A twist on multiplication flash cards.

I read a short ebook this week by Renee Ellison called Teach Math Faster.  (Quick read, only $0.99 on Amazon Kindle.)  She has some very good hands-on ideas for helping children with math basics.  One of them was a twist on the tried-and-true multiplication flash cards, in which she added a visual representation of each quantity instead of simply writing the numerals.

Why is this helpful?  Because the more often a child can use concrete ideas when learning math, the better he will understand what the numbers are doing. Math educators call this "number sense."  For example, instead of rotely memorizing the words and symbols of
6 x 4 = 24,  the child using Ellison's flash card design sees an array of 24 blocks, arranged in four rows of six blocks each.

This is the front of a
Teach Math Faster flash card.

Now of course, children who are old enough to learn multiplication will have long mastered the concept of number symbols standing for quantities.  However, many children have only a shallow understanding of what multiplication actually means.  They will use "times" as a verb-- "I can times six an three"-- in the same way that my children used to think that verse was a verb meaning "to defeat"-- as in, Godzilla versus ("verses") Megalon.  (They would say, "I'm going to verse those bad guys!")  But when the front of the flash card shows a graphic representation of what  "six times four" means, the concept is reinforced at the same time that the memorization is taking place.

An added dimension of concept reinforcement occurs when a range of multiplication facts is studied-- while "8x7=56" uses the same number of digits as "2x5=10," there is a clear difference between the space occupied by ten squares compared to fifty-six squares.  The child's number sense increases when she sees just how much bigger 56 is than 10.

On the back of the flash card, Mrs. Ellison repeats the array, this time without the numbers, and provides the product:

This is the back of a Teach Math Faster flash card.

So when the child is using this flash card, he is reading the numbers on the front and multiplying them in his head just like a traditional flash card.  He checks for the correct answer on the back, just like a traditional flash card.  But his brain is experiencing something totally new: the abstract symbols are gone, replaced by a picture of what's happening in the problem.

Genius!

Now for the bad news:  as far as I know, this style of flash cards is not commercially available.  Like the other manipulatives in Mrs. Ellison's little book, they have to be home made.  She recommends using graph paper to keep the size of the squares consistent-- otherwise, there is less visual impact in the size difference that I explained above.  So you-- or your child-- write the numbers in the squares and cut the arrays out, then glue them onto index cards.

One suggestion I would make:  if you are using color-coded math manipulatives such as Cuisenaire rods or Math-U-See blocks, match the flash cards to whichever set you're using.  It's as simple as adding lines of the appropriate colors to the edges of the array on the front side of the card, like this:

This array matches Cuisenaire rods, with
forest green for 6 and magenta for 4.
Math-U-See would use purple and yellow.

Cuisenaire rods and Math-U-See blocks
use different colors to signify quantities.

Be sure not to color-code the back side of the card, however, because that would too quickly give away the numbers that the student is supposed to be memorizing.  The array itself already shows the answer if the student counts the blocks, but he still has to work for it. Color-coding the front side makes sense because the number symbols are also written in, so nothing is being given away.

As mentioned in a previous post, adding color provides one more level of connection for the brain.  If your child already associates certain quantities with specific colors, tapping into this association will increase his number sense as he learns his multiplication facts.

Color can add hands-on interaction to assignments

Not every assignment that your child is asked to do will be as "hands-on" as she might need. Some worksheets and study guides tend to be so non-engaging that they end up being more busy-work than instructional aids. However, there are specific modifications you can make to the assignments which can increase her whole-brain involvement without affecting the intent of the assignment (or the grading of it).  One of the easiest modifications is to add extra concept-bridging steps to the assignment through the use of color.

No, I am not talking about coloring pages here.  Handing a child a picture to color rarely engages his brain in any way that reinforces learning.  On the other hand, asking a child to add colors to a diagram, text or drawing in a way that requires analysis of the words or pictures will make a significant impact to his learning.

A net and its solid.

Add color.  Highlighters, markers, crayons, and colored pencils are indispensable when it comes to helping your child sink her brain into a pencil-and-paper assignment.   In a recent math lesson, one of my students was having trouble visualizing the net of a geometric solid.  He couldn't see how the faces of the 3D drawing matched the faces of the net diagram. Ideally, we could have cut out a series of nets, then folded and taped them together to build paper models of the solids-- and we did do one just to make sure he understood the concept-- but that's very time consuming and we had many different solids to figure out.  This was a case where "hands-on" had to be a little more creative, and color-coding became the hands-on bridge from looking at a simple assignment to actually being able to work with it and analyze it.

