## Taking Time to Understand Time

For years, teachers have observed students’ frustration as they grappled with learning to read an analog clock (as opposed to a digital clock). I remember being frustrated myself and not fully understanding why learning to tell time was so difficult for my students and wondering how to help.

I began to realize that there are two aspects of time that have to be distinguished in teaching time: firstly, one must try to develop a concept of time in a child, and secondly, one must teach the child to “tell the time” (teaching clock time).

Teachers of young children generally concur that their students learn mathematical concepts best when they construct understanding through concrete experiences. When we remember that time can be neither seen nor touched but experienced and measured only indirectly with such tools as clock, we can begin to understand why time-related concepts are difficult for our students to learn.

From the body of research available, as well as from our own firsthand teaching experiences, we know that everything to do with understanding and using time concepts develops rather late. I will go so far as to say that most children do not really fully understand the intricacies of telling time until about the third or fourth grade.

What usually happens in the classroom is that developing the concepts of time is skipped. In keeping with the admonition that children must actively develop concepts of time, I have included a few of the math activities I began to use in my classroom:

Time Intervals

Just How Long Is a Minute?

Have your students close their eyes and you time one minute. Have them keep their eyes closed and put up their hand when they think one minute has passed. Call time at the end of the minute. Now try it again. The more you do it, the better sense of a minute they will have.

Also try some of the following activities:

How many times in one minute do you think (make an estimate) you can:

1. Sing “Happy Birthday”? Estimate ____ Actual _____

2. Touch your toes? Estimate ______ Actual ______

3. Hop on one foot? Estimate ______ Actual ___

4. Do jumping jacks? Estimate ______ Actual ______

5. Write your first name? Estimate ______ Actual ______

6. Run around the basketball court? Estimate ______ Actual ______

7. Draw stars? Estimate ______ Actual _______

8. Recite the alphabet? Estimate ______ Actual _______

9. Snap your fingers? Estimate ______ Actual _______

10. How high do you think you can count in one minute?
Estimate _______ How high did you go? ________

More or Less Than a Minute? Homework

Here are some things you do everyday. For each one, guess whether you think it will take more than one minute or less than a minute to do it. Now try each thing while someone keeps time.

1. Put on your socks and shoes.
Guess ______________ It really took _____________
more or less than 1 minute / more or less than 1 minute

Guess ______________ It really took _____________
more or less than 1 minute / more or less than 1 minute

3. Eat a banana.
Guess ______________ It really took _____________
more or less than 1 minute / more or less than 1 minute

Guess ______________ It really took _____________
more or less than 1 minute / more or less than 1 minute

5. Pledge Allegiance to the Flag.
Guess ______________ It really took _____________
more or less than 1 minute / more or less than 1 minute

Now make up a short list of things you think will take about one minute, and give them a try.

There are some great games which help children understand time!

## Why Use a Math Game?

I am a believer in the many benefits of using math games in the classroom. Besides the fact that games can motivate and engage students in thinking about and applying concepts and skills, math games can foster mathematical communication as students explain and justify their moves to one another.

Games afford students an opportunity to communicate their ideas and justify their thinking. In using games, the teacher plays an important role in encouraging students to explain their thinking and in keeping students focused on mathematical ideas. Requiring students to explain and justify their moves during a sample round of the game played as a whole class models the type of thinking and communicating that is important for students to use later when they play the game in pairs.

The ability to pose questions that elicit, extend, and challenge students’ thinking is essential to creating a classroom environment in which intellectual risks, sense making, and deep understanding are expected. In daily lessons, teachers must make on-the-spot decisions about which points of the mathematical conversation to pick up on and which to let go, and when to let students struggle with an issue and when to give direction.

While the students are playing a game, it is the teacher’s responsibility to ask probing questions, such as:
• What card do you need?
• Which cards would not be helpful?
• Prove to me that a ____ is what you need.
• Why do you think that?
• How did you know to try that strategy?
• How do you know you have an answer?
• Will this work with every number? Every similar situation?
• When will this strategy not work? Can you give a counterexample?
• Who has a different strategy?
• Do you agree or disagree with your classmate’s idea? Why?
• What can you do to help yourself? Use your fingers to count? Count the dots on the dice or cards? Use manipulatives to figure it out? Draw a picture? Start with something you already know?

