Published by Trigger Memory Systems https://school-workbooks.com/
By Jennie Winters and Marillee Flanagan
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Assessed by Michael Leppert
Pet Math is an ingenious way to present and reinforce math facts in an enjoyable and useful way through “story” problems. Most households have pets; children are typically drawn to dogs and cats, especially, so creating a math program revolving around pet care and feeding is a win-win product. Pet Math provides a real-life application of math, making the concepts easy to grasp and remaining with your child for life!
The story problems in the textbook provide instructions to use one of the pamphlets to obtain relevant price information to use in answering numerous questions. For instance, in one pamphlet are 3 receipts for Grooming services. Referring to the pamphlet’s price list, the child calculates the total cost of the 3 receipts; then the child calculates what change s/he would receive from paying the receipts with 3 $20 bills; then change from a $100 bill and finally, if you paid for receipt #3 with a $200 gift card, how much money would remain on the card. Math does not get any more “real” than that and your child will develop a sense of the actual value of money from working through Pet Math.
But Pet Math is not just about money. The next situation presented is about pet care at home. Brian brings home a new dog and a new cat from the pound. He will be scooping the waste from each animal and there is a chart to help the student calculate (1) how many piles Brian will scoop for each animal in 2 and 4 week durations; (2) using another pamphlet price list, the student calculates the cost of one month’s supply of dog waste bags and cat litter, necessary to accomplish the work and then, 3 month’s worth of the same supplies. For a child with a first-time pet, this lesson provides insight into pet care matters, as well as math-related issues.
Pet Math includes graphs and charts, multiplication and division problems, area and perimeter problems and data analysis.
All of Pet Math’s 48 pages follow this method. It is well-established that learning via fun is vastly more effective than learning via rote; there is no doubt that your child will be eager to do Pet Math every day and will retain the thinking and facts contained – some lessons might require a bit of review — but the enjoyment of using the Pet Math program will lead you and your child to apply the same lessons to your own pet care and costs, quietly reinforcing the concepts and math skills set forth in the program. Visit the website for complete cost and ordering information. MjL
]]>The United States is about to eliminate cursive writing from its teaching curriculum! If implemented, we will be taking an enormous gamble that cursive in today’s environment is obsolete. As of today, 40 states have adopted the Common Core curriculum which supposedly establishes the requisite skills necessary for mastery in college and career. This document ultimately phases out teaching cursive writing. Is the U.S. ahead or behind other countries who stress foundational skills like cursive handwriting to develop their leaders and workers? Is Common Core curriculum just another educational experiment like “The New Math” where mindless memorization of facts was deemed unnecessary and “Word Recognition” was a revolutionary method to replace phonics? The problem with this gamble is that once cursive is gone from our culture, it is not recoverable.
We may have already reached that point of no return. Elementary teaching graduates are not taught how to teach cursive in college, so when they get their first teaching assignment, many have no idea what to do. Students are telling their teachers they have to print the assignments on the chalkboard because “they can’t read cursive.” Many high school students have to ask their parents to read their teacher’s notes on their tests and compositions because “they don’t read cursive.”
Historians and political scholars often require greater insight into the birthing of our Republic. Original depictions of the drafting of the Constitution and the negotiation letters of Jefferson, Madison and Hamilton are often-used resources that were written using cursive. Some say we don’t need to read cursive; just find alternative transcriptions in print form. But when cursive is gone and only a few people can read it, we will be solely dependent on the select few to interpret the laws of our land.
Learning cursive writing helps children’s brain synapses develop, because it requires fluid movement, hand-eye coordination and fine motor development. But, the most important reason for learning cursive writing is to be able to READ cursive.
In 2006, only 15% of SAT takers used cursive on the written essay part of the test but, they averaged measurably higher scores. Why? It was because they had been made to stretch their total learning range of skills the other 85% had not been required to learn or no longer used. They were better note takers; they could write faster and could organize their thoughts on the fly.
Patrick O’Neill, assistant principal of academics at St Francis High School in Sacramento said cursive is a necessary skill. “If our students can’t read cursive there will be parts of the world they will not be able to access. Students must be able to access all forms of communication available today.”
