math storybook

Math Storybooks

If you’re looking for a creative project that involves math, here’s an idea.

One of the little girls in my Saturday math group enjoys writing stories. So this past weekend, we decided to write our own math story. Our plan is to work on it for 10 or 15 minutes every week, until we have a completed story. I thought I’d describe the beginning of our creative process here. I’ll tell you about the completed book in a future post.

Why read a math storybook? Two reasons. First, it has been shown that reading improves math performance. Second, math storybooks are a fun way to introduce mathematical concepts.

Why write our own math storybook? Just one reason. It is a great way to make math fun.

The Creative Process

We began by perusing a couple of math storybooks that I own. They are:

the grapes of mathif you were a fractionThe Grapes Of Math by Greg Tang and Harry Briggs

If You Were a Fraction (Math Fun) by Trisha Speed Shaskan and Christianne C. Jones

Then we talked about how the books use pictures and words to present math concepts. I gave the books to my student to take home and read from cover to cover.


Next we approached the idea of characters. I tried to be a good facilitator, asking lots of questions but not providing any answers, with one exception. Since the story was born on March 14, 2015 (the Pi day of the century!), I suggested that one of the characters have the name Pi. My student thought that was a fine idea. Maybe she was just humoring me, but in any case, she agreed.

Here’s how the conversation proceeded, minus the “uhms”.

    Me: What kind of creature is Pi? A human? An animal? A tree? Something else?
    Student: A human.
    Me: Is Pi a boy or a girl?
    Student: A girl.
    Me: What is Pi like?
    Student: She has short, curly, brown hair and she is 10 years old. (My student has long, straight, brown hair. We females always seem to want the opposite of the hair we have.)
    Me: Does Pi have any friends?
    Student: She has a friend who is a girl, and her name is Candy.
    Me: Does you have any male friends?
    Student: Yes. His name is Edwin.
    Me: Where does the story take place? It can be a real place or an imaginary place, whichever you prefer.
    Student: At a big fair.
    Me: When does it take place? In the past? The present? The future?
    Student: Twelve years from now.
    Me: What year will that be?
    Student: 2027.

What’s Next?

And so, we had the beginnings of our math storybook. I asked my student to think about how the story would start, and about how Pi got her name. We stopped there, since we had lots of math activities to get to. We’ll continue to work on the story every week. Once it’s completed, we’ll illustrate it. Then we’ll share it with all of you. Stay tuned!

p.s. Here are some other wonderful math storybooks. Many of them are available “used” on Amazon.

The Multiplying Menace Divides (Math Adventures) by Pam Calvert and Wayne Geehan

Sir Cumference and the Viking’s Map (Charlesbridge Math Adventures) by Cindy Neuschwander and Wayne Geehan [Check out the whole series of Sir Cumference books!]

Tyrannosaurus Math by Michelle Markel and Doug Cushman

A Place for Zero (Charlesbridge Math Adventures) by Angeline Sparagna LoPresti and Phyllis Hornung

Zero the Hero by Joan Holub and Tom Lichtenheld

the cow jumped over the moon

Mnemonics Can Be Good … or Not

The use of mnemonics dates back at least to the time of Plato and Aristotle, and some people seem to think they have outlived their usefulness. While there is the potential for overuse, and they should never take the place of teaching conceptual understanding, a good mnemonic can be invaluable. A great mnemonic can be retained in memory for a lifetime. Remember “Every Good Boy Does Fine”? Because of that oldie but goodie I will never forget the notes on the treble clef, even though it’s been years since I played the piano.

But not all mnemonics are memorable. In fact, I find that some of them are harder to remember than the number or rule they are intended to evoke. Personally I find it easier to remember the first 7 digits of π than to recall the phrase “May I have a large container of coffee?” An alternative phrase, “How I wish I could calculate pi” is better, but once I do recall it, I have to write it down and count the letters in each word to come up with 3.1415926. And would you believe that someone wrote a poem to help us remember the first 740 digits? The same author later wrote a longer version, for the first 3,835 digits of π! I don’t often have a need to know more than the first few digits of π, but I have to give lots of points for creativity on the poems.