The power of questioning is in the answering. As teachers, we not only need to ask good questions to get good answers but need to ask good questions to promote the thinking required to give good answers.

While playing a math game, students’ abilities to learn from, and work with, others should expand. They should become more skilled in speaking to one another and in convincing or questioning their peers. The discourse should focus on making sense of mathematical ideas and on using mathematical ideas effectively in modeling and solving problems. When thinking is discussed regularly in the classroom, students feel comfortable describing their thinking, even if their ideas are different from the ideas of their peers. Discourse is not a goal in itself; rather, the value of mathematical discussions should be judged by whether students are learning important mathematics as they participate in them.

Effective teaching involves observing students, listening carefully to their ideas and explanations, and using the information to make instructional decisions. Through their teaching, teachers can also continue to deepen their understanding of the mathematics they teach, by learning with and from their students and then reflecting on that learning. The ability to reflect on and refine mathematical understanding as well as instructional practice is essential to achieving the vision of an effective mathematics classroom, whatever the grade.

## Engage and Motivate Your Students in Math

As a veteran elementary teacher and math specialist, I am absolutely sure that what is important in helping children develop a positive attitude toward math and become confident mathematicians is the power of an effective teacher. Finding those engaging “hooks” to draw children into the math is the challenge.

Many students feel like math is just memorizing facts and processes and then repeating it on a test. Certainly doable, but not very entertaining. Not much real learning is going on.

Overcoming math terror is a job teachers face, and it’s true more often than we would like. How can teachers get students past that terror and into a love of mathematics. What might that “hook” be?

Effective teachers seem to rely on proven approaches, including high expectations, engagement, motivation, and support. All are worthy, but I would like to speak to engagement and motivation.

Research has demonstrated that students learn more if they are actively engaged with the math they are studying. Constance Kamii, a world renowned expert on how children learn math puts it this way, “Children who are mentally active develop faster than those who are passive.”

Active learning is, in short, anything that students do in a classroom other than merely listen to a teacher’s lecture. There are several ways of doing this. Playing math games is a particularly useful one.

Games can provide an atmosphere where children are encouraged and motivated to:
• share their ideas with other children
• come up with interesting ideas, problems, and questions
• have confidence in their abilities to figure out things for themselves
• speak their minds with confidence

Games are engaging (maintain interest); dittos or workbook pages rarely are. In the process of playing the game, students may develop initiative, interest, curiosity, resourcefulness, independence, and responsibility. Would that happen with a ditto or workbook page?

Children learn math best when they participate in games that are relevant to them, hold their attention, and require them to “make meaning” for themselves.

Teaching methods that stress rote memorization of basic number facts or algorithmic procedures are usually boring and do not require learners to participate actively in thought and reflection. In other words, they are not engaged or motivated.

Engage and motivate your students in any of the many math games that I have tried and loved.

## Make March Math Awareness Month!

The main goal of Math Awareness Month should be to “demystify” the subject, alleviate the “math anxiety” some students experience, and show students that math can be interesting, challenging, engaging, and fun.

How you, as teachers, encourage and promote your student’s math learning, from preschool to high school, can be pivotal to their attitude toward math and their achievement in this subject area. In an effort to deliver the fundamentals of math in new and interesting ways, teachers should organize a fun-filled month of educational math activities.

Notice math in the world. You can help your students see the usefulness of math by pointing it out wherever you see it. Math is a part of everyday life. Students need to see that math is practical and useful. The more closely you align your teaching with the real-life activities of your students, the more learning will resonate with them. Math is one of the easiest subjects to connect to real-life activities.

Mary Ellen Bafumo in her article Making Math Meaningful suggests trying the following:

“Distribute empty cereal boxes to small groups of students. Practice the four operations via word problems built around preparing a class breakfast. Students use portion info on the side of the box to complete math examples. How many boxes are needed to feed the class? What is the cost per serving? How many gallons of milk are needed? The class votes, via a bar graph with each cereal represented, about which to serve in class. Students measure cereal and milk servings and enjoy!