A hole will exist in the student’s total learning spectrum by neglecting to teach cursive. Primary grade-aged children have an enormous capacity to absorb new skills. It is important to teach cursive when children have the most time and desire. In this day and age they still consider cursive a “rite of passage” and are as excited as ever to learn it. My class used to beg me to teach it on their first day in third grade. That window of opportunity closes quickly, as more demanding curriculum is introduced and their maturing interests expand. Why not teach cursive when the investment of time for the teacher and student is so small, usually ten minutes a day for 56 days? No expensive workbooks or worksheets, just one teacher’s manual, a pencil, lined paper and a kid.
The richness of our culture is being diminished by the disappearance of the handwritten word. Whether you are a historian or grandmother, the poignancy of the written word cannot be denied. Many cherished recipes by mothers and grandmothers or letters from home to lonely service men are saved, read and reread. How many times have you looked at a letter or note from a deceased loved one? Grandmothers have saved and treasured every “Thank you” note written to them over the years by their grandchildren. It is a wonderful record of their growth. Often, it is the handwritten letter offering words of encouragement that becomes your special voice. We should take pride in taking the time to sit down and hand write a letter to friends and family. It demonstrates to them we care and our relationship is meaningful.
On a more mundane level, can your 16-year-olds write a cursive signature? Many states require a cursive signature on their driver’s license application. How do you sign and endorse checks without a signature? The clunky, longhand scratchings of today’s cursive-illiterate person can be much more vulnerable to forgery.
If you need to relay information immediately and have just a half second to grab anything, maybe a napkin, cursive is dependable. It doesn’t rely on batteries or A/C current. It’s like breathing; it’s always with you and you won’t shut down or crash.
One of my clever third graders once told me that since he now knows how to write in cursive, he knows how to read it — even his older sister’s diary.
About the author:
Linda Corson taught elementary education for 36 years in the public schools system. She recently authored a scripted teacher’s text on teaching cursive writing for new or reentry teachers and homeschool parents.
Linda L Corson Teaching Cursive, This Method Works!
linda@teachingcursive.com
By Michael Leppert
This company in its original form was founded by former animators from the Walt Disney Company. In collaboration with professional educators they have developed a unique one-of-a-kind, interactive art program that is not to teach drawing, but to teach seeing and observing.
These creators wanted their own children to learn how to “see” the way artists do and experience how artists for centuries, have communicated visually. As you would expect, the program’s videos and hands-on activities are professionally animated (illustrated) using a polar bear and two penguins – the ‘Art Guides’– to connect with children and animate the lessons and activities.
Creativity Express Online is a comprehensive Visual Arts program that includes:
• 16 lessons that inform the elements and principles of art, including light and shading, use of
color choices, shapes and show how math and art work together
• 32 hands-on art projects with integrated lesson plans
• Artist Cards and puzzle pieces as rewards for lesson completion
• a personal portfolio for your child’s work to be viewed online
• a glossary of art terms — and more.
Creativity Express Online offers a homeschool package that includes the Teacher Account Center, where the student’s progress can be tracked and Lesson Plans in pdf that can be downloaded. Art is one of the first budget cuts made in public schools, rendering entire school generations ignorant of the important cultural and intellectual values that art imparts. Using Creativity Express Online, whether at home or in an institutional school setting, guarantees that children will not have to be art illiterate anymore. Visit the website today and see all that Madcap Logic has to offer your family!
]]>By Joan A. Cotter, Ph.D.
For years geometry was not considered a topic that primary students could or should study, even though they are surrounded by geometrical shapes. I had no geometry in school until 10th grade. Sometimes teacher education programs provide little guidance on how to teach geometry. I know an elementary teacher who graduated about ten years ago with no geometry in either high school or college.
Geometry and Math
Some second graders I worked with didn’t consider their geometry work to be math until they saw numbers. Yet, geometry is more than quantity; geometry is about measurements and relationships. To determine length, we need a linear unit. The same unit is used to determine area, or surface space, by finding the number of squares having sides 1 unit long that fill the area. Likewise, the same unit is used to determine volume, or space, by finding the number of 1-unit cubes that fill the space.
