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Encouraging Mathematical Reasoning in the Classroom

In this standards-based assessment world in which we educators find ourselves, little thought is given to the development of mathematical reasoning skills. Instead, the focus has become test performance. No longer do we ask students to think. This lack of thinking skills has caused a lessening of enthusiasm in teachers’ and students’ attitudes – about school in general and mathematics specifically.

I think there needs to be a shift in the school culture toward promoting engagement through inquiry-based learning opportunities. I do not think any teacher needs convincing on this point.

One of the best ways to ensure active student engagement in math is the use of games. Good games for the classroom are engaging, fun, and create opportunities for students to explore concepts and develop mathematical reasoning.

Playing a math game is the first step on the road to mathematical reasoning. Teachers need to create opportunities for students to explore mathematical ideas by planning questions that prompt students to reflect on their reasoning during and after the playing of a game. When we carefully consider the questions we ask and plan an appropriate level of competition, students will focus on the mathematics and not just the game.

Questioning
While the students are playing the game, the teacher’s job is to move from group to group listening to their conversations. 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?

Too often the other player is willing to give his/her partner the answer, thus making it possible for that player to do no thinking whatsoever. Not good! Your (and the 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.

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.

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?
Click here to find math games at every elementary grade level.

Children Love Math Games

Another elementary school (this one is in Knoxville) has joined the ranks and begun to use math games in the classroom to motivate and engage students in meaningful mathematics.

“Often times, mathematics is viewed by our society as cruel and unusual punishment. That makes it very difficult for teachers to teach math and for students to invest the time and energy it takes to learn math. Through games, you have the motivational factor that helps move learning along.”

Classic games are finding their way into classrooms as educators creatively use the games to reinforce math, language and critical thinking skills.

I have found that games have a multitude of benefits:
• Meets Mathematics Standards
• Easily Linked to Any Mathematics Textbook
• Offers Multiple Assessment Opportunities
• Meets the Needs of Diverse Learners (UA)
• 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

Whichever grade level you teach, there are many games that your students can play which will be effective, useful, and fun!

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.

Give a math game a try. Find many math games at your child’s or student’s grade level by going to this page.

Math Games and Computation

The New York Times has another article about an ideal elementary classroom.

In their theoretical classroom, children would spend a short period of time each day practicing computation — adding, subtracting, multiplying and dividing.

As a veteran elementary teacher, I have found that math games are an effective and useful way to practice that computation.

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 – games require active involvement. Actively engaged children learn more quickly.

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.

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. Games require a variety of problem-solving skills, such as making and testing hypotheses, creating strategies (thinking and planning ahead), and organizing information. Plus, as children play, they further their development of hand-eye coordination, concentration levels, visual discrimination, memory, and their ability to communicate and use mathematical language.

Play games with your children/students. My grade-level specific games would be an ideal addition to your home/classroom!

Math Games Can Motivate Students

“Games can motivate students, capture their interest, and are a great way to get that paper and pencil practice”, says Marilyn Burns, world-renowned mathematics expert.

Games offer teachers a way of practicing and reinforcing arithmetic and other math skills, as well as supplementing a sole diet of drills and practice-problems with workbook pages or dittos.

Not only do games engage students, they also present the opportunity to present “high level” math concepts in a colorful and simple way.

In my experience, students are more engaged when we connect the mathematics they are going to learn with something that excites them (e.g., games).

Despite those benefits, some teachers and parents are reluctant to use board games and similar activities. Those critics tend to regard them as activities that cut into time spent on practicing problems, when in fact games should be used as another form of math practice.

Research on the link between games and math learning has implications not just for educators, but also for parents.

Turning off the television and engaging children in a simple card or dice math game just a few times a week can greatly improve their comfort in math.

There’s a huge amount of math in card and dice games that is not on television and video games.

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

_______________________ ______________________

Having Fun with Math Games

The old-fashioned method of solving problems from a textbook was not getting the job done at Hanover-Horton Elementary School.

So the school took a new approach to math education Wednesday – games.

Take a look at how Hanover-Horton Elementary School students are using board game to improve math skills.

This Michigan school is typical of most elementary schools – they are looking for ways to energize their math curriculum and engage their students. Math games are an enormously effective way of doing both.

Multiplication Games

More and more in my teaching career, I see that children struggle to memorize their multiplication tables. I’ve even worked in many 6th grade classrooms where it was perfectly evident that a majority of the 6th graders had not yet fully memorized their multiplication tables.

Simple multiplication is usually introduced into the math curriculum in 2nd grade, so it is important that 2nd and 3rd graders begin to get a strong handle on their multiplication facts. If they don’t, understanding division becomes ever more difficult.

I’ve found that multiplication games are wonderfully useful. Multiplication 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 the multiplication facts over and over.

Teachers and parents are partners in this process, and both can offer greater opportunities for their students/child to succeed in memorizing the multiplication tables. Multiplication games fit the bill wonderfully!

One of my favorite multiplication games is Multiplication Fact Feud. It’s a great way target and practice certain facts.

Multiplication Fact Feud

What you need:
2 players
deck of cards

Teacher or parent decides the particular multiplication fact to practice
(i.e. x7, x4, x8, etc.) Once the constant factor is determined, that card is placed between the two players. Players then divide the remaining cards evenly between themselves.

Each player turns over one card and multiplies that card by the constant in the middle. Players must verbalize their math sentence. The player with the highest product collects both cards.

Example: Player #1 Player #2
4 5 7
“4 x 5 = 20” “7 x 5 = 35”

Player #2 would collect both cards.

In the event of a tie (i.e. both players have the same product), each player turns over one more card and multiplies that by the constant factor. The player with the highest product wins all four cards.

When the cards are all used up, the player with the most cards wins the game.

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