you know there are two kinds of mathematical tangents not even remotely related? A line just touching a curve is called a tangent line. And in a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side.

You have heard of right angles, but what about left angles? Actually, the original meaning of *right* meant correct or acceptable. As far back as the twelfth century, a right angle was thought as the angle formed by the intersection of horizontal and vertical lines. The word upright also reflects this meaning. Later, the right hand was so named because it was considered the correct, or proper hand. So, no, there are no left angles.

You probably thought a billion was always a billion. Although today it represents one thousand millions; originally, one billion was equal to one million millions. The meaning changed in the U.S. in the 1800s, but Britain officially didn’t change until 1974.

__Sum__**mary**

Yes, the word *summary* is derived from sum. It means we are summing up all our points. Introduce new words when needed. For example, nobody needs the terms numerator and denominator in order to begin learning about fractions. Use examples for new concepts, rather than a definition, especially for younger children. Therefore, watch your language.

*By Joan A. Cotter, Ph.D.*

Measurement is an application of math that is an important part of everyday life. We measure length, area, volume, mass (weight), temperature, time, angles, and many other attributes. For more than simple comparisons, we need numbers and a basic unit. Sometimes, two or more units are combined. For example, we speak about speed as miles per hour, fuel economy as miles per gallon, pressure as pounds per square inch, and grain yields as bushels per acre.

The very young child is interested in which of two items is bigger. RightStart introduces measuring length in the first year. The child is asked to determine the length of an object by finding how many 1-in. tiles it takes to equal the length. A little later the child is asked to repeat the activity measuring with an edge of a centimeter cube.

Doing this exercise with two different tools highlights a situation unique to U.S. children. They need to become “bimeasural,” that is, proficient in two systems, US customary and metric. It is estimated that half a school year is devoted to making students proficient in both measuring systems. President Thomas Jefferson suggested the fledging nation go metric, but to no avail. Even though science, medicine, and the military have all adopted the metric system, the United States officially still uses the US customary system. However, I like to say, “We are going metric inch by inch.”

Although Canada officially converted to metric in 1975, thirty years later, in 2005, teachers began teaching the rudiments of the US customary to their students to enable them to work in industries that didn’t entirely change over.

**The Basics of the US Customary System**

Length is measured in increasingly larger units. Periods are optional for abbreviations except for in. (inches) where it is needed to avoid confusion with *in *the word. Inches are divided into fractions: halves, quarters, eighths, and so forth. Larger units are shown below.

1 ft (foot) equals 12 in.

1 yd (yard) equals 3 ft

5280 ft (feet) equals 1 mile

Area in the US customary system uses squares formed by the same linear units. Thus, a square inch is a square that is 1 in. on the sides. Also, there are 620 acres in a square mile. Likewise, volume is measured with cubes made from the same linear units.

Capacity is the amount that a container can hold. Gallon is the basic unit, which is divided into half-gallons and quarts. Quart gets its name from quarter of a gallon. Further divisions include pints, cups, fluid ounces, tablespoons, and teaspoons.

Mass, often thought of as weight, measures the quantity of physical matter. Usual units are pounds (lb) and avoirdupois ounces (av oz) with 1 lb equal to 16 oz. Also a ton equals 2000 lb.

Some people find it surprising to learn that there are two types of ounces. To tell them apart, “fl” is often written before the fluid ounces, although the “av” is usually omitted before the ounces measuring weight. Eight fluid ounces of water weigh 8 ounces, but 8 fluid ounces of honey weigh 12 ounces. Ice cream is sold by volume, so the more air whipped into it, the less the carton will weigh.

**The Basics of the Metric System**

About two hundred years ago, French scientists devised a new simpler system of measurements, known as the International System of Units (SI), or the metric system. Multiples and divisions of the basic unit are always based on tens, not fractions, and indicated by appropriate prefixes. For example, the basic linear unit is the meter, which is a little over 3 ft; other linear measurements include:

1 m (meter) equals 100 cm (centimeter)

1 m equals 1000 mm (millimeter)

1 km (kilometer) equals 1000 m

Note that the prefix *centi* means one-hundredth just as one cent is one-hundredth of a dollar. The prefix *milli* is related to the millipede with its somewhat exaggerated thousand feet. A kilometer is about 0.6 of a mile, so 10 km is about 6 miles.

Another basic metric unit is the liter, pronounced *leader*. Its abbreviation L is usually capitalized. A liter is the size of a cube 10 cm on an edge and holds a little more than a quart. Another basic unit is the gram, the unit of mass. A kilogram (kg) is 1000 g and is 2.2 pounds.

**Measurement in the RightStart Math Curriculum**

Measuring in both the US customary and metric system is taught in all levels of RightStart Mathematics. Measurement is an application of math that is concrete rather than abstract; relevant to everyday life; necessary for other subjects, especially science; and leads to more advanced math concepts, such as exponents.

Because of its importance, measurement needs to be incorporated into math instruction at all levels. Occasionally, instruction on telling time and calendar activities is found in social studies texts and instruction on the metric system is found in science texts. They belong in the math class.

One misstep that some older textbooks advocate is having the child learn to measure starting with paper clips. The justification was that the children would come to realize the need for a standard unit. That concept is lost on the primary child. Instead, they need to become very familiar with inches, feet, and centimeters.

