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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Contact Chris

borlandeducational.com

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

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

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

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

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

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

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

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

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