The Homework Inequality: 1 Great Problem > 50 Repetitive Exercises

Yesterday, my sons Alex and Eli were completing their homework on fraction operations, which included 39 problems in 4 sets, wherein each problem in a set was indistinguishable from its neighbors.

The worksheet contained 12 problems of fraction addition, 9 problems of fraction subtraction, 9 problems of fraction multiplication, and 9 problems of fraction-times-whole-number multiplication. That’s 39 problems of drudgery, when 10 problems would’ve been sufficient. Here’s a link to the worksheet they were given, if you’d like to torture your children or students in a similar way:

Lesson 0.1: Adding and Multiplying Fractions

Frustrated by the monotony of the assignment, I told the boys they didn’t have to do all of the problems, and they could stop when they felt that they had done enough from each set.

“No,” said Alex. “We’re supposed to do them all.”

My sons are responsible students, but I’m frustrated by teachers who take advantage of their work ethic. Just because they’re willing to complete 50 exercises for homework doesn’t mean they should be assigned 50 exercises for homework.

My colleague at Discovery Education, Matt Cwalina, puts it this way:

Some say that a picture is worth a thousand words. I say,

A great problem is worth a thousand exercises.

Personally, I would much rather have students think deeply about one challenging problem than mindlessly complete an entire worksheet. Luckily, my sons take after their daddy and love number puzzles, so I spontaneously created one.

Find three fractions, each with a single-digit numerator and denominator, that multiply to get as close to 1 as possible. Don’t repeat digits.

Eli started randomly suggesting products. “What about 4/5 × 6/7 × 9/8?” He’d work out the result, say, “I think I can do better,” then try another. And another. And another. Finally, he found a product that equaled 1. (No spoiler here. Find it yourself.)

Alex eventually found an answer, too. At the bottom of his homework assignment, he added a section that he titled “Bonus” where he captured his attempts:

I don’t know exactly how many calculations Eli completed while working on this problem, but I know that Alex completed at least seven, thanks to his documentation. Wouldn’t you agree that completing several fraction computations while thinking about this more interesting problem is superior to doing a collection of random fraction computations with no purpose?

There is a preponderance of evidence (see Rohrer, Dedrick, and Stershic 2015; definitely check out Figure 4 at the top of page 905) that massed practice — that is, completing a large number of repetitions of the same activity over and over — is counterproductive. Unfortunately, massed practice feels good because it results in short-term memory gains, which trigger a perceived level of mastery; but, it doesn’t lead to long-term retention. Moreover, students who learn a skill by practicing it repeatedly get really good at performing that skill when they know it’s coming; but, two months down the road, when they need to use that skill in an unfamiliar context because it’s not on a worksheet titled “Lesson 0.1: Adding and Multiplying Fractions,” they’re less likely to remember than if they had used more effective practice methods. One of those more effective methods is interleaving, which involves spacing out practice over multiple sessions and varying the difficulty of the tasks. Whether you’re trying to learn how to integrate by parts or how to hit a curve ball, be sure to make your practice exercises a little more difficult than you’re used to. Know that interleaving your practice will not feel as good as massed practice while you’re doing it; but later, you’ll feel better due to improved memory, long-term learning, and mastery of skills.

Interleaving is one of the reasons I love the MathCounts School Handbook, which can be downloaded for free from the MathCounts website. The topics covered by the 250 problems in the School Handbook run the gamut from algebra, number sense, and probability, to geometry, statistics, combinatorics, sequences, and proportional reasoning — and any given page may contain problems of any type! Veteran MathCounts coach Nick Diaz refers to this mixture as “shotgun style,” meaning that students never know what’s coming next. Consequently, similar problems are not presented all at once; instead, students are exposed to them several days or perhaps weeks apart. Having to recall a skill that hasn’t been used for a while requires more effort than remembering what you did just five minutes ago, but the result is long-term retention. It’s doubtful that the writers of the first MathCounts School Handbook knew the research about interleaving and massed practice… but they clearly knew about effective learning.

