Showing posts with label Mathematics. Show all posts
Showing posts with label Mathematics. Show all posts

Saturday, June 01, 2024

Best Local Stargazing and Dark Sky Sites

Nothing says applied mathematics like physics, and nothing says physics stars and cosmology.

Following is a list of best places in the SF Bay Area and around California to find dark, clear skies ideal for star gazing.

A good pair of 7x50mm binoculars makes stargazing even more amazing. Click here for a great deal on an excellent set for beginners.

Monday, April 01, 2024

Trivium and Quadrivium

The reason I've always been captivated by the Trivium and Quadrivium is almost certainly that these ancient western educational models happen to coincide with six main interests of mine: math, music, astronomy/cosmology, logic, writing, and debate. 

Moreover, philosophy, another one of my main interests, was considered such an obvious part of classic liberal arts training that it wasn't included in the list of subjects for either the Quadrivium or Trivium.

From the Wikipedia article on Quadrivium:

"From the time of Plato through the Middle Ages, the quadrivium (plural: quadrivia[1]) was a grouping of four subjects or arts—arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in the trivium, consisting of grammar, logic, and rhetoric. Together, the trivium and the quadrivium comprised the seven liberal arts,[2] and formed the basis of a liberal arts education in Western society until gradually displaced as a curricular structure by the studia humanitatis and its later offshoots, beginning with Petrarch in the 14th century. The seven classical arts were considered "thinking skills" and were distinguished from practical arts, such as medicine and architecture."

One has to wonder what our society would look like if schools prioritized these essential subjects in grades K-12.

St. Ann Classic Academy is a school trying to implement such a curriculum.

For an excellent book on the Quadrivium, try Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology. 

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Monday, January 01, 2024

Desmos – the New Standard

The online
Desmos graphing calculator is fast taking over from the venerable Ti-84 series of handheld calculators as the default calculator tool in secondary education. Desmos is now included as an integral part of the digital SAT, and acquiring intermediate-level Desmos skills is fundamental to maximizing math scores.

[Familiarity with the Ti-84 Plus CE handheld graphing calculator is still crucial to optimizing math scores on the ACT.]

I'm not aware of any succinct, comprehensive exposition of Desmos skills required for use on the dSAT (I'm working on it).

At this point, the best one can do is to peruse the various official materials linked in the "Desmos First Steps" and "Desmos Graphing Calculator" sections below. 

Check out each link, read the information provided, and do the sample exercises until you've covered all topics presented (search Google for additional help with particular topics).

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Desmos First Steps

User Guide

Quick Start Guide

Getting Started: Desmos Graphing Calculator

Getting Started: Creating Your First Graph

Getting Started Articles

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Desmos Graphing Calculator

Graphing Calculator

Graphing Calculator: Essential Skills

Graphing

FAQ: Graph

FAQ: Student Graphing

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Desmos Geometry

Geometry

Geometry Tool

Transformations

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Other Desmos Calculators

Scientific Calculator

Matrix Calculator

3-D Calculator

Friday, September 08, 2023

The Backward-Forward Method

Unfortunately, little or no time is spent in most math classrooms discussing "heuristics," the art of problem solving. 

This often leaves students grasping at straws, struggling even to know where to begin when staring down an unfriendly, unfamiliar math question.

George Pólya's How to Solve It is a classic on this subject, required reading for all serious math students.

In another classic, How to Read and Do Proofs, author Daniel Solow advances a powerful problem solving approach he calls the “Forward-Backward Method.” 

I’ve found it helpful in my own teaching and mathematical work to reverse the method, first thinking backward from the ultimate goal to various subgoals which, if achieved, would enable direct progress to the original objective.

Whether writing complex proofs or tackling simple algebra problems, this “Backward-Forward” process provides students with a simple yet powerful structure for solving problems.

Just as an archer would prefer to move the target closer, so can a math student make a problem easier by finding a nearer target to shoot at. The next step would be to think further backward, to find another even closer target tied directly to the first, and so on. These subgoals are set by repeatedly asking the same question: “What would I need to know to find that?” And then “What would I need to know to find that?” Subgoals should be written down, to keep the trail clear.

After looking backward as far as possible, it’s time to reason forward from each given fact, with the last subgoal in mind. A different question governs the forward process: “What can I imply from that fact?” And then “What could I imply from that?”

