Tuesday, December 01, 2015

Fraction Toys, Aids, Activities

Aside from "word problems," nothing mathematical perplexes and dismays students more than fractions.

Most of this unfortunate and unnecessary dread stems from a lack of "fraction sense," an insufficient familiarity with fractions, what they are, and how they work.

There are numerous ways to help students acquire fraction sense. Following are several of my favorites:

Everything's Coming Up Fractions with Cuisenaire Rods

Rainbow Fraction Squares

Rectangular Fraction Decimal Tiles

Rainbow Fraction Circles

Rainbow Fraction Circles (Marked)

Transparent Fraction Circle Rings, Opaque Fraction Circle Rings

Fraction Angle Circles, Fraction Decimal Circles

Decimal Squares

Decimal Squares Playing Cards

Fraction Bars

Fraction Bars Playing Cards

Fraction Towers

Fraction Tower Card Game

Fraction Flash Cards

Fractions Dominoes Cards

Fraction and Decimal Dominoes Game

Math Stacks Fraction Equivalence Game


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

Sunday, November 01, 2015

Alphabet Toys, Aids, Activities

From a very early age, parents can make a game of learning the ABC's.

There's the song, of course, which is a great way to start. Even better if sung slowly while reading a favorite alphabet book and touching your child's index finger on each letter, in order, as sung. Refrigerator magnets can be ID'd and traced using your child's index finger, to help with recognition, memory, and (later) proper letter formation when writing. Once the alphabet is mastered, young children can learn sounds typically associated with letters along with a few words beginning with each letter and sound.

Immersion is a great way to learn anything, and so it goes with reading, writing, math, or anything else. Having your home filled with fun and interesting learning activities is one good way to make learning a basic family value.

Learning the alphabet is game number one! Below are some ideas to help get things going:

Alphabet Bulletin Board Cards

Self-Correcting Alphabet Wooden Puzzles

Alphabet Puzzle

Alphabet Jumbo Floor Puzzle

Spot it Alphabet

Alphabet Slap Jack


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

Thursday, October 01, 2015

Make Learning a Basic Family Value

Children will generally follow any regular pattern set by their parents.

They won’t crave sweets if their diet doesn’t contain them. Kids will be active if they see their parent exercise regularly. They’ll love books if you read lots of good books with them, drawing/writing if parents draw/write with them daily, and doing math if you regularly count, add, and solve simple story problems with them.

It’s up to parents of young children to decide which values they wish to pass on, which rules and activities will support those values, and how best to habituate those family rules and activities.

My strong recommendation is to establish learning as a basic family value.

An excellent way to do this is to strengthen reading, writing, and arithmetic skills by instituting, from a very early age, special times to engage in fun math games, drawing/writing activities (and later keyboarding), and of course reading great books with your young children. Organized activities involving science can even become a fun part of the daily routine.

The younger the child, the shorter the attention span, and the more important it is to keep kids interested and happily engaged. Keep sessions quite short, especially at first (five minutes max), and measure success by how much fun you and your child have during that time, not by “how well” or “how quickly” the child is catching on. Keep the atmosphere light and easy, slowly introducing reading, writing, and math as neat things to do with Mom or Dad, presenting new skills as games to play and activities to enjoy together, with plenty of happy silliness and loads of lighthearted fun.

A regular schedule works best in habituating learning times; same time each day is ideal. These early “lessons” are all play, and should be tailored to the personality and preferences of each child. Although you may have a list of teaching goals to loosely follow, this isn’t about checking off boxes, it’s about fun! Model enthusiasm for the activities. Follow your child's lead, respect your child’s attention span, and watch for signs of waning interest. Less is more; quit early, and leave them wanting an encore. Daily regularity and having fun matter far more than length of sessions.

Start as young as your child can enjoy math, writing, and reading-related activities. As soon as he or she can sit up on the floor, you can hold your child in your lap while you read a story book, do some fun scribbling together, and practice saying numbers and counting to three or higher, always with a big smile and lots of giggles, hugs, and high-fives! Just a minute or two at a time is plenty, to begin with. Later, you’ll be able to spend 5-10 minutes per day per child on each key area: reading, writing, and math.

This kind of special daily parent-child bonding time deepens relationships and cements learning as a basic family value. “In our family, we enjoy learning together.” Essentially, you’re taking time to play with your child very day; you just happen to be doing things that are at least loosely academic at the same time.

If possible, it's wise to set up a dedicated "Family Learning Area" with a bookcase, table, chairs, and necessary learning materials. It's important to stay well organized. Invest in plastic tubs and containers to provide homes for all the various items, and teach your children that putting things away is an essential part of playtime.

