Helping Your Child with (Mental) Math

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.

The following are some important things you should know and do:

1. Problems can be solved in different ways. While some problems in math may have only one solution, there may be many ways to get the right answer. And remember, the way you solve a problem may not be the way your child solves the very same problem. Learning math is not only finding the correct answer, it’s also a process of solving problems and applying what you have learned to new problems. If their way of solving the problem gets the job done, let them give it a try.

2. Wrong answers can help! While accuracy is always important, a wrong answer could help you and your child discover what your child may not understand. The wrong answer tells you to look further, to ask questions, and to see what the wrong answer is saying about the child’s understanding. It is highly likely that when you studied math, you were expected to complete lots of problems using one, memorized method to do them quickly. Today, the focus is less on the quantity of memorized problems and memorized methods and more on understanding the concepts and applying thinking skills to arrive at an answer.

3. Doing math in your head is important. Have you ever noticed that today very few people take their pencil and paper out to solve problems in the grocery store, restaurant, department store, or in the office? Instead, most people estimate in their heads, or use calculators or computers. Using calculators and computers demands that people put in the correct information and that they know if the answers are reasonable. Usually people look at the answer to determine if it makes sense, applying the math in their head (mental math) to the problem. This, then, is the reason mental math is so important to our children as they enter the 21st century. Using mental math can make children become stronger in everyday math skills.

In terms of mental math, here are some questions you might ask your 3rd through 6th graders (no pencils and paper allowed):

Start off easy with –

98 + 47

51 + 99

146 – 101

5 x 99

150 + 199

137 – 99

99 + 49

4 x 24

58 + 16

65 – 19

Then increase the level of difficulty with –

You buy an $80. dress which has been reduced 20%. How much did it cost?

What is 3/8’s of 40?

6 ½ – 2 ¼ =

What is 75% of 32?

What is 10 squared divided by 5?

You get the idea. Now think of real-life questions that you face every day.

Teachers Taking Time for Math Games

As an elementary school teacher, you probably feel like you don’t have enough time to teach all of your content within the course of a school year. Why on earth would you ever want to add more material in the form of math games when you can’t seem to finish your assigned math textbook? Turns out that making time to incorporate math games in your classroom can lead to rich results.

One of the most immediate benefits of using math games is increasing student engagement. Games are engaging and maintain interest. Dittos or workbook pages rarely are. 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. Research has demonstrated that students learn more if they are actively engaged with the math they are studying.

Contrast this with the reaction that many students have toward the textbook: either a lack of interest or an assumption that the assigned math/problems will be too difficult.

Incorporating math games also allows you to differentiate instruction. Using math games which better match students’ abilities can help them build content knowledge and interact more successfully with the required text.

Because math games require active involvement, use concrete objects and manipulatives, and are hands-on, they are ideal for all learners, particulary English language learners. Games provide opportunities for children to work in small groups, practice teamwork, cooperation, and effective communication. Children learn from each other as they talk, share, and reflect throughout game times. Language acquisition is meaningful and understandable.

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.

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 skills 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 to you of what your students know and can do than a workbook page.

Memorizing the Basic Facts with Math Games

Frank L. Palaia, PhD, is a science teacher in the Lee County School District and at Edison State College. As a guest columnist for the News-Press.com of Ft. Myers, Florida, he had this to say about students in his classes, “Most students today have not memorized basic math facts in elementary and middle school. Each year there will be otherwise intelligent junior or senior students in my high-school classes who asks a question like, “What is eight times seven?”

As an elementary math specialist, I see that children no longer memorize their addition facts or multiplication tables. With the math curriculum as extensive as it is, teachers cannot afford to take the time to ensure that students learn the basic facts (sad, but true).

Parents are partners in the process, and can offer greater opportunities for their child to succeed in math if they support the learning of the basics at home. Math games fit the bill wonderfully!

Math games for kids and families are the perfect way to reinforce and extend the skills children learn at school. 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 (I’m sure you remember memorizing 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.

First graders and second graders need to have the addition facts to 10 in long-term memory. When they hear 6+4, they immediately know (without counting fingers) that the answer is 10. Using fingers to count is a good, early strategy but with practice, those facts should be automatic.

Third graders and fourth graders need to have all of the multiplication facts to automaticity.

Methods such as flash cards, dittos, and workbook pages stress rote memorization of basic number facts and are usually boring and do not require learners to participate actively in thought and reflection. They do not go easily or quickly into long-term memory.

