Showing posts with label Articles. Show all posts
Showing posts with label Articles. Show all posts

Wednesday, May 01, 2024

Mischievous Engineers

Easter Eggs aren’t just for kids, and they aren’t found only on Easter – but they’re always hard to find and never fail to spark joy. 

Hiding “Easter Eggs” in software began in earnest in the 1970’s and continued through the Atari era into the modern age of computing. Engineers with too much time on their hands would deliberately program all kinds of surprises (little games, silly graphics and animations, text info, etc.) into their software projects.

Although Easter Egg grinches like Steve Jobs and Bill Gates banned the practice within their own companies, comedic Google engineers have managed to continue the tradition.

A Business Insider article gives a partial rundown of hidden tricks and treats to be found within the Google search bar. Give some of these a try! 

Easter Eggs provide a window into the minds of bored code monkeys, and furnish fatigued students and professionals a way to punctuate their day with diverting amusement.

A Wikipedia entry provides historical context and further info.

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

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.

Friday, March 01, 2024

U.S. News State Educational Rankings

U.S. News is famous for its annual college rankings, but it also ranks states on a number of educational benchmark
s. Their recent College Readiness Rankings are a revelation.

Not surprisingly, Northeastern states occupy half of the first 10 spots. 

California? Number 49. Ouch.

When I was schooled as a boomer kid, California public schools were the best in the country – and, therefore, the best in the world. Number 49 is very hard to take.

Annual U.S. News college rankings (and others) are notorious as less-than-stellar indicators of college caliber. But they do, at least, provide a sense of relative quality. Similarly, one should probably take these overall educational rankings of the 50 states with a large grain of salt.

Nevertheless, if you'd like to see them, click here for the complete state rankings.

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

Thursday, February 01, 2024

Teachers are Quitting

It seems our schools are always in trouble for this reason or that one. 

This was not at all the case when I was a boy in the 1960’s. California public schools, K-College, were unquestionably the best in the world, and all were levels were free or very low cost. So it’s especially hard for boomers like myself to see the kite hit the ground like this.

As a private academic coach for more that 45 years, I’ve tried to figure out what went wrong. I still don’t know, after all these years.

And now, a sobering new trend is emerging. Teachers are leaving K-12 schools in droves.

The reasons are many and various. But what I hear over and over again is that they boil down to several core issues: low salary, poor mental health, toxic work cultures, unsupportive administrations, lack of respect inside and outside school, lack of authority in the classroom, and out of control students and parents.

I have no answer, but I do have suggestions. At a minimum:

Pay teachers like doctors, require the same achievement, give them the same respect inside and outside schools, and weed out poor teachers (the Finnish approach); give teachers back-up they need and deserve within schools hierarchies; return classroom authority to teachers; allow sensible grading and disciplinary procedures, including compassionately but unapologetically holding back students who don’t meet grade level standards; a teacher-aid in every classroom; effective protection against parent-zillas think their little Johnny can do no wrong; make it unnecessary for teachers to buy learning supplies their students need; and provide enough public funding to pay for all this, realizing that doing so is less expensive than not.

For further information, I recommended the following:

I quit teaching for better mental health: former teachers share the jobs they got after teaching

Why Teachers Quit: Lack of respect, abominable working conditions, and more.

After Teaching For 11 Years, I Quit My Job. Here's Why Your Child's Teacher Might Be Next.

I quit my job as a teacher after 6 years to work in tech sales. I make $20,000 more ...and am so much happier now.

Teachers Who Quit Are Sharing The Moment They Realized It Wasn't For Them

Why Teachers Quit + Top Signs Quitting Teaching Is The Right Move

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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.

Tuesday, June 01, 2021

ACT Perfect Scores Have Been Blowing Up

In the two decades between 1998 and 2018, the number of perfect scores on the ACT increased 34x from .01% to .34%.

Have students preparing to enter college suddenly become 34 times smarter? Unlikely

This explosion in perfect test scores is chronicled in an article on the PrepMaven site:

These are still very low numbers – in 2021, only 4,055 students taking the ACT earned a perfect score. However, the percentage of students getting a 36 on the ACT has gone from less than 0.01% to 0.34% in 2020! (There’s a small dip for 2021, which we can likely attribute to the Covid-19 pandemic.)

The makers of the ACT have assured us that this isn’t because the test is getting easier!

