Explorations In Math

This isn’t a rhetorical question but, rather, the title of a ‘must read’ math book, whether you’re a parent (or other caregiver) or working in a classroom with students in math. The first thing you need to know about the book is that it is not a dry, scholarly work full of theories and equations. Quite the contrary. What’s Math Got to do with it is a thoroughly enjoyable read. The author, Stanford mathematics education professor Jo Boaler, followed the progress of middle- and high-school students over a number of years, observing teachers who could engage their students in math and just how they did it. She also observed classes where the students were not engaged and in her book points out the differences between those who do engage their classes and those who do not.

What she found should be required reading for all of us. A partial list of chapters is indicative of how much ground she covers: What’s Going Wrong in Classrooms? Stuck in the Slow Lane. Paying the Price for Sugar and Spice. Giving Children the Best Mathematical Start.

She provides many enlightening examples of how children think about math and there are math puzzles and problems (and solutions!) for the reader to solve. For example:

Given a 5-liter jar and a 3-liter jar and an unlimited supply of water, how can you measure out exactly 4 liters of water?

Or this one:

Race to Twenty is a game for two people. Play starts at 0, Player 1 adds either 1 or 2 to 0 and announces the sum. Player 2 adds either 1 or 2 to that sum and announces the new sum. Play continues this way. Whoever gets to 20 is the winner.

Check out the book here.

BTW, don’t be put off by the fact that she followed middle- and high-schoolers. Just about everything in her book applies equally to elementary students as well.

Dave Gardner, Mathematician in Residence

This week’s guest post is from Laura Larson. Laura serves as Explorations in Math’s Board Chair and founded Explorations in Math around a kitchen table 9 years ago.  If you would like write a guest post, please contact Dave Gardner.

Last week I saw an ad for tutoring services that made me laugh out loud.  A teenage son is doing homework and we hear him calling loudly, “Mom, I need help with my math homework”.  Panic stricken, she turns around, runs out of the house and down the street…until at last she reaches the tutoring center, where help is found.  I think that ad tapped into a (seemingly) fundamental parental truth: ask me about anything you want, just don’t ask me to help with your math homework.

When my children were quite young, I read an article discussing an experiment.  Scientists had created a glass floor for babies to crawl across, parent waiting on the other side.  They discovered that if parents were encouraging and confident, the babies would crawl across. If, on the other hand, the parents expressed worry and concern, the babies refused to cross and started to cry.

As a parent you may or may not have the math skills to help your child with their homework, be it 3^rd grade or advanced algebra.  But what you do have is the ability to confidently express your belief that your child will be successful.  Our children look to us, more than we realize sometimes, for clues on what to expect.  “Is this something I can do, or not,” they wonder.  Is math for some people, but maybe not me?   Maybe you’re not sure.

But one thing we do know is everyone can do math.  Difficulties and even getting things wrong the first time, are the mental challenges upon which all learning is built.  Your voice, your assurance that yes, this problem has a solution and one way or the other, success will be theirs, can make the difference between a child who gives up on their chances and one who’ll stop at nothing.  If persistence is the key to success, our children must know and believe that they are capable of reaching the other side.  Remind them that the harder the problem, the better they’ll feel once they find the solution.  Sometimes the best help of all is your unwavering belief in them.  Math matters to every child’s future.  And your belief in their ability to succeed at math, matters too.

We all know there are scores, hundreds, thousands of math-related websites out there. You can spend hours searching and then exploring different sites. Here’s a shortcut: periodically, I’ll be sharing some good websites with you including a summary of the content, whether or not it has ads, whether or not I think it’s worthwhile and, of course, the URL. I have checked them all recently. Meanwhile, here’s the first go-round:

http://www.mathsisfun.com/

This site lives up to its name: it offers math in fun ways for students, teachers and families. It is aesthetically inviting and user-friendly. You can choose from a variety of math topics from the “Math Menu,” such as Numbers, Algebra or Geometry. After selecting Geometry, for instance, you can choose from a well laid-out, long list of sub-headings (Angles, Plane & Solid Geometry and Using Drafting Tools), which offer explanations, diagrams, formulas, etc. The site includes a comprehensive “Illustrated Math Dictionary,” lots of “Math Puzzles” and cool interactive, online “Math Games,” as well as “Math Worksheets” for students. You can also participate in the “Math Forum,” where users have posted/responded to math questions, math resources, puzzles/games and even age appropriate jokes. This site is a very engaging educational resource for teachers and students alike. You will have to sign up for the Math Forum but that’s quick and simple. The site carries no ads.