Because the lengths being measured aren’t always whole numbers, fractions and decimals were needed. Also, some relationships introduced other new numerical values. There is no unit that will produce whole numbers—not even fractions or decimals—for both a side and a diagonal of a square. The diagonal in a square equals the length of a side times the square root of 2, which is approximately 1.4142135.
Another example lies in a circle. The diameter fits around the circumference a little over 3 times. Since this ratio is not an exact fraction or decimal, it is given a special name, pi.
Teaching Geometrical Names
Today, most standards for the earliest grades recommend teaching the names of common geometrical figures and objects. This is not as simple as it sounds. Take for example a rectangle. Does the longer side need to be at the base, or bottom? No. Can the four sides be congruent (the geometrical word for equal)? Yes. However, the curriculum for a well-known preschool program refers to a rectangle as “long and low.” That’s one definition that will have to be unlearned. And yes, a rectangle can be a square—a special rectangle.
Triangles frequently suffer the same fate as rectangles. Children are often presented with an equilateral triangle as a model. Andy, 5, was adamant that a triangle must have all the sides equal. Also, triangles are often shown as having one edge parallel to the bottom of the page or table. Children need to see all types of triangles in all types of orientation. And 5-year-olds enjoy learning about equilateral, isosceles, scalene triangles, as well as acute, right, and obtuse triangles.
A more difficult problem for some children is distinguishing between two- and three-dimensional figures. I showed Jonathan, 4, a paper plate and reminded him it was a circle. Then I showed him a ball and told him it was called a sphere. He knew they were both round, but he really had to think about the sphere.
Until I was about 9 years old, because of an eye problem, I lived in a 2D world. I remember asking people how can bricks turn a corner; I didn’t understand that bricks had depth as well as width and height.
Babies are born into a 3D world. But, electronic screens and television along with workbooks flatten children’s perceptions into a 2D experience, especially when they start school. To understand cylinders, prism, and pyramids, children need to touch and explore these objects just like Jonathan did with the sphere.
Drawing Board Geometry
Children learn by manipulating and creating new objects. Since the time of Euclid, to make geometric constructions, traditional geometers use only a straightedge and compass. No measuring was permitted. Complying with these restrictions made it impossible for children to delve into the wonders of geometry.
As a freshman taking an engineering drafting course, I was introduced to the drawing board and tools: T-square, triangles, and compass. When my daughters were around 7 and 8, I located the equipment, substituted a simpler compass, and showed them how to make some constructions. To add some color to their work, they did the following. Before erasing the unwanted lines, they traced over their work with colored markers and then they erased all the pencil lines. Even the lines under the colored portions disappeared.
A drawback was that the size of the equipment was simply too big for children. The board and T-square were 2 feet wide. When daughter Connie took a drafting course, she was provided with an additional smaller portable drawing board and tools for doing homework on standard-size paper. When I saw this equipment, I asked her if I might borrow it and try it with my Montessori kindergartners.
First, I showed the 5- and 6-year-olds how to construct equilateral triangles. Next they divided them into halves several different ways and wrote the fraction on each half. The children continued by dividing equilateral triangles into thirds, fourths, sixths, eighths, ninths, and twelfths. Some children also did 18ths and 36ths. One girl, Stephanie, actually divided that triangle into 256 equal parts!
The children also constructed hexagons, stars, squares, and inscribed squares. At a number of conferences, I demonstrated these activities to hundreds of teachers.
RightStart and Geometry
These geometry activities are incorporated into RightStart Grade 2 and Level C. The student needs to be able to explain how they know where to draw every line. Lines are not drawn willy-nilly. This practice of justifying every step is consistent with the whole field of mathematics.
After a number of years, teachers asked if I could write more advanced lessons. About the same time, I realized that much of the math taught at the middle school level could be approached either algebraically with lots of equations or geometrically with visual representations. Since most preteens prefer visual information, I spent four years writing the RightStart middle school curriculum. These lessons are written to the student. The intent is to encourage them to learn how to read a math textbook and work through the mathematics more independently. Ω
]]>Choosing a math curriculum for your classrooms can be a challenge for any charter school and it should not be chosen lightly. Following the latest education fad can feel like gambling with the future of the students entrusted to you, especially when these programs usually have little to no proof of any measurable growth of student knowledge. Saxon Math has been used by the charter and homeschooling communities for decades and has only grown in its popularity, due in large part to its success. Instead of force-feeding larger concepts, such as critical thinking skills, Saxon Math is meant to teach students how to discover these concepts through their own learning process. The key to success in Saxon Math is to provide each student with the support they need to not only master the drills, but to learn how to use that mastery to analyze and solve real world problems. Although this support is necessary, it may also be difficult for educators to provide to every student one-on-one.