All that aside, I’m willing to bet that for every clunky math mnemonic out there, there are ten useful ones, and one or two clever ones. What? You aren’t familiar with any math mnemonics? Read on. In the style of David Letterman, I’m saving the best for last, so read all the way to the bottom!

My Top Ten Math Mnemonics

math mnemonic 10) I don’t use this one very often, but it is useful for remembering the lesser known metric prefixes (hecto-, deca-, and deci-). The units go from largest at the top of the stairs to smallest at the bottom, and each step represents a power of ten. The metric system makes so much sense! To be honest, I did not use hecto-, deca-, or deci- in 30 years of engineering in the U.S., unless you count engineering exams, but students are still required to learn them. (image from

Application: Prefixes of metric units (kilo-, hecto-, deca-, units, deci-, centi-, milli-)
Mnemonic: King  Henry Doesn’t Usually Drink Chocolate Milk
Alternative Mnemonic: King Henry Died Unexpectedly Drinking Chocolate Milk

the man with the big nose9) This one comes in handy for all algebra students. It offers a structured approach for multiplying monomials.

Application: How to multiply two monomials
Mnemonic: First Outer Inner Last (FOIL)
Visual Mnemonic: The Man with the Big Nose (image from

8) Lots of students have trouble with word problems, and anything you can do to help with approaching a problem is good. I especially like the idea of crossing out irrelevant information in the problem statement. Plus, it gives students something to do when their brains would otherwise be freaking out over having to solve a word problem.

Application: Steps for solving word problems
Mnemonic: Box in the question, Underline relevant words, Circle relevant numbers, Knock out (that is, strike through) irrelevant information (BUCK)

pemdas mnemonic7) I use this one with middle and high school students. But it’s important to make sure they understand that multiplication and division are performed in the order given (you don’t do all the multiplication and then all the division). The same goes for addition and subtraction.

Application: Order of operations: Parentheses, Exponents, Multiplication or Division, Addition or Subtraction
Mnemonic: PEMDAS, or Please Excuse My Dear Aunt Sally
Alternative Mnemonic: GEMS (Groups, Exponents, Multiplication or division, Subtraction or addition)

6) All my life I’ve mixed up isosceles triangles and scalene triangles, so I was glad to discover this little memory aid. It orders the types according to the number of equal sides.

Application: Types of triangles (equilateral has 3 equal sides, isosceles has 2 equal sides, scalene has no equal sides
Mnemonic: Eat Ice Slowly
From: Lynn Greenwade and students

By the way, there is a really cute triangle music video here.

5) This method of remembering what to do when dividing by a fraction is useful. (My 5th grade student and I prefer to do a little dance rap with “Flip and multiply.” I tried to get a male 11th grader to try it; he chuckled and moved on to the next question.)

Application: Dividing by a fraction
Mnemonic: Kentucky Chicken Fried (Keep the first fraction, Change to Multiplication, Flip the second fraction)

quadrant sine cosine tangent
I’ve found that many students have trouble remember the signs of sines, cosines, and tangents in the quadrants of a circle, so this one comes in handy.

Application: The trigonometric functions that have a positive value in a given quadrant
Mnemonic: All Silver Tea Cups

    In quadrant I – all functions are positive
    In quadrant II – only sine is positive
    In quadrant III – only tan is positive
    In quadrant IV – only cosine is positive

3) I’ve used this one since I was in high school.

Application: Relationships among sine, cosine, and tangent of a right triangle

    Sine = Opposite over Hypotenuse
    Cosine = Adjacent over Hypotenuse
    Tangent = Opposite over Adjacent

2) Now we’re really having fun. This is a really cute little ditty!

Application: Circumference and area of a circle (image from

    Tweedle-dee-dum and tweedle-dee-dee,
    around the circle is pi times d.
    But if the area is declared,
    use the formula pi “r” squared.

