Building Schema in Preparation for Division of Decimals

As you may know if you read the August blog entry Introducing Equations with Multiple Variables, I have been very interested in increasing my understanding of how children learn fractions, decimals, and percents, particularly how they learn to use these numbers in operations. Often students cite their 5th/6th grade years when these concepts were introduced as the beginning of their mathematical woes. The topic has been of particular personal interest to me because my oldest daughter is in 5th grade and I am homeschooling her in mathematics this year. As mentioned previously, we are using a Japanese curriculum as a primary guide for content. The Japanese curriculum introduces division of decimals in 4th grade, unlike the U.S. where it usually occurs in late 5th or 6th grade. As I looked at the text, I became concerned that Kylie was not yet ready for the manner in which the text approached the topic, given her limited schema for the topic. This seems like a typical problem that teachers face--approaching a new topic for which students have potentially insufficient background knowledge. This blog entry explores how I am dealing with this problem in the context of division of decimals. (I'm sure there are multiple strategies that might be effective here above and beyond those that I describe. I would love to hear them!) Instead of abandoning the topic, I decided to plan an approach that would involve skipping ahead to a geometry unit for our core topic, and use our minilessons (5-10 minute lesson starters) and our "something different" Friday classes to build schema in preparation for the unit.

I chose minilessons that would focus on building her whole number division skills. This was really essential because I don't want division of decimals to seem like a "brand new" concept to her. She needs to anchor new knowledge on previous schema. Whole number division will play a pivotal role in the process as she extends it to decimal numbers. Therefore, I wanted to extend her in this area. She has only used invented algorithms to solve division. These algorithms have gained considerable efficiency since 3rd grade, but I wanted her to make greater use of the distributive and associative properties. I also wanted her to develop her notation skills. This goal involved both algebraic notation and standard representation. Although this developed over a period of some weeks, I will give you an example from her recent work. I gave her the following string of exercises and asked her to solve them mentally and then write a number sentence which would illustrate her mental strategy. She did not have to write a number sentence if she already had that fact in her schema. The actual string was written with a division symbol, but I'm finding that symbol hard to recreate in this post.

The string and her number sentences:
28/7 = 4 ; (14 +14) / 7 = 4
70/7 = 10
98/7 = 14 ; (70 + 28) / 7 = 14
168/7 = 24 ; (70 + 98) /7 = 24
170/17 = 7
187/17 = 11 ; (170 + 17) /17 = 11
340/17 = 20 ; (170 + 170) / 17 = 20
357/17 = 21 ; (340 + 17) / 17 = 21

Initially, I introduced standard representation of division by taking problems she had already solved mentally and then represented her own strategy in that format. I should point out at this point that I am discussing standard representation, not the standard algorithm of long division. At first glance, there are similarities, but long division has a set procedure, so all students who carry out the procedure will essentially show the exact same work on their pages. Standard representation looks similar, but incorporates students own strategies and they use facts that they possess in their schema. Another difference is that long division does not maintain place value, but Kylie's representation does. Note that she solved 187 / 17 by decomposing 187 into 170 and 17, two numbers that she could easily divide by 17. Therefore, her standard representation looks like that shown below. The answer 11 is placed on top as her final step when she calculates how many 17s she used. (I apologize for the appearance of this--it should look like long division, but I can't make the notation look right on the blog).


187
-170 (10)
17
-17 (1)
0

Up to this point you see my strategy is to ensure that she understands division with whole numbers, has developed efficient strategies that make use of her schema, and that she has appropriate systems of notation. However, I simultaneously want to build her understanding of division of decimal numbers. I am taking a two-pronged approach with this. First, I want her to develop an understanding of situations where division of decimals are required. To this end, I give her occasional homework problems with fairly easy division of decimals that do not require sophisticated algorithms to solve. An example is shown below.

You have 7.5 cups of sugar in the cupboard. One of your favorite recipes takes 3 cups of sugar to make (very sweet!). How many batches can you make?