Art Reed has more than twelve years classroom experience teaching John Saxon’s math books from Algebra ½ through Calculus. In addition to his classroom teaching, Mr. Reed became known as an experienced curriculum advisor for Saxon Math and a sought-after advisor to homeschooling families and traditional schools alike. After several decades of experience, Mr. Reed has developed a library of online video courses, each one perfectly supplementing the Saxon Math book series: Math 76 (4th Ed), Math 87 (2nd or 3rd Ed), Algebra ½ (3rd Ed), Algebra 1 (3rd Ed), Algebra 2 (2nd or 3rd Ed), Advanced Mathematics: Geometry with Advanced Algebra, Advanced Mathematics: Trigonometry and Pre-Calculus. Although these courses are designed to accompany the editions specified, they can be used to support virtually any math curricula available. However, please note the Saxon math editions noted next to each course. Recent changes to the traditional Saxon Math books have meant that Geometry is now being sold as an independent course. Mr. Reed goes into great detail regarding why he does not support many of these changes, including the unnecessary separation of the Geometry content, and why he has chosen not to support these new editions. Charter schools that choose to use the new Saxon Math editions can still easily use Mr. Reed’s online video courses with some minor adjustments. Charter schools can purchase a 2 year subscription to each course for the bargain price of $59.95. And this technology is hassle-free: the videos will work on any device that has access to wifi.
Mr. Reed is both kind and encouraging throughout the videos, while keeping a great sense of humor. These videos are the perfect supplement to the Saxon Math books because of Mr. Reed’s patient and clear teaching style, but also because they help to “fill in the gap” that some kids fall in while using Saxon Math. Saxon Math has been extremely popular in the charter school community for a number of reasons. Ready-to-use learning resources (such as solution keys and ready to print tests), widely available copies of the texts (Saxon Math is so popular that most local libraries have at least one book), and a strong track record of success are only a few of those reasons. Even some public school districts have taken notice – implementing Saxon Math as an alternative to the “reform math” programs that had been unsuccessful in raising student math scores on standardized tests. And although most of these schools have seen an increase in test scores, some have run into an issue familiar with those who implement Saxon Math in their classrooms.
Unlike many “reform math” programs that emphasize critical thinking and higher-level analysis, Saxon Math focuses on an incremental approach, in which students learn complex math concepts in smaller, “bite-sized” increments. A new increment is presented each day, along with a few practice problems on the new material, and then the majority of the homework reviews concepts covered in previous lessons. Unfortunately, this is where some kids get lost or overwhelmed in the “gap” formed when the curriculum moves on to cumulative review before the student has internalized the new content. This emphasis on cumulative review in Saxon Math can sacrifice those students who need some additional explanation and/or practice with the new material before being challenged to recall the material learned from the previous day(s). And this is especially true for those students without an educator experienced in teaching Saxon Math, or an educator that is unable to provide the one-on-one support some students will need. Art Reed’s video courses bridge that “gap” while challenging students to apply what they learn.
Throughout all of the online courses, Mr. Reed’s lessons are user-friendly and allow students control over their own learning by reviewing the steps of a specific problem or a particularly challenging concept at their own pace. Not only do students receive the additional practice and examples for new content, but it is being delivered by a professional educator with more than a dozen years of extensive classroom experience with this material. These online videos are more than a solution for the “gap” kids that need extra support, Mr. Reed shows kids how the new lessons are connected to the larger picture, including how to apply their knowledge in the real world. They are perfect for all charter schools using Saxon Math – ensuring each and every student walks away with the most valuable lesson from the series: to learn how to learn. The courses are priced to fit into the tight budget of a charter school and are a bargain when compared to the expense of private tutoring. Also, Mr. Reed is available through email for those families that need some free advice or assistance with their Saxon Math teaching. For more information, including pricing and free lessons from each video course, please visit the website at https://teachingsaxon.com.
