Summer Math with your Children: Parent Newsletter 6

Summer Math with your Children: Fun and Rewarding
By Linda Levi

Soon schools will be closing for the summer. Many of us are thinking of ways to include reading into our children’s summer routines. It is also important to think of ways for our children to engage in mathematics over the summer. Research shows that children who do not engage in mathematics can lose about a month of learning over the summer. Over 6 years of elementary school, this can add up to 6 months or about 2/3 of an academic year.

Engaging in mathematics with your children can be just as rewarding as reading with them. This doesn’t mean you should go out and buy math workbooks for your children to do this summer. Even if you would enjoy working through workbooks together, they are unlikely to provide the rich experience that you as a parent can provide for your child.

Before I offer some of my favorite ways for families to engage in mathematics, here are two important things to remember when doing math with your children.

Allow children to do the math in their own ways. If your child is having trouble, encourage him or her to get some paper and pencil to draw or write something that might help. Children may also benefit from having counters (such as pennies or small blocks) that they can use to act out the problem. It is often tempting to say, “Your way of solving that problem was good, now let me show you another way.” Do your best to resist this temptation. Children are typically very proud of their solution strategies. This pride is very empowering and encourages them to take risks when trying new strategies. Having an adult show you a better strategy can diminish a child’s pride in her or his strategy.

After children solve a problem, ask them to explain what they did. We all learn from reflecting on our thinking. Explaining your ideas to an interested adult is an excellent way for elementary school children to reflect on their thinking.

Here are some ways to do math with your children:

Make everyday situations into math story problems. It is impossible to get through the day without using mathematics. Here are some examples of how you might turn everyday situations into math problems.

Let young children set the table without telling them how many forks they need; try this sometime when you are having company.

If you decide to give your child money to spend on something, increase the level of difficulty your child may have in figuring out what he or she can buy. Figuring out what you can buy with a handful of change is far more challenging than figuring out what you can buy with a dollar bill. For younger children, give them only dimes and pennies.

If you decide to give your child an allowance, it need not be a nice round number. An older child might get $3.37 each week. Ask her how long it will take before she has $10.

Make the way you talk about time challenging for children. Once your child has the basics of telling time, you can say, “We’re going to visit Aunt Tessa at 5:00, go look at the clock and tell me how much longer will that be.”, or “You can leave on your light to read for another 20 minutes, what time will it be then?” (This last one might be difficult if it is 8:53.)

When driving in the car or riding on the bus, ask children mathematical questions, “If we buy 5 gallons of gas at that gas station we just passed, how much would it cost?”, “If we count all the eyes of the people on this bus, how many would there be?”, “If we count all the fingers of the people in the car, I wonder how many there would be.”

Children often need to do some mathematics to fully understand the books they read. You might say, “Charlie has to walk 2 ½ miles just to get to school. Your school is about half a mile from our house. How many times would you have to walk to school to walk 2 ½ miles?”

Play math games with your children. Some of my favorite math games are: Mille Bornes, Clue, Set, Connect Four, Mastermind and traditional card games like Rummy. Other games that involve good mathematics include: Battleship, Yu-Gi-Oh, and Pokemon. There are many video or computer games that involve more than hand eye coordination. If your children play video games, try to include some problem solving games in their repertoire.

Provide plenty of building toys. Children learn problem solving and spatial skills when they play with blocks or other construction toys. Interlocking blocks like Lego and K’Nex provide additional challenges. Mud and sand are great summer building materials. Ask children to draw what they have made. Ask, “Can you draw what the other side would look like without going over there to look at it?”

Have fun! The most important thing to remember when doing math with your child is to make sure that the activities are rewarding for everyone. If your child isn’t engaged or is getting too frustrated with a particular activity, let it go. You can always try something new on another day.

Linda Levi is an Elementary School Mathematics Consultant and Researcher and Developer of Cognitively Guided Instruction. Dr. Levi has researched the factors that enable children to learn math with understanding and is currently studying how the teaching of mathematics in elementary school can prepare children for success in algebra.

Algebra in Elementary School: Parent Newsletter 5

Algebra in Elementary School
By Linda Levi

Parents have many questions when it comes to their children’s math instruction. The math we learned and the way we learned it is often different from what we see happening with our children. Algebraic reasoning in the elementary school is an example of one of these differences. Most of us learned arithmetic in elementary school and didn’t engage in algebraic concepts until middle or high school.

Educators throughout the nation have come to the conclusion that if children are to learn algebra with understanding, algebraic reasoning must be an integral part of elementary school mathematics. University of Wisconsin-Madison researchers Thomas Carpenter and Linda Levi along with a core group of elementary school teachers throughout the Madison Metropolitan School District (MMSD) have been involved in this pioneering work of understanding how to include algebraic reasoning in the elementary grades. This work has been supported by major grants from the National Science Foundation and The US Department of Education. Results from this research were used in a study involving over 300 teachers in The Los Angeles School District.

