Borland Educational News and Views: 2019

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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Contact Chris

borlandeducational.com

Friday, November 01, 2019

Fox, Chicken, Grain

"Fox, Chicken, Grain" is one of my favorite logic puzzles for younger students.

Here are the rules.

A farmer wants to successfully ferry a fox, chicken, and sack of grain across a river. He can take several trips back and forth, drop things off, and pick things up on each side of the river between trips. But he can carry only two things in his small boat, other than himself, at one time.
Unfortunately, the farmer has a few problems.

If left alone together on the same side of the river, the fox will eat the chicken, and likewise, the chicken will eat the grain. But the fox will not eat the grain (and, of course, the chicken will not eat the fox).

How to get everything safely across the river?

I usually use coins of different sizes to represent the fox, chicken and grain, and a small torn piece of paper as the boat (rowed by the farmer).

It helps to use a sheet of paper with a "river" drawn on it to simulate the situation. You an download a printout here.

Similarly, there's a puzzle called "Cannibals and Missionaries," in which three missionaries are trying to get themselves across a river filled with man-eating fish. They have to contend with three cannibals, as well, who also must get across the same river in the same canoe at the same time.

Here are the conditions:

Only two people can occupy the boat, and multiple trips can be taken in either direction. People can get off the boat, once it gets to one side of the river or the other, and can re-enter the boat, at will. Unfortunately, if the cannibals outnumber the missionaries at any time on either side of the river, they will eat the missionaries. If this happens, the games ends unsuccessfully.

Complicating matters, only one missionary knows how to paddle the canoe, although all the cannibals know how to do so.

How to get all six people across the river in one piece?

In this case, I use three coins of one type to represent the missionaries (only one of which is heads up, indicating he can paddle the canoe), and three of another type as the cannibals.

Likewise, it helps to have the piranha-infested river drawn on a sheet of paper as a backdrop for the game, and a small piece of torn paper to use as the boat. Click here to download the backdrop.

In both games, the analogy to mathematics is found in the fact that, at each and every stage of the game, only one move makes sense and isn't obviously problematic. Patient, mindful examination of each potential move at each decision point leads one inevitably in the right direction – just as it does in solving math problems.

Can you do it? Give these a try!

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Contact Chris

borlandeducational.com

Tuesday, October 01, 2019

Test Your Mindfulness

How strong are your mindfulness muscles?

To find out, try your hand at these deceptive, simple-looking tasks.

In the first case, you job is to simply read the sentence contained within the triangle. As you read the sentence, write it down on a piece of paper. With luck, you'll see what's going on, here.

In the second case, try to say the COLORS of the words written, NOT the words themselves, in order, while reading at a normal pace. If you mess up, start again. Do not slow down to a snails pace in order to succeed in getting to the end of the list (although even that might not work).

Mindful, attentive concentration is a critical prerequisite for success in mathematics, and for quite a number of other activities, as well. Mindfulness is a capacity that can be trained, and just like any other training regimen, one begins with weakness but gets gradually stronger with practice and determination.

To find more puzzles for practice, search the internet.

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Contact Chris

borlandeducational.com

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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Contact Chris

borlandeducational.com

Thursday, August 01, 2019

100 Key Math Concepts for the ACT

First produced a number of years ago, Kaplan's 100 Key Math Concepts for the ACT still has relevance for ACT test takers today.

Each core math concepts presented is elemental and should be easily understandable to any high schooler signed up to take the test. Nevertheless, a great many will have trouble with at least several of those listed.

Students should use the document as a checklist, as follows:

Beginning with the first item, grade each concept A, B, or C depending on your current level of understanding. If the topic is completely understood, give the item an "A" grade. If the item is not at all understood, give the item a "C" grade. Somewhere in between? Give the item a "B" grade.

Your task, of course, is first to master any concepts graded "C," and then all those graded "B." The best way to accomplish this is to use a friend, tutor, or Google to help you learn to handle each concept and/or skill presented.

The list is a bit dated, however, and the ACT writers have since updated the content of the test's math section to include more advanced concepts.

Click here for the original document and here for a list of the most recently added ACT math topics.

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Contact Chris

borlandeducational.com

Monday, July 01, 2019

Slow Down to Speed Up

High-stakes timed tests like the SAT and ACT are inherently stressful experiences. Reducing stress, in practice and on test day, is a primary goal in preparation. A primary driver of this stress is time pressure.