The other reason I like the MathCounts School Handbook is the difficulty level of the problems. Sure, some of the items look like traditional textbook exercises, but you’ll also find a lot of atypical problems, like this one from the 2017-18 School Handbook:

If p, q, and r are prime numbers such that pq + r = 73, what is the least possible value of p + q + r?

That problem, as well as the fraction problem that I created for Alex and Eli, would both fall into the category of open-middle problems, which means…

• the beginning is closed: every students starts with the same initial problem.
• the end is closed: there is a small, finite number of unique answers (often, just one).
• the middle is open: there are multiple ways to approach and ultimately solve the problem.

Open-middle problems often allow for implicit procedural practice while asking students to focus on a more challenging problem. This results in a higher level of engagement for students. Moreover, it reduces the need for massed practice, because students are performing calculations while doing something else. You can find a large collection of open-middle problems at www.openmiddle.com, and the following is one of my favorites:

Use the digits 1 to 9, at most one time each, to fill in the boxes to make a result that has the greatest value possible.

It’s a great problem, because random guessing will lead students to combinations that work, but it may not be obvious how to determine the greatest possible value. Consequently, there’s an entry point for all students, the problem offers implicit procedural practice, and the challenge of finding the greatest value provides motivation for students to continue.

I have a dream that one day, in traditional classrooms where 50 problems are assigned for homework every night, where procedural fluency is valued over conceptual understanding; that one day, right there in those classrooms, students will no longer think that math is simply a series of disparate rules with no purpose, but instead will experience the joy of attempting and solving challenging problems that inspire purposeful play and, as a side benefit, encourage students to practice the skills they will need to be successful learners.

There are myriad resources available so that teachers and parents can encourage their students to engage in these kinds of problem-solving activities, so it is my hope that this dream is not too far away.

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As a special bonus for reading to the end, check out this Interleaved Mathematics Practice Guide that Professor Doug Rohrer was kind enough to share with me (and now, with you).

Math Problems for 2016

“What homework do you have to do tonight?”

I ask my sons this question daily, when I’m trying to determine if they’ll need to spend the evening doing word study or completing a math worksheet, or if we’ll instead be able to waste our time watching The Muppets or, perhaps, pulling up the animated version of Bob and Doug Mackenzie’s 12 Days of Christmas on Dailymotion.

When I asked this question last night, though, the answer was surprising:

We have to do our reading, but we already completed your math problem.

My problem? I had no idea what this meant. So they explained:

It’s not a problem you gave us. It’s one we got from [our teacher], and it says, “This problem was written by Patrick Vennebush.”

I was puzzled, but then it dawned on me. I asked, “Does it have a monkey at the top with the word BrainTEASERS?”

“Yes!”

“Which problem?”

“It’s about the word CAT.”

I knew the problem immediately. It’s the Product Value 60 brainteaser from Illuminations:

Assign each letter a value equal to its position in the alphabet (A = 1, B = 2, C = 3, …). Then find the product value of a word by multiplying the values together. For example, CAT has a product value of 60, because C = 3, A = 1, T = 20, and 3 × 1 × 20 = 60.

How many other words can you find with a product value of 60?

As it turns out, there are 14 other words with a product value of 60. Don’t feel bad if you can’t find them all; while they’re all allowed in Scrabble™, the average person won’t recognize half of them.

You can see the full list and some definitions in this problem and solution PDF.

This problem resurfaced at the perfect time.

With 2016 just around the corner, no doubt many math teachers will present the following problem to students after winter break:

Find a mathematical expression for every whole number from 0 to 100, using only common mathematical symbols and the digits 2, 0, 1, and 6. (No other digits are allowed.)

And that’s not a bad problem. It gets even better if you require the digits to be used in order. For instance, you could make:

• 2 = 2⁰ + 1⁶
• 9 = 2 + 0 + 1 + 6
• 36 = (2 + 0 + 1)! × 6

But that problem is a bit played out. I’ve seen it used in classrooms every year since… well, since I used it in my classroom in 1995.