Distribute flyers from office stores. Pairs of students “shop” for a complete computer station for home. They figure cost, tax and shipping, then respond to word problems. On a \$150 monthly budget, how long will it take to pay for the equipment? If you pay off the balance in three, four or five payments, how much is each installment? Students then develop a word problem structured around the task to share with the class.

Distribute travel ads. Small groups of students plan a dream vacation. They calculate transportation, accommodations, meals and incidentals, then multiply by their group members. Ask your class the following questions. If the PTA provides \$2,500 for the trip, how much will each group member have to raise? If airfare is donated, how much will the trip cost, etc.?

Create scenarios based on the interests of your students. Use advertisements (movies, video games, cds, bicycles, etc.) that spark their enthusiasm and watch math take on new meaning.”

Another great thing to try is math games. I’ve been teaching math to children for many years, and I’ve found that math games are, from a teacher’s point of view, wonderfully useful. Math games put children in exactly the right frame of mind for learning. Children are normally very eager to play games. They relax when they play, and they concentrate. They don’t mind repeating certain facts or procedures over and over. Games incorporate the ways children best learn mathematics: through the use of physical manipulatives within the context of developmentally appropriate practice – games require active involvement.

Children throw themselves into playing games the way they never throw themselves into filling out workbook pages or dittos. And games can, if you select the right ones, help children learn almost everything they need to master in elementary math. Good, child-centered games are designed to take the boredom and frustration out of the repetitive practice necessary for children to master important math skills and concepts.

Games teach or reinforce many of the skills that a formal curriculum teaches, plus a skill that formal learning sometimes, mistakenly, leaves out – the skill of having fun with math, of thinking hard and enjoying it.

Try a math game in March! Need some ideas at your grade level?

## Math Games Effectively Meet Math Standards

Your state’s mathematics standards are intended as a statement of what students should learn, or what they should have accomplished, at particular stages of their schooling. The goal of every state’s math standards is to engage students in meaningful mathematical problem-solving experiences, build math knowledge and skills, increase students’ ability to communicate mathematically, and increase their desire to learn mathematics. Those are the goals for math games, too!

Specific content knowledge will vary according to the game students play and the connection to school-day learning and the state standards. A major goal for students in the elementary grades is to develop an understanding of the properties of and the relationships among numbers. One of the very effective ways teachers can reinforce the development and practice of number concepts, logical reasoning, and mathematical communication is by using math games. They are great for targeted practice on whatever standard the children need to meet.

You will meet significantly more of your state’s grade- level mathematics standards by having your children play a game than will have been met by having them complete a ditto or a workbook page.

At all my teacher trainings, I begin by giving the teachers a quiz using a ditto with many three-digit addition problems. We then proceed to look at the mathematics standards, and the teachers decide which standards (or parts of each standard) were met by doing the ditto.

We then play a three-digit addition game, and, again, look at the standards. The teachers decide which standards were met by playing the game. Here are the results:

Standards met when doing the ditto:

Number Sense

1.1 …write whole numbers to 1,000…

2.2 Find the sum… of two whole numbers up to three digits long.

Mathematical Reasoning

2.2 Make precise calculations…

Standards met when playing the game:

Number Sense

1.1 …write whole numbers to 1,000 and identify the place value
for each digit.

1.3 …compare whole numbers to 1,000…

2.2 Find the sum … of two whole numbers up to three digits long.

Mathematical Reasoning

1.1 Determine the approach … and strategies to be used.

2.1 Defend the reasoning used and justify the procedures selected.

2.2 Make precise calculations…

As you can see, not only did we meet more standards by playing the game, but many of the standards were met more fully! Many teachers are surprised at this result, but once they begin to use games in their classrooms to help their students learn and reinforce math skills, they are convinced.

## Fractions Activity and Game

In 2006, the National Math Panel reported that knowledge of fractions is the most important foundational skill for algebra that is not developed among American students.

Research shows that fractions are one of the most difficult topics for students to understand in elementary school. I think the problem lies in the fact that children are expected to be passive receivers of information rather than be actively involved with the subject matter.

CGI (Cognitively Guided Instruction) has been stressing for many years that the best way to help children really understand fractions is to begin with “fair shares”.

