Monday, January 03, 2011

CGI Conference 2011

It has been a long time since I posted something up here! Susan Empson and I have just finished a book titled, Extending Children's Mathematics: Fractions and Decimals. It will be published by Heinemann and available in a few months. You haven't heard from me because I been writing the book rather than writing the blog.

I wanted to make sure that everyone knew that the 2011 CGI conference will be held June 22 - 24 in Little Rock Arkansas. I will post the url for registration and information when it becomes available. I hope to see you there!

Tuesday, March 17, 2009

Cognitively Guided Instruction Conference July 2009

If you haven't heard already there will be a Cognitively Guided Instruction Conference, July 30 - August 1, 2009 in San Diego, CA.  This is a great opportunity to talk with people from around the country who are interested in children's mathematical thinking.  To learn more, please go to

I hope to see you there!

Monday, September 08, 2008

Learning Fractions with Understanding - Repeated Decimals

I had the following conversation with Andy, who was in the last month of 6th grade. We had this conversation without and paper and pencil. I have added some notation in green to help the reader follow his ideas.  (Please note that I am writing divided by rather than the division sign because I can’t get the division sign to appear in this blog.) 

Andy: I was just thinking that point 9 9 9 9 and on and on forever must be the same thing as one.

.99999…. = 1

Ms L: Why were you thinking that?

Andy: Well, you know with 1/3. If you wanted to write 1/3 as a decimal you would think about how many tenths it was the same as. One is the same as 10 tenths.  One third would be 10 tenths divided by 3. Ten tenths divided by 3 is 3 tenths with one tenths left over. 

1/3 equals 1 divided by 3

1/3 equals 10/10 divided by 3

1/3 = 3/10 R1/10

(R stands for Remainder) 

You could take that one tenth left over and again divide it by 3. One tenth is ten hundredths. Ten hundredths divided by 3 is 3 hundredths with a hundredth left over. So 1/3 is 3/10 plus 3/100 with 1/100 left over. 

1/3 = 3/10 + 3/100 R1/100

You could do it again with the left over 1/100.  1/100 is the same as 10/1000.  10/1000 divided by 3 is 3/1000 with 1/1000 left over.

1/3 = 3/10 + 3/100 + 3/1000 R1/1000

It would just go on like that forever. There would always be one piece left over and you would divide that again by 3. You get three with again one piece left over. I think that is what it means when they write .33333…..

1/3 = .33333….

Ms L: What got you thinking about this?

Andy: My teacher told me that one third as a decimal was point 3 3 3 3 and on and on forever and I wanted to know why. So I started thinking about how many tenths 1/3 was, and how many hundredths it was.  Is my idea what it means when you write .33333….?

Ms L: Makes sense to me. But you said something about point 9 9 9 9 9 and on and on being the same as 1.

Andy: Well, I already told you that.

Ms L: You did?

Andy: Yes, 1/3 + 1/3 + 1/3 = 1. Of course. Point 3 3 3 3 and on forever plus point 3 3 3 3 3 and on forever plus point 3 3 3 3 and on forever is point 9 9 9 9 and on forever which means that point 9 9 9 9 and on forever must be the same as 1.

1/3 + 1/3 + 1/3 = 1

1/3 = .3333

.3333…. + .3333…. + .3333…. = .9999….

.9999….. = 1

Ms L: Are you sure of that?

Andy: Yes, I just proved it.

When I taught high school more than 20 years ago, I taught a proof for .9999…. = 1. My daughter’s babysitter, Mary, is taking Pre-Calc and was taught this same proof a few weeks ago. The proof typically taught in high school is very different from the one generated by Andy.  The typical high school proof is a series of equations that proceed logically.  Even though many high school students can understand how one equation logically follows from the previous one, the proof seldom convinces students that .9999…. = 1. When I asked Mary whether or not .9999…. equaled one, she said, “We proved it in class a few weeks ago. I understand the logic of the proof, but I’m still not convinced that .9999…. equals 1.  I think it is very close but just a little bit less than 1.”

In my professional development work with teachers, I have been asked to talk about .9999… = 1. Before talking with Andy, I hadn’t thought of a way to talk about the relationship between .999…. and 1 other than the typical formal proof. Since I didn’t know what else to do, I presented the formal proof to the teachers. After the presentations, teachers were left feeling pretty much like Mary.  There were no “aha’s!” coming from the group.

