Explorations In Math

Why Homework?

Published on December 20 2011

This week’s post, while not about math specifically, is relevant because homework assignments often include math. Here are my thoughts, based on my years in the classroom.

Many parents and educators question the need for, and the value of, homework. They believe six hours a day in school is enough, that children need time to relax and play after school, and besides, their after-school schedule is already crammed with activities: music lessons, dance lessons, soccer practice, Little League baseball, gymnastics, and often some combination of these. Where is the time for homework?

On the other side are those who argue that more is better. Homework is simply an extension of the school day and gives students a leg up on the ladder to success. They’re spending more time studying and working in the different subject areas and they’re learning the value of responsibility, hard work and perseverance.

There is merit to both arguments. Nonetheless, when the proper balance is struck between too much and none, homework becomes an effective part of teaching/learning. Here‘s why:

• Homework provides an additional opportunity for your child to practice a skill being worked on in class;

• Homework lets you know on a daily basis what’s being covered in a given subject;

• Homework provides an opportunity for you to see how well your child is learning or understanding the material;

• Homework is an opportunity for you and your child to spend some positive and productive time together

• Homework helps instill a sense of responsibility as well as accomplishment in your child. It’s the child’s responsibility to bring the work home, do it, and return it the next day.

How can you help? Here are some tips:

  • Ask about your child’s homework every day
  • Have a designated time and a special place for doing homework
  • Offer to help if your child is struggling, but don’t insist
  • Don’t allow your child to turn in unacceptable work

Got some homework tips to share with Math Matters readers? Please send them to me at davega@eimath.org.

Dave Gardner

EIM Mathematician in Residence

To drill? Or not to drill?

Published on December 8 2011

To drill? Or not to drill? It’s a question that teachers and parents ask themselves and it’s an important one. Is drilling kids on their basic arithmetic facts effective? Or is it counterproductive? The answer depends on how we define “drill” and how we do it. If drilling is nothing more than regular rote recitation and memorization of facts, then I believe it’s counterproductive, although in an unexpected way. The child may, indeed, memorize the facts but this brings up two concerns. The first, as I’ve noted before, is that we don’t want our children to memorize the basic facts; we want them to know the basic facts, and there’s a world of difference. Second, we run a good risk of killing math spirit in children when we drill in this way. It becomes something boring and repetitious, a drudgery. (It’s called “drill and kill” for a reason.) And this feeling then begins to color all areas of math.

So how can we avoid these pitfalls? How can we change “drill and kill” into “drill for skill”? The overarching answer is to make learning basic arithmetic facts something children want to do, something they look forward to. Here are some ideas on how to do that.

Flash Cards

These cards have the problem on one side and the problem and answer on the other. The child is shown the problem side and states the answer. There are many ways they can be used productively. You can set incentives for learning new facts or for getting so many in a row correct. Kids can quiz their parents or teacher. For those who are competitive, competition is a way to engage them.

Verbal Drills

This is for when you’re with your child in the car, or at the breakfast or dinner table, or out for a walk. Simply give them a fact (e.g.: 5 + 4) and they give you the answer. It makes it much more engaging, though, if you alternate with your child: you give them a fact and then they give you one.

Mental Math

This is another kind of verbal drill, but instead of presenting problems individually, a series of numbers and operations are strung together. The child does the math mentally and responds with the correct answer. For example, for a 4th grader the string might go like this: 8 + 3 x 2 – 12 = ?  For younger students, limit the number of terms and limit the operations to adding and subtracting.

Math Games

Yes, math games are a good way of drilling and there are many good commercial math games available and many non-commercial math games that utilize only cards, dice or paper and pencil.

Another question is how frequently should you drill? I believe that if you engage your children/students as they learn their facts, you can drill every day because they’ll enjoy it and look forward to it. And guess what? They will know them.

Do you have any drill activities you’ve found to be effective? If so, please send them to me to share with other readers. Thanks.

Dave Gardner

Mathematician in Residence

Lessons from Kaprekar

Published on November 30 2011

This week’s guest post is from Mark Taylor. Mark teaches at Coe Elementary in Seattle, is on EIM’s advisory council and has been a long time EIM volunteer. If you would like write a guest post, please contact Dave Gardner.

In the third grade curriculum I teach, we try for three things:  1. Math based on exploration, 2. Ways to increase math fluency and 3. Social Justice – that is, math discoveries are not just for European men before the year 1800.