Start with situations of 2 or 4 children, as children’s earliest partitions are based on halving:
4 children share 4 cookies so that each child gets the same amount.
4 children want to share 10 brownies so that each child gets the
same amount.
4 child want to share 22 apples so that each child gets the same
amount.

Move to situations with more sharers:
3 children want to share 7 candy bars.
6 children have ordered blueberry pancakes at a restaurant. The
waiter brings 8 pancakes to their table. If the children share the
pancakes evenly, how much can each child have?
Matthew has 13 licorice sticks. He wants to share them with 8
friends.
20 friends are sharing eight cakes.

Ask your child or your students to solve the problems using a strategy that makes sense to them. Strategy is the primary dimension of development because student-generated strategies can (and I believe should) serve as the foundation for mathematics instruction. A focus on student-generated strategies allows a teacher or a parent to begin with, and build on, what children already know, and it allows children to participate in instruction by making contributions that are personally meaningful.

Give children pencils and paper and access to any kind of manipulative they find helpful and allow them to work out the problem by themselves.

Once the task is completed, children need to be able to demonstrate to each other what they did and the answer that was found. The more students are encouraged to contribute the intact products of their own thinking to class discussions, the more likely they are to identify themselves as understanding math – no matter the level of the thinking.

The key in fraction instruction is to pose tasks that will elicit a variety of strategies and representations. Equal-sharing tasks are not the only problems that can do that, but many teachers, like myself, have found them to be a definite source of variety in thinking. Children learn from each other, and the teacher begins to get a picture of what each child knows.

Another great way to help your child or your students to understand fractions is to play a fraction game. I have found that Fraction War can be highly effective. The first level begins simply, and it is probably best to start here, even with older children. Once you are sure they understand this concept, move to the next concept level.

Fraction War

Materials:
 One deck of cards
 Fraction War Game Board (following)

Game:
Players draw cards and create a fraction. The player with the fraction with the greatest value wins a point for that round. The player with the most points when all the cards have been used is the winner.

Variations:

Concept 1:
Each player finds and places a one in the numerator position on his/her game board. This card remains in place until the end of the game. Each player draws a card and places it in the denominator position. The player with the greatest fraction wins the point. Play continues until all cards have been used.

Concept 2:
Place a one in the denominator position and play as above.

Concept 3:
Decide on a number between 2 and 10. Each player places that number in the denominator position. Play as above.

Concept 4:
Place the same number in the numerator position. Play as above.

Concept 5:
Each player draws 2 cards. The first is the denominator, the second is the numerator. Play as above.

Fraction War Game Board
Player #1 Player #2

_______________________ ______________________

## Teaching Math at Home

Many parents don’t feel comfortable with math, or they assume it takes special expertise to teach it. Remarks like “I never was any good at math” or “How can I help my child with math? I can’t even balance my checkbook!” are common. However, even parents who feel this way use mathematics all the time. They hand out lunch money, cut sandwiches into quarters, calculate how much paint or wall paper they need to buy, estimate how much a trip will cost, read and interpret graphs, talk about the probability of rain, and decide that it’s time to fill the gas tank. Some of them knit, piece quilts, measure wood for cutting, decide how many cups of spaghetti sauce they need to make for 6 people, and use metric tools to work on their cars. The list goes on and on.

Many adults also feel they aren’t doing things the right way, that they aren’t really using mathematics, because their approaches, even though they work, are not the methods they learned in school. There are, in fact, many ways to do mathematics, and more than one can be right. People who devise their own strategies for finding answers to mathematical questions, far from being mathematically incompetent, are often excellent independent problem solvers. They are using mathematics creatively.

1. You have a great deal of important mathematical knowledge to share.
2. Children learn best from the people who most accept and respect them.
3. Learning is more lasting when it takes place in the context of familiar home experiences.
4. Children must see that math is not just a subject studied in school but is used constantly in everyday family life.

The home is an ideal place in which to learn mathematics because the problems encountered there are real, not just paragraphs in textbooks.

Making Math Part of Your Family’s Life

It’s common knowledge that young children whose parents read to them have a tremendous advantage in school. But did you know that you can also help your child learn mathematics by doing and supporting math at home?