1) And now, for my favorite math mnemonic. Not only is this incredibly useful (it’s hard to remember these definitions), it’s as cute as pie. You can listen to a 6th grade class singing it here.

Application: Definitions of median, mean, mode, and range of a set of data
Mnemonic: (Warning: You are going to have some version of Hey Diddle Diddle stuck in your head for days.)

hey diddle diddle mnemonicI’d love to hear about your favorite math mnemonics!

M&M Math

M&M abacus

Candy is a Great Incentive

skittlesIn my experience there are two things that will get the attention of even the most bored student: money and candy. The only time I bring money to a tutoring session is when I’m planning to teach how to count money. And all of it returns home with me. Candy is a different story.

Sweets are a great incentive for kids, as we all know, and they certainly make doing math fun. I’ve found that M&M’s and Skittles are great for math because they come in many different colors. When you line them up in rows, they even look like an abacus, so surely they were intended to be used to learn math. (By the way, lining up M&Ms is not easy … they are surprisingly nonuniform in size!)

Healthier Alternatives

Fiber One fruit piecesI know all you health-conscious parents would like to reach through the computer and hand me an article about the detrimental effects of candy, but not to worry! You can substitute fruit candy pieces, raisins (black, golden, and Craisins) or a mixtures of nuts for candy. The only requirements are that: 1) there is variety, so you divide the pieces into groups of like things; and 2) your kid likes them. Let’s face it, prunes will never work as an incentive for kids to learn new math concepts.

M&M Math

Now for M&M Math. There are any number of ways to make math games using candy pieces. You can start with counting, literally using the pieces like you would an abacus. How many orange pieces are there? How many red? Then move up to adding and subtracting. If I subtract 2 blue pieces from the 9 blue pieces, how many blue pieces are left? If I add the red pieces to the yellow pieces, how many will I have? Then multiply and divide. If I have two groups of green candies, with 4 in each group, how many total green pieces are there?

These are all fun games to play with candy, but the very best use of M&Ms, Skittles, or fruit candy is for learning fractions. Here are a couple of ways to play, but you can make up your own versions. The game on the left combines counting with whole number addition and fractions. The one on the right focuses on fractions.

Give it a try! Your kids will love it, and you’ll have an excuse to eat M&Ms … guilt free!

M&M Fractions Games

multiplication, past and present

The Common Core Blues

So, you thought you knew how to multiply one number by another. You even thought multiplication was easy. Then your kid comes home with “arrays” or “lattices,” and wants you to help him, but you can’t make sense of the new methods. You’ve got the Common Core blues.

These days, you would be hard put to find an education site that isn’t showcasing an argument for or against the Common Core (CC). I admit that it doesn’t take much to get me on my soap box, railing against some of the inane (in my opinion) methods that are being promoted in North Carolina’s Common Core curriculum. (You should not confuse the Federal standards with the various curricula for teaching those standards. North Carolina and some other states have shelled out lots of $ to consultants to create brand new curricula.)

Okay, I’m stepping away from the box.

Multiplication Past

I thought it might be instructive to look at some of the specific approaches to multiplication that are being taught to elementary school kids. I actually find most of them to be very helpful in teaching kids the meaning of multiplication. However, anything can be taken to extremes. In the interest of full disclosure, I do not have extensive experience in using the new methods with my students because those that attend public school are in the higher grades while my younger students are home schooled. But I do run into the new math while working with a little tutoring group that I lead on weekends, so I had to educate myself on the CC methods.

Okay, let’s get started! As an example, let’s use 11×12. This is a simple problem, and the fact that both numbers have two digits makes it useful for demonstrating various methods.

old school multiplicationIn the “old school math,” there were two ways to do this problem, at least as far as I know. One is to extract the answer from the memory bank (analogous to the RAM in a computer) that resides in each of our heads, because we all memorized the multiplication tables up to 12×12. So we old folks can come up with 11×12=132 without much thought.