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

Multiplication Games and Activities

Traditionally, instruction in multiplication has focused on learning the multiplication facts using flash cards, dittos, workbook pages, and timed tests. However, it is becoming apparent to many that these methods are woefully ineffective, and children continue to struggle to memorize their multiplication tables.

So what can parents and teachers do to help their children/students learn these multiplication facts? The following are some very effective math games and activities that not only work, but are lots of fun! When was the last time you or your children said that about multiplication?!

1. Numbers and equations are far more interesting when they represent real-life specifics. For example, the problem “What is 3 x 4?” can be posed as “If there are 3 pods with 4 whales in each, how many whales are there altogether?” As kids begin to visualize whales swimming through the ocean, the math becomes much more specific, rich, and understandable.

When my granddaughter was in the 3rd grade, we would use travel time in the car to practice our multiplication facts. First, I would make up a problem (7 tricycles, how many wheels?), and she would have to give me the complete equation (7×3=21). And then I would ask, “Why isn’t this a 3×7 problem?” Too many times all we say is 7×3 is the same as 3×7. That can be very confusing.

Then it would be her turn to make up a question (5 cars, how many rear-view mirrors?), and I would have to come up with the entire equation, plus justify why it wasn’t a 3×5 question.
Sometimes we would discuss what might make a good 4×7 question, or a 9×6 question, etc.

The following are just a few of the situations we used:
• 3 weeks – how many days?
• 9 cans – how many round bottoms?
• 12 noses, how many people?
• 5 cows, how many legs?
• 8 sleeves, how many shirts?

2. Play “What Am I?” Say to children “Seven is one of my factors. The sum of my digits is 6. What am I?” (42). Repeat this activity with other numbers.

3. Use a blank multiplication chart. Ask the children to enter the multiplication facts that they are sure of. Then have pairs of students exchange charts and quiz each other on the facts that are on the chart. If a child misses a fact, ask the partner to make a small mark by the fact to indicate that they need to practice it further. Marking missed problems with a highlighter is a strategy that may benefit some students. Keep these multiplication charts around and continue to add to them and test each other.

4. Most children struggle with multiplying by 6, 7, 8, and 9. These are the ones that need the most practice. The following is a way to work on these factors:

Provide students with paper and crayons and ask them to draw six blue vertical lines on the paper. Now ask them to draw four red horizontal lines intersecting the vertical lines. Ask them to circle in purple each place there is an intersection and count the number of intersections. Challenge them to identify what multiplication fact they have just demonstrated. Tell them that in this model, the number of rows is given first. [4 ×6 = 24.] Ask them to turn their papers a quarter turn and name the multiplication fact now modeled. [6 ×4 = 24.]

Encourage them to generate other facts where one factor is 6, including 6 × 0 and 6 × 1.

Repeat with 7 as a factor.

It may be helpful for students to visualize the vertical lines as city streets, the horizontal lines as roads, and the intersections as marking where a stoplight is needed.

5. Distribute index cards to each pair and ask each student to make a set of 10 cards numbered 0 to 9, one to a card. When they have finished, ask them to shuffle the two decks together and stack them face down. Tell them to take turns turning over the top card, multiplying the number drawn by 6 and then saying the product. As each card is used, it should be returned to the bottom of the deck. Give students time to play, and then ask the class to skip count in unison by 6. Encourage them to do so without looking at the game board.

Repeat for 7 as a factor.

6. Number Drawings – great for helping to memorize skip counting!

What you need:
paper, pencil, and crayons

Give each child a blank piece of white paper. Tell the children that today they are going to be skip counting by 4’s to 40 and each of them would be making their own unique drawing.

Tell them they are going to start by putting the number 4 anywhere on their paper and putting a little dot beside it. The object is to scatter the numbers all over the page. Now what number comes next if we are skipcounting by 4’s? Keep going until you reach 40.

Now connect the dots starting at 4, going to 8, and so on. When you reach 40, connect it back to 4.

Now color the inside of your drawing.

Make a Number drawing for 2’s, 3’s, 4’s, 5’s, 6’s, 7’s, 8’s, 9’s, 10’s, 11’s, 12’s and so on.

7. Play a game.

Rectangles

What you need:
2 players
2 dice
12×12 grid or graph paper for each player
pencils and crayons

During a series of rounds, players toss the two dice that determine the length and width of rectangles that are constructed on 12×12 grid or graph paper. Points are scored by finding the areas of the rectangles.

