Mrs. Wright, my generous and kindly second-grade teacher, once told my parents that I was good at arithmetic. But I felt she was wrong. Yes, I could add and subtract better than some of my peers, but I was often flummoxed by relatively simple word problems (also known as “story” problems). To my eye, the typical story problem always looked a lot like this:

“Blah blah blah 6 elephants; blah blah blah 2 elephants. Blah blah blah blah blah?”

Seeing story problems in this way, I didn’t have a clue whether I was supposed to add, subtract, multiply, or divide. It wasn’t that I had trouble reading. Even when I could successfully read all of the words out loud, I often had trouble making a reliable connection between those words and the various arithmetic operations I’d been taught to perform.

Over the many years since I left Mrs. Wright’s class, I’ve given a lot of thought to what, if anything, she or my parents could have done to help me. The lessons that make up Adventures in Arithmetic are the result of that thinking. They are designed to teach the usual computation skills (adding, subtracting, multiplying, and dividing), but the main goal throughout is always to help kids solve story problems.

Kids are usually given a lot of practice with adding and subtracting, but they don’t often get enough experience solving story problems. My lessons contain lots and lots of story problems, though most of them at first glance don’t look like traditional “word” problems. I’ve disguised them (and hopefully made them more engaging) by replacing a lot of the usual printed words with pictures. And I’ve replaced some of the “imagining” that story problems normally require with actual “doing” — coloring, cutting, pasting, drawing. But even when disguised, they are still story problems, and they still pose the same “how many” questions that story problems have always asked. For kids to correctly answer these “how many” questions, they will need to actually add, subtract, multiply, or divide in one way or another, even if at first they may not be fully aware that this is what they are doing.

Ultimately, in order to solve story problems proficiently, kids will need to have a working model in their heads of what an addition or subtraction or multiplication or division situation typically looks like, and they will then need to be able to match up the situation in any given problem with the appropriate model. And that’s not easy, because even though there are only four basic operations, the story problems that match up with those four operations come in more than four “flavors.”

For instance, consider the two division problems below, and note how very different they might seem from a child’s point of view:

“George has 12 loose blue socks. How many pairs of 2 blue socks can he make?”

“Emma has 12 dog-treats. If she gives the same number of treats to each of her 2 cocker spaniels, how many treats will each dog receive?”

Similarly, here are two subtraction problems that have distinctly different flavors:

“Fred had 10 pet hamsters, but 4 of them ran away. How many does he have now?”

“Linda has 10 pineapples. Mary has only 4. How many more pineapples does Linda have?”

Even addition problems can come in slightly different flavors:

“Lulu has 3 bananas. If somebody gives her 2 more bananas, how many will she have then?”

“Lulu has 3 green bananas and 2 yellow bananas. Altogether, how many bananas does she have?”

The difference between the two addition problems immediately above is so subtle that an adult might not even notice it. But that subtle difference might be enough to confound a child. There is action in the first problem — you could make a movie out of it, and the image in the last frame of that movie would look different from the image in the movie’s first frame. There is no action whatsoever in the second problem — you could illustrate everything that’s going on in that second problem with a single still photograph. The two groups of bananas in the second problem (the green ones and the yellow ones) still look as different from each other at the end of the story as they did at the beginning, and in order to answer the problem’s “how many” question, the reader must ignore the different colors and consider that for counting purposes, all of these bananas can be considered identical. A child who expects that addition situations will always entail some sort of action — a group of new arrivals joining up with a pre-existing group — might have trouble identifying my green and yellow banana story problem as an addition problem.