So here are two versions of a problem — the first one being for younger folks — using the year and based on the Product Value 60 problem above:

How many words can you find with a product value of 16?

How many words can you find with a product value of 2016?

There are 5 words that have a product value of 16 and 12 words that have a product value of 2016 (spoiler: those links will take you to images of the answers). As above, you may not recognize all of the words on those lists, but some will definitely be familiar.

A Father’s Day Gift Worth Waiting For

Alex made a Father’s Day Book for me. Because the book didn’t make it on our trip to France, however, I didn’t receive it until this past weekend. It was worth the wait.

The book was laudatory in praising my handling of routine fatherly duties:

I loved when you took me to Smashburger.

I appreciated when you helped me find a worm.

I love when you read to me at night.

I love when I see you at the sign-out sheet [at after-school care]. It means I can spend time with you.

But my favorite accolade — surprise! — was mathematical:

I liked the multiplication trick you taught me. Take two numbers, find the middle [average], square it. Find the difference [from one number to the average], square it, subtract it. (BOOM! Done!)

Priceless.

The trick that I taught him was how to use the difference of squares to quickly find a product. For instance, if you want to multiply 23 × 17, then…

• The average of 23 an 17 is 20, and 20² = 400.
• The difference between 23 and 20 is 3, and 3² = 9.
• Subtract 400 – 9 = 391.
• So, 23 × 17 = 391.
• BOOM! Done!

This works because

,

and if you let a = 20 and b = 3, then you have

.

In particular, I suggested this method if (1) the numbers are relatively small and (2) either both are odd or both are even. I would not recommend this method for finding the product 6,433 × 58:

• The average is 3,245.5, and (3,245.5)² = 10,533,270.25.
• The difference between 6,433 and 3,245.5 is 3,187.5, and (3,187.5)² = 10,160,156.25.
• Subtract 10,533,270.25 – 10,160,156.25 = 373,114.
• So, 6,433 × 58 = 373,114.

Sure, it works, but that problem screams for a calculator. The trick only has utility when the numbers are small and nice enough that finding the square of the average and difference is reasonable.

Then again, it’s not atypical for sons to do unreasonable things…

Son: Would you do my homework?

Dad: Sorry, son, it wouldn’t be right.

Son: That’s okay. Can you give it a try, anyway?

I’m just glad that my sons understand math at an abstract level…

A young boy asks his mother for some help with math. “There are four ducks on a pond. Two more ducks join them on the pond. How many ducks are there?”

The mother is surprised. She asks, “You don’t know what 4 + 2 is?”

“Sure, I do,” says the boy. “It’s 6. But what does that have to do with ducks?”

My Son’s New Joke

My son is doing his math homework — he’s in first grade, so it involves writing a certain number, spelling that number, and finding all occurrences of that number in a grid of random numbers called a “Number Hunt.” Based on today’s number, he came up with the following joke:

What number is mostly even but not even?
Eleven.

Not a great joke, to be sure… but as good as most jokes on his dad’s blog, and he’s only six years old.

The homework was frustrating (for me), because my sons are capable of much more.

When my sons ride their bikes through the parking lot, they solve problems involving parking space numbers, the digits on license plates, and other numerical things. They ask me to create “math challenges” for them to think about as they ride. Yesterday, they solved the following three challenges:

1. Which license plate has the greatest product if you multiply its four digits together? (The license plate format in Virginia is LLL-DDDD, where L is a letter and D is a digit.)
2. How many different license plates  are possible with the format LLL-DDDD?
3. Each of the three rows in our parking lot has a different number of cars. If our parking lot had a fourth row, how many cars would there be in the fourth row?

For Question 1, Eli realized that the license plate with {9, 7, 6, 5} would have a greater product than the license plate with {9, 7, 6, 3}, since 5 > 3. But then he realized that {9, 9, 8, 2} would be even greater, and he correctly determined that the product is 1,296.