Rather, our hypothesis is that the rise in students getting perfect scores is because more high-achieving SAT “superstars” are now also taking the ACT, whereas before they would have only taken the SAT.."

Plausible? SAT superstars migrating to the ACT?

Perhaps.

In another treatment of the subject, Hannah K. Sparling and Dan Horn of the Cincinnati Enquirer put it this way:

"If the test is essentially the same, why are so many more students acing it?

The most likely answer is a booming test-preparation industry that’s built on the hopes and fears of students and parents who are willing to work – and pay – to get an edge.

They see the investment of a few hundred or even a few thousand dollars in a test prep program as worthwhile if it helps land their child at an Ivy League school or secures a big financial aid package."

Plausible? Anxious parents, hyper-competitive students, and increasing billions spent on test prep?

Definitely could be.

On the same subject, the Applerouth Team said the following:

"Perfection can be a tricky thing. Too much of it diminishes value. Think of little league where everyone got a trophy. Or that Earth Science class where the teacher promised an A+ if her students simply did the work. Not everyone deserves the label of “perfect,” and the same is true for the ACT. A composite of 36 should be reserved for a select few, and too much perfection should raise red flags over the difficulty of the test..."

It's worth noting that, despite a recent dip in perfect scores correlating with the Covid years, numbers of perfect ACT scores (at last count) are still 30 times larger than they were a generation ago.

I concur completely with Applerouth. When grades and test scores are no longer seen as trustworthy, they become worthless as means of assessing merit and rewarding talent and hard work. A 30x increase in perfect test scores is unsustainable and unacceptable.

Grade inflation is real problem, and has been for decades. Although merit certainly isn't everything, life is inherently competitive, whether we like it or not – and refusing to hold back failing students is neither helpful nor compassionate.

I hope we come to our senses sooner rather than later. Denial of reality is never a good long-term solution.

Quite the opposite, in fact.

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

Monday, March 01, 2021

Patti Smith at Wesleyan 2016

My daughter graduated with degrees in mathematics and computer science from Wesleyan University in 2016. She's now a senior software engineer at Meta, well married, with a healthy and happy first baby, our first grandchild.

Can you tell I'm proud?

I'll never forget her graduation ceremony. Sitting in the audience, I was nervously flipping through the program, not paying much attention to anything in particular. Time passed slowly, as more and more parents and loved ones filled the chairs facing the stage in front of the beautiful Wesleyan library. Eventually, things got under way.

And then ... that voice!

I recognized it instantly as that of a musical hero of mine, the venerable Patti Smith. Not more than a few words into her talk, I knew it was her. What more fitting a way to mark my daughter's college graduation than with the spoken words of this particular poet extraordinaire:

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I was dreaming in my dreaming 

Of an aspect bright and fair

And my sleeping, it was broken

But my dream it lingered near.

In the form of shining valleys

Where the pure air rarified

And my senses newly opened

And I awakened to the cry.

That the people have the power,

The people have the power.

And where there were deserts,

I saw fountains and like cream the waters rise

And we strolled there together

With none to laugh or criticize.

And the leopard and the lamb

Lay together truly bound

I was hoping in my hoping

To recall what I had found.

I was dreaming in my dreaming

God knows a purer view

But as I surrender to my sleeping

I commit my dream to you.

That the people have the power

To redeem the work of fools

Upon the meek the graces shower

It’s decreed the people rule.

And I believe that everything we dream

Can come to pass

Through our union we can turn the world around

We can turn the earth’s revolution.

For the people have the power,

The people of the power.

Graduates, you are the future, and the future is now.

Congratulations.

Thank you.

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Read the article in The Wesleyan Connection and watch the video 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.

Tuesday, December 01, 2020

History of the SAT and ACT

Test nerds and prep professionals already know about this site, which outlines in some detail the history of the SAT and ACT from the late 19th century to the present day. 

Information provided covers the founding of the College Board, the fits and starts and many early and late iterations of the SAT, the inception of the test prep industry in 1946 (Stanley Kaplan), the beginnings of the ACT in 1959, and a great deal more.

It’s a fascinating read for anyone with more than a passing interest in American college entrance tests.

Bibliographic notes are found at the end of the article (though most are not linked).

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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.

Friday, May 01, 2020

Episodic Memory vs. Semantic Memory

Memory is mysterious.