The second “old” approach to multiplying 11 by 12 is shown in the image to the right. This approach came in handy when we needed to multiply numbers bigger than 12, since we didn’t have those solutions in our memory banks. It is quick and efficient, but I have to admit that it is possible to just memorize this method without really understanding what is going on. This is the issue the Common Core standards and curricula are trying to address.

Dot, Dot, Dot

Okay, out with the old and in with the new. Let’s start with what the younger kids are learning. It is called the “dot” method. We can start our problem by drawing 11 rows with 12 dots in each row. Of course, it would take a while to count them, not to mention the time it takes to draw them by hand. Photoshop solves the drawing problem (for me), but what to do about the counting problem?

We can break the dots up into groups, creating two or more easier multiplication problems. In the image below I divided them into two groups, one with 10 rows of 12 dots and another with 1 row of 12 dots. Now the problem is easier, as long as we know how to multiply by 10. The problem 11×12 becomes 10×12 + 1×12, or 120 + 12 = 132.

dot method of multiplication

If a child has not yet learned to multiply by 10, he can break these groups into smaller groups but I’ll illustrate that in the next section. My take on the dot method is that it is very useful for younger children because it offers the option of brute force counting, as a starting point, and it helps kids learn that no matter how they group the dots, they’ll get the same answer. This is an important concept. The problem with the dot method is that some teachers assign problems with numbers that are way too large, as in our example. I’ve read stories about kids being brought to tears by having to draw hundreds of dots for homework!

Arrays Make Sense

Let’s move on to the “array” method. This approach can be developed naturally from the dots method. Instead of drawing dots, we draw boxes to hold the dots. So the second of the dots images above (i.e., 10×12 + 1×12) is represented by the first of the array images below. At this point we can break the problem up into as many groups as we like, as you can see in the three other array images. The only rule is that the lines dividing up the array have to extend fully from left to right or top to bottom so that we maintain rows and columns.

The lower right image below shows a more formalized version of the array method, one that represents another progression in the learning process. In this approach, we always divide up the numbers according to place values. So we would divide 12 into 10 + 2 (tens place and ones place), and 11 into 10 + 1 (which we did before). Then we have 100 + 20 = 120 for the upper row, and 10 + 2 = 12 for the lower row. Finally, we add 120 + 12 to get 132.

array method of multiplication

I really like this method too. I find that it offers a natural progression from addition to multiplication, and I’ve started using it with my younger students. But there is one multiplication method that I think is completely inane.

Lattices Do Not

Welcome to the “lattice” method. Most parents absolutely despise this approach–I’ll show you why.

Our example does not make a great demonstration problem for the lattice method, but we’ll muddle through. The image below walks us through the process.

Step 1. Set up a table for which the number of cells is determined by the number of digits in the numbers to be multiplied. In our example, both numbers have two digits, so we create a 2 by 2 table.

Step 2. Write our two numbers across the top and down the side, taking up one cell width or height for each digit, as shown.

Step 3. Divide each square cell into two triangles, using diagonal lines, as shown.

Step 4. Multiply all the possible combinations of digits. For us that means: 1×1 = 1, written as 0 and 1 in the upper left cell. The zero represents the tens place while the 1 represents the ones place. We go on to multiply 1×1 again and write 0 and 1 in the upper right cell. Then we multiply 1×2 and write 0 and 2 in the lower left cell. Finally we multiply 1×2 again and write 0 and 2 in the lower right cell.

I’m guessing that at this point, you are thinking something like, “What the heck is going on?” I feel your pain. But let’s keep going.

Step 5. Add the digits along each diagonal, so we have 0 for the top left, 0+1+0 for the next diagonal down, 1+0+2 for the next, and 2 for the last. And there you have it!