I explained Andy’s proof to Mary she immediately said, “Oh, I get it! Now I am convinced, .9999… is the same as 1” and added that she was going to tell her friends about the proof when they went bowling that evening. (She later reported that her friends were now convinced as well.)

Before I proceed, I want to state that I am not advocating we abandon logical sequence of equations as proofs in high school.  The typical proof of .9999…. = 1 probably should remain in the high school curriculum.  However, wouldn’t it be nice if students had an informal understanding of .9999…. before working with a formal proof that involved .9999….?  To state the question more generally, wouldn’t it be nice if students had an informal understanding of repeated decimals and their relationship with fractions before working abstractly with repeated decimals?

How did Andy gain an understanding of .9999….? Andy had been in a CGI class for 1st – 5th grade. I had often worked in his class when he was in elementary school. Andy wasn’t an unusual student.  Unlike some of his classmates, he wasn’t put in a math class with older kids.  He tended to master concepts a little earlier than some of his classmates, but he always had peers in his classroom who were working at the same level that he was.

Andy’s CGI teachers had engaged with Susan Empson’s work on learning fractions with understanding.  Susan is a member of the CGI research and development team who specializes in the learning and teaching of fractions.  She has developed a framework of problems and strategies that link learning fractions to children’s understanding of whole number operations.  I can’t provide a full explanation of her framework in this blog, but I encourage you to check out her blog at

In elementary school, Andy’s learning of fractions was integrated into his learning of whole number operations in many different ways.  He developed an understanding of many relationships between fractions and whole number operations.  In this proof, we see a strong understanding of the relationship between fractions and whole number division in that he has a deep understanding of one third as one divided by three.

Throughout elementary school, Andy’s CGI teachers posed problems to help their students to develop an understanding of the relationship between fractions and whole number division.  For example, in first grade, he solved problems like this,

 4 kids were sharing 13 cookies, how many cookies would each kid get?

In third grade, his teacher used problems like the following to develop an understanding of the relationship between fractions and whole number division:

  4 kids were sharing 3 small cakes.  How much cake would each kid get?

The problems in fifth grade were more sophisticated:

  8 kids were sharing 21 sandwiches.  How many sandwiches would each kid get?

Andy was not taught fraction in isolation from whole number operations until he got to sixth grade. He was not shown pictures of circles and asked to color in ¾.  He did not make fraction strips or use commercially produced fraction manipulatives. His teachers used their students’ understanding of whole number operations as a foundation to teach fraction concepts. Fractions were never an isolated unit.  They were never something brand new.

Elementary and middle school teachers have the potential to teach a great deal of important mathematics when they teach fractions.  Fraction instruction can help students understand the relationships between fractions and whole number operation.  Fraction instruction can help students develop a foundation that allows them to learn later mathematics with understanding.  I believe the best way to do this is to integrate fraction instruction with whole number instruction and to use what children understand about how number operation as a foundation for learning fractions.

Friday, September 05, 2008

If you are interested in learning what Cognitively Guided Instruction (CGI) is, I suggest you listen to this podcast:

In this podcast, CGI Teacher Bobby Norman interviews my friend and colleague Dinah Brown.  Dinah is a CGI instructor with Teachers Development Group.


Tuesday, February 26, 2008

Students' Grades

My friend Sandy (not her real name) called me this weekend because she was upset about her sixth grade son Noah’s (also not his real name) mid term grade in social studies. Last term, Noah’s social studies grade was 87%. Now, only 6 weeks later his grade is 68%. Noah is upset by his grade, and Sandy was confused by it. When Sandy asked Noah why he thought his grade had gone down in social studies, he said, “I have no idea, I have been working just as hard and learning a lot. I’ve really learned a lot about Japan.” (Japan is what he is currently studying in social studies.)

If you have read this blog before, you know that for me learning is all about students’ understanding of important concepts. I asked Sandy if she had noticed any change in Noah’s understanding of the concepts they were studying in social studies. Sandy thought that Noah understood the concepts well. In the last month or so, there were many times when Noah talked about Japan in relation to something he was doing. For example, when at a Japanese restaurant, Noah wondered aloud if the restaurant was run by the Japanese work place philosophy. Noah explained this philosophy and tried to find evidence by looking at how the workers interacted at the restaurant. When the family was talking about what would make a good president, Noah explained how the ancient Japanese believed that their rulers were direct descendants of the Gods and how these rulers had great power because of this belief. He also talked about the current Japanese government’s charitable works and how these works compared to those of the US government. Sandy was able to relate other stories which show that Noah has a good understanding of many important concepts about Japan.