Eventually, forward progress enables us to hit the target. All that’s left is to follow the string of subgoals up the ladder to the desired result.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Saturday, July 08, 2023

Well Begun is Half Done

Aristotle’s admonition to make a good start on any journey counts doubly on difficult SAT/ACT math problems. 

Beginning is often half the battle, and almost anything you can do to get yourself going will probably be helpful.
 
This is sometimes easier said than done, but there are several things you can generally try. It's good to keep in mind a few tips to help grease the wheels when stuck at the beginning of a tough question.

Primary among these is to reread the question slowly and carefully, at half-speed. Many times, you’ll find you simply missed something, and can now solve the problem. Easy as that.

To get a feeling for what’s going on, experiment with simple, realistic numbers in place of unknown quantities. Let the cost of the sweatshirt be $20, for instance. Use that number in the problem, and see what happens. Based on what you learn, the solution may reveal itself.

You can try making up a “simpler similar problem.” Solve that simpler problem, and apply the same approach to the more complex one you’re tackling.

For multi-part questions, pick the easiest part, and work that out first. With such a “jump start,” you may find you’re able to make progress and find your way to the answer.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Wednesday, March 01, 2023

lizardpoint

Lizardpoint is a wonderful all-around early learning site for people of all ages. Students can use lizardpoint to learn geography, flags, world leaders (current and past), art, and math.

The site is great for adults as well as for children.

Sections on geography help students master finding countries and provinces, naming their capitals, topographical features, airport codes, and more.

The section on art features quizzes on major western artists and their most well-known works, fine art terminology, and important movements.

Math offerings include on-screen and downloadable/printable worksheets (with answer sheets) for arithmetic operations, decimals, percents, and fractions.

Quizzes on trivia, important definitions, and more round out learning activities presented.

New stuff is added regularly, collected in What's New.

Like Freerice.com, lizardpoint can be addicting, but in a good way. These two educational sites keep interested students learning and growing while having fun.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Tuesday, November 01, 2022

Wolfram Alpha

Stephen Wolfram and his engineers, makers of Mathematica, are at it again.

Wolfram Alpha is Stephen Wolfram's groundbreaking computational knowledge engine, one of the earliest such applications online. 

Type a question into the calculation field, and Wolfram will input the question and output the answer on-screen.: "How many goats in Japan in 1986?" Alpha knows the answer (~47,500).

Input z=x^2+y^2, and Alpha will output a 3-D graph along with detailed information about the relation.

Also of interest is Wolfram Mathematical Functions, the largest repository of mathematical functions ever to exist. 

Math fans, especially, will find Wolfram Alpha a fun site to peruse.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Thursday, September 01, 2022

Gödel, Escher, Bach

Douglas Hofstadter's mind-bending tome about self-reference and meta-thinking published in 1979 is a classic in recreational mathematics. Getting all the way through it is a tantalizing, formidable intellectual challenge.

Self-reference applies to ideas that loops inward (outward?) on themselves. 

Some examples:

This sentence is false. Seeing one's own eyeballs (without using a mirror). Brushing the bristles of a brush with that same brush.

Hofstadter compares the works of three geniuses: Kurt Godel in the domain of pure mathematics, M. C. Escher in the world of fine art, and Johann Sebastian Bach in the realm of western classical music. The similarities are, indeed, surprising and impressive.

All three masters dealt with the concept of circular self-reference, but in different ways. Godel proved the illogical nature of mathematics (which is, itself, based on logic); Escher was famous for stairs that climbed upward to the bottom of the stairs and identical tessellating foreground and background images that seem, somehow, to "cause" each other; Bach would take a short series of notes, and then invert the same motif, play it backwards, string these versions together, etc.

The book is highly intriguing, almost addictive. But it isn't for the faint-of-heart or faint-of-mind. I tried to finish it. Wasn't able to. Made me dizzy. Read it at your own risk.

Buy GEB here.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Monday, August 01, 2022

Get a Head Start

A head start in life is a great advantage, and nowhere is this more clear than in the educational realm.

For example, any parent with a good high school education can teach his or her child to read at a very early age. See: Teach Your Child to Read in 100 Easy Lessons.

Likewise, any child who can draw stick figures with a crayon can learn to form letter shapes.

Simple mathematical concepts can be taught as soon as a child can count. Counting objects is the first step in acquiring "number sense," a visceral feel for numbers essential to mastering mathematics. The goal of all early math education should be the development, strengthening, and maintenance of number sense, along with the ability to do accurate mental and paper/pencil calculations and estimates. These aims should be ardently pursued and achieved in grades K-6.