Players take turns. During a turn, a player tosses the dice and constructs a rectangle by making its length on a horizontal line on the graph paper according to the number thrown on one die, and marking its height according to the number thrown on the other die. The player then outlines the entire rectangle, writes the equation within the rectangle, lightly colors it in, and calculates his score by determining the number of squares within the rectangle.

The rules for placing rectangles are as follows:
• All rectangles must be placed entirely within the graph.
• The edges of rectangles may touch (but do not have to).
• Rectangles may not overlap each other.
• No rectangle may be placed within another rectangle.

Players drop out of the game and calculate their cumulative score when their throw of the dice gives them a rectangle that will not fit on their graph. The game ends when all players have dropped out. The player with the highest score wins.

Real-Life Math in Elementary School and Beyond

Elementary school students in three of Kingsport, Tennessee’s four high school zones took some weekend time this school year to learn practical, hands-on applications of math in the “real world”.

My question is, why isn’t their regular, everyday math curriculum talking about math in the “real world”?

Many educators contend that children must go beyond memorizing rules—they need to know when and how to apply the rules in real-world situations. Many also argue that realistic problems can serve as a powerful motivator in the mathematics classroom. They go on to conclude that the curriculum should consist of real-world problems because students will naturally learn mathematics by solving such problems.

The basics are changing. Arithmetic skills, although important, are no longer enough. To succeed in tomorrow’s world, students must understand algebra, geometry, statistics, and probability. Business and industry demand workers who can-

solve real world problems

explain their thinking to others

identify and analyze trends from data, and

use modern technology.

The mathematics students do in school should prepare them for the new basic skills necessary for their futures.

Instead of problems done with no context using worksheets, dittos, and workbook pages, students should be working on problems to investigate that are related to real life, such as investigating salaries, life expectancy, and fair decisions, for example.

Giving students opportunities to learn real math maximizes their future options.

Using money, counting change, etc. is a real-life skill that children need to learn. Play the following game with your second graders, third graders, and fourth graders.

Money Race

What you need:
2 players
1 die
pennies, nickels, dimes, and quarters
sturdy paper plate for “bank”

The following coins (which equal $1.00) are placed in the “Bank” between the two players. A paper plate makes a great bank.

10 pennies, 5 nickels, 4 dimes, and 1 quarter

Each player also takes the same combination of coins for a total of $1.00.

Money Legend:
1 – subtract a penny and put it in the bank
2 – subtract a nickel or 5 pennies and put it in the bank
3 – subtract a dime or a combination of coins that equals 10 cents
and put it in the bank
4 – subtract a quarter or a combination of coins that equals
25 cents and put it in the bank
5 & 6 – choose any one coin from the bank

Player #1 rolls the die and either adds or subtracts the appropriate coins.

Player #2 does the same.

Play continues in this manner until both players have completed 10 rolls. Players total their own coins. The player with the greatest amount wins.

Teaching Math with Games

Do your students like to play math games? If so, do you think of games as time fillers or part of your educational program?

In my classroom, teaching math with games was a serious educational activity. The value of math games can be enhanced or decreased depending on what teachers/adults do. The following are three of the most important principles of teaching that I followed while students were playing games:

• Do not show students how to play at a higher level; instead, encourage them to do their own thinking.
• Do not reinforce “correct” behaviors or correct “wrong” ones.
• Play with individual children whenever possible.

Most of us have been taught that the way to teach mathematics is by showing children what to do. Extensive research into how children learn mathematics shows that children construct mathematical knowledge by doing their own thinking. Therefore, we must encourage them to figure things out rather than obeying and mimicking their teachers.

Also, most of us were told that the role of the teacher is to reinforce “right” behaviors and correct “wrong” ones. A teacher’s occasional expression of pleasure is not harmful, but when the teacher says that an answer is correct, all thinking stops! I know this is a radical thought, but I truly believe that students should be encouraged to come to their own conclusions based on debate among themselves. The nature of mathematical knowledge is such that if children argue long enough, they will agree on the correct answer (unless the question is too hard for everybody in the group).

I have always believed that assessment is much easier to accomplish when using a math game, rather than a workbook page. Teachers find out much more about children’s thinking by playing with individual children or a small group than by merely observing them. Therefore, playing with them whenever possible is desirable.