It gave Sandy some relief to realize that Noah was indeed learning. Still, Noah was upset that his grade was low and Sandy was perplexed with how to talk with him about his low grade. Noah’s school has an on-line grade book that parents have access to. Sandy and I took a closer look at Noah’s specific grades. There were 8 grades that factored into Noah’s mid-term grade. All of these grades came from low-level cognitive tasks. For example, there was a test on Japan where all of the questions relied purely on memory. (E.g. The highest elevation in Japan is _______.) Noah couldn’t remember some of the answers and also missed some points because he spelled some words wrong and forgot to capitalize some proper nouns.

The tasks that made up Noah’s 87% grade from last semester were a mix of high-level and low-level cognitive tasks. One of the high-level tasks was a research project on The Great Wall of China. Noah researched, wrote a paper, made a poster and presented his work to the class. Noah got an A on this project. Last term, his grades on the low-level cognitive tasks tended to be lower than his grades on the high-level cognitive tasks.

So, how is Noah doing in social studies? He seems to be learning what we would want students to learn about social studies. He can research a topic, write about it and present his ideas to his peers. He can make relationships between another culture and his own. His teacher is clearly doing a great job of exposing her students to important social studies concepts and Noah is clearly doing his part by engaging in these concepts in a way that leads to understanding. Not all social studies teachers teach in a manner that fosters the level of understanding that Noah has. As someone who cares about Noah, I am grateful for what he has been able to learn.

What about Noah’s low marks on the low-cognitive level tasks? For example, what about the fact that Noah couldn’t remember and spell correctly all four of the major islands of Japan? I would guess that most educated, economically successful, well adjusted adults don’t know the names of the four major islands of Japan. I would also guess that these adults could find the names of these islands in less than 30 seconds if they had access to the Internet. It doesn’t distress me that Noah didn’t get these answers right on the test.

It does however distress me that Noah is getting a D in social studies. One might think that a D as a mid-term grade in sixth grade wouldn’t have much impact on Noah. No college or scholarship agency is ever going to see this grade. Noah however sees this grade. As adults we might be able to take the perspective that this D isn’t a reflection of his understanding. Most middle and high school kids just don’t have the reasoning skills or the maturity to take this same perspective. It would take an unusual adolescent to receive a D and draw the conclusion, “well, actually I am strong a social studies, it is just that I don’t memorize things well.”

This grade has the potential of sending some pretty harmful messages to Noah. One message that this grade might convey is that Noah isn’t good at social studies. Part of the reason that Noah was able to learn so much about Japan was that he engaged in his social studies class and readings. I wonder if Noah would continue to be as engaged if he continues to receive D’s in social studies. Another, equally harmful, message that Noah could take from this D is that the only things that are important to learn in social studies are things that can be assessed by low-level, fact recall tests. Most kids can learn better study skills, spend more time memorizing facts and eventually figure out how to get better grades on low-level tasks. It would be a shame if Noah, in an effort to improve his grade, paid less attention to important social studies concepts so that he could spend more time memorizing facts. Learning time is too precious to spend much of it memorizing information.

What can we, as teachers, learn from Noah’s midterm grade? As teachers we need to be careful that our assessments and grades reflect what we view as important learning. Kids and parents believe that grades are reflections of how well students have mastered concepts and skills that we value. We can’t really fully embrace learning with understanding if our assessments aren’t aligned with learning with understanding. Something is clearly wrong if kids who understand can do poorly on our assessments or kids who don’t understand can do well on them. I have worked with many ninth grade students who got mostly A’s in middle school math and struggled in Algebra 1. These students don’t struggle with the new content that appears in Algebra 1 but rather struggled with the mathematics they should have learned in middle school. These students and their parents were not getting the information they needed to get from their middle school grades.

As I said earlier, I am distressed about Noah’s grade and I really hope that Noah is able to maintain a positive concept of himself as a learner and hold on to productive beliefs regarding the type of learning that is important. I am distressed about Noah but I am doubly distressed about students like Noah who don’t have parents like Sandy. Sandy is highly educated and has many supports in her life. Not all parents will be able to provide support to Noah as Sandy will.