One of my favorite lessons is an exploration of the Kaprekar constant.  Consider would happen if we took the following steps:

  1. Take any three-digit number, using at least two different digits.  In other words, the digits are not all the same.
  2. Arrange the digits in ascending and then in descending order to get two three-digit numbers.
  3. Subtract the smaller number from the larger number.
  4. With your new difference, go back to step 2.
  5. Continue until you see a pattern.

What will happen?  We try an example together.

Start with 281.

Descending order: 821
Ascending order:   128


Now repeat with new number 693 – so subtract 369 from 963:


Repeat with new number 594 – so subtract 459 from 954


Repeat with 495 – wait – we’re done!  It will repeat now.

What if we start with different three-digit numbers?  Assign each student a different three digit number.  They will ALL eventually end up with the same pattern – assuming students do their math correctly.

An awesome aspect of these calculations is that they ALWAYS require regrouping in the 10s and 1s places (do you see why, dear reader?), so it’s great practice for subtraction with regrouping.

Next up: is there a 4-digit Kaprekar constant?  Assign as homework the mission of finding the 4-digit Kaprekar constant.

Is there a 5-digit Kaprekar constant?  Hmmm.  There actually isn’t!  But it’s interesting to see what happens for different 5-digit numbers.  There is room for research here.

Spoiler:  For numbers of 5 or more digits, there are various different numbers that students end up with.  In other words, there is no Constant for numbers of 5 or more digits.

Why are they called Kaprekar constants?  They are named after mathematician Dattaraya Ramchandra Kaprekar (1905 – 1986).

From Wikipedia:

Kaprekar received his secondary school education in Thane and studied at Fergusson College in Pune. In 1927 he won the Wrangler R. P. Paranjpe Mathematical Prize for an original piece of work in mathematics.

He attended the University of Mumbai, receiving his bachelor’s degree in 1929. Having never received any formal postgraduate training, for his entire career (1930–1962) he was a schoolteacher at Nashik in Maharashtra, India.

Yes, these discoveries were made in the 20th Century, by an amateur mathematician with no formal training – you too can make discoveries in math one day, brave third grader!

Mark Taylor



Published on November 15 2011

A change of pace this week – enjoy!

Q: What did the zero say to the eight?
A: Nice belt!

Theorem: A cat has nine tails.

Proof: No cat has eight tails.

Since one cat has one more tail than no cat, it must have nine tails.

  • Trigonometry for farmers: swine and coswine…
  • Some engineers are trying to measure the height of a flag pole. They only have a measuring tape and are quite frustrated trying to keep the tape along the pole: It falls down all the time.  A mathematician comes along and asks what they are doing. They explain it to him.
    “Well, that’s easy…” He pulls the pole out of the ground, lays it down, and measures it easily.  After he has left, one of the engineers says: “That’s so typical of these mathematicians! What we need is the height - and he gives us the length!”
  • In a class, a math professor claims that he can prove everything under the assumption that 1+1=1. A student challenges him: “Then prove that you’re the pope!” He ponders for a moment and then replies: “I am one, and the pope is one. Therefore, the pope and I are one.”
  • One evening Rene Descartes went to relax at a local tavern. The bartender approached and said, “Ah, good evening Monsieur Descartes! Shall I serve you the usual drink?” Descartes replied, “I think not.” and promptly vanished.
  • Three men are in a hot-air balloon.  Soon, they find themselves lost in a canyon somewhere.  One of the three men says, “I’ve got an idea. We can call for help in this canyon and the echo will carry our voices far.” So he leans over the basket and yells out, “Helllloooooo! Where are we?” (They hear the echo several times.) Fifteen minutes later, they hear this echoing voice: “Helllloooooo!  You’re lost!!” One of the men says, “That must have been a mathematician.”  Puzzled, one of the other men asks, “Why do you say that?” The reply: “For three reasons.  1) He took a long time to answer, 2) he was absolutely correct, and 3) his answer was absolutely useless.”

You can find a lot more math jokes and humor. Just type in ‘math jokes’ or ‘math humor’ in your search engine and you’ll find tons of ‘em.

Dave Gardner

Mathematician in Residence


Published on November 7 2011

Wait, parents! I know, this sounds like something for teachers only, but read on. There’s useful information here for you, as well.