What? You don’t see it? The answer is the concatenation of the numbers that we just wrote around the perimeter of the table: 0132, or leaving off the zero, 132.

lattice method of multiplication

This is what I call a parlor trick. When a little girl in my Saturday group showed it to me, my jaw dropped. As she explained it, she did not refer to ones places or tens places or any other terms that might indicate her understanding of the process. She said, “You draw squares, then you write the numbers across and down, then you draw diagonals, then you …….” She really was performing a parlor trick. Do all these steps and there’s your answer!

Sorry. I’m putting away the soap box again.

I hope my explanations are clear enough to enlighten some of you parents who have run across the new ways of doing multiplication, because I know that you really do want to help your kids with math. I’m sorry if you have to deal with the lattice method. Please be sure that your child also learns the array method, so that he or she will develop an understanding of multiplication, rather than memorizing a series of steps to get an answer.

Calculators are Good … and Bad

Wait … I almost forgot! There is yet another way to multiply numbers–you can use a calculator. I have loved calculators and computers for as long as they’ve been available to the general public, and I spend many of my waking hours using them. Kids love them too–they make life so easy! Recently I was reviewing the previous week’s work of a high school student, and he mentioned that the section on complex numbers had been taught by a substitute teacher. I asked him whether he had understood the material. His answer was along the lines of, “Well, the teacher just showed us how to do the problems on our calculators.” Needless to say, my student and I went back over that section, sans calculator. Calculators are very helpful, but they are not a substitute for mathematical thinking. There is no substitute for that.

Get a Glyph!

glyphs represent information

What’s a Glyph?

If you don’t know about glyphs, you’re in for a treat and your kids with love them! Glyphs are pictures that represent specific pieces of information. They are a fantastic way to introduce students to graphical representation of data.

The glyphs you see at the top of this post are what I call “Let’s get to know each other” glyphs. I modified an example from Susan R. O’Connell’s book, Glyphs! and I use it at the beginning of my initial session with each of my elementary age students.

I like to introduce glyphs by first asking students how they usually answer questions. I give examples such as “Do you like ice cream?” In this case they answer with a word (usually the word “Yes!”). Then I ask “How many grade levels are in your school?” And they answer with a number.

Enter glyphs. Glyphs are just another way to represent answers to questions, but instead of relying on words or numbers, we use pictures. Depending on the ages of the students, I sometimes mention the hieroglyphs of ancient Egypt at this point. Then we start making our own glyphs.

How to Make a Glyph

Each glyph is created based on answers to multiple choice questions. The glyphs above represent the answers given by three students to a series of questions about themselves, as shown in the table below.

information for a glyph

With our table of questions and possible answers, combined with a picture, we can tell a lot about a person! For example, the student that is represented by the center picture: likes math (yay!), is a girl, is the youngest of four children, is 8 years old, and loves cats. That’s a lot of information contained in a sketch of a face!

Make Your Own Glyphs

There are lots of free resources for glyphs on the Internet. Try Mrs. Jackson’s Class, MathWire, or Sweet Tea Classroom. Or you can make up your own using questions about favorite snacks to create a gingerbread man glyph, or favorite holiday pastimes to create an elf glyph, or any topic you want.

So put on your creative hat and give it a try! A silly little picture could be the start of your child’s career as a data scientist, a field that is in ever-increasing demand!

monkey minds have a hard time focusing

We All Lose Focus Sometimes

sponge bob in school


It happens to all parents, tutors, and teachers. You’re working with a child in reading or math, and you think things are going pretty well. You’re even feeling proud of yourself … you are inspiring a child to learn!

Then your intrepid student says something like, “I think Sponge Bob is on right now.” Sigh.

What to do?

I have wondered this myself, many times. My approach has always been to respond with something like, “That’s nice! Now, what did you say is the answer for six times six?” I admit. This is a short-term solution.

Stopping Monkey Thoughts in Their Tracks

Then, last week someone in a tutoring group on Facebook mentioned a blog post she had read, in which the author suggested introducing students to the idea of the “monkey mind.” It turns out that the original post was on one of my favorite sites, The Tutor House. It was a guest post by educational therapist, Anne-Marie Morey.