Parents can work with eager children at home to provide them an early start. My own daughter was reading at age four, mastering math facts at five, operating with negative numbers at six, typing at seven, and doing algebra at eight.

Not all parents are professional academic tutors, as I am. Still, it doesn't take much forethought for most parents to provide their children a leg up in their schooling.

For those who wish to have professional guidance, I offer private consulting for parents of young children. In addition to providing general information about schools, early education, and tutoring at home, I help parents design and implement individualized, age-appropriate academic head start plans tailored to each child's particular needs and interests. You can contact me here for more information about this unique service.

An good head start is a great gift, one of the best parents can offer their children.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Friday, July 01, 2022

Mad Minute

Mastery of the basic facts of arithmetic is the first step toward mathematical fluency, and is a central early academic goal. 

A "Mad Minute" is a simple, daily exercise for elementary school students striving to memorize their basic math fact "tables." Mad Minute worksheets contain several dozen questions using the same operation (addition, subtraction, multiplication, or division), and are an excellent way to provide kids with daily practice.

To complete an exercise, students are given one minute to answer as many questions as they can. Each correct answer earns one point. After the minute is over, the number of correct answers is counted and written atop the page. Speed, therefore, is essential to a high score. However, students stop earning points after the first mistake! So accuracy is even more important. 

[Note: Mad Minutes introduce and reinforce two very important rules in studying mathematic – speed is important but accuracy is even more important – and are therefore doubly beneficial.]

Mad Minutes should be assigned at home each day (one for each operation, if possible, according to the student's current ability).

Mad Minute exercises are also great ways to track progress, since each completed exercise acts as a current assessment of mastery. Students might enjoy recording their scores in a 2-column table, and visualizing them using a 2-D chart with dates along the horizontal axis and scores along the vertical – an excellent way to introduce the concept of numbers as data. You can find large-format graph paper here and here.

Mad Minute books, providing multiple worksheets for each operation, have been around for decades. Click here to buy the latest on amazon.

Free downloadble/printable basic number fact worksheets, along with online drills and simple games, plus more worksheets and drills covering advanced work with operations, decimals, percents, and fractions, and simple mental exercise games like Make a Match and Find It, can be found on lizardpoint.com here.

Hint: Downloaded files are easier to work with, since workbooks eventually fill up and need to be re-purchased. Mad Minute exercise files, on the other hand, like those available on lizardpoint, can be printed over and over again, as needed.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Tuesday, February 01, 2022

Freerice

At
freerice.com students of all ages can learn everything from basic arithmetic to art history, world geography, anatomy, and English grammar and vocabulary, while earning grains of rice to donate to the United Nations' World Food Program (WPF).

"Rice" is used as a metaphor for donations generated by users of the site that sponsors then fulfill. Each correct answer stores 10 grains of virtual rice for donation to the WFP. Sponsors then give the monetary equivalent of all rice collected to the WPF to fund its charitable work around the globe.

From the site:

"The United Nations World Food Programme (WFP) is the 2020 Nobel Peace Prize Laureate. We are the world’s largest humanitarian organization, delivering life-saving food assistance in emergencies and working with vulnerable communities to improve nutrition and build resilience."
Freerice is a great way to learn while helping others in need throughout the world. The site generates billions of "rice grains" each year, a total of 224 billion to date! Caring students get smarter while making the world a better place in a concrete way."

For anyone who loves learning, it's hard to imagine a better way to have fun.

FAQ here.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Friday, October 01, 2021

Mathematical Logic

The trunk of the tree of mathematics divides into two main branches: applied mathematics, and pure mathematics.

Applied mathematics is concerned with calculation. Getting the right answers. Building things. Making sure the probe lands safely on Mars, that the bridge can withstand high winds, that revenue will exceed expenses. Utilitarian math.

I've always been interested in pure mathematics: the study of numbers, purely, for no other reason. Useless math, in other words. Math for the sake of math, only. Utterly non-utilitarian math.

The purest of pure math is logic, the foundation of mathematics. Mathematical logic is "meta-mathematics," the software running the machine, the engine under the hood.

One my favorite undergrad courses was an upper-division class in mathematical logic. Not long ago, I decided to take out some old textbooks, and summarize what I'd learned decades ago. The result was a set of simple notes for doing "Truth-Tree" proofs, one my favorite class activities.