I strongly encourage you to open your grade book and look at what your grades are based on. If your grades aren’t based on what you think is important, try to figure out what changes you can make to better align your grades with the learning you value. I challenge you to think of one small thing you can do this week to move your grades one step closer to being reflections of the learning you value.

End Note: Many schools are currently using Standards Based Report Cards. Standards Based Report Cards give students and parents information on the specific concepts each student knows and understands. My own children’s elementary school uses Standards Based Report Cards with great success. It takes a great deal of time, effort, resources and especially teacher professional development to design and use Standards based report cards well. It also takes a good amount of well-designed parent education. Standards Based Reports Cards have great potential for helping teachers and schools move towards teaching for understanding and ensuring that all students learn important skills and concepts.

Tuesday, November 20, 2007

Preparing students for success in algebra 1: The role of elementary and middle school mathematics

My friend’s son, Kurt, was having trouble with his ninth grade algebra homework. He was given the equation,

P = -300 + 15N

and was asked to solve for N. The story in the textbook that went with this equation was,

“Jo has a lawn mowing business. This equation shows the relationship between the profit that Jo makes, P, and the number of lawns that Jo mows, N.”

I started by asking Kurt some questions to see if he could link the number sentence to the story about Jo. Kurt was able to tell me that Jo started off 300 dollars in debt, that N would be the number of lawns that Jo mows and that Jo charges 15 dollars for each lawn she mows.

We then moved on to solving for N. I asked Kurt if he had any ideas about how to start. Kurt said, “N is being multiplied by 15, so I could start by dividing by 15.” Kurt’s new equation was,

P÷15 = -300 + N

I responded as I imagine many algebra teachers would respond, “You divided the N by 15 and the P by 15, but what about the -300?” Kurt replied, as many algebra students will reply, “Oh, I have to divide -300 by 15 also.” His new equation was,

P÷15 = -20 + N

I said, “That was a great way to solve for N. With the numbers in this number sentence, starting by dividing by 15 worked well. Sometimes starting with division will be a little messy. Is there any other way you could have solved for N?” After some time, Kurt replied, “I could have added 300.” I asked him to do this. His new equation was,

P + 300 = 315N

The number sentence that Kurt wrote showed me that he did a good job generalizing the rule that I had just told him. I had told him that when he divided by 15, he needed to divide everything by 15. Now he that he is adding 300, and he thinks he needs to adds 300 to each term as well.

Why is it that

(-300 + 15N) ÷ 15 = -300÷15 + (15÷15)N
when you divide by 15 you need to divided each term by 15

where as

(-300 + 15N) + 300 ≠ -300 + 300 + (300 + 15)N
when you add 300 you don’t need to add 300 to each term?

Mathematicians would explain the difference between dividing by 15 and adding 300 in terms of the distributive versus the associative property. You can think of dividing by 15 as multiplying by 1/15 and since multiplication distributes over addition both the -300 and the 15N need to be multiplied by 1/15 in order to preserve the equality relationship. When you add 300 to -300 + 15N, the distributive property no longer applies because addition does not distribute over addition. In this case we use the associate property,

(-300 + 15N) + 300 = (-300 + 300) + 15N = 15N

My response to Kurt when he forgot to divide the -300 by 15 was not at all helpful. It just another rule that Kurt needed to remember and he didn’t know when the rule applied and when it didn’t.

I knew Kurt as a third grader when he was in a CGI class. If I had asked him what 465 + 300 was, he would have said something like, “400 plus 300 is 700 plus 65 is 765.” He would have implicitly used the associative property of addition to solve this problem. Although he would not have written it like this, the idea that he used could be represented as,

465 + 300 = (400 + 65) + 300 = (400 + 300) + 65 = 700 + 65 = 765

He would not have added 300 to both the 400 and to the 65 as he did when he added 300 to -300 + 15N to get,

(-300 + 15N) + 300 = (-300 + 300) + (15+300)N = 315N

If I had asked third-grade Kurt to solve a problem like, “15 kids are sharing 180 jelly beans, how many jelly beans would each kid get?” He might have said something like, “150 divided by 15 is 10, there are 30 left, 30 divided by 15 is 2, so each kid gets 17 jelly beans.” This idea could be represented as,

180 ÷ 15 = (150 + 30) ÷ 15 = (150÷15) + (30÷15)

He would have known that both the 150 and the 30 would have to have been divided by 15. He would not have only divided one of the addends by 15 as he did when he first divided -300 + 15N by 15 and got,