For Question 2, Alex thought it would be 144. His argument was that there would be 6 ways to arrange the letters and 24 ways to arrange the digits, and 6 × 24 = 144. We talked about this, and I pointed out that his answer would be correct if we knew which three letters and which three digits we were using (and they were all different). He and Eli reconvened and eventually claimed there would be 26³ x 10⁴ possible license plates… and being the good father that I am, I let them use the calculator on my phone to find the product.

For Question 3, the number of cars in the three rows was 2, 5, and 8. They extended the pattern and concluded that there would be 11 cars in the non-existent fourth row.

So you can understand why I’d be frustrated that Alex’s homework involved writing the number 11 repeatedly. I thought about telling him not to do it, but then I imagined the following conversation:

Alex: Would you punish me for something I didn’t do?
Teacher: Of course not, Alex.
Alex: Good, because I didn’t do my homework.

Or perhaps he’d just fabricate an excuse:

I thought my homework was abelian, so I figured I could turn it in and then do it.

And finally — should abelian be capitalized?

Giving Thanks

The end of the year is a good time to reflect and be thankful for all that we have. I have two fantastic, five-year-old sons who love math and their daddy — what more could a man want?

Eli is thankful, too. This is the note he wrote to his teacher for the Math Enrichment homework she asked him to complete during the holiday break:

When I asked why he was thankful for homework, he said, “Because this was fun!”

Checking sales on Amazon Author Central tonight, I was thankful to the 299 folks who bought a copy of Math Jokes 4 Mathy Folks from December 17‑23, making it the best-selling week for my silly joke book yet. In fact, between Thanksgiving and Christmas, an astounding 928 people bought my book; people who, apparently, are unaware that they could have gotten not one but two venti, decaf, sugar-free, non-fat, vanilla soy, extra hot, no foam, mocha cappuccinos with three shots, light whip, extra syrup, cinnamon and sprinkles at Starbucks for the exact same price. Oh, well… their loss.

These numbers represent a sales increase of nearly 40% compared to the 2011 holiday season. My financial planner previously predicted that I’d be able to retire at age 65; but, if this trend continues, I might be able to retire at age 64 9/10.

Allow me to take this opportunity to thank all of you, whether you read my blog posts religiously in 2012, stopped by only once in a blue moon, bought Math Jokes 4 Mathy Folks from a local, independent bookstore, or stole a copy from your local library. I appreciate your support, in whatever form it takes.

Wishing you peace, joy, and happiness in 2013, y’all. May you occasionally laugh so hard that milk comes out of your nose.

Back to Pencils, Books, Dirty Looks

The fall semester is underway. Here are some jokes for you, no matter your level.

For professors…

Mathematical conferences are very important. They demonstrate how many faculty a department can operate without.

For graduate students…

Why is grad school like a hot bath?
Because after you’ve been there for seven years, it ain’t so hot anymore.

For undergraduates…

An undergraduate student said to his statistics professor, “You know, I hate being a full-time student and mooching off of my parents. I’d really rather have a job.”

The professor says, “You’re in luck! I just heard that the President of the University is looking for a bodyguard and chauffeur for his beautiful daughter. You’ll be expected to drive her around in his Mercedes, accompany her on overseas trips, and satisfy her sexual urges. He’ll provide all meals and supply all of your clothes. You’ll be given a two-bedroom apartment above the garage, and the starting salary is $75,000 per year.”

The wide-eyed student says, “You’re kiddin’ me?”

The professor replies, “Well, yeah… but you started it.”

And for high school kids…

“Why don’t you work on your math homework with Sarah anymore?” a mother asks her daughter.

“Would you do your homework with a lazy slug who just copies all of your work?” says the daughter.

“Well, no, I suppose I wouldn’t,” says the mother.

“Yeah, well, neither will Sarah.”

Free Copy of My Book

A few days ago, a seventh‑grade math teacher and assistant baseball coach sent me the following request:

I would love to have a copy of Math Jokes for Mathy Folks, but I am financially unable to purchace it right now because my wife is unable to work and hasn’t been approved for disability.