You'll find the notes here.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Monday, February 01, 2021

Cart Before Horse

Students in marginalized groups are not doing well in math, and this is hampering upward mobility. But the solution provided in the recently adopted California Mathematics Framework (CMF) doesn’t serve the laudable goal of improving mathematical fluency in under-represented populations. 

In a 11/29/21 article by Joe Hong in Cal Matters, Tom Loveless, a retired math education expert puts it in a nutshell: “The way you get social justice in mathematics is to teach the kids math … not by dressing up mathematics in social justice.”

In the original draft of the CMF, central author and instigator Jo Boaler actually went so far as to write that mathematical talent isn't a thing, that it doesn't exist. Of course, to say such a thing is nuts. It's ridiculous. Nevertheless, differentiated advanced instruction for gifted students was condemned, at best, in early drafts of the CMF. Palo Alto parent Avery Wang makes the point clear. “Holding back high achievers makes them achieve more? That’s exactly the same philosophy that’s being promoted in the math framework.”

But this is the inevitable, illogical conclusion of woke ideology in math education, and woke ideology, generally: “Our personal differences are of central importance, but despite real differences, we should all achieve equally.”

The CMF is the culmination of a decades-long crusade within math education circles to teach math “constructively" by "discovery” and make math “more relevant” and “more fun." As a long-time member of the National Council of Teachers of Mathematics, a trade group for math educators that took a leading role in promoting constructivisism the late 1980s, I’ve watched this movement gain steam in recent decades.

Music Theory is not “Music Appreciation.” Those are two entirely different courses. Likewise, “Precalculus” and “Recreational Mathematics” are utterly dissimilar in purpose, method, and scope.

In the article, Michael Malone, parent and math tutor, puts it well: “They’re changing math to make it math appreciation. A part of math is learning things that are not authentic to life.” He then opines that the CMF “does a disservice to historically marginalized student groups by offering them a simplified version of math that fails to prepare them for the challenges of a career in science, tech, engineering or math.” Finally, Malone correctly concludes: “Math is gonna be hard for students who don’t enjoy it as much.”

[Thank you, Captain Obvious.]

In a separate Cal Matters article, UC professor Svetlana Jitomirskaya expresses her exasperation by the decision of the authors of the CMF not to seek much input from from STEM experts who naturally have first-hand awareness of the level of mathematical maturity and training incoming undergrads must have. “The process should have definitely involved STEM faculty from top CA universities with direct knowledge of what is needed for success as STEM majors,” she emailed. “It is absurd this was not done.” 

Jitomirskaya further criticizes the CMF for emphasizing “exploration at the expense of skills development,” and says there’s a “mountain of evidence that similar ideas have consistently failed when implemented at scale, and a rigorous approach — teaching students to back up answers with logic — is the only method known to decrease the [mathematics achievement] gap.”

This has been my own experience as a professional math educator for 45+ years. Sure, it would be nice if each student could reinvent the wheel, and in an ideal world, constructivism would be the best approach to take in teaching mathematics. In a very small class with a genius teacher, highly motivated and gifted students, and two math periods a day, constructivism could play a key role and be an important and highly productive part of the mix. But how many American math classrooms does this describe? As the professor points out, the constructivist idea doesn't scale. It's good in theory, bad in practice. Although both are important ideally, in reality, acquiring mathematical skill is more important than discovering mathematics.

Citing CMF-styled math exercises, Professor Jitomirskaya shows how these problems are illogical, poorly formed, and could introduce "a wrong idea of what it means to solve a problem — something that college professors struggle to undo."

Jitomirskaya states “It is irresponsible to make the entire state a laboratory for very controversial educational theories ..." and concludes that "Social justice, while desirable and necessary, will not come about by abandoning mathematical rigor."

I couldn't agree more. If one is concerned about upward mobility, getting good grades in math isn't what counts. What does count is having genuine mathematical skill and intuition, which cannot be developed simply by watering-down curricula so that struggling students are able to show better marks on their report cards. Masking problems doesn't make them go away. Pretending students are accomplished isn't helpful, and it isn't compassionate. Eventually, the rubber will actually hit the road. Again, quoting Tom Loveless: "The way you get social justice in mathematics is to teach the kids math."

Whatever one thinks of the "social justice" movement, math is still math and chemistry is still chemistry. The derivative of 6x^2 and the atomic weight of boron have nothing to do with “diversity, equity, and inclusion.”