Today mathematics is more critical to school success than ever before. Modern occupations now require a firm foundation in mathematics – and that’s true for almost any type of job your child will consider in the future.

How you encourage and promote your child’s math learning, from preschool to high school, can be pivotal to their attitude toward math and their achievement in this subject area. Children are taught math in school, but research shows that families are an essential part of this learning process. In other words, by doing math with your child and supporting math learning at home, you can make a great difference.

There are many ways to make math part of your family’s life. Consider the following checklist of key ideas:

• Always talk about math in positive ways. Regardless of your own math background, let your child know that learning math is very important. Communicating a positive, can-do attitude about math is the single most important way for you to ensure that your child is successful in math. Never tell your child that math is too hard or that you hated it or weren’t good at it when you were in school.

• Make math an everyday part of your family. Find math at home. Spend time with your child on simple board games, puzzles, and activities that involve math. Involve your child in activities like shopping, cooking, and home fix-it projects to show them that math is practical and useful.

• Notice math in the world. You can help your child see the usefulness of math by pointing it out wherever you see it – not just in your home. What shape is that building? How many more miles before we get there? How many glasses of milk are in a carton? How much will you save if you buy a combo meal at McDonald’s?

## Parents as Math Teachers

Parents, Children, and Math Games

Math games for kids and families are the perfect way to reinforce and extend the skills children learn at school. They can also be a dynamic, motivating mathematics instructional tool for homeschoolers. They are one of the most effective ways that parents can develop their child’s math skills without lecturing or applying pressure. When studying math, there’s an element of repetition that’s an important part of learning new concepts and developing automatic recall of math facts. Number facts (remember those times tables?) can be boring and tedious to learn and practice. A game can generate an enormous amount of practice – practice that does not have kids complaining about how much work they are having to do. What better way can there be than an interesting game as a way of mastering them?

Games are fun and create a context for developing children’s mathematical reasoning. Through playing and analyzing games, children also gain computational fluency by describing more efficient strategies and discussing relationships among numbers.

Games offer a pleasant way for parents of homeschoolers to energize the memorization of basic math skills.

Here are some other benefits of using math games in the homeschooling context:

• Offers many opportunities for you to discover your child’s strengths and weaknesses
• Meets the needs of diverse learners, such as English-language learners and special needs children
• Supports concept development in math
• Encourages mathematical reasoning
• Engaging (maintains interest)
• Repeatable (reuse often & sustain involvement)
• Open-ended (allows for multiple approaches & solutions)
• Easy to prepare
• Easy to vary for extended use such as making any game harder, easier, or just meeting the needs of your child
• Improves basic skills, i.e., addition and multiplication facts
• Enhances number and operation sense
• Encourages strategic thinking
• Promotes mathematical communication
• Promotes positive attitudes toward math

## Teachers Search for Ways to Energize Math

As an independent, elementary mathematics consultant, I work with elementary teachers all the time as they search for ways to motivate and energize their students in math. I just read the following article about Indiana elementary teachers on that very quest. Take a look at how a grant brought together ISU professors and Vincennes teachers.

One of the instructional strategies they are trying are math games, an effective, hands-on way to teach math concepts. No matter which textbook your district uses, games can easily be incorporated into instruction. Some textbook companies are “seeing the light” and have begun to implement games as a part of each unit.

Even if your textbook does not incorporate games, identify a skill need almost all your students have, and give a game a try. I guarantee it will be more of a learning experience for the students and more informative for you of what your students know and can do than a workbook page.

Here are some of the many benefits of using math games in the classroom:

• Meets your state’s elementary mathematics standards
• Easily linked to any mathematics textbook
• Offers multiple assessment opportunities
• Meets the needs of diverse learners
• Supports concept development in math
• Encourages mathematical reasoning
• Engaging (maintains interest)
• Repeatable (reuse often & sustain involvement)
• Open-Ended (allows for multiple approaches & solutions)
• Easy to prepare
• Easy to vary for extended use & differentiated instruction
• Improves basic skills
• Enhances number and operation sense
• Encourages strategic thinking
• Promotes mathematical communication
• Promotes positive attitudes toward math
• Encourages parent involvement