Why do we recall some facts and not others? These event but not those?

How does human memory work?

What can we do to optimize memory in various situations and settings?

30-year primary teacher Clare Sealy has written a fascinating acting article in Education Next contrasting Episodic Memory with Semantic Memory and discussing the implications in educational settings.

The article begins:

When we look back on our own school days, our strongest memories are probably a mix of big occasions—field trips, plays, and sports days alongside more personal events tinged with strong emotion. Things that happened that were really funny or sad, or that made us feel excited, interested, exhilarated, or angry. We don’t tend to remember vividly, if at all, actually learning the substance of math or English or design technology. We might remember amusing anecdotes from lessons gone awry, or still bristle at past injustices — “but I wasn’t talking” — or have a vague impression of sitting in the science lab, with fleeting snippets of memories of this or that experiment. All of which leads to us making the entirely reasonable hypothesis that if we want students to remember what we teach them, then we need to make our lessons more like the spectacular one-off special events, or, at the very least, involve something specially selected because it’s exciting and possibly unusual. Memorable events, in this view, should form the template for creating memorable lessons.

As reasonable as this seems, this is a myth. It is a myth because human memory works in two different ways, both equally valid but one of which is much better at enabling us to transfer what we have learnt to new contexts. This transfer is an essential prerequisite for creativity and critical thinking.

The two forms of memory are known as episodic and semantic memory. Episodic memory is the memory of the ‘episodes’ of our life—our autobiographical memory. This takes no effort on our part, it simply happens. We don’t have to try consciously to remember what happened yesterday. Those memories just happen automatically. But there is a downside. Episodic memory is “easy come, easy go.” If you try to remember what you had for lunch yesterday, you will probably remember. If you try to remember what you had for lunch a year ago today—unless that happened to be some very significant date and some particularly noteworthy lunch—you will have no idea.

Semantic memory, on the other hand, involves much harder work. We have to expend effort to create semantic memories. This is the kind of memory we use when we consciously study something because we want to remember it. Unlike episodic memory, it does not just happen. The upside, however, is that the effort involved results in a long lasting memory.

Have you ever been in a course where you have really enjoyed listening to the speaker, found the subject matter interesting and the presenter amusing and engaging. Yet when you try to explain to someone the next day what the course was about, all that is really left is a vague impression of your emotions during the day, tinged with the odd snippet of content? You know the course was really good yet can’t really explain what it was actually about beyond the most general of assertions.

[Continue reading here.]

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

Wednesday, August 01, 2018

The Matchless Enthusiasm of Martin Gardner

The hyper-competitive struggle for survival against the old Soviet Union made American math and science education a top priority during the Cold War decades of the 20th century.

Many Baby Boomers (like me) remember with fondness the fascinating articles published monthly in Scientific American magazine. One of the most interesting features of the magazine in our time was Martin Gardner’s column on recreational mathematics, which ran for 25 years. 

Among the many necessary qualities of truly great teachers, enthusiasm might be listed first. An instructor’s genuine, overflowing enthusiasm is that which excites students' souls and convinces them that the required academic work and sacrifice will be amply rewarded. The etymology of the word “enthusiasm” (en-theos: literally, "in God") points straight at the Divine, and no one could excite the soul with the beauty of mathematics like Gardner could.

A 1998 article by the master preserves for modern readers the flavor of Gardner’s contagious enthusiasm and gold-medal exposition that so characterized his column, presenting to Gardner fans and neophytes alike the pure noetic joy that accompanies deep dives into the realm of creative mathematics.

Reflecting the timelessness of the subject, the article reads as if it were penned yesterday, fresh and new. It’s not long, and is well worth a bit of your time:

A Quarter-Century of Recreational Mathematics.

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

Tuesday, May 01, 2018

The Game of LIFE

John Conway's 1970 coding masterpiece mimics life's basic processes in an utterly simple yet strikingly lifelike way. 

Conway's Game of Life quickly became a sensation, inspiring a cult following among mathematicians and programmers. 

Seed shapes are born, live, reproduce, and die. In the process, fascinating animated patterns and "social" interactions occur on the computer screen. Beginning with a seed created by the user, a mesmerizing drama unfolds. Sometimes these living worlds expire in a moments, seconds, or minutes; others continue for eternity.