Now, I like to think I’m a generous guy, but I am unable to send a free copy of my book to everyone who asks for it. A little‑known fact about the publishing industry: The majority of authors actually have to pay for copies of their own book. It’s an interesting percent problem. I pay 50% of retail price plus shipping to purchase copies of my book, but I then receive a 15% royalty on the discounted price of every copy I purchase. (You can do the math to figure out how many books I could give away for free before going bankrupt.)

So I sent the following reply:

Send me your favorite joke(s), mathy or not, and I’ll send you a copy of the book.

My correspondent responded quickly with three jokes, two of which I had never heard before. A copy of my book is in the mail to him, and his jokes are pasted below for your reading pleasure.

What’s a seventh grader’s favorite excuse for not doing homework?
I have a solar‑powered calculator, but yesterday it was cloudy.

The student’s second semester seemed so much like her first that she hoped she could graduate sooner by combining like terms.

How is an indecisive third‑base coach like multiplying or dividing by a negative integer?
In both cases, the sign changes.

Incidentally, you can download one chapter of Math Jokes 4 Mathy Folks for free by clicking the following button:

Father’s Day Reflections (and Other Transformations)

I just got a new stepladder. Don’t get me wrong, it’s a fine stepladder. I just wish I had gotten to know my real ladder.

I had the privilege of knowing my real father.

At age 15, my father faked a birth certificate and joined the Navy. When he was 18, he received a dishonorable discharge — after allowing him to fight in Japan during the last two years of World War II, the Navy finally learned that my dad was under age when he enlisted. So, what did he do? He joined the Army. Before he was 21, he had been to each of the 50 states and had traveled around the world 4 times.

My father had only a sixth-grade education, but he believed in the power of school and learning. It was not easy to get my dad to part with his money. When I was in third grade, my teacher asked me, “If you have two dollars, and you ask your father for another three dollars, how much will you have?”

“I’d have two dollars,” I told her.

“Young man,” she said, “you don’t know your arithmetic.”

“No, Mrs. Wargo,” I said, “you don’t know my father!”

But he often gave me $20 for a good report card, and I was the first kid in my school to have a Commodore 64 with a disk drive. When I graduated high school, my family was subsisting on my father’s disability pension, and I considered working for a year to save money before enrolling in college. “You’re too damned smart,” he said. “Send in the forms. We’ll make it work.”

My father passed away in December 1994. The last words he said to me were, “You’re my pride and joy.” Father’s Day is always a little rough for me, but it’s a good time to reflect. I continually ask myself, “Am I a man that my father would be proud of?”

Are you kidding? I have to believe my father is smiling down from Heaven, saying, “That’s my boy! Yeah, that geeky one there! He’s the author of a math joke book and math joke blog, ya know.”

For all you math dads (and sons, too), here’s some humor for today:

Son: Dad, can you do my homework for me?
Dad: I’m sorry, son, it wouldn’t be right.
Son: That’s okay. Can you try anyway?

I spent today with my twin four-year-old sons, hiking, doing KenKen (more on that later), playing the anagram game, and helping them figure out how the number of “cheers” we do with our glasses at the dinner table is related to the triangular numbers. What a great day. Happy Father’s Day!

Back to School

Even though parents, teachers, and kids in Alabama aren’t happy that their summer ended several weeks ago, school has started or will be starting soon for kids across the country. Best of luck to all those returning to the classroom, regardless which side of the desk you’re on.

Here are a few rib-ticklers about (and appropriate for) the classroom.

Father: Did you learn a lot in math class today?
Son: Apparently not! They want me to come back again tomorrow!

Why did the student eat her homework?
Because she thought it was “a piece of cake.”

A young boy asked his grandmother for help with his math homework. “I need to find the least common denominator,” he told her. 
“My goodness,” his grandmother replied. “I can’t believe they still haven’t found that. They were looking for that when I was in school!”