Math Games and Effective Teaching

I have spent many years of my elementary teaching career building a community of learners who approach math with eagerness. I’ve learned that students need to be mentally engaged in a challenging and worthwhile mathematical task that emphasizes the conceptual aspects of the mathematical topic and promotes the formation of mathematical connections if they are to learn skills with meaning and be able to use those skills to solve problems.

It is not so critical whether students “discover” everything for themselves but it is critical that students are allowed to do some genuine mathematical work on their own. If teachers do all the work and students are left only to copy and imitate and practice what the teacher has done, they are less likely to make sense of the material, remember it later, transfer it to new situations, or do well on standardized tests.

The most important thing with respect to student learning is the nature of the learning task students engage in. Students need to be encouraged to think and persist with respect to the mathematical task, and the teacher should refrain from stepping in too early to provide students with answers or tell them exactly what steps they should use. Rather, the teacher can support students by asking them questions that guide them toward mathematical learning. This can be effectively done in a range of instructional settings from the most student-centered to the most teacher-centered.

Reform-minded teachers pose problems and encourage students to think deeply about possible solutions. They promote making connections to other ideas within mathematics and other disciplines. They ask students to furnish proof or explanations for their work. They use different representations of mathematical ideas to foster students’ greater understanding. These teachers ask students to explain the mathematics.

I have found that math games are an engaging and mathematically challenging task that can effectively offer students the opportunity to be engaged and talking with one another, and where they are encouraged to question and think about the mathematics and mathematical relationships. Have you ever used a ditto or workbook page that could make that claim?

Of course, it is not enough to teach students the game and let them play it. It is the teacher’s responsibility to move from group to group, listen to the conversations, and ask probing questions. My most-used question is, “Can you convince your partner and me that you are correct?” After hearing the question several times, the students usually begin to ask the question of their partners without my prompting.

Other questions that are well worth asking are:
• 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?

Too often the teacher or the partner is willing to give the other player the answer, thus making it possible for that player to do no thinking whatsoever. The teacher’s or partner’s questions to that player should be:
• 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.

Math Games and Understanding Equality

I contend that one of the big reasons why U.S. students lag behind their peers in many European and Asian countries in mathematics is because we are lax in helping children develop critical thinking skills.

Critical thinking skills that require students to apply content knowledge to real-world problems is of great importance. It’s very clear that if students can recall discrete content knowledge but cannot apply it, they’re going to be in trouble.

Here’s an example. By the time students have mastered rudimentary math, elementary-school pupils should understand that the numbers on either side of the equal sign are equivalents. Many students drilled in rote memorization don’t always grasp the concept of equivalency. I’ve frequently seen sixth graders who still believe that the equal sign means “the answer goes here”.

Equivalence/equality is undoubtedly one of the most important, connecting ideas in school mathematics. Developing this concept of equivalence calls for lots of experiences with materials as students are developing their conceptual understanding of numbers and operations. More important, it calls for teachers to help students connect their experiences with the mathematical idea(s) they are developing, in this case, equivalence or equality.

One of the experiences elementary teachers can use to help develop this understanding of equivalency is math games. The following is one of my favorites, and I use it with first through sixth graders.

Balancing Act

What you need:
2 players
deck of cards, face cards removed
cut a 3×5 card in thirds. On two of the thirds write a + sign. On the last third write an = sign.

Shuffle the cards and deal six cards to each player. Stack the rest of the cards facedown in a pile.

Each player chooses four cards from his/her hand. The object is to balance the equation by arranging the cards into two addition problems with equal sums. A player earns one point for balancing the equation.

Example: a player could place a 7 and a 1 on one side of the equation and a 3 and a 5 on the other (7+1 = 3+5)

A player can also place two cards of the same value on the equation to balance it (4+0 = 0+4).

At the end of a round, the cards played are placed at the bottom of the deck. The dealer shuffles the cards and gives six more to each player. Play continues in the same way.

The game ends when one player reaches ten points.

Variation: Children can play a similar game using subtraction or addition and subtraction. Change your “operation cards” so that children can create various balancing equations.

Math Games and English Language Learners

As an elementary math specialist, I talk about math all the time. The moment I begin a conversation, a wall comes down, and so many children (and adults) quickly blurt out that they dread math and say they have never been good at it.