Let’s start with this question: What is Bloom’s Taxonomy? It’s a way of measuring the complexity of the questions and tasks teachers and parents pose to children. Using the taxonomy is a way to encourage children to higher levels of thinking. The important thing to keep in mind is that the first three levels are review of existing knowledge; the last three levels are where new learning occurs.  Those are the levels to strive for when discussing new learning. Here’s a summary of the six levels of questioning:


Knowledge, the lowest level, is characterized by simple recall of facts:

-  What’s 9 + 8?

-  What’s 4 x 6?

Comprehension includes compare and contrasting tasks:

-  Which is greater, 301 or 310?

-  What’s the definition of a square?

Application involves solving problems and using knowledge:

-  If there are 7 days in a week, how many days are there in 3 weeks?

-  It’s 100 miles to grandma’s house. If we drive at 50 MPH, how long will

it take to get there?

-  Do you have enough money in your hand to buy a 50 cent cookie?


Analysis asks students to look for patterns and organize parts.

-  5,  10,  15,  20,  ??, 30,  ??

-  3 x 37 = 111,  6 x 37 = 222,  9 x 37 = 333,  12 x 37 = ???

Synthesis is where new learning takes place, connecting existing knowledge and ideas to formulate new ones and to bring together knowledge and facts from different areas.

-  If  6 + 8 = 14, what would 8 + 6 equal?

-  If the formula for the area of a square is L x W, what would the formula

be for the area of a triangle, which is one half of a square?

- What would happen if…?

Evaluation is assessing what has been presented, including one’s own ideas.

This is the highest level of thinking. It asks the child to think about and explain the reasoning that led to an answer. The most important math questions you can ask your child elicit this thinking:

-  Why do you think so?

-  Tell me how you did that.

-  Is there a better way to solve the problem?

The taxonomy, of course, applies across all subjects. A great opportunity for parents to use it is reading aloud to your children. Make sure some of the questions you ask as you read fall into the last three levels. For example: What might have happened if Alice had caught the White Rabbit? Why do you think the Red Queen is always so angry? If you were Alice, what would you do? These kinds of questions will stimulate thinking, and that’s what we want.  Want more? Click here.

Dave Gardner

Mathematician in Residence


What We Mean by Sustainable Math Culture

Published on November 1 2011

As we at EIM talk with our different constituencies, there are questions that come up regularly. These are thoughtful questions from people interested in our success. Here’s an interesting question that came up recently; I hope you will take the time to respond and to ask additional questions.

Q: According to your website, your mission is “to build a sustainable math culture in elementary school communities.” Why “communities” instead of “schools”? And just what is a “sustainable math culture”?

Math, like reading, has to be more than just school-focused. For many years, reading has involved both families and the wider community in many ways, which is what we want to do with math. As examples, families read to their children regularly, libraries sponsor summer reading programs, schools hold family reading nights, businesses offer reading incentives, and there’s a nationwide yearly reading contest called Global Reading Challenge. Our mission is to do the same for math, take it beyond the classroom and the school and create an appreciation for the importance and enjoyment of math community-wide.

As for “building a sustainable math culture,” I’m going to draw an analogy once again to the culture of literacy that exists in virtually every elementary school in the nation. This culture comprises many elements. A partial list would start with the school library and would include reading assemblies, book giveaways, book fairs, extensive professional development, family reading nights, reading aloud to students and summer reading programs. Reading is far more than just what goes on in the classroom between 10:00 AM and lunch.

We would like to start a math movement that, over time, will develop into a comparable math culture in schools across the country. We’d like to see math assemblies, math tournaments, more math professional development, family math nights, summer math camps (EIM offers four summer math camps), and a math resource center in every school. The key to this is the word “sustainable.” EIM can work with schools to help develop a math culture but in order for it to grow and stick, it has to be engrained into everyday activities and unique to their students and educators so it will continue long after EIM the formal partnership has concluded.

Starting the Math Period

Published on October 25 2011

Teachers, here’s something to consider doing at the start of your math period: open with some mental math and/or a quick math game. Doing this accomplishes three things. First, because kids enjoy doing mental math and math games, you’ll get their attention quickly and easily without having to clap, whistle, ring a bell or any of the other things we routinely do. Just start saying the problem in a normal tone of voice and students quickly settle down. Second, you’ll have them thinking and doing math from the beginning of the period, and doing it in a way they enjoy. Finally, when the lesson starts, they’ll be in a math-positive frame of mind.