Research into Conway's game is ongoing and extensive. Could life itself be this simple? Why not?

Create virtual life! Experiment with The Game of Life here. FAQ here. Stanford discussion here. LifeWiki:About here.

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

Friday, December 01, 2017

MOOCs are Coming of Age

At their inception several short years ago, MOOCs (Massive Open Online Courses) were an unproven concept with passionate advocates on opposite sides of a great debate. MOOCs were going to revolutionize higher ed, or destroy it. No one could tell which it would be.

A decade later, top MOOC providers like Coursera and EdX have grown and prospered. Top-notch course offerings by the best universities in the world have attracted millions of students world-wide. Legions of online pupils of all ages have completed courses, some earning coveted professional certificates and even fully-accredited graduate degrees online. Low cost has made high quality higher ed available to a much wider, world-wide audience.

Though forms are still evolving and the precise roles to be played by MOOCs are still uncertain, both the radically new concept and the traditional educational landscape have survived and even thrived as a result of the introduction and mainstreaming of MOOCs.

Today, MOOCs and associated certificates/degrees are legitimate educational alternatives.

See links below for further info:

Massive Open Online Course (WikiPedia)

By the Numbers: MOOCs in 2017

The Future of MOOCs

Coursera

Coursera Professional Certificates

Coursera Undergrad and Grad Degrees

EdX

EdX MicroMasters Certificates

EdX Professional Certificates

EdX Series Programs

MIT Open Courseware

Stanford Online

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

Sunday, October 01, 2017

Children's Books for Teaching Math

Great story books abound, and children love them ... but there are also plenty of fun, well-crafted nonfiction books for kids, and these are just as important to include during family reading time.

Elyse Mycroft at proudtobeprimary.com has compiled an excellent list of wonderful books to use in introducing the panoramic world of mathematics to children.

Early math topics from numbers and counting to patterns and sorting, fractions, measurement, time, basic operations, and financial literacy are introduced and explored.

Children's Books for Teaching Math

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

Wednesday, July 01, 2015

Math Exploration Station

Playing with particular objects in particular ways is the best way to learn early math concepts and develop instinctive number sense crucial to success in the study of mathematics. 

Building a “Math Exploration Station” at home is a great way to help children hone critical math skills while playing! All you need is a small table and bookcase, manipulatives and other math-centric toys like Tinker Toys, blocks, Legos, etc.

Don’t forget good ol’ paper, pencils, felt pens, and crayons for drawing and designing. We had a family friend who was an early computer scientist, and I remember going through reams of surplus wide-format dot-matrix printer paper as a young child in the early 1960’s on which I happily drew everything from complex forts to rockets to abstract free-form doodles.

The best place for early learning is in the home, and the best way is when the mind is opened through play.

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

Monday, June 01, 2015

Math Makes Sense with Manipulatives



Mathletes and early computer scientists had recently won World War II, Soviet scientists were threatening to militarize Earth orbit after Sputnik in 1957, and so the goal was set in the late 1950's that the United States would aim to quickly boost the quality of mathematics education.

Emphasis was shifted from wrote memorization of “facts and formulas” to discovery-based experiential learning. The goal was for K-16 students to understand deeply the math they were learning, rather than merely repeating facts or algorithms robotically without knowing why. Grade school math homework now involved mathematical theory in addition to traditional drills in basic operations. Parents were left scratching their heads at homework that involved the abstractions of set theory rather than simple, unadorned arithmetic. In answer to the question “Why does 2 plus 2 equal 4?,” it was no longer acceptable to respond with an exasperated “BECAUSE!”

New Math was designed to literally “make sense” to young students by involving their senses and whole brains as they learned mathematics rather than mentally photographing and filing pages of facts “just because.” Objects were counted and distances measured when learning to add and subtract. Rods of equal length were arranged in rectangles to prove multiplication facts.

“No doubt about it: 2+2 does equal 4; five two's are in fact ten."

Students learned not only to memorize facts of arithmetic, but to understand arithmetic processes themselves. And this new comprehension was viscerally anchored, which meant it was deeply understood, and could be more easily and productively connected to other knowledge the student had (or would later have).

Physical objects used to reify numbers and clarify mathematical concepts and processes later came to be called Math Manipulatives. Manipulatives are still are the best way to teach young children basic arithmetic and early mathematics. The result of manipulative-based learning is an indispensable instinctive “feel” for numbers called “number sense.” *Legitimate criticisms of "New Math" notwithstanding, I simply can’t imagine learning arithmetic any other way.