My second goal is to build Kylie's schema related to place value patterns present in decimal division. She has been exposed to decimal numbers in 4th grade and has had a previous unit this school year on multiplication of decimal numbers. At the beginning of the school year, she explored the impact on numbers if they were multiplied or divided by factors of 10. I decided to extend these ideas by having her carry out some calculator investigations that would allow her to learn how to manipulate decimal numbers in formulating strategies for division of decimal numbers. Shown below is the first calculator investigation I designed, along with Kylie's answers:

Investigation A--Decimal position in the divisor
420 / 28 = 15
420 / 2.8 = 150
420 / 0.28 = 1500
420 / 0.028 = 15,000 <--- 1st Predict;

Patterns you noticed: I notice when the decimal moves to the left on the divisor the number gets bigger--10 times bigger. ;

Investigation B--Decimal position in the dividend
420 / 28 = 15
42 / 28 = 1.5
4.2 / 28 = 0.15
0.42 / 28 = 0.015 <--- 1st Predict;

Patterns you notice: I noticed when the decimal point on the dividend moves to the left, the quotient gets smaller--10 times smaller. Then I gave her 2 additional problems and challenged her to use the patterns she had just discovered to solve them: 42 / 2.8 and 4.2 / 2.8 Notice that the calculator allowed numerous calculations to be carried out with ease, allowing her to focus her attention on unfolding patterns. She actually really enjoyed this investigation and was thrilled that she was able to successfully predict answers and solve the challenge problems. The next calculator investigation will build on this one, but focus her attention a bit more on these patterns. It is shown below:


Calculator Investigations A and B:
Complete each number sentence by filling in blanks with x10, x100, x1000, etc. OR /10, /100, /1000, etc. After each investigation, write what you learned.

Calculator investigation A--Different decimal positions in dividend
3650 /25 = 14.6 ________
365 / 25 = 14.6
36.5 /25 = 14.6 _________
3.65 / 25 = 14.6 ________ <--- 1st Predict;

Calculator investigation B--Different decimal positions in divisor
365 / 2500 = 14.6 ________
365 / 250 = 14.6 _________
365 / 25 = 14.6
365 / 2.5 = 14.6 _________
365 / 0.25 = 14.6 ________ <--- 1st Predict;

Calculator Investigation C--What happens when both the dividend and the divisor increase or decrease by the same factor of 10?
25 /5 =
(25 x 10) /(5 x 10) = 250 / 50 =
(25 x 100)/(5 x 100) = 2500/500 =
(25/10)/(5/10) = 2.5/0.5 =
(25/100)/5/100) = 0.25/0.05 =

What happens to the quotient?
Use what you learned in this investigation to turn the following problem into a whole number operation you can easily solve: 2.4/0.12

It is possible to build schema for division of decimal numbers, allowing students to learn with comprehension and develop flexible strategies which personally make sense.

Fostering mathematical thinking/Motivating students in mathematics

As I mentioned in a previous entry, I have just recently begun homeschooling my daughter Kylie. Naturally, I have a very strong interest in her mathematical development. I want her to feel that all career options are available to her and the truth is that the lack of mathematical understanding is all too often the cause of career "doors" slamming shut. I want her to feel as if, should she choose, she could be a mathematician, a chemist, a physicist, an economist, an architect, a computer scientist, a pharmacist...you get the idea. Math plays an important role in many, many jobs. Those of you who have had me as an instructor know that I believe that if you are not running into situations that make you stop and really think about what you know and how you might use that information, then you aren't really learning in a way that is fostering intellectual development. So, we began the school year with what I would describe as "challenging, yet achievable" problems. Much to my surprise, Kylie was not very happy. Kylie has always loved math and enjoyed tackling difficult problems, so I had not expected this response. Part of the difficulty arose from the transition from school to homeschool. She was used to strategy shares when she could spend more time pondering other people's strategies than in sharing her own thinking. She is a bit of an introvert, so initially she found the idea of the "spotlight is just on me" to be overwhelming. Most of these issues have worked themselves out, but I have spent some time lately thinking about how to keep her love of math alive through the "tough spots."