]]>The first charter school opened in 1992, and they have been controversial since then. They have provided a healthier educational environment for minorities than typical public schools, where racial bias and discrimination are often prevalent. Since 1992, charter schools enjoyed bipartisan support from liberal Democrats to conservative Republicans.
But this Democrat support has been fading for a few years – among white Dems – while black and Hispanic Democrats still favor them as much as ever. This division by race is significant and will be an important aspect in the presidential elections of 2020. Representative groups like the NAACP and Hispanic groups will be voicing their opposition to the failing support coming from the Big 3 Democrat candidates – Sen. Bernie Sanders, Sen. Elizabeth Warren and form V.P. Joe Biden – all white.
Teachers who work in charter schools teach for the same reasons as other teachers – a desire to impart knowledge to children and serve their communities in meaningful ways. Charter Schools provide an important resource to minority children who are typically underserved in mainstream public schools and the criticism by Democrats is a bit confusing and muddled. Only time will tell what the next phase of charter school-dom will be.
]]>by Michael Leppert
In the infancy of television, one man fully understood its value as an educational tool and put it into play. Don Herbert aka “Mr. Wizard”, provided weekly science lessons in a proactive, hands-on style that made science come alive for millions of children and adults, alike. Beginning in 1951 in Chicago, “Watch Mr. Wizard” went on to become the most popular educational film of its day. Mr. Wizard made science interesting and exciting, thereby holding a young viewer spellbound (me) for the entirety of the show – and hungry for more!
In the inaugural year, Mr. Wizard produced 28 live shows, but from 1952 to 1964, he produced 39 lives shows every season, filling an entire generation’s consciousness with the wonders of science and no doubt, inspiring dozens of then-children to become grown up scientists themselves.
Mr. Herbert/Wizard was relaxed and completely natural on camera and he came by acting honestly, having acted, directed and created legit summer theatre in college, while majoring in English and General Science. He even acted with Nancy Davis (Reagan) shortly after graduation. In World War II, he flew a B-24 bomber in 54 missions over Italy, Germany and Yugoslavia. He worked in radio and began developing what would become “Watch Mr. Wizard” in 1950. He pitched it to a number of potential advertisers, all of whom passed on it! Fortunately, a group called The Cereal Institute underwrote the first year of 29 shows. In 1952, his television presence increased with a spot on the General Electric Theater every Sunday night and the main show grew in popularity. In 1954, he won the Peabody Award and produced a color version of the show for schools.
From 1955 to 1964, his show aired on NBC and then switched to the forerunner of PBS – the National Educational Television (NET).
In 1959, his first book, Mr. Wizard’s Experiments for Young Scientists, was published by Doubleday and was eventually translated to Spanish, French and Japanese. Through the 1960s, he continued to produce science shows and was featured on virtually every late-night talk show of the day, including the Tonight Show, Today, Mike Douglas and Merv Griffin. It was the first time a science teacher had so captured the imagination of mainstream America.
Mr. Wizard worked tirelessly throughout his life, producing countless classroom science series, a complete color series of “Watch Mr. Wizard” in Canada, produced a series called Experiments for NET, had a second book published, Mr. Wizard’s 400 Experiments in Science and in 1980, wrote a second book Mr. Wizard’s Supermarket Science, published by Random House and presented with an award from the National Science Teachers Association as the Outstanding Science Book for Children. This book showed the reader how to create fascinating, simple experiments using household/grocery store ingredients, explaining the science behind them.
Since Mr. Wizard originally aired, there have been other notable televisions science teachers such as Bill Nye, the Science Guy, Carl Sagan, right up to Neal deGrasse Tyson of the present day, but Mr. Wizard still reigns supreme as the pioneer of T.V. science teaching. DVDs of his shows are available from the website http://www.mrwizardstudios.com/ and they can continue to spellbind young viewers for decades. Do your children a favor and check them out – and if you are old enough, breathe in some nostalgia as you watch with them! MjL
]]>Which one would your students take? The SAT or ACT? Well, twenty years ago, making the choice was easy as pie. Back then, it all boiled down to where you wanted to go to college: You sat the ACT for colleges in the north and mid-west; and the SAT for the rest of them (colleges in the South, and on both the east and west coast). Nowadays, basically every university in the United States widely accepts both SAT and ACT results. Even if a school prefers one over the other, admissions officers usually convert the scores interchangeably.