In addition to traditional physical math manipulatives, screen-based "virtual manipulates" are now available for use on computers and other devices to aid in teaching and learning early math. These on-screen versions are superior to physical math manipulatives in many respects. Virtual manipulatives are easier to use, harder to lose, don't need to be picked up and put away after use, they go anywhere a tablet device goes, and can save tons of money and space.

Though virtual manipulatives shouldn't completely replace traditional physical manipulatives at home or school (especially during the first stages of a child's introduction to numbers and mathematical ideas, when sensory information involved in handling physical manipulatives is such an important aspect of developing critical early number sense), virtual manipulatives are well worth considering as an adjunct to traditional manipulatives.

For parents eager to set up a Math Exploration Station at home, I’ve listed below what I believe to be the most generally useful math manipulatives and virtual manipulatives along with selected guidebooks on how to use them in teaching math at home:

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Physical Manipulatives

Colored Plasticine

Cuisenaire Rods

Cuisenaire Rod Track

Jumbo Cuisenaire Rods

Centimeter Gram Cubes

Centimeter Snap Cubes

2cm Creative Color Cubes

Inchworms

EAI Master Fraction Ruler

Measuring Tapes, Tape Measure: 30m/100ft

Animal Counters, Dinosaur Counters, Family Counters, Transportation Counters

Two-Color Bean Counters

10 Frames

Hundred Numbers Boards

Hundreds Pocket Chart with 100 Number Cards

Place Value Disks

Play Money

Square Color Tiles

Wooden Pattern Blocks

Fraction Toys, Aids, and Activities

Tangram Toys, Aids, and Activities

Pentomino Toys, Aids, and Activities

Wooden Geometric Solids

Fillable Geometric Solids

AngLegs: 72 Piece Set

Math Dice, Polyhedral Dice: Set of 7 (D&D)

5-Bead Soroban Abacus, Guide Book

Algebraic XYZ Bosse Tiles Set

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Virtual Manipulatives

National Library of Virtual Manipulatives

Matti Math

BrainingCamp Virtual Manipulatives

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Guidebooks and Resources

Why Teach Math With Manipulatives?

Manipulative Glossary

Activity Math: Using Manipulatives in the Classroom

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* Parents today are experiencing a similar frustration with "Common Core," which could be characterized as "New Math 2." There's good reason for this: For most students, "Discovery-Based Math Instruction" only works well when closely paired with traditional algorithm-based "Direct Instruction." Although currently out of fashion, memorization, algorithms, drills and other traditional approaches to math instruction are at least as important and useful to most students as the "New Math" methods in vogue since the Math Standards movement picked up steam in the 1990's.

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

Wednesday, April 01, 2015

Top Lower-Income Students Should Aim High

While it’s normally true that elite private colleges and universities cost families far more than state schools, this isn’t always the case.

When family income is below certain limits, schools like MIT, Harvard, and Stanford (and even public institutions like UCB) will waive all tuition for admitted students. Excellent students from families of modest means will often get an admissions boost and pay less for a diploma from an elite college than for a similar credential from a less prestigious in-state public university – sometimes far less.

Many poor families erroneously assume that top-flight schools are beyond their financial reach, and that the only realistic option is junior college followed by a transfer to a state university. In fact, for low-income students sharp enough to gain admission, attending an elite school can often be their least expensive option.

Click here for an article listing schools offering free rides and “no-loans grant-only” aid to top students from lower-income families.

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

Thursday, January 01, 2015

Return of the Daily Quiz

A regimen of low-stakes testing in class – and self-quizzing (reciting notes from memory) while studying – vastly improves learning outcomes and makes high-stakes testing far less daunting.

So says professor Henry L. Roediger III, author of the well-reviewed "Make It Stick: The Science of Successful Learning" and an intriguing NYT article on the subject.

This certainly reflects my own experience as a student and teacher. The effort required to recall information does seem to exercise critical brain functions, improve intelligence, and promote academic success in a number of important ways. Well-established learning models like SQ3R and Cornell Notes are classic implementations of this idea.

Though out of fashion pedagogically for some time, memorization is an essential element of learning and an important part of what students should be doing during school hours and while studying at home.

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