I have just finished reading Jo Boaler's What's Math Got to Do With It? ( Subtitle: Helping Children Learn to Love Their Least Favorite Subject--And Why It's Important for America). In this book she mentioned that many famous mathematicians did not develop their love for the subject in school, but rather from someone who shared mathematical puzzles with them. The mathematical puzzles were intriguing and pulled them in to mathematics in joyful ways. So, I introduced Kylie first to the container problem (see sample puzzles at the end of this entry). She was riveted, frustrated, and ultimately thrillingly victorious! I do realize that math instruction cannot consist of just a steady diet of puzzles. However, I have come to see the need for them to play a role in the big picture of mathematical instruction. I don't pretend to have the "perfect" plan for approaching math, and I'm sure that this will evolve, but currently we have a 3-phase math time. Initially, we have a series of mathematical exercises that have been specially chosen to highlight certain mathematical relationships, suggest efficient, sense-based strategies, and/or see the use of particular mathematical models. If you are a former student, you may recognize this as the "computational minilesson" that Fosnot/Dolk talked about in Young Mathematicians at Work (by the way, if you liked those books, did you know that they now have supplemental curriculum available? Check it out at http://www.heinemann.com/ ). Following this time of strategy development, we do the concept-development phase of instruction. For this portion, I am using the Japanese curriculum that I mentioned in an earlier entry as a general guide. I do supplement it depending on Kylie's needs. The final portion of the class is usually choice time. I have given her a selection of mathematical puzzles and extended problems to explore. She chooses which ones she wants to do, and approaches them in any order, depending on how they intrigue her. Adding this component has had a hugely positive impact on her attitude. I also give her a "menu" of open-ended problems related to our current topic part way into the unit. She has to have these done by the end of the unit, but this still allows plenty of time for her to explore other problems.

These three phases allow Kylie to: 1) develop her strategies and grow in her understanding of number relationships and mental math; 2)move her forward into newer concepts in a way that builds on her previous schema, and; 3) allows her to see herself as a true mathematician tackling intriguing problems.

Container Puzzle:
Given a five-liter container, a three-liter container, and an unlimited supply of water, how do you measure out four liters exactly?

The Rabbit Puzzle: (also taken from Jo Boaler's book)
A rabbit falls into a dry well thirty meters deep. Since being at the bottom of the well was not her original plan, she decides to climb out. When she attempts to do so, she finds that after going up three meters (and this is the sad part) she slips back two. Frustrated, she stops where she is for the day and resumes her efforts the following morning--with the same results. How many days does it take her to get out of the well?

The Chessboard Problem:
How many squares are on a standard 8 x 8 chessboard? (Keep in mind that this is referring to squares of all sizes). How can you know for sure if you have found all of the possibilities? Can you generalize a method for finding out the number of squares of any chessboard of with different edge lengths? (For Kylie, I printed off sheets of empty chessboard clipart that she can use to write on, if she wishes. This is the problem she is currently working on. I haven't seen her work. She prefers to do her thinking on these problems on her own as much as possible.)



For some online mathematical puzzles, check out these links. The first one is interactive:

http://www.math.com/students/puzzles/puzzleapps.html

http://www.jimloy.com/puzz/puzz.htm

http://thinks.com/puzzles/loyd/loyd.htm

What does it mean to be a professional?

Teaching is considered to be a profession. We certainly expect to be treated like professionals. I propose that the varied uses of that word have obscured the true expectations we should hold ourselves to and the rights that we should advocate for as professionals. Some of the problem can be blamed on one of the most common uses of the word--to "act professionally." Certainly, we are (and should be) expected to dress appropriately, speak appropriately, and interact with others appropriately. But this is not what makes us a professional. If a receptionist is chewing gum, blowing bubbles behind her desk, and texting her friends before waiting on me, then I may say that she is not acting very professional. However, being a receptionist is not a professional job. So clearly, acting professionally and being a professional are two separate things. To explore the meaning of the word "professional" as it applies to teaching, I will use a metaphor to compare it to being a doctor--a career for which we have high expectations of professionals.

I suppose it is possible to be a "technical" doctor--one who is more comfortable following prescribed protocols than in fostering patient's overall health. Such a "technical" doctor might listen to patients only long enough to do a "protocol match", "Oh, this is the problem, prescribe X." She might treat isolated symptoms rather than thinking and problem solving through the complexities of multiple symptoms. She might see all problems as variations of issues that were previously learned about in medical school; therefore, there would be no perceived need for on-going learning and research. She may not see any urgency in keeping up with the latest research. If her prescribed protocols don't work and the patient continues to experience problems, she may become frustrated and imagine that the patient is probably just depressed or a hypochondriac (stupid or not trying). I call this a "technical" doctor, because this is not what should be expected of a professional. Rather, this doctor is a technician--someone who follows procedures without demonstrating deep professional understandings. How does this look in a teaching setting? Technical teachers follow prescribed protocols when they follow curriculum blindly without considering carefully the needs of their students. Who is ready for this lesson? Is it a rich enough lesson that it will offer some potential for growth in all of my students or will it really only address the needs of the "middle"? The isolated symptoms that are treated are the isolated skills that are addressed. People's health (and their minds) are complex. The symptoms do not always (or even usually) stand in isolation. We cannot assume that treating all the isolated bits will address the problems of the "whole".