The net and solid, color-coded.

For this assignment, I had him color-code the faces of the solid figure and the corresponding faces on the net; he was then able to make the connection between the 2D and 3D figures.  Once he could see the way the faces related to each other, where they connected, and the shape of each one, he could more easily match the solids to their nets.






Another use of color can help students understand math processes. For example, some of my students have struggled with long division or multi-digit multiplication, getting lost with all the different numbers that have to interact with each other.  Here we see a number multiplied by 23.  The digits 2 and 3 are color coded to match their products as the number is multiplied out:

In this way, a student can see that 1353 is the complete product of 3 x 451, while 9020 is the complete product of 20 x 451.  Notice the "placeholder zero" is colored an almost invisible gray to reflect the "invisiblitity" of the ten's place zero that makes 23 = 20 + 3.    Division can be analyzed the same way, writing each digit of the quotient in a different color so that the student sees what is happening in the process:

Color coding is equally effective in studying spelling rules.  Students who cannot identify the individual phonograms in a word are stuck with memorizing the unique spelling of every word they encounter, which severely limits their ability to spell. What they need is a way to see the phonograms in a word as they study its spelling. While my favorite hands-on spelling instruction is done by manipulating movable letter tiles, or physically cutting up words into syllables and letter combinations, this is not always practical. Instead, students can be shown how to color code the various phonograms in their spelling words.

Color-coded phonograms.

In this assignment, for example, a student might have been told to simply copy his spelling list.  That's easily done without involving much of the brain, even if he has to "copy each word five times each." Instead, modifying the assignment to add in some color-coding will require the student to analyze and interact with the words in a more hands-on way, which can increase the likelihood of actually learning to spell the words.



For the example above, the student may the words himself, changing the color of his pencil for each phonogram.  He could also write them in regular pencil and then underline the letters in different colors.  Highlighting or using colored pencils to circle the phonograms in a pre-printed list could achieve the same purpose for students who have diffficulty with writing.

The directions for the above example could go something like this:
1. Separate each word into syllables.*
2. Write single-letter vowel sounds in black, single-letter consonant sounds in blue.
3. Write double letter (same letter) consonant sounds in red, two-letter consonant sounds in green, two-letter vowel sounds in purple.
4. Underlined silent e..
5. If a separate, single-letter consonant or vowel sound follows another one, give it a different color (e.g., the "c" in escape, or the "a" in creation).

(*Note: I was taught that syllables are officially divided in the middle of double consonants: bel-low.  However, for the sake of identifying phonograms, the sound of a double letter such as /l/ occurs once in the word, so we can treat "ll"  as a single phonogram spelled with two letters, just like ph or sh, which are never separated.)

Color-coded roots, suffixes and prefix.

Besides learning the spelling of words, adding color can help children learn the meaning of words.  Vocabulary study often emphasizes prefixes, suffixes, and roots from Greek or Latin.  Color-coding can be very helpful for this as well.  Simply highlighting prefixes in one color, and suffixes in another, can help a student focus on the base words and analyze the meanings of words.  Or in a list of words that share certain roots, highlighting each root in its own color can call attention to the shared meaning of the words.

This concept can be easily extended to have students color-code the roots to match the corresponding key words in their definitions:

As you can imagine, analyzing words this way gets the hands, eyes, and brain involved in a manner that simply copying the words over and over cannot do. Similarly, color can be used to analyze text.  People have used highlighting to mark important information for years.  Why not use a variety of highlighting colors more intentionally?  If your child's assignment involves reading for information, he may use color to match information in the text to individual questions, either before he writes his answers, or afterwards.  While it may seem redundant to mark up a text in this way, the process actually increases reading comprehension by engaging the brain in a more concrete way than simply writing an answer.  Some schools teach this technique of "justifying" a response, requiring students to mark the information in the text that supports their answer:

This worksheet becomes more effective when the student

 uses color to match textual information to the questions.

Adding the targeted use of color to an assignment is limited only by the imagination.  A science diagram can be color coded to indicate the function of various structures. A history article might have facts highlighted that correspond to opposing political views. Whatever the student is asked to learn, color can help add hands-on interaction to even the most black-and-white of worksheets.