To be perfectly honest, as a student, I struggled with math. I didn’t understand why it came so naturally to some students, but not to me. Looking back, however, I realize that I had an advantage that I wasn’t even aware of — I understood the language in which the problems were written, even if I didn’t understand how to solve them!

I can imagine what it must be like for English language learners (ELLs). Although it is easy to assume that many ELLs will excel in math because math is a “universal language”, and students may have had prior educational experience that included mathematical instruction, that assumption can lead educators astray.

Young children, whether ELLs or native English speakers, need to work with more than just worksheets to learn and understand math concepts. Utilizing multiple learning modalities will help all students to develop a deeper understanding of number concepts and relationships, but is especially helpful for English language learners.

If your goal is an excellent mathematics program for every child, then for these students, successful teachers need to find ways to make math understandable, relevant, and familiar. It is imperative that teachers utilize multiple instructional approaches.

The use of pairs or small groups is an instructional strategy that can be very effective for ELL students. By grouping students, you can encourage communication and interaction in a non-threatening and more relaxed setting.

Because math games require active involvement, use concrete objects and manipulatives, and are hands-on, they are ideal for all learners. Games provide opportunities for children to work in small groups, practice teamwork, cooperation, and effective communication. Children learn from each other as they talk, share, and reflect throughout game times. Language acquisition is meaningful and understandable.

Understanding and Mastering Multiplication

Many children struggle to memorize their multiplication tables, and many adults have bad memories of trying to learn them.

Why should children learn the multiplication facts? Because children without either sound knowledge of their facts or a way of figuring them out are at a profound disadvantage in their subsequent mathematics achievement. Students without multiplication-fact fluency spend more time determining routine answers and less time on more meaningful applications. Students who know their facts build on these fundamental concepts which ultimately benefits their later mathematical development.

For years, learning to compute has been viewed as a matter of following the teacher’s directions and practicing until speedy execution is achieved. There has been little or no emphasis on understanding the concept. Memorize 7×6=42, and so on.

When skills such as multiplication facts are taught for conceptual understanding and connected to other mathematics concepts and real-world meaning, however, students actually perform better on standardized tests and in more complex mathematics applications.

Numbers and equations are far more interesting when they represent real-life specifics. For example, the problem “What is 3 x 4?” can be posed as “If there are 3 pods with 4 whales in each, how many whales are there altogether?” As kids begin to visualize whales swimming through the ocean, the math becomes much more specific, rich, and understandable.

When my granddaughter was in the 3rd grade, we would use travel time in the car to practice our multiplication facts. First, I would make up a problem (7 tricycles, how many wheels?), and she would have to give me the complete equation (7×3=21). And then I would ask, “Why isn’t this a 3×7 problem?” Too many times all we say is 7×3 is the same as 3×7. That can be very confusing.

Then it would be her turn to make up a question (5 cars, how many rear-view mirrors?), and I would have to come up with the entire equation, plus justify why it wasn’t a 3×5 question.

Sometimes we would discuss what might make a good 4×7 question, or a 9×6 question, etc.

Keeping with the idea of making multiplication facts understandable, you might try a math game such as Bubbles and Stars.

Bubbles and Stars (Beginning Multiplication)

What you need:
2 players
1 die
paper and a pencil for each player (fold it in quarters)

Player #1 rolls the die and draws that many bubbles (as big as he/she can in one of the quarters).
Example: Player #1 rolls a 5 and draws 5 bubbles.

Player #2 rolls the die and draws that many bubbles.
Example: Player #2 rolls a 1 and draws 1 bubble.

Player #1 rolls the die and puts that many stars inside each bubble.
Example: Player #1 rolls a 3 and draws 3 stars inside each of his/her 5 bubbles

Player #2 rolls the die and puts that many stars inside each bubble.
Example: Player #2 rolls a 6 and draws 6 stars inside his/her 1 bubble.

Both players record how many bubbles and stars they drew and then record how many stars they have altogether.
Example: Player #1 – 5 bubbles x 3 stars = 15 stars
Player #2 – 1 bubble x 6 stars = 6 stars

Player #1 rolls the die one last time.
If the roll is odd – 1,3,5 the player with the most stars wins.
If the roll is even – 2,4,6 the player with the least stars wins.

Parents and teachers who use these kinds of activities with their children, will help them master their multiplication facts. Resultingly, these students will have a more positive attitude about their mathematics abilities and further mathematics experiences. Teaching for understanding equals a formula for success.