Her idea is simple. Explain to kids that all of our brains spit out random thoughts on a pretty regular basis, like monkeys jumping from tree to tree, making it hard for us to focus on one thing. Tell them that a great way to rein in the monkey mind is to recognize when it is jumping to another tree. This is called a “monkey thought.” If we recognize the monkey thought as it is happening, we can bring our attention back to the subject at hand before we get derailed.

monkey thoughts derail focusThat sounded great to me. So I modified a clipart image of a monkey to include a thought bubble, where my students and I could keep track of our monkey thoughts. Then I printed the page and placed it in my tutoring folder. I couldn’t wait to try it.

Fast forward to this morning. I introduced my 10-year-old autistic student, Tom, to the concept of the monkey mind and monkey thoughts. He was intrigued, asking whether I have monkey thoughts too. I told him that I have them all the time, but that when I see them coming, I do my best to stop them in their tracks and get back to what I was doing. I proposed that we keep track of our monkey thoughts during the session, and that maybe, just maybe, over time we could both get to the point where we wouldn’t have any monkey thoughts for an entire session! Tom was on board.

Right away I caught myself thinking about something I wanted to introduce in the following week’s session. I reported my monkey thought to Tom and recorded it on the sheet. Tom was next. In the middle of a hot game of division checkers, he started talking about the Mona Lisa (Tom is an avid fan of da Vinci). We looked at each other and he said, “Aw, that was a monkey thought.” I placed a mark in Tom’s column.

And so the session went. Once Tom had 3 marks on the page, he started negotiating. “If I win multiplication dominoes can I take 2 marks off?” I agreed, saying that this was, after all, our first time trying to control our monkey thoughts.

When the session was wrapping up, I proposed that starting next week, we should not take off any marks once they are recorded. Then we’ll be able to see how well we’re doing from week to week. Tom grudgingly agreed.

We’re In This Together

giving a hand upOne thing that I feel is very important in this approach to helping a student improve his or her focus is to make it a “we” thing, as opposed to a “you” thing. I volunteered myself for the first mark on the sheet, so Tom did not feel that I was pointing fingers when I gave him a mark for changing the subject. We were in it together, and that made it more comfortable for him.*

I’m looking forward to seeing how this works over time, but I was very encouraged by our first try! I’ll keep you posted!

* I borrowed this idea from a story I once heard about Winston Churchill. I have no idea whether it is true but it illustrates a great approach for working with other people. At some state dinner or other, Churchill was informed by the hostess that Lord something or other had just pocketed one of her silver spoons, and she wasn’t sure what to do. Without a word, Churchill picked up a spoon from the table and slipped it into his pocket. Approaching the would-be spoon thief, Churchill pulled the spoon out of his own pocket and said something to the effect of, “The game is up — they’re on to us. I guess we’ll both have to return our spoons.” The stolen spoon was returned and Lord whatever-his-name-was saved face.

Twister GameYou remember the Twister game. Someone spins and a player places his right foot on a blue circle. The person spins again and the next player places her left hand on a red circle. This continues until … everybody gets twisted.

Math Twister puts a new twist, or if you prefer, a new spin, on the old Twister game. The number of ways it can be played is limited only by your imagination. It can be played indoors, using a commercial or homemade Twister mat, or outdoors, using sidewalk chalk, and it’s great for any group of kids, whether they are in kindergarten, elementary school, after school, or home school.

For each version of the game, you will pick a “base number” that will be used in every turn. Then you will write a number on each of the positions on the spinner. Finally, you’ll write the answers in the circles on the mat. One person will spin and call out the problems. You can have all of the players place their hands or feet on the right circles for every problem, or you can have one player move for each spin. (Note: I’m using wet-erase markers for the spinner and the mat, but do a test with your markers to make sure the numbers will wipe off with water.)