(-300 + 15N)÷15 = -300 + N

Now it is 6 years later and what has happened? Why is it that the informal understandings that Kurt had as a third grader aren’t connected to the algebra he engages in as a 9th grader? According to Kurt and his mom, he has been in a lot of math classes where he has been expected to learn procedures that his teachers have demonstrated for him. Kurt often tells his mom that he doesn’t understand his math lessons. This is frustrating to Kurt because he works hard in school and he really wants to understand. Although Kurt’s mom has been concerned about Kurt’s lack of understanding, Kurt appears to be a good math student. He received mostly A’s in his middle school math classes and was recommended for algebra as an 8th grader (he took 8th grade math instead). To make matters even more complicated, Kurt’s middle school used a highly regarded reform math curriculum.

There are several things that elementary and middle school teachers can do to prepare students to learn algebra with understanding. Perhaps the most important is to encourage students to use their own strategies to solve problems. This is true for complex problems as well as basic computation problems. We know that when students generate their own strategies to solve problems, they understand these strategies. Students may or may not understand the strategies we teach them. It might not matter whether we teach students standard algorithms or the more intuitive algorithms found in reform math curricula. Students can use the intuitive algorithms that they don’t understand just as they can use standard algorithms without understanding them. If a student learns to use a strategy that we have taught them, it is very difficult to assess if the student is using the strategy with understanding. When students use strategies that they have generated, we know they understand these strategies. The only strategies that can become a foundation for future learning are the strategies that students use with understanding.

A second way to provide students with a foundation for learning algebra with understanding is to recognize when students intuitively use the fundamental principles of mathematics and highlight their use of these principles and draw attention to the use of these principles. The fundamental principles to watch for are: the associative, distributive, and commutative properties as well as the inverse relationships between operations. (See Thinking Mathematically: Integrating Algebra and Arithmetic in the Elementary School, (2003) by Carpenter, Franke and Levi, published by Heinemann, Chapters 4, 8 and 9 for a discussion of these principles.) If you see a third grader using the associative property of addition, I don’t recommend saying, “You just used the associative property of addition, and let’s learn this property.” I do recommend that you draw attention to what the student has done to reinforce the idea for this student and others. If a child solves, 438 + 300 by saying,

“400 plus 300 is 700 plus 38 is 738”

we can draw attention to this intuitive use of the associative property in several ways. We can have the students share and make sure the rest of the class hears this strategy. We can ask questions such as, “why did you just add the 300 to the 400 and not also to the 38?” We can write the number sentence,

438 + 300 = 400 + 300 + 38,

on the board and ask other students if they think this number sentence is true.

It is possible for all students to learn high school algebra with understanding. Part of the responsibility for learning algebra with understanding clearly lies within the algebra class. Elementary and middle school teachers also play an important role in students’ learning algebra with understanding. How elementary and middle school teachers teach basic computation has a great impact, for better or worse, on students’ ability to learn algebra with understanding.

Tuesday, December 19, 2006

What Students Learn From The Standard Algorithm

What Students Learn from Instruction on
The Standard Subtraction Algorithm
Linda Levi, ©2006

Two third grade boys, Andy and Terry, were playing one of those popular trading card games. These boys were in a non-academic setting. Although I watched them play through the lens of a math teacher, the boys were clearly focused on playing the game. They were very interested in the cards they had collected and the points and powers of the monsters in their decks

At the start of the game each boy had 6000 points. On the first move, Andy’s monster attacked with 347 points which needed to be subtracted from Terry’s initial 6000 points. Each boy figured out how many points Terry would have; not surprisingly, they got different answers. I knew Andy; he was in a CGI classroom. I didn’t know what type of math instruction Terry had, but I knew that he and Andy attended different schools. After the boys argued a bit about what 6000 – 347 was, Andy said, “why don’t we just show each other how we got our answers?” Terry showed Andy his work; he had done the standard subtraction algorithm. He got the right answer and the marks on his page indicated that he had performed all the standard steps. They then discussed Terry’s work.

Andy: What is that?

Terry: This is the real way to do subtraction, I learned it at school. And see, I got 5,653.

Andy: You have to show me what you did.

Terry: Ok, this is the real way to do subtraction. You have to write it like this. (Writes 6000 with 347 lined up below it.) Now you have to start with the ones. When do subtraction you always have to start with the ones.