Measuring manipulatives used in RightStart for teaching measurements include geared clock, 1-in. tiles, cubic centimeters which weigh 1 g, AL Abacus where each bead is 1 cm in width, 4-in-1 ruler, folding meter stick with a yard on the reverse side, math balance adapted for measuring weight, and goniometer for finding angles.

To be proficient in measurement, you need to know how to convert to other measurements. RightStart teaches changing units within the same system in Level E. Converting between the US customary and the SI systems is taught in Level F. A process called dimensional analysis makes the task straightforward.

Measurement will continue to be part of mathematics, part of a good math curriculum, and an increasing part of everyday life. Ω

]]>As a young child, I was exposed to the daily routine of adults working from home, so I would imitate what they did as if it were a game. What I remember most from these days was the adrenaline rush that came with the successful completion of a bright idea.

Playing “pretend” for me was drawing up a new business logo for my imaginary company or turning my bedroom into a studio apartment. On the weekends, my neighbor friends and I would use our play cash register to sell lemonade on the corner of the block, then we would come back to my house and divide up the profits. This was also the ’90’s, so I could pretend my bicycle was a motorcycle and be on my way, as long as I was home before the porch light came on.

My days were filled with dance classes, language lessons, rock climbing competitions, nature walks and adventure camps. I learned lessons, such as self-confidence, empowerment, commitment and responsibility — tools to maneuver through my life independently. I

knew that the world was just a yellow-page phone call away. I was taught that if I wanted dance lessons, I could call around town and compare rates. We would go to different studios to see which teacher was the best fit and I could make the final decision. I gained a sense of empowerment, knowing that I was capable of doing all the research. There was also the weight of accountability on my shoulders because I knew that once I committed to a studio, we were paying for ten lessons, which I had to attend, regardless of my mood that day.

On the flip side, some of my friends in the neighborhood went to public school and I was curious about what I might be missing out on. I would meet them at the bus stop and walk with them back to each of our houses to hear all of their stories. Sometimes I wondered why my parents hadn’t chosen to send me to that school as well, but then the thought of waking up at 5:00 a.m. during Colorado winters quickly made me feel better about their decision. As we approached middle school, many of our parents were getting divorced or moving away, and our childhood clan slowly ripped apart.

It was at this point, at ten years old, when I realized that I was ready for something new. I was approaching an age where the unschooling options were limited in my community and it was inevitable that I was going to have to make new friends. So I chose to move to a boarding school — for fun — just because it was the polar opposite of anything I had ever known. There was a girls’ dorm and a boys’ dorm and we would all walk to the main building for school. Over time, the campus became my home and I had all of my friends with me. It surprisingly felt similar to homeschooling, except that I shared the experience with seventy other kids. I learned to go to class and do the homework, but my real education was learning to negotiate the jungle called “school”.

During this time, I observed that I still had a curiosity for learning and my peers did not. They were trained to do exercises 10 – 20 in their textbooks as fast as possible, rather than absorb the content that was being presented. On the other hand, they learned skills, such as time management, because they managed to get the assignment done in five minutes in order to have another ten minutes of fun time. They learned how to check all the boxes on the teachers' list as fast as possible so that they could maximize their time that actually stimulated their hearts and minds.

I continued to attend school until I was fifteen and I started catching myself in class staring out the window, thinking about the variety of things I could be doing if I went back to unschooling. I had been to my share of school dances, sports events, and other typical school activities that always looked so appealing in the movies. Yet I didn’t feel that the time I spent inside the classroom outweighed the social experiences that people will tell you is only possible by “going to school”.

In a nutshell, I started feeling like the grass is always greener on the other side.

In the blink of an eye I evolved from a curious, adventurous child to a confident yet confused teenager. I could feel the essence of who I was slowly be molded by the encroaching expectations of society. My mom pulled me aside after making some questionable

decisions (as most teenagers do) and she asked me, “If you had a million dollars, to be anywhere or following any dream, what would you be doing?” — this is the unschooling mindset that she raised me with. Growing up in a multi-cultural household with multilingual parents, I had always envisioned myself traveling and experiencing other cultures as I got older. Her question prompted me to bring out my old foreign language workbooks, read about art history and study the world maps.

As I researched the various countries and cultures that I was most drawn to, it was fulfilling to channel my creative energy in a positive direction. After discussing many options with my family, we decided on an international school in Costa Rica that my sisters had also attended as teenagers. Preparing and planning for the trip was an empowering learning experience and I reaped the rewards when I arrived in Costa Rica. I loved the challenge and satisfaction that came with breaking language barriers as my knowledge increased. I engaged in the cultural customs by learning about cuisine, fashion, and the various perspectives regarding the countries’ history. Being fully immersed in another culture was exhilarating and traveling became my passion.

Through these experiences, I understood the value of unschooling on a profound level; it gave me the freedom to create myself and supported my innate abilities. It also gave me many opportunities to learn resourcefulness, gain confidence, and the ability to know that my options are limitless. I was able to experience the feeling of excitement and confidence that came with figuring things out for myself, as well as the trepidations that come with not knowing if your decision will end up being the best one. It’s the price unschoolers pay for the freedom to direct their education. I want my kids — and all kids — to know they have the freedom of choice, that they are never locked into just one method of growing up.

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