Your child will soon associate positive, close, familial feelings with the activities. Over the long term, these feelings will expand to include learning and studying in general, which is the ultimate goal.

At that point, you’ve done your job as your child's first teacher. Your children love to learn.


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

Tuesday, September 01, 2015

Math is Everywhere – and it's Fun!

Whether professional educators or not, a child's parents are his first and most important teachers.

Young children take pride and have fun doing practically anything that earns them a big smile and lavish love and approval from parents. We teach our kids by our reactions what is good and what isn't; they watch our every move for help in sussing out the ever-new world around them.

One of the most important things to show your young child is that math is everywhere, and it's fun!

After learning to recite number names (one, two, three, four ...), toddlers can be taught to count objects by touching them as they deliberately recite the number names, in order, till they get to the last object.

Suddenly, a world of countable objects opens up! Socks can be counted while doing laundry. Stairs can be counted even and odd while a child ascends or descends with Mom or Dad. Toes and noses can be counted. Apples in the fruit bowl can be counted. So can petals on a flower, flowers in a vase, and vases in a cupboard.

Geometry, too, is all around us. Straight lines and curves dominate a child's landscape. Rectangles abound. So do circles. Triangles are harder to find, but all the more treasured for their rarity when they do show up. Pots and pans are small, smaller, smallest, big, bigger, biggest. Patterns on kitchen tiles can be described ("White, blue, white, ...").

Until children reach first grade, at-home learning is ALL about fun (then, till third grade, it's almost all about fun). It's not about accomplishing, or milestones, or checklists, or any of that. Parents should focus entirely on making daily math time, reading time, and writing/drawing time a fun family activity that includes plenty of praise, love, and laughs. There'll be lots of time later to focus on results and home-work.

To keep the emphasis on fun, structured times should be quite short when teaching preschoolers at home: a few minutes is fine, 5-10 minutes max. Try to find multiple opportunities throughout the day to improvise 5-second learning tasks that children can easily accomplish successfully ("How many chairs at that table? Can you find two blue sweaters?"). Quit early, and leave them wanting more!

The most effective learning happens when both students and teachers are enjoying themselves. At first, and for quite some time, it's nothing but home-play, with success measured in giggles, smiles, hugs, and high-5's.


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

Saturday, August 01, 2015

Start Early with Internships and Career Development

Long gone are the days when good jobs could be had with nearly any college diploma.

Making the most of summer vacation months should no longer be considered optional for serious high school and (especially) college students. Nowadays, it’s increasingly important to carefully and thoughtfully use summers preceeding grades 10-16 to gain valuable work experience, do job and career research, and/or build work-related skills that will help students stand out when applying for highly competitive internships and other valuable academic and professional opportunities.

Initially, in high school, almost any job will do. But as time goes on, try to focus on significant internships in careers you find intriguing. Use these work experiences to build your resume as you check out interesting fields from the inside, narrowing the list to a few good options before you begin college. Showing this initiative early on will make your college applications stand out, help you make the right choice of college major, and put you well ahead of the pack when applying for important college internships.

Haven’t found a great summer job or internship? No problem!

Learn Excel. Gain CPR and first aid certification. Earn a Python coding credit. Contact professors at local colleges and ask if you can assist them in their research.

The test is this:

The proposed summer activity should be significant and targeted enough to stand out as a useful line on your resume.

*Note: NTAs (Non-Teenage Activities) earn extra credit; these are activities not normally done by teenagers (e.g. professional skills certification, research with local professors, significant leadership or business experiences, etc.).

The bad news is that American students no longer rank head and shoulders above students from other countries in terms of educational, economic, and career opportunities. International competition is stiff, and unstructured, playful summers after middle school are a thing of the past for forward-thinking American teenagers.

The good news is that this isn’t really bad news. "Competition" can be energizing. “Fun” is at least half attitude. Even the most mundane chores can seen as fun, and done with alacrity, from the right point of view and with the proper attitude. Well-crafted, productive summers can be fascinating, purposeful, life-changing experiences.

Starting in high school, finding and completing meaningful summer work and career development experiences can be as at least as interesting and invigorating as most things most students actually do during summers off.

"The early bird catches the worm," and many are the advantages for enterprising students disciplined enough to get an early start on life goals.


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.

Establishing a “Math Exploration Station” at home is a great way to provide young children opportunities to hone critical math skills while playing!

All you need is a small table, chairs, and bookcase in which to store 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.


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 it was decided in the early 1960'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:


Physical Manipulatives

Modeling Clay, Colored Plasticine

Cuisenaire Rods

Cuisenaire Rod Track

Jumbo Cuisenaire Rods

Centimeter Gram Cubes

Centimeter Snap Cubes, Building Base

2cm Snap Cubes

2cm Creative Color Cubes


EAI Master 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

120 Number Board

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 Books

XY Coordinate Pegboard

Algebraic XYZ Bosse Tiles Set


Virtual Manipulatives

National Library of Virtual Manipulatives

Matti Math

BrainingCamp Virtual Manipulatives



Guidebooks and Resources

Why Teach Math With Manipulatives?