We need these standardized tests so that we can compare the abilities of students across the country—fairly. For example, a 4.0 GPA at one school can mean something entirely different to a 4.0 earned at another. How else can we make up for obvious differences between student knowledge, teaching aptitude, degree of difficulty across different curriculums, and just plain old marking biases? And that’s where standardized tests like the SAT and ACT come in, as they help compensate for these differences by leveling the playing field. Interestingly, a student’s scores also help predict what kind of academic success they’ll have in their first year in college.
The creators of both the SAT and ACT are guided by very similar philosophies: Their aim is to design an instrument to assess a student’s critical thinking and problem-solving skills. And the similarities go much deeper.
In both tests, students will find questions that are objective and have only one correct answer. Sections dedicated to math, vocabulary, and reading comprehension assess the learners’ “innate abilities.” Tricky and confusing phrasing is purposely used to determine skill level. This also has the effect of checking how a student performs under pressure and their ability to identify exactly what is being asked of them. It isn’t necessarily measuring comprehension on a specific subject, but of course does cover basic high school material, by default. What the examiners are more interested in is how well a student can critically think through a problem—considering they are given roughly one minute per question—and then move on.
Now that the SAT has been redesigned, the format is very similar to the ACT. When the new president of the College Board was appointed, he hired ACT writers to create the redesigned SAT. The resemblance between the instruments is good news to any college hopeful. Both have four long sections, require a student to understand basic test-taking techniques, and need them to answer the questions quickly. The best score a student can receive on the SAT is 1600 and 36 on the ACT.
Here is a summary table of the main similarities and differences between the SAT and ACT:
SAT | ACT | |
Reading | 4 answer choices | 5 answer choices |
Writing | Tests grammar, style and analysis | Tests grammar, style and analysis |
Math | Trigonometry, geometry, algebra, contains geometry formulas | Trigonometry, geometry analysis, no grid-in questions |
Science | No Science section | Science questions similar to SAT Reading section |
Essay | Analytical response required, duration 50 minutes | Persuasive writing required, duration 40 minutes |
Scores | Scores are not averaged | Sections are averaged |
Annual frequency | Offered 7 times per year | Offered 6 times per year |
Permitted attempts | Unlimited | Limited to 12 attempts |
Best possible score | 1600 | 36 |
Website | www.collegeboard.org | www.act.org |
What is crystal clear is that learning critical thinking skills will benefit students whether they sit either or both tests. And there are plenty of other standardized exams where these skills are completely transferable. These include the popular PSAT/NMSQT test, which when taken in a student’s junior year could yield incredible scholarships like full tuition, free room and board, graduate school money, study abroad stipends, and more. The list of other exams that will benefit from learning test-taking skills include AP, Subject Tests, GRE, CLEP, LSAT, ISEE and so on.
When you boil it down, the SAT and ACT largely examine the same aspects of a student’s capabilities, in similar ways, yielding similar results that can be converted to suit the institution you or your student is applying for. The question I find people are asking now is “if they are so similar, is there a benefit in taking both tests?” The answer is “Yes.” Despite the incredible similarities, it does seem prudent to consider doing just that. While the tests both fulfill the same role in the admissions process of college, some colleges do give a better scholarship based on their preference for using the results of one test over the other. Nowadays, many students are considering taking both the SAT and ACT so they can stack the cards in their favor, showcase their abilities and receive more money.
If you’re looking for a program that will help you or your student ace the SAT (and other standardized tests that could make a huge difference to your students’ futures) then take the time to check out the College Prep Genius programs. Thousands of students swear by Jean Burk’s system and yours will too! www.CollegePrepGenius.com
]]>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.
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.