Twister spinner with numbersI’ll give you an example for the multiplication version, borrowed from the CCafterschool channel on YouTube. This is going to sound confusing, but the video makes it clear, so please do watch it. Let’s pick “6” for our base number because we want to work on multiplying 6 times 6, 7, 8, and 12. The spinner will have the number 6, 7, 8, or 12 written on each spin position, with all four numbers represented in each quarter of the spinner. The mat will have 36, 42, 48, or 72 written on each circle, with each number being written on one of each color of circle. Now let’s say the spinner person spins: left foot, green, 7. Then all the players (or a single player) move their left foot to a red circle with a 42 (6×7) on it. Then it’s time for another spin.

There are some obvious variations on Multiplication Twister, including Division Twister, Addition Twister, and Subtraction Twister. Here are some other possibilities I’ve thought of and would like to try.

Twister MathSign Twister. This can be based on any of the four variations above, except that the game would include positive and negative numbers. What a great way to practice: -6 x -7 = 42!

Fractions Twister. This could work just like any of the four variations above, except that instead of whole numbers, the spinner would contain fractions. The base number could be a whole number or a fraction. The answers on the mat could represent either the direct answer, e.g., 1/4 + 1/4 = 2/4, or the equivalent (reduced) fraction, 1/2.

Exponents Twister. Ah, now we’re talking! In this version, the base number could represent either the base of the problem (e.g. the 7 in the problem “What is 7 to the power 2″) or the exponent (the 2).

Roots Twister. This one would work just like Exponents Twister except that the base number would represent the order of the root (square root, cube root, fourth root, …) and the spinner would contain the numbers inside the radical (inside the root symbol). So for each spin, you would ask questions like, “What is the cube root of 27? Or, “What is the cube root of 125?”

I’m sure there are other ways to use this idea, and I’m very excited to have the opportunity to try them out, starting this weekend! This will be my first weekly session with a group of 4th to 6th grade girls, and I plan to introduce them to several math games, including Twister. Games are a great way to show kids that math can be fun. Whether you use Twister or some other game, you will see a difference in your child’s attitude about math.

fun with math signs

We’ve all been there. We’re trying our best to help a child with math homework and we just can’t seem to capture his attention. The lesson needs to be taught, or the homework needs to be done, but the one who is supposed to be learning and doing is interested in everything but that paper in front of him. What to do?

You have two options. One, you can be the boss … remind your child that he has to do the work, no ifs, ands, or buts. Tried that already! How’d it work? In my experience, that’s a sure way to heighten resistance and make a child resent doing homework.

The other option is to let him exercise his free will, that is, give him choices. Here are a couple of examples from my tutoring practice.

4th Grader

Tom is 9, autistic, and home schooled. When I show up at his house, I have not one, but two, bags full of math-related stuff in hand. One is packed with hands-on activities; the other contains books. I also come prepared with a rough lesson plan and—here’s the key—each item in the plan has at least two options for addressing that topic. So, every step of the way through a session, I offer Tom choices. The conversation goes something like this:


    Do you want to do geometry or fractions first?
    Great! Do you want to use the GeoBoard or Play-Doh?
    Would you rather play Pizza Fractions or Dominoes Fractions now?

And so it goes until we’ve done at least one activity in every area that I want to cover. And Tom enjoys every activity because he chose it.

Now let’s looks at a conversation with an older student. Mark is 13 and attends public school, so he usually has math homework that must be done. I’m also working with him on specific areas in which he needs extra work, such as percentages, so I bring my own worksheets for these areas. A conversation with him goes more like this.

7th Grader


    Do you want to tackle homework problems or other areas first?
    Homework. (He always says “homework” because he likes to get it out of the way, but I still let him choose it.)
    (Later … while we’re doing homework, Mark sometimes gets a bit bored. When I see that, I give him a choice again.)
    Would you rather skip to some other problems and come back to these?
    No problem! We can finish these later.
    (Now the homework problems are done.)
    Do you want to work on percentages first or move on to a geometry review?
    Let’s do percentages.
    Okay. I have word problems and some regular problems. Which do you want to start with?
    Regular problems.
    Great! We’ll work on the word problems after that.