I sometimes hear objections to doing this. “I can’t do mental math!” is one. Nonsense! All it takes is a little practice. After a while, you’ll be able to reel off strings of numbers and operations and keep track of the answer. And when you do make a mistake (I certainly have!), it’s good for the students to catch and correct it.

“I don’t know any good, quick games to play!” is another objection. The solution to that is easy: visit our fun math games website and you’ll find several math games for grades K-5. These games are easy to learn, quick to play and require no elaborate set-up or clean-up.

“There’s just not enough time!” is a third objection, and this one is harder to argue with—there is simply not enough time in the day or days in the week to accomplish everything you need to do. But, as I noted above, starting with four or five minutes of mental math or with a quick game brings the class together quickly and productively and starts them off with a positive math attitude. You haven’t lost any time and may even have gained a little.

And, of course, the big benefit: kids are learning that math is fun and challenging, not boring and hard.

Dave Gardner

Mathematician in Residence

MathFest is Coming!

Published on October 19 2011

In 2 weeks the community will be gathering for our 5th annual MathFest at the Rainier Community Center. If you’ve been to MathFest in the past then you know what to expect: math games and math excitement. Last year’s MathFest drew nearly 1,000 folks who spent three hours enjoying Airplane Race, Function Machine, Multiplication Bingo, Mathemagic Mind Reader, Sudoku, Shut the Box, Puzzles and much, much more, including a raffle and prizes. These games and others, as well as some new ones, are a great way for parents, families and kids to discover the fun in math together. In addition, the MathFest Spirit Award is given annually to the Explorations in Math Partner School with the greatest percentage of staff and students attending.

There’s more! Admission is free and refreshments and a special area for tots are included. Doors open at 5:00pm on Thursday, November 3rd. If you’ve not been to MathFest in the past, make it a point to be there this year - you’ll be glad you did! Given the number of people attending, it’s wise to avoid the registration line by pre-registering.  You can register and learn more here.  We hope to see you at the Fest!

Dave Gardner

Mathematician in Residence

P.S. Congratulations to the 4th grade class at St. George Elementary in south Seattle for correctly answering both puzzles from two weeks ago. You’ll be getting a math game for your classroom! Well done!


Math Story Time

Published on October 10 2011

One of the ways we instill a love of reading in our children is the nighttime ritual of reading to them. Doing this communicates the importance we attach to reading, shows our love of reading, models good reading skills, opens opportunities for questions and discussion, and it increases our children’s vocabularies. There’s a way to capitalize on all this and also include a love of math as one of the reasons for reading aloud: read math-related story books. When you start looking for them, I think you’ll be surprised at how many good books there are, and at all grade levels. Talk to your school librarian or the librarian at your public library branch; they’ll be happy to help you. Meanwhile, here’s a great list to get you started.

“Sugar-Coating” Math?

Published on October 3 2011

As we at EIM talk with our different constituencies, there are questions that come up regularly. These are thoughtful questions from people interested in our success. Here’s an interesting question that came up recently; I hope you will take the time to respond and to ask additional questions.

Q: I know EIM believes that making math fun is key to helping students do better in math. But doesn’t “sugar-coating” math perpetuate the perception that math is hard and boring? After all, we don’t “sugar-coat” reading.

The phrase “sugar-coating” is mistaken, we sugar-coat things that are unpleasant, burdensome or unpalatable. Math is none of these things and part of our mission is to help teachers, families and students understand and appreciate that.

I think we all agree that if we enjoy doing something, we’ll do more of it and, over time, become better at it because of the regular practice. When we make math interesting and enjoyable (“fun” if you will), student perceptions of math begin to change from “hard and boring” to “challenging and fun.” Games are a very effective way of doing this. Students enjoy playing them and they develop math skills doing so.

As for reading, one of the reasons reading scores are so much higher than math scores is that we have made reading fun, not so much with games but in other ways: cut and paste activities, coloring, skits, and making dioramas to name just a few. If reading were taught the way we’ve taught math for so many years, reading scores, too, would be abysmal.

None of this is to deny the fact that math is a rigorous discipline that penalizes carelessness and sloppy thinking. We believe that games are a way of instilling math competence and confidence in our children which will, in turn, lead to the kind of rigor and understanding that math demands.

Dave Gardner

Mathematician in Residence