Manipulative Glossary

Math Tasks Teacher Guides: K-8

Activity Math: Using Manipulatives in the Classroom


* 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.


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

Friday, May 01, 2015

The Junior College Transfer Option

Why pay full-fare for four years?

Transferring to the in-state flagship school after living at home and taking JC courses achieves the same credential for half the cost.

For low and middle-income families, the junior college transfer option is looking better each year. Although students taking this path forgo the "real college experience" for a couple of years and may feel a prestige hit, the drastically reduced financial burden could more than make up for these rather insubstantial benefits.

Junior colleges often have transfer agreements with particular state universities that make it easier for successful JC students to attend a top state school. For instance, Santa Rosa Junior College, our local JC, has a transfer agreement with U.C. Berkeley, arguably the best public school in the nation.

Our daughter graduated high school in 2011, took a gap year, and then matriculated at pricey Wesleyan University in Connecticut, graduating in 2016 with a double major in math and computer science. Wesleyan seemed like a good idea at the time, considering she had originally planned to major in English ... but given the STEM degree she ultimately earned at Wesleyan, she would actually have been better off transferring from SRJC and graduating with a math/compsci degree from UCB, one of the most highly regarded STEM schools in the world.

She did fine. Our daughter was offered a good scholarship to attend Wesleyan, and though her liberal arts math/compsci degree offered her fewer opportunities than the same degree from UCB would have afforded, she landed a great job soon after graduation as a software engineer at Amadeus North America in Boston.

One significant disadvantage of the JC transfer option is the lower-quality student body JC's typically attract. Although JC instruction is often excellent, top students who attend JC's won't find the same high level of intellectual interaction with peers as those who attend good four-year schools.

But if money is a family's biggest consideration, the JC transfer option could make very good sense, even if the student involved is capable enough to gain admission to big-name schools.


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.


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

Monday, March 02, 2015

Favorite Story Books

One of the joys of childhood is reading time with Mom and Dad.

Each of us would recite a different list of favorite titles, but the experience would be the same: reliving moments when sparks of imagination and curiosity were struck, igniting our young souls.

Below are my top 50 favorite books and series for youngsters:

A Child's Garden of Verses

Alice in Wonderland

Robin Hood

On the Night You Were Born

Goodnight Moon

Curious George


Dr. Seuss

The Three Investigators

A Christmas Carol

Harry Potter

The Chronicles of Narnia

Charlotte's Web

Winnie the Pooh

The Adventures of the Muddle-Headed Wombat

Where the Wild Things Are

The Hobbit


Are You My Mother?

The Wizard of Oz

The Very Hungry Caterpillar

The Tale of Peter Rabbit

The Gingerbread Man


Charlie and the Chocolate Factory

Stuart Little

Make Way for Ducklings

The Story of the Three Little Pigs

Frog and Toad

Phantom Tollbooth

Huckleberry Finn

A Bear Called Paddington

Mike Mulligan and his Steam Shovel

Go Dog Go!

Love You Forever

The Runaway Bunny

Pippi Longstocking

The Little Engine that Could

The Story of Babar

Aesop's Fables

Jonathan Livingston Seagull (Original Edition)

The Little Red Hen

Children of the Lamp

Where's Spot?

Nancy Drew

Mother Goose

Little Golden Books

Caps for Sale

The Story about Ping

Stone Soup


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

Sunday, February 01, 2015

Hire a Good College Admissions Consultant

Test prep, course tutoring, college visits ... now college consulting?

Where does the list of miscellaneous higher ed costs end??

Long gone are the days when motivated high school grads could easily apply, matriculate, and work their way through college without outside assistance.

No pre-college expense is more worthwhile than that involved in hiring a good college admissions consultant. Applying for a seat in an institute of higher learning these days is practically a Herculean effort, fraught with costly mistakes to make if one is unfamiliar with the territory, inexpert in assessing various pros and cons, or even slightly disposed toward disorganization, procrastination, anxiety, or forgetfulness.

As is generally the case in hiring professionals, a trusted personal recommendation is worth gold. In the "College Admission" section of the Professional Referrals page in the Resources section of my business site are listed several excellent college counselors whom I endorse. One can also find a college consultant though HECA and NACAC, two reputable national professional groups.

It's best to begin work with a college admissions counselor during a student's sophomore year in high school (or earlier); if possible, parents should research various options and have a consultant lined by the end of freshman year.


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.


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