The point I’m trying to make with all of the above examples is that having a single, fairly rigid and limited idea of what an addition story problem or a subtraction story problem ought to look like will not suffice. To become proficient with story problems, a kid must be exposed to many problems of varying types so as not to develop too limited an expectation of what sort of situations call for which mathematical operations. It’s a little like trying to teach a child the difference between dogs and cats. If you show the child only gigantic dogs with curly hair and floppy ears, how can you expect the kid to correctly identify a chihuahua as a dog rather than a cat? It wouldn’t be practical to teach such a child about every possible breed or mixed-breed of dog or cat, but it would surely be useful to introduce the kid to a whole lot of dogs of various shapes and sizes to encourage the formulation of useful generalizations about when a given animal is probably a dog. It’s the same with story problems: what I’m trying to do is offer kids enough experiences for them to form reliable generalizations suggesting when they should add and when they should subtract, when they should multiply and when they should divide.

Now, I’m by no means claiming that this represents a fool-proof way to make word problems easy. Word problems are never going to be easy. All I’m saying is that I’m aware that they are difficult, and that by providing lots of hands-on experiences and lots of gentle guidance, I’ve tried my level best to make them easier, more comprehensible, and less intimidating.

My lessons are dedicated to the memory of Mrs. Wright. Second grade was a very long time ago, but I remember her vividly and with great affection.

What is this, and who is it for?

These free printable pages of arithmetic lessons are to be used at home by a parent or tutor working with a child.  Much of the material covered is typically taught to second graders.  However, the lessons would also be appropriate for younger children who can count and write numerals up to 100, or for older children who have seemingly learned to add, subtract, multiply, and divide mostly by rote, without absorbing the underlying concepts which make those computational skills both meaningful and useful.

These are not “practice” pages that can be used by an unsupervised child.  Instead, these are pages that require frequent interactions between the child and an adult, whose presence is needed to read the directions for various activities and to provide both encouragement and verification that the child is successfully completing each new challenge.

What will you need to get started? 

You will need access to a color printer.  For some lessons, the child will need crayons, for others a pencil, for still others scissors and either glue or some other adhesive.  All of the materials needed for each specific group of lessons are listed along with special printing instructions for each downloadable file.

How many pages should be attempted in each session?

Some pages can be completed more quickly than others, and different children (or the same child on different days) will bring varied levels of concentration and enthusiasm to the project.  For these reasons, it is difficult to predict ahead of time how many pages to print out for any given session, but a good guess would be somewhere between 6 and 10 pages.  The child’s demeanor should tell you when it’s time to call it quits for the day, and it’s always better to err on the side of stopping too soon, before the child’s attention has begun to wander.

Is the order of activities and questions important? 

Yes.  Each question is ever-so-slightly more challenging than the preceding question.  Answering the questions in the prescribed order is a little like climbing a staircase – each step up leads naturally to the next step.

Is it okay to continue if the child does not complete a given page correctly? 

No, that is not a good idea.  Have the child try again with a little more help this time (print out fresh copies of the page as needed).  Then, if you’re unsure that the child could now complete the page without help, turn the page face down, distract the child for 30 seconds, and ask the child to attempt the task yet again.

Is it okay to skip the most advanced addition lessons and go straight to the most elementary subtraction lessons? 

Yes, that is absolutely okay.  Skipping an individual question or task within a block of lessons is NOT a good idea, because that can disrupt the flow of the lessons, making the subsequent tasks more difficult.  But skipping an entire block of lessons is a different matter.  Kids are likely to benefit from the introductory lessons for a different operation (such as the lessons in Division Part One) even if they are not yet ready to comfortably tackle the most challenging questions in a preceding section (such as the advanced material in Multiplication Part Two).  Anytime a child has trouble with several tasks in a row within a block of lessons, it is a good idea to move on to a different block of lessons.  Blocks of lessons that turn out to be too challenging should probably not be revisited until a significant amount of time has elapsed – time for the child to grow and accumulate useful real-world experiences.

Is there anything else that you’ll need to know before starting the first addition lessons?

No, absolutely nothing.  But if it makes you feel uncomfortable to proceed without a better understanding of what these materials are designed to do, and why they look the way they do, click here for an explanatory interview with the author.

ABOUT THE AUTHOR:  Jonathan Cerf wrote and illustrated these lessons.  He taught fifth and sixth grades in the New York City public school system and later worked on math textbooks for Random House and for Harcourt Brace Jovanovich.