http://www.figurethis.org/index.html

Wonderfully accessible resources for teachers, students and families:  clicking on the math index directs the user to math content, where a number of math problems and activities can be accessed.  This website is also helpful because it cites the specific content/math reasoning used for each problem. It’s bright and colorful, and greatly appeals to students. It’s easy to navigate and all information is easily accessible and is completely free of charge.  Additionally, this website can be accessed entirely in Spanish. There is also a “Family Corner” with tips for parents and challenges for the whole family. The site is geared to upper elementary and middle school. This is an NCTM (National Council of Teachers of Mathematics) site. It is ad-free.

You can find additional websites that our team recommends here.  If you have some websites to recommend, please leave that information in the “comments” box for others to read. Thanks!

Dave Gardner

Mathematician in Residenc

One of my colleagues in the office the other day coined the term “mathitude” to indicate a positive attitude toward math, whether in children or adults. There was a perfect example of that last week in a first grade class at an EIM member school. The students were playing a game and at one point a girl, with a big smile, said to her partner, “I love math!” and her partner, also smiling, nodded in agreement. Clearly, the teacher in this classroom is doing far more than just instilling math competence in her students; she is also instilling math confidence by making sure her students are engaged in math and enjoying the experience.

Contrast that with this situation. We were having dinner at some friends’ the other night. After dinner, we decided to play Apples to Apples®, a game EIM uses and recommends. I’d played it before and told the others that it uses the same kind of mental skills that math requires: flexible thinking and making connections. At the word “math,” one woman blanched and sputtered, “I hate math! Math scares me!”

We need to ask ourselves: Do we want our children, our students, to grow up with this kind of paralyzing feeling about math? Or do we want to instill in them the “mathitude” shown by the first grade girls above? Another first grade boy who, as I was saying my goodbyes to the class after a fun session last year, hastily scrawled a sign and held it up for me to read: “Math Rocks!” I took a photo of him which you can see below.

Let’s hear it for MATHITUDE!

Our children and our students can be easily intimidated by math and can start developing math anxiety, even math phobia, at a young age. There’s no question that this will seriously hinder their ability to learn, let alone enjoy, math as they go through the grades. One of the things we can do, as parents and as teachers, is to build in safety nets when we work with our children and students. Here are some ideas, things I’ve found to be effective:

• If a child gives a wrong answer, I tell her that that’s OK—I like wrong answers. They’re not as good as correct answers, but here’s why wrong answers are also good: They tell me three good things about her: she’s listening, she’ s thinking and she’s trying, and what more can we ask? Any child who listens, thinks and tries is going to succeed.

• In a similar vein, when a child gives a wrong answer I tell him that it’s OK to be wrong. All of us are wrong sometimes but we learn from our mistakes. I give the example of falling off a bike when learning to ride. Every time you fall off a bike, you’ve done something “wrong,” but you learn from that, get back on and do better.

• Sometimes I’ll ask a question and the student won’t respond. Usually that means he doesn’t know but doesn’t want to say so for fear of appearing “stupid.” In this case, after a few seconds of silence, I simply say, “You know, it’s OK not to know. Nobody knows everything and we’re all learning.”

• Don’t tell your child or your students that math is “hard.” A much better word is “challenging.” In a student’s mind (and for a lot of adults), “hard” is too closely associated with boring and failure. “Challenging,” on the other hand, can be presented as a way of the child testing him or herself. “Challenging” carries it with the possibility of being interesting, rewarding, even fun.

In short, let’s do all we can to instill in our kids the idea that math is, indeed interesting, rewarding, even fun. And, above all, doable.

Dave Gardner

Mathematician in Residence

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? 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 4^th 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

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

821
-128
693

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

963
-369
594

Repeat with new number 594 – so subtract 459 from 954

954
-459
495

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

Mark Taylor

Teacher

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

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:

REVIEW (OLD LEARNING)

• 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?

NEW LEARNING

• 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