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

2. Brush your teeth.
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

4. Read a page from your favorite story.
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!

Math Games and Math Anxiety

Parenting is arguably the biggest thing you can do in your life, and guess what? No manual. Everything else has a manual. I just brought home a plant, and it had instructions pasted on the side. I purchased a bookshelf, and there was an entire pamphlet of detailed instructions and pictures on how to put it together – in three different languages!

As a veteran elementary teacher, I have found that many parents are eager to help their children with their school work, but don’t always know what is best or where to begin. What I, also, know is that a parent’s involvement in a child’s education is the single most important factor in that child’s academic success.

Decades of educational research tells us that an involved parent contributes overwhelmingly to his/her child’s grades and test scores, school attendance and quality of homework, positive attitudes and behavior at school, likelihood of graduation, and desire to enroll in higher education.

I bet you’ll never guess which subject raises the greatest consternation in parents – you’re right, math! Math anxiety is rampant in the world, and yet no one comes out of the womb with a stamp on their head that says, “I am math-anxious”.

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.

“You have what you need to help your child with math because:
• You have a great deal of important mathematical knowledge to share.
• Children learn best from the people who most accept and respect them.
• Learning is more lasting when it takes place in the context of familiar home experiences.
• 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.
How can parents foster math skills and mind-sets so that their children are confident mathematicians? Make math games a family ritual!

Games offer a pleasant way for you, as parents, to get involved in your child’s mathematics education. You don’t have to be a math genius to play a game. You don’t have to worry about pushing or pressuring your child. All that you have to do is propose a game to your child and start to play.

Number of the Day

One of my favorite math activities for any age child is Number of the Day.

This is a great activity for anyplace you happen to be! It will give your child lots of computation practice, be a good deal of fun, and everyone (even you) will be forced to “prove” that they are correct!

Let’s say that our “number of the day” is 6. Everyone has to think up one way to make 6. Young children will probably begin with simple addition.

Example: 4 + 2 = 6

Ask your child to “convince you” (prove) that 4 + 2 = 6.

Everyone has to come up with an equation that equals 6, and each one has to be different.

After gaining in confidence, encourage your child to think of 2 different things that equal 6.
Example: 3 + 3 and 5 + 1

Then ask them to find 3 things that equal 6
Example: 1 + 2 + 3 = 6

See how many different ways everyone can find to make the number of the day. Write it all down if pencil and paper are handy.

Depending on your child’s age begin to encourage the use of other operations such as:
• subtraction 9 – 3 = 6
• addition & subtraction 8 – 4 + 2 = 6
• multiplication 3 x 2 = 6
• multiplication & addition 2×2+2 = 6
• division 24 ÷ 4 = 6
• all 4 operations in one equation
(50 ÷ 2) x 3 – 70 + 1 = 6
• coin values – 1 nickel and 1 penny =
6 cents
• fractions 4 ½ + 1 ½ + 6
• decimals 2.4 + 3.6 = 6 or 12 x .5 = 6
• integers – positive 10+negative 4 = 6

Family members can take turns choosing the number of the day. What about the day of the month, someone’s age or weight, number of windows in your home, the sum of your telephone number, etc. Try a variety of numbers, including large ones (such as 555 or 62,437) and small ones (they can be just as challenging as large ones).

Well, you get the idea! Dad might be coming up with 4 x 25 – 80 – 14 = 6!!
Does he have to prove it??!! Absolutely!

Helping Children Master the Basic Number Combinations

Educators generally agree that it is very important that children master the basic number combinations. In fact, the National Research Council (NRC) concluded that attaining computational fluency is an essential aspect of mathematical proficiency.

Children typically progress through four stages when mastering these basic number combinations.

Stage 1
Elena, a Kindergartener, determines the sum of 6+5 by getting out 6 cubes and then adding 5 more and counting the total.

Stage 2
Kevin, a first grader, determines the sum of 6+5 by saying “six” and then extends five fingers (one at a time), and counts “Seven, eight, nine, ten, eleven”.

Stage 3
Theresa, a second grader, tackles 6+5 by mentally reasoning that if 5+5 is 10, and 6 is 1 more than 5, then 6+5 must be 1 more than 10, or 11.

Stage 4
Sam, a third grader, immediately and reliably answers, “Six plus five is eleven”.

How do we, as teachers and parents, best get our children to stage 4 – mastery (efficient, fast, and accurate production of answers)?

Too often it is thought that memorizing basic facts by rote through extensive, time-tested drill and practice is the most efficient way to help children achieve mastery. This approach makes learning the basic number combinations unduly difficult and anxiety-provoking and undermines interest in mathematics and confidence in mathematical ability. Many children give up on learning all the basic combinations. They may appear inattentive or unmotivated or otherwise fail to learn the combinations.

I believe that one of the best ways to get this kind of practice on the road to mastery is with the use of 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.

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 those basic number combinations. Good, child-centered games are designed to take the boredom and frustration out of the repetitive practice necessary for children to master these important math skills and concepts.

Playing math games is even more beneficial than spending the same amount of time drilling basic facts using flash cards. Not only are games a lot more fun, but the potential for learning and reasoning about mathematics is much greater, as well.

The following is a simple game, but one of my favorites, for mastering the basic number combinations:

Turn Over 5

What you need:
2 players
cards 0 – 5, 4 of each

The object of this Concentration-type game is to capture pairs of cards that add up to 5.

Mix up the cards and lay them face down in four rows of six. Players take turns by choosing two cards to turn over, trying to find a combination that adds up to 5. If they find one, they keep (capture) that pair. If they do not, they turn the two cards back over for the next player. When all matches have been made, the player with the most cards wins the game.

Variation: This game can be made more challenging by using higher cards and a different sum, such as 6, 7, 8, 9, 10, 15, 15, etc.

There are many such games that teachers and parents can play with their students/children which will happily put them on the road to mastery.

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?
• How is your answer like or different from another student’s?
• Can you repeat your classmate’s ideas in your own words?
• 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
• be alert and curious
• 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.

Helping Children Learn Mathematics – Count Collections!

As a veteran elementary teacher, I have found that most kindergarten children and many first graders come to school able to rote count to ten, or twenty, or higher. Even though the counting sequence seems to be in place, these children often have difficulty counting objects accurately past five or ten.

After more than 20 years of being an elementary math specialist, I have found that the most important thing parents can do to support their children’s mathematical growth at this age is to count things.