Paradoxically, sometimes the best thing one can do to work quickly, confidently, and competently on timed tests is to do untimed practice.

Untimed practice allows students to 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 learn the ins-and-outs of the types of questions found on the test and figure out the best ways to get to the right answers most quickly and easily.

It's also ironic that getting stuck on super-hard questions during such untimed practice work is actually a good thing. By spending way too much time on certain very tough questions, students develop a "sixth sense" about questions that are just going to waste their time. By learning to quickly and reliably recognize "Nightmare Questions," students can save valuable time and energy on test day. [Strategy: eliminate, guess, move on.]

Once basic understanding and familiarity with test material has been established, students return to timed practice testing, and work on speeding things up. Students now know what you need 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.

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 the beginning, deliberately sacrificing speed for the sake of improved concentration, accuracy, and control is the best way to accelerate progress.

This is counter-intuitive, however, often uncomfortably so. But there are multiple points along the way at which this advice must be employed. Doing untimed practice testing, in the beginning, especially on difficult sections, is certainly one of these.

Slow Down to Speed Up!

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Contact Chris

borlandeducational.com

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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Contact Chris

borlandeducational.com

Wednesday, May 01, 2019

Perspective

You are a ghost driving a meat-covered skeleton made from stardust riding a rock floating through space.

FEAR NOTHING.

[My daughter's photo of a sign stuck in the ground at Outside Lands, circa 2013.]

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Contact Chris

borlandeducational.com

Monday, April 01, 2019

What’s Going On in this Graph?

Reading – i.e. the ability to understand and derive meaning from textual information – will always an essential academic skill.

But today we live in the age of data, and basic numeric and statistical proficiency are also likely to be de regueur throughout the 21st century.

Academic goals and curricula are already changing to reflect this trend. Among the most pressing of these new number-based skills is the ability to understand charts and graphs, to wean pertinent information and draw relevant conclusions from visual information (e.g. ACT science scores depend on this skill).

In partnership with the American Statistical Association, the New York Times is aiming to make its own contribution to the data literacy of America’s collective student body by presenting a new feature throughout the 2018-19 academic year:

What’s Going On in This Graph?

Each week, a new professional-grade NYT infographic will be presented with questions to aid in analyzing, understanding, and questioning the information it illustrates. The stated purpose of the educational project is to “teach students how to read, interpret and question graphs, maps and charts,” and is intended to support math and stats teachers across the country.

Information presented graphically is going to become more and more a part of daily life as time marches on. Data visualization, and visual communication generally, are on the rise as essential academic and life skills.

This effort by the NYT and ASA is to be applauded. I look forward to checking out the featured infographics each week, and hope you will do so, as well.

Click here for the latest graph in the series.

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Contact Chris

borlandeducational.com

Friday, March 01, 2019

White House Fails English

The recent experience of retired English teacher Yvonne Mason echoes the exasperation felt by many of us who spent the first two years of high school English doing nothing but arcane grammar exercises out of a workbook.

It may be too much to ask in this age in which English teachers no longer feel the need to teach grammar (and English majors aren't even required to study it), but shouldn't we expect better than this from the office of the highest governmental official in the land?

Argh. [Facepalm.]

I might suggest these English language resources.

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Contact Chris

borlandeducational.com

Friday, February 01, 2019

ratemyprofessors.com

Until you declare a major, take highly-rated professors, not classes!

Think of ratemyprofessors.com as Yelp for college professors. Search for your school (or prospective schools), and get reviews and ratings by real students of instructors in all departments. Use the site to find life-changing teachers and avoid duds.

Some reviews are more helpful than others, and as is the case with other review sites, ratemyprofessors.com doesn’t tell the whole story. But with quantified measures like "Overall Quality" and "Level of Difficulty" (among other indicators) it’s a whole lot better than having no idea at all as to which teachers are likely to be golden and which should probably be avoided like the plague.

Most/all established professors at are listed at each institution, and university-wide averages give you some idea as to the quality and collective personality of various faculties.

Schools are rated by students according to other important factors, as well (e.g. reputation, happiness, food, facilities, location, social life, etc.), providing useful comparative data.

As an example, click here to see data for Wesleyan University.

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Contact Chris

borlandeducational.com

Tuesday, January 01, 2019

Math-as-Art Blog

I had planned to store the overflow from the "Math as Art" project in a single post on this blog, but it quickly became apparent that the project would require a site of its own.

Here it is:

Math-as-Art.

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Contact Chris

borlandeducational.com