A major focus of this algebra work has been on fostering an understanding of the fundamental principles of mathematics. Consider, for example, the following problems:

3476 + 524 – 523 = n

98 + 325 + 102 + 175 = n

n = 38 + 8x38 + 38

From an arithmetic standpoint, the above problems are difficult to solve. The numbers involved are large and there are multiple opportunities for errors. However, if you understand some fundamental principles of mathematics, each of these number sentences has a fairly simple solution. For example, in the first problem, 523 is one less than 524. If the first thing you do is subtract 523 from 524, you can simply add one to 3476 to figure the value of n. This type of understanding is integral to learning math in the elementary school and paves the way for success with algebra in later grades. What are some ways you could solve the other two problems using algebraic reasoning?

Linda Levi is an Elementary School Mathematics Consultant and Researcher and Developer of Cognitively Guided Instruction. Dr. Levi has researched the factors that enable children to learn math with understanding and is currently studying how the teaching of mathematics in elementary school can prepare children for success in algebra.

Learning Number Facts: Parent Newsletter 4

Learning Number Facts
By Linda Levi

Learning number facts holds an important place in today’s elementary school mathematics class. When children encounter number facts, they should be encouraged to figure them out in whatever way makes sense to them. When they first solve a problem like 8x6, they may draw a picture of 8 groups of 6 circles and then count all the circles. Doing many problems like this in the early grades gives children a foundation for understanding multiplication. As children get more sophisticated, they might solve 8x6 by adding 8 sixes. This strategy gives children an understanding of the relationship between multiplication and addition. As children continue to mature, they will start to use multiplication facts they know to figure out those they don’t know. A child might say, 8x6 = 6x6 + 2x6 = 36 + 12 = 48. Another child might say, 8x6 = 10x6 – 2x6 = 60 – 12 = 48. This strategy gives children a solid understanding of the distributive property. Eventually children will just know that 8x6 = 48. When I was in elementary school, number facts played a minor role in my education. Every so often I was given a set of flashcards to memorize and a timed test to assess how well I had done. I was not encouraged to figure out number facts and did not learn the big ideas of mathematics in the process of learning my facts. When I taught high school algebra, I had many students who learned number facts as I did. Many of these students did not understand the relationship between multiplication and addition and most of them did not understand the distributive property. It was very hard for these students to learn algebra. In the end, students who use the big ideas of mathematics to help them learn number facts are as efficient and accurate as students who do a good job of memorizing their facts. They, however, have a great advantage in that they understand the big ideas of mathematics and are well prepared to learn further mathematics. Students who memorize number facts miss an important opportunity to develop an understanding of the concepts that they will need to succeed in mathematics.

Linda Levi is an Elementary School Mathematics Consultant and Researcher and Developer of Cognitively Guided Instruction. Dr. Levi has researched the factors that enable children to learn math with understanding and is currently studying how the teaching of mathematics in elementary school can prepare children for success in algebra.

Direct Modeling: Parent Newsletter 3

Direct Modeling: A Window into Children’s Mathematical Thinking
By Linda Levi

Choose one of these problems to pose to your child. Remember to let your child use his or her own strategy to solve the problem.

A. Tylesha has 5 bags of marbles with 3 marbles in each bag. How many marbles does Tylesha have altogether?

B. Patrick has 12 peace cranes. How many more cranes would Patrick have to make to have 21 cranes altogether?

C. Mrs. Richards’ class is taking the bus on a field trip. They are riding in a min-bus that has 11 seats. Altogether 27 people are going on the trip. How many people can sit 2 to a seat and how many have to sit three to a seat?

D. Mr. Wu bought 22 animals for his students to take care. He knows he is getting some lizards, some chicks and some beetles. He also knows that altogether his creatures will have 100 legs. What might he have ordered? Is there anything else he might have ordered?

Children’s initial conceptions of mathematics are quite different than adults’. Adults would solve problem A by multiplication, problem B by subtraction, and, if they remember their high school algebra, problems C and D by using systems of linear equations. Children will first be able to solve these problems by direct modeling the situation in the problem. For example, for problem B they may draw 12 lines and keep drawing more lines until they get 21. They then will go back and count the extra lines they added. For problem D, they may take 100 pennies and make groups of 4, 2 and 6 until they end up with 22 groups.