]]>By David Chandler, Math Without Borders
I just got out of an email exchange with a long-time friend who quoted the following rant and challenged me to justify why algebra should be included in the high school curriculum.
The quote: “I cannot see that algebra contributes one iota to a young person’s health or one grain of inspiration to his spirit… It is the one subject in the curriculum that has kept children from finishing high school, from developing their special interests and from enjoying much of their home study work. It has caused more family rows, more tears, more heartaches and more sleepless nights than any other school subject.” –Arthur Dean, MIT graduate, engineer, math teacher, writing in an item published on March 27, 1930.
My answer: “Mathematics allows one to ask quantitative questions of nature. It allows us to see the world in a fundamentally different way and it promotes what someone has called ‘empowered curiosity.’ To do Algebra 1 in isolation, and then stop, is like learning scales but never playing a song, or learning the alphabet without ever reading a book or writing a story. Mathematics is a fundamental form of creativity.” … I could have continued on for quite some time.
Consider one topic introduced in Algebra 1: graphing. In Algebra 1 you learn to plot points as ordered pairs, like (2, 3), over 2 and up 3, and graph linear equations which, unsurprisingly, turn out to be straight lines. The idea that an equation (which is an algebraic object) can be associated with a graph (which is a geometric object) has some profound implications in itself. It means you can learn to “picture” what you are doing when you do algebra. Pretty much anything you can do algebraically you can do geometrically, and vice versa.
When an algebraic equation is graphed, each variable is plotted on its own axis. If there are two variables in a problem, you can represent them as the horizontal and vertical directions on normal 2-dimensional graph paper. If there are 3 variables you would need to picture the graph in 3-dimensional space. What if a problem has 20 variables? You would need to “plot” it in 20-dimensional space. The physical world is limited to 3 spatial dimensions, but mathematicians don’t limit themselves to 2 or 3 dimensions. They commonly talk about n-dimensions, where n can be as large as you like. How could this ever be meaningful in practice?
Some years ago I was the adviser for a computer club at a private boys’ school. Across town there was a girls’ school and the two schools jointly sponsored dances a few times per year. Our computer club took on as a project writing a “computer dating” program so the boys at our school and the girls at the other school could easily find dates for the dance with similar interests.
We wrote out a questionnaire where each question was rated on a scale from 0 to 10. We asked questions like, “How much do you like reading?”, “How much do you like sports?”, etc. If we asked 20 questions, we could represent the interests of the student as a point in a 20-dimensional “interest space.” Instead of being an ordered pair, a point would be a “20-tuple,” like this: (8, 4, 2, 9, 6, 4, 2, 0, 8, 4, 10, 7, 3, 8, 0, 10, 5, 4, 1, 6). You can’t visualize 20-dimensional space, but you could visualize 2 or 3 of the scales at a time. Apart from visualizing, working with 20-tuples is just about as easy as working with ordered pairs.
In Geometry you learn the “distance formula” for the distance between two points on an x-y graph. You find the difference of the two x-values, the difference of the two y-values, square each of these differences, add them together, and take the square root. The differences tell how far over and how far up you have to go to get from one point to the other. If you try this on graph paper you will see you have traced out a right triangle. The “distance formula” is just the Pythagorean Theorem. In three dimensions you would do the same, except you would use the difference of the x’s, the difference of the y’s, and the difference of the z’s. Square each of these, add them up, and take the square root for the distance in 3-dimensional space. It’s like finding the diagonal distance from opposite corners in a box.
Back to our 20 dimensional “interest space.” The goal is to find people with similar interests, so it makes sense to measure the degree of similarity as the distance in this 20-dimensional space. In the computer we would take the difference on the reading scale, square it, take the difference on the sports scale, square it, etc. for all 20 scales. Finally add up the squares and take the square root of the answer. This would be the “20-dimensional distance” between these two people. The computer would match each boy with each girl and come up with a list of the 5 closest matches for each participant.
The moral of the story is mathematics is not just “about” solving a particular kind of problem. It gives you tools for thinking, and the ability to apply the tools to wildly different problems. The tools themselves are creations of the human mind. Each tool may have been inspired by a particular problem, originally, but the resulting mathematics is not limited by the original context. Mathematics is truly a fundamental form of creativity.
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