Andy: What do you mean you have to start with the ones? You can start with the hundreds, the thousands – you can start anywhere.

Terry: No, you can’t, when you do subtraction the real way, you have to start with the ones. That is what my teacher told me.

I had never thought carefully about exactly what we tell children when teaching the standard algorithm until I heard this discussion. When we teach the standard subtraction algorithm, we tell children to start with the ones. Of course, you don’t have to start with the ones when you subtract. For example, in a problem such as 5,000 – 3,002, it makes much more sense to first subtract 5,000 – 3,000 (the thousands) and then subtract 2 (the ones). We want children to know that 5,000 – 3,000 – 2 is the same as 5,000 – 2 – 3,000. Children with a strong understanding of subtraction know that they don’t need to start with the ones. Understanding how subtraction works will help students solve algebraic equations such as 3x – 48 – x = 49 or 3x – 48 = 49 – x.

Terry and Andy’s discussion then went on,

Andy: Ok, with that way you have to start with the ones. Then what did you do next?

Terry: Well, you can’t subtract seven from zero so I had to borrow.

Andy: What do you mean you can’t subtract seven from zero! Don’t they have negative numbers at your school?

Terry: Well, yeah, we have negative numbers. Like with the temperature or the number line, but you can’t subtract seven from zero.

Andy: Of course you can, zero minute seven is negative seven.

Terry: Well, I guess.

I again thought about the directions we give when teaching the standard algorithm. Perhaps Terry’s teacher didn’t tell Terry, “You can’t subtract seven from zero,” and said something like, “We have more ones in the number we are subtracting from than the number we are starting with so we are going to need to borrow.” However, what Terry learned was you can’t subtract seven from zero. It is not atypical for children to say something like, you can’t subtract a bigger number from a smaller one when explaining how they compute with the standard subtraction algorithm. This misconception may interfer with students' learning about negative numbers.

Andy: Ok, so now what did you do?

Terry: Well, I needed to borrow, and since there were no tens in six thousand I needed to go the hundreds, and there are no hundreds in six thousands so I needed to go to the thousands.

Andy: What do you mean there are no tens in six thousand? I don’t know how many tens are in six thousand, but there are a lot of tens in six thousands. There are ten tens in one hundred, so there has to be lots of tens in six thousand.

Again, even though Terry’s teacher might have stated the borrowing rule without stating misconceptions what Terry took from this instruction was that there were no tens in six thousand.

At this point Andy seemed to give up and listened as Terry went on through his explanation of the standard algorithm. When Terry was finished, Andy explained his strategy.

Andy: Well first I did 6,000 minus 300 and that was 5,800. Oh no, I did that wrong, it should be 5,700. And then it would be 5,700 minus 40 that would be 5,660 and then minus 7 is 5,653. Oh we got the same thing.

Terry: I was right, 5,653.

I later asked Andy if he had ever seen the way that Terry solved that problem. He said he thought he had, that some kids in his class showed that way during math times. I asked him if he had ever been taught that way and he said, “No. My teacher wants us to solve problems in our own ways, she doesn’t teach us how to solve problems.”

Knowing Andy’s teacher, I wouldn’t say that she doesn’t teach her students how to solve problems. She is careful in choosing problems for her students that will help them progress in their understanding of number and operation. She also carefully scaffolds her conversations with students and facilitates conversations among students to lead them to more sophisticated understandings of number and operation. As their understanding grows, their solution strategies become more efficient. Andy’s teacher does not teach standard algorithms.

It is very difficult to consider the possibility of not teaching the standard algorithms. These algorithms are part of our culture. Most elementary school teachers spend many hours teaching these algorithms to their students. Many elementary school teachers who are committed to teaching math for understanding throw in some instruction on the standard algorithms as another way of solving problems.

In deciding whether or not to teach the standard algorithm, we need to think about what our students learn from our instruction on standard algorithms. We want all students to leave elementary school with a strong understanding of addition, subtraction, multiplication and division. We also want students to leave elementary school with efficient strategies for computation. One of my goals is to help teachers learn to increase students’ computational proficiency in conjunction with their understanding of number and operation. Terry was quite proficient in performing the standard algorithm; he was efficient and accurate. However his proficiency came at the expense of his understanding of number and operation. We can teach children to be proficient with computation at the same time as we help them develop their understanding of number and operation. I will be writing more about this in future postings.

Thursday, August 17, 2006

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.