Professional doctors see patients as individuals with unique situations, while also being able to relate their profile to a database of known information about patient health. One of the things that especially drew me to Cognitively Guided Instruction was the combination of these two features--a rich database of children's cognitive development in mathematics to help guide your instructional decision-making as you listen to and carefully observe your individual student(s). The professional doctor recognizes that diseases may not fit a "one size fits all" description in the manner that it manifests itself. She listens to all symptoms, asks questions to find out more about the internal workings of their bodies, generates multiple potentially valid hypotheses about the condition(s), and (rather than following a prescribed protocol automatically) considers carefully whether such protocols will be effective in this situation, whether adaptations need to be made, or whether totally new protocols need to be developed. The professional doctor does not assume that apparent health means that there is no need for action, just as a professional teacher does not assume that gifted and high-ability students can be left to their own devices. She is intrigued with "mystery" cases and works hard to help these patients progress. "Working hard" doesn't mean trying out treatments willy-nilly, but rather doing individual research, consulting with colleagues, and finding out more information about the patient and listening more closely to them. The longer she practices, the more aware she is of what she doesn't know, and the greater her need to develop herself professionally. In the teaching world, this need to develop oneself professionally goes far beyond attending inservices and even beyond going back to earn a master's degree. It involves a deep personal commitment to be involved in professional organizations, read professional journals and books, and to seek ways every year to improve our practice in ways that distinctly benefit our students' intellectual development.

How many teachers do you know who truly are professionals by the above standard? I have been privileged to meet some amazing teachers who do fit this description. Seeing them in practice only highlights what a rare phenomenon this really is (I believe the majority of teachers are wonderful, caring individuals. As you read further, you will see that this is not about teacher-bashing. Rather, I am decrying the fact that teachers are not often given the opportunity to be all that they are capable of being) . I find myself wondering why there are so few of these highly professional teachers. Teachers are not paid as well as doctors, but most teachers I know did not go into it for the pay, so I don't buy that argument. I have two hypotheses that seem quite reasonable to me, although I suspect that it is one of those complex problems with many influences. One likely reason is that the teachers are not only not expected to behave like this; they are often treated as if they should NOT be truly professional. That sounds like a radical statement, but it usually manifests itself in a couple of familiar ways. An administrator may tell a teacher that she has to follow a particular curriculum to the letter. This is like telling a teacher not to consider the needs of their students in making instructional decisions. Teachers may also feel pressure from other teachers, particularly those with more status, to conform to "the way we do it here." If there is no overt encouragement for teachers to be researching and applying best practice, then the environment is anti-professional (or a more favorable interpretation is that they are NOT pro-professional). Another likely reason follows from this one. Because many schools are not actively encouraging best practice, it turns out that many teachers have not seen it in practice. It is difficult (although not impossible) to put into practice what you haven't experienced personally. Sadly, many teachers have spent long enough in this professionally-deprived environment that they have forgotten that they are capable of so much more.

So what advice might one give a teacher who truly aspires to be a "professional"? First and foremost is to read, read, read about best practice. It is easy to say that you don't have time. Students laugh, but my advice for professional reading is to: 1) find a publisher who you know is sound when it comes to best practice (http://www.heinemann.com/ is one of my favorites); 2) keep a wishlist handy--offer it to family members for Christmas and birthdays or use it if you get an Amazon gift card; 3) keep your professional books in the following places--the headboard of your bed, the back of your toilet, and in your car. If you have insomnia, read. If you like to read in the bathroom, you're in business! If you have a doctor's appointment or an oil change or you are killing time waiting for your daughter's dance lesson to end, you have something decent to read; 4) push yourself to try at least one new instructional approach each year; ideally one that you have targeted to address a particular need you have noticed in your students. Expect that things will not go smoothly right away. Plan to spend some time problem solving the difficulties that naturally arise when you try something unfamiliar.