Direct Modeling is a powerful strategy that provides a foundation for the more advanced strategies that follow. If you cannot model a problem, you cannot solve the problem. When your child struggles with a math problem, avoid referring to the operation (addition, subtraction, multiplication or division). Kids who are struggling should be reminded to think about what is happening in the story and find a way to show it. As children grow in their sophistication, they will no longer need to and should not be expected to directly model every problem. Adults use direct modeling to solve complex problems throughout our lives. Elementary school children naturally directly model problems they find challenging; we should encourage children’s use of this strategy.

Linda Levi is an Elementary School Mathematics Consultant and Researcher and Developer of Cognitively Guided Instruction. Dr. Levi has researched the factors that enable children to learn math with understanding and is currently studying how the teaching of mathematics in elementary school can prepare children for success in algebra.

Listening to Children's Math Ideas - Parent Newsletter 2

Listening to Children’s Math Ideas
By Linda Levi

The first step in fostering children’s mathematical understanding is allowing them to use their own strategies to solve problems. This article focuses on the second step in fostering children’s mathematical understanding: listening to children as they explain how they solve problems.

Choose one of these problems for your child. Try to pick something that will challenge, but not overwhelm your child:

5 + 6 = 
17 + 9 – 9 = 
27 + 35 = 
25 + 18 – b = 25
82 – 67 = j
228 + 49 = m + 226
76 – 39 = 77 – k
87 x 19 = 87 x 20 - p

After your child solves the problem, ask, “How did you get that?” Even though these problems are fairly traditional, the strategies that children use to solve them can be quite innovative. Children’s understanding of mathematics will grow if they are given opportunities to reflect upon their solution strategies. A great way for elementary school students to engage in reflection is to tell someone their ideas. It is common for a child to realize a mistake or notice a more efficient strategy when telling someone how he or she solved a problem. If you don’t understand the strategy, ask questions. Children deepen their understanding when they provide further explanations. If your child solved the problem with a traditional procedure, you might ask, “Can you solve this problem in a different way?” before moving on to another problem. You can ask this question even if your child used an innovative strategy. As children’s mathematical understanding develops, they are able to use multiple strategies to solve a problem.

Have fun listening to your child’s own strategies!

Linda Levi is an Elementary School Mathematics Consultant and Researcher and Developer of Cognitively Guided Instruction. Dr. Levi has researched the factors that enable children to learn math with understanding and is currently studying how the teaching of mathematics in elementary school can prepare children for success in algebra.

Doing Math with Your Child: Parent Newsletter 1

Doing Math with Your Child
By Linda Levi

Here are some problems to try with your children this month. Choose a problem that you think might be appropriate for your child.

Ms Jones has 4 children. Each child has 3 stickers. How many stickers do they have altogether? (For older children try 9, 26, 149, or 247 stickers for each child.)

19 children are taking a mini-bus to the zoo. The bus has 7 seats. How many children can sit 2 to a seat and how many children have to sit three to a sit?

Leon bakes pies and sells them for $13 each. The cost of the flour, sugar, butter and cinnamon is always $2 per pie. He also has to buy 3 pounds of apples for each pie, but the price of apples varies. How much can he pay per pound for apples if he wants to make at least $5 profit on each pie?

The most important thing to remember when doing math with your child is to allow your child to solve problems in his or her own way. Since we all want to help our children learn math, it is often tempting to say, “The way you solved that problem was great, but now let me show you a faster way.” Unfortunately, this can give children the message that our strategies are better than theirs. Children will choose to use strategies that enable them to solve problems with understanding and will adopt more efficient strategies as their knowledge increases. If they are shown efficient strategies that they don’t understand, they may be able to replicate them, but this replication comes at a cost. Children might start using strategies they don’t understand. They also might develop the belief that someone else has to show them how to solve problems. Believing that you are a person who understands mathematics and can generate ways to solve problems is essential to success in the mathematics one encounters throughout life. These beliefs are crucial outcomes of our children’s elementary school mathematics education.
If you pose one of these problems to your child and she or he can’t solve it, put it aside for now. Next month’s math article will be devoted to some things you can do when your child can’t solve a problem.

Linda Levi is an Elementary School Mathematics Consultant and Researcher and Developer of Cognitively Guided Instruction. Dr. Levi has researched the factors that enable children to learn math with understanding and is currently studying how the teaching of mathematics in elementary school can prepare children for success in algebra.

Newsletters for Parents -- next 6 posts

A few years ago, the curriculum coordinator of our local school district asked me to write a series of articles to help parents understand the mathematics instruction that was happening in our schools. These articles were distributed to elementary school principals who often included them in their monthly newsletters. Since I have received several requests for these articles, I thought that they would be a good way to start my blog. Perhaps some of you who visit my blog to see my parent articles will develop an interest in my log and keep reading.