In the end, we finish every single thing that I planned to do, but we do it in an order that appeals to Mark because he chose it. It’s a win-win!

A little flexibility goes a long way in a math session, and having choices is empowering, even for kids, or maybe especially for kids. Give it a try, and let me know how it goes!


1 x 1 = FUN


When was the last time you ran across a math problem that looked fun, or amazing? I’ll bet it’s been a while … or maybe you’ve never encountered one. Most of the math textbooks present problems that I’m sure are meant to be interesting, but I rarely find them so, and neither do my students. In last week’s post, I talked about changing problems to include specific interests of each student. This week, I’ll show you a problem that I hope will elicit a “Wow!” from you and your child.

It goes like this. Find the squares of each of the first nine numbers that contain only ones, that is, find 1×1, then 11×11, then 111×111, and so on, up to 111111111×111111111. Sure, you can do this on a calculator, but it would take a while and it is much more fun to build a number pyramid using the pattern created by the solutions.

Ask your child to find the first 3 products. These are easy to do by hand. Then start building a pyramid, starting from the top and lining up the equal signs, like this:

Do you see the pattern? Does she? If needed, keep going with calculations (now she can resort to using the calculator) until she sees the pattern. Then have her fill in the remainder of the pyramid without the aid of electronics. Here’s the result:

Amazing right? And beautiful, I think.

I showed the 1’s pyramid to two students yesterday. One of these students is a 9-year-old with high-functioning autism; the other is 14 and gifted. Both their faces broke out in smiles when they saw the pattern.

The point of this exercise is to awaken the sense of wonder that is buried somewhere inside each one of us—to surprise students who didn’t know that math can be amazing. It probably won’t happen for every kid, but you won’t know until you try it.

Let me know how it goes!

butterfly speed to bullet speedI began working with Mark last summer, after he had failed the state end-of-grade (EOG) test, along with several of his classmates. Then we continued our weekly sessions after school started. Mark’s math skills and test grades have improved but I’m concerned that he may perform poorly on the standardized test again this year. I know that isn’t the end of the world, but Mark is smart and capable, and I’d like to see his scores represent his ability.

The main reason I’m concerned about the EOG is that I don’t think Mark’s teacher is going to complete all of the areas that are required by the Common Core (CC) standards for his grade level. I say this based on the pace of progress thus far. I discussed my concerns with Mark’s parents, and they agreed that it would be a good idea to begin working ahead. So I started bringing problems from CC worksheets and sample tests for us to work through after we finish with Mark’s regular homework.

After the first couple of sessions, I could see that it was not going well. While Mark was very motivated to finish his homework, his enthusiasm took a nose dive whenever I pulled out those extra problems. I couldn’t really blame him. Most pre-teen boys are not interested in the flight speed of butterflies, the area of a kitchen floor, or sales at a clothing store. I had to try something else.

So before our third session with the CC problems, I rewrote an entire set of questions to make them relevant to Mark. He is an avid hunter (not something we have in common), so in the revised worksheets, butterfly flight speed became bullet speed, kitchen square footage became designated hunting area, and clothing discounts became gun sales. It took some thinking, but I was able to change every single problem into something that Mark could relate to.

During our next session, when it was time for the extra problems, I told Mark that I had some new worksheets. He shrugged and waited for me to get them out of my folder. I watched in anticipation as he read the first problem. Just a few seconds later he smiled. Bingo!

We worked through several problems, and along the way Mark shared stories about hunting. I was thrilled to see his enthusiasm and I felt a sense of connection between us that had not existed before. I know he enjoyed talking about his interests and I’d like to think that he appreciated my efforts. I definitely let him know that I appreciated his.

So the next time your kid comes down with a case of math boredom, try changing the problems. You don’t necessarily have to type up new problems—you could just verbalize them. Use any subject that your child thinks is interesting or silly or outlandish. If you get him engaged, his mind will be more open to learning and he’ll have more fun with math.