If we're smart as a society, we'll make sure the horse precedes the cart.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Wednesday, July 01, 2020

The Centimeter Grid

Use of a "Centimeter Grid" is a wonderful, multi-sensory way to teach basic math facts: addition, subtraction, multiplication, division.

Using the grid, students color-in squares to represent numbers, and then count the end result.

For instance, to learn 2 + 6: 

The student first colors two squares the same color, labeling them with a "2," and then six more in the same line using a different color, labeling these with a six, and finally, after counting up all the colored squares, labeling the entire set of colored squares with an "8." By this demonstration, it's clear that 2 + 6 = 8. [It's also clear that 6 + 2 = 8, 8 – 6 =2, and 8 – 2 = 6, thus completing a "fact family" cementing the  addition/subtraction relationship of the numbers 2, 6, and 8].

After discovery of each math fact, students "collect" the facts by writing each one on a flash card for later games of "flip the card" to help with memorization (Triangle Cards can speed up the process considerably by emphasizing fact family relationships).

But memorization should only be attempted after discovery. Students must first discover the math fact experientially, preferably physically, in multiple ways, by repeatedly demonstrating the fact for themselves. Then they record the fact for purposes of memorization. 

The order here is critical: discovery first, then recording, and finally memorization.

Consistent with the Scientific Method, it's best if students use more than one method, and repeat the experiment several times, to confirm results before recording them (e.g. first using a Centimeter Grid, then a Hundred Numbers Chart, and then counting pennies). This helps ensure the development of "number sense," a core mathematical capacity without which memorization is an empty exercise, at best. Memorization of math facts without corroborating discovery robs students of the intuitive "feel" for numbers they'll need to be successful in advanced courses later on.

Only if the student knows, experientially, by his own experimentation and record keeping, that 2 + 6 does in fact make 8, will he be able to make "sense" of that fact and integrate it with other ideas. This is a crucial distinction: the difference between mere belief and actual experience; between mastery and connectable knowledge on the one hand, and isolated, disassociated, meaningless memorization on the other.

Download your own copy of a Centimeter Grid here.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Monday, June 01, 2020

Math and the Master

The phrase "Renaissance Man" is epitomoized by Leonardo da Vinci, the master of masters, founder of the High Renaissance. Geometry infused Leonardo's work, and was a particular obsession of his (e.g. The Golden Ratio, perspective, knots, fractals).

An article published by The Mona Lisa Foundation goes into some detail about the geometric underpinnings of Leonardo's design thinking.

It begins:

The important relationship of mathematics to art cannot be [overstated] when discussing Leonardo’s later work, and in numerous documents, letters and notes, the relevance of this is well documented. At times, he seems obsessed with these issues: while working on Mona Lisa for example, Leonardo is reported by Fra’ da Novellara to be concentrating intensely on geometry.

“Non mi legga chi non e matematico.”

“Let no one read me who is not a mathematician.”

-- Leonardo da Vinci

[Continue reading here.]

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Wednesday, January 01, 2020

Early Math Materials and Pedagogy

For several years, when I was a young father, my daughter and I took an enormous amount of pleasure in working together on early math.

In the process, I used and developed a series of graduated activities to enable her to learn increasingly advanced concepts at her own pace. 

The primary goal was always to develop "number sense," an intuitive feel for numbers and how they behave. Efforts at memorization came only after concepts made "sense" and were fully internalized.

Now that I'm a proud grandfather of another little girl, I recently reviewed early outlines of these activities, principally for my own recollection, but also so that I could recreate these happy experiences with my daughters daughter.

Interested parents or grandparents are welcome to download my rough notes here and here for their own use in beginning or supplementing early math education within their own families.

I hope you and your young ones have as much fun exploring early math as we did.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Sunday, December 01, 2019

Flatland

This late 19th century science fiction work about single and multi-dimensional worlds was penned by actor, minister, and headmaster Edwin Abbott to interest his students in geometry – and also to mock some of the many vagaries of Victorian society.

Written in inimitable 19th century British style, the book is a classic novel that brings to life core concepts of mathematics and physics that would be otherwise inaccessible to most mere humans.

An active imagination is all that's required to absorb the core concepts presented. It's easy to enter Abbott's 1-D, 2-D, and 3-D worlds, and, by extension, into those of even higher dimension. 

After all, the book was written with a middle school audience in mind!