The second piece of advice I have is for you to believe in yourself and your ability to change. You will very likely find yourself in a place where change is not being encouraged. Believe in yourself, and (just as importantly) know why you believe what you do and be prepared to defend it in polite, well-reasoned, well-supported ways (now you see the reason for that reading above). Of course, teaching this way is hard work, but it is very rewarding in terms of fulfillment and self-efficacy. If you do find resistance, consider who your support system is that will give you the courage to keep going. If there is no one nearby or in your school, consider a listserv. Speaking from personal experience, I can tell you that dramatic improvement in your professional practice is possible, even in work environments where your colleagues would just as soon that your teaching approach failed spectacularly!

The third (and most important) piece of advice is to listen to your students. What do you see as their needs? Think deeply about this and identify the gaps in your own understanding. Those "gaps" will be the best impetus ever to drive your desire to be a professional in the truest sense of the word. I wish you all the best in this endeavor.

Subliminal Messages Sent to Children About Mathematics

My oldest daughter is entering 5th grade. For the first time (at least formally) we are homeschooling her. Given that math education is one of my fields, I have quite a large collection of resources and curricular materials to draw from. Actually, making a decision was quite challenging, because there were many good options. As you may notice, one of the blogs I enjoy reading is that of Tad Watanabe. He has advocated for a curriculum that is more coherent, and which builds conceptual understanding as well as procedural understanding. He is one of the supporters of Global Education Resources (http://www.globaledresources.com/) which sells English translation of Japanese textbooks. They are fairly reasonable, so I decided to order the 5th grade textbook. The first thing I noticed is that the year's curriculum comes as two fairly thin paperbacks about the size of a 5 x 7 picture on the front. There is no teacher's manual, at least not of the kind that we are accustomed to seeing here in the U.S. that lays out some prescribed approach (without knowing the needs of the students). Instead, you are advised to purchase the Teachers' Guidelines document, which is on a CD. It lays out specific, grade level target objectives and provides guidelines, such as ensuring that students have opportunities to model relationships and operations with concrete objects before beginning to introduce algorithms. It is a very useful document for anyone wishing to teach a child mathematics, regardless of their country of origin.

When I opened the front cover of the text, I was immediately taken with the introductory paragraph directed to students. It seemed to reflect vastly different attitudes and ideas about mathematics. It really drew a contrast in my mind between the kinds of attitudes that are fostered in Japan and the subliminal messages that American children receive. Shown below is that first paragraph from the Japanese text in bold print, along with the italicized comments, indicating the contrasting subliminal message I think U.S. children pick up.

There are many children who think that studying mathematics is interesting. Math is boring. Very few children like it, unless they are "math geeks." That is probably because, in mathematics, you can usually find the answer if you think persistently and diligently. Some people (maybe many) just can't do math. Perhaps they didn't inherit the math gene. It is like climbing up stairs one by one, although it may take time. You either know how to do it or you don't.

In mathematics, there is not only one way to find an answer. You have to do it exactly as the teacher shows you or it will be wrong. If you figure out another way, that is like cheating. So it is important to listen to your friends' ideas and discuss the ideas with them. You shouldn't discuss your ideas and answers with anyone else, or else you could get in trouble for cheating.

Let's work hard and do challenging mathematics together. If you don't know how to do something, ask for help from the teacher right away. They are supposed to "make it easy".

These subliminal messages play a significant role in the "mathematical health" of our nation. One study that I read on problem solving contrasted the differences between how Japanese first graders responded to a challenging problem compared to American first graders. The Japanese students worked on the same problem for 45 minutes, which is when the teacher concluded the lesson. The American students all gave up on the problem within 3 minutes. Clearly, those Japanese children believed that they could make sense and solve that problem, if they were just persistent. How do you suppose they developed that belief? Being told that persistence is important is a starting point, but I suspect those students had had plenty of opportunity to exercise that persistence without a teacher coming in at the first sign of confusion to "save the day."

What subliminal messages did you receive as a child about mathematics? How might you "break the cycle" of negative attitudes and foster a healthier, more productive, problem-solving attitude? These are questions well worth pondering.

Place Value and Standard Algorithms

For some years now I have been thinking about the role of standard algorithms (carrying, borrowing, long division, invert and multiply, etc.) in mathematical development. Constance Kamii has research evidence to show that students "unlearn" place value as they work with the standard algorithms. I well remember the first time I read some of her research findings. I actually thought I must have misread, so I had to read it a second time! Since that time, I have spent a lot of time thinking about the role of standard algorithms in math education. Should they be taught at all? If so, when? I don't think these questions have easy answers, and I believe that we will be better teachers if we grapple with these questions. We need to seriously consider what criteria we would use to make such a decision. In this blog entry I will share my experience in an interview with Anna during the summer after she had completed third grade. I had seen Anna during the school year. She had been with a teacher who was using Cognitively Guided Instruction principles to inform math instruction. My observations of Anna led me to believe that she was a very capable third grade mathematician. Her teacher told me, however, that Anna's parents had been trying to get her to use the standard algorithms. The teacher felt that her meaning-making processes had suffered as a result. During my interview with Anna, I noticed that she typically used the standard algorithm for addition, but relied on an open number line for subtraction. Interestingly, she seemed to illustrate better number sense for subtraction than she did for addition. This makes sense, since she her approach to subtraction required her to think and make decisions about the numbers, whereas she was following a procedure for addition. The following three anecdotes from the interview illustrate this point. The original problems were in context, which she conceptualized well, creating an appropriate numerical expression.

Subtraction: 301 - 199
***Anna drew an open number line and showed a jump of 1 to get to 200, a jump of 100 to get to 300, and a jump of 1 to get to 301. Then she said, "It's 102, because 100 + 1 + 1 = 102." In this situation, Anna used what she knew about the numbers to quickly generate a solution. Although she drew out the number line, this method was more efficient than the standard algorithm, and has the advantage of easily translating to a mental math strategy in the future.

Addition: 2015 + 1587
***Anna solved this problem using the carrying algorithm.
A: 5 + 7 is 12 (she wrote down "2")
K: Oh, are you splitting up the 12?
A: Yes, into a 1 and a 2
K: Doesn't 1 and 2 make 3, not 12? How did you split the 12?
A: (uncertainly) 6 and 6?
K: I don't see any 6.
Long Pause
K: I see you took off 2. Then what did you have?
A: Oh, 10...pause...yes, this 1 up here is the 10.

Anna was not in tune with her place value understanding when she was using the standard algorithm. She was able to talk about it with meaning only after probing and her response was far from automatic. It is important to remember that Anna was a girl who had been generating her own strategies and using place value understanding to guide that process. I was dismayed at how quickly she gave that up when introduced to the standard algorithm. The next example was from later in the same interview.

Addition: 1999 + 1999
***Anna solved the problem with carrying, but struggled a great deal and there were many crossed out marks. At times she had to stop what she was doing and model parts of the problem with base 10 blocks in order to figure out what she was doing well enough that she could explain it to me (I was asking her to explain her thinking to me, so blindly following steps wasn't really an option). Finally, she came up with her answer: 3998. Then I rewrote the numerical expression, 1999 + 1999, in a clean space on her paper.

K: What do you know about the number 1999?
A: It's close to 2000.
K: Could you use that information to solve the problem?
A: Yes, I could make the 1999s into 2000s and then add those and then at the end just take off 2. That would have been a lot faster than what I did actually.

Anna's place value understanding was so much more in evidence when she was using her own thinking. The place value is not very transparent in the standard algorithm, and clearly she wasn't keeping her own place value understanding paramount when she was using the standard algorithm. The discouraging aspect here is that Anna has good place value understanding. With probing, she was able to finally figure out what was happening in the algorithm. This would be much less likely to be the case if she had been introduced straight away to a procedure without having to use her own place value understanding to make sense of personal algorithms.

To be honest, I still struggle with the question of whether standard algorithms are ever necessary, and (if so) when. Based on this experience (and others) as well as the research findings, I suspect that students need more time than we might anticipate to solidify their understanding before being introduced to the standard algorithms. However, I am also willing to admit that the way that the standard algorithm was introduced to Anna may have contributed negatively to her sense-making. Her parents probably introduced it as a procedure to follow, not as an algorithm to explore and find out why it makes sense. Her parents probably referred to all numbers by their digit name rather than their actual value. They probably merely expected her to follow steps rather than make sense of them. Anna had built up some outstanding personal strategies for addition by the end of third grade. Perhaps if her parents had shown her how they used to do it and challenged her to make sense of why it worked, maybe some of her difficulties may have been minimized. Likewise, if they had referred to all numbers by their actual value, she may have been able to see how to use her own understanding of place value to make sense.

My daughter enters 5th grade this fall. So far she has never seen any of the standard algorithms. I cannot see at this point that it has done any damage at all. She is mathematically very competent. Still, this is the year when typically standard algorithms get really crazy, because the procedures and rules vary according to each of the operations and whether you have whole numbers, mixed numbers, fractions, decimals, percents... I will continue to ponder the issue of when and how for introducing standard algorithms and hope that you will also.

Introducing Equations with Multiple Variables

A little over a year ago, I was privileged to attend a conference session led by Linda Levi (see her blog listed to the right). In it she showed students' work in developing place value understanding, which extended into an understanding of decimals. This understanding was further extended into experiences with algebraic expressions and equations with multiple variables. The sample student work was impressive and I was particularly fascinated by the high level of understanding in one fifth grader's work as he found multiple solutions for the variables. At the time my daughter was nearing the end of her third grade year and I knew that she was definitely not ready for this work yet. Still, I was fascinated by the question of what it would take so that she would be ready by fifth grade. What would need to happen? How does one make that "leap"? My learning journey in regards to this question is still in process, but one of the things I did was to purchase and read, Thinking Mathematically: Integrating Arithmetic and Algebra by Thomas P. Carpenter, Megan Loef Franke, and Linda Levi. I highly recommend the book. I will be mulling it over for some time to come.

I belong to a group that gets together each summer to explore aspects of Cognitively Guided Instruction. Before we begin our 2 weeks together, we articulate our goals and expectations for the experience. This year, my focus was on the development of algebraic reasoning and sense-making tied to fractions, decimals, and percents. In this blog entry, I will be describing an experience in which I attempted to use what I had learned from Linda Levi's conference session and the book Thinking Mathematically to plan and conduct an interview with an incoming fifth grader that would introduce him to algebraic expressions (including decimals!) for the very first time. His classroom teacher was my co-interviewer and she told me that decimals had been introduced during the previous year, but that it had been "rough going."

In planning the interview, I knew that I wanted to begin with ideas that were in his schema. He was a student in a school that fostered the use of Cognitively Guided Instruction (for further reading in this area, get Children's Mathematics: Cognitively Guided Instruction by Thomas Carpenter et al). Therefore, the children were accustomed to a problem-solving curriculum and they had been exposed to a lot of problems that fostered place value understanding. In addition, they were accustomed to formulating their own strategies for problem solving, based on their personal sense-making. I believe this is critical in understanding the student's success in this interview. If he had been a student in a passive learning environment, he could have quite easily simply told me that "we haven't done problems like that yet, so I don't know how." After beginning with two problems involving multiplication with groups of tens (6,10, and 13 leftovers) and a measurement division problem with groups of ten (87, 10), I left the contexts behind for a time and asked him how many tens were in 57 and 243. Thus far, none of this was a problem for him. He easily answered that 243 had 24 tens. Initially, he described a trick you could use, by covering up the three and then you would find the answer. We probed further asking him how he could convince me that it would work, using mathematical reasons, rather than tricks. He was able to do so, saying "Ten fits into 100 ten times, so it fits into 200 twenty times, and then the four tens in forty." His teacher now made an unplanned suggestion to our interview protocol, which I think was critical to his success later in the interview, because it caused him to spend considerable time "grappling" with place value concepts outside of his familiar number range. She said, "How many tens are there in 1,243?" He knew that there were 24 tens in the 243, so he just had to figure out how many tens were in 1,000. This took him some time. He was then able to tell us that it was 100, but it took him further think time (and building a base 10 representation) to be able to explain it with mathematical reasoning, saying, "There are ten hundreds in a thousand and ten tens in a hundred and 10 x 10 is 100."

After this initial opportunity to bring forth his place value understanding (this was two months after the end of the school year), we moved into fractions. He was more familiar with fractions than decimals, having worked with them longer. In addition, it allowed us to get him to think about tenths, which would translate easily to decimals. "Sam eats 1/10 of a pound of fudge a day. How many days would it take him to eat 3 1/2 pounds of fudge?" After a pause, Chris said "How many tenths are in 3 1/2. I think that's the problem." He started to model it by drawing 3 circles and a semicircle on his page, which he told us represented the pounds of fudge. He started to partition the first circle into tenths, but before he completed it, he told us that it wasn't necessary because it would obviously be 10 