Middle school 150 years ago was a different animal than it is today. Nevertheless, Flatland remains both an accessible recreational math primer and marvelous short work of fiction and satire. Less than 100 pages long, Flatland can easily be read in a single sitting.

I used to buy this book by the dozen, and give them away as gifts to my most curious math students.

You can read others' impressions of Flatland and order you own copy here (or get the ultra-inexpensive edition here).

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Sunday, September 01, 2019

Hundred Numbers Chart

A 10x10 grid of numbers written from left to right, top to bottom, starting in the upper-right corner with 1 and ending with 100, is called a "Hundred Numbers Chart." 

It's a highly effective tool to teach eager youngsters how to add and subtract two digit numbers, instinctively, using the chart as an aid. Eventually, the chart becomes internalized, and students can do the calculations rapidly and accurately entirely in their heads.

To begin using the chart:

First, make sure the child can already count by 10s. Then have the child learn from experience that the numbers in the chart simply represent counting, and that counting forward 10 squares can be done more easily by starting on any square and moving straight down to the square immediately below.

To demonstrate 23 + 35 = 58:

Start with your finger on 23. Move straight down three rows (each move represents adding 10, so three such moves represents adding 30). Now, count right five spaces. The answer, clearly, is 58. A similar process can be employed to carry out subtraction. Borrowing and carrying is handled by wrapping around the ends of rows as one executes the process.

Parents should give this a try themselves, first, and become masters at utilizing the tool before attempting to use it to instruct their children. After a very short while, using the chart becomes second nature for adults.

Children will take more time to achieve the same level of skill, of course. But with patience, practice, and plenty of good energy, encouragement, and hand clapping, they'll soon be performing difficult mental calculation with ease and accuracy.

A Hundred Numbers Chart is also useful in teaching young kids to count by twos, fives, tens, threes, and fours (a precursor to learning multiplication). 

For example, to learn to count by threes:

Have your student start on three, circle that number, then count spaces three at a time, circling the number in colorful crayon each time they land on a new square, and saying each new number out loud. Verbalization is critical as an aid to memory. The colored numbers will form a clear geometric pattern that will eventually make memorizing the sequence easier. After a while, counting by twos, three's fours, etc. will become second nature.

Learning to count by threes, for instance, is a great way to introduce multiplication. One 3 is 3; 2 threes are 6; 3 threes are 9, etc. Again, doing this activity out loud and with colored crayon is super important. Before you know it, your child will have mastered the threes.

Click here to download a Hundred Numbers Chart to print and use at home.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Monday, July 01, 2019

Slow Down to Go Faster

High-stakes timed tests like the SAT and ACT are inherently stressful experiences, and reducing stress is a primary goal. A chief driver of this stress is time pressure.

Paradoxically, sometimes the best thing one can do to improve speed on timed tests is to do untimed practice.

In untimed testing, students take all the time they need to fully understand questions and find correct answers. Without the pressure of the clock, it's much easier to master various question types and discover the best ways to find right answers most quickly and easily.

It's also ironic that getting stuck on super-hard questions during untimed practice is actually a good thing. By spending way too much time on impossibly difficult questions, students learn to quickly recognize "nightmare questions" they can't answer and just going to waste their time on test day (the strategy: eliminate, guess, move on).

After gaining everything possible from untimed testing, students then return to timed practice, and work on speeding things up. Now they know what to do – they just need to do it faster.

This approach has been useful to a great many students of mine over the years, particularly on the ACT science section, which is famous for being a time-burner. Unfortunately, its counter-intuitive nature can make this miraculous study tactic uncomfortable to use at first. Once regularly employed in practice, however, improved results generally dispel any initial doubts or fears.

Building muscles slowly in the early stages of any strength-building process just makes sense. The same goes for building test-taking muscles.

It's a maxim that applies to disciplines as widely varied as musical performance, athletics, academics, and more. At all stages of the learning process – and especially at the beginning – deliberately sacrificing speed for the sake of developing concentration, accuracy, and control is the best way to optimize progress.

Slow down, to go faster!

Saturday, June 01, 2019

Cosmic Eye

This epic science video takes viewers from the realm of everyday human experience to macrocosm, microcosm, and back, by the power and magic of exponential growth. 

It's a brief 2:29 excursion that could make your day.

Powers of 10 was one of the first such "logarithmic zooming" films, released in 1977 by ground breaking mid-century designers Charles and Ray Eames (see Wiki article here). Orders of Magnitude is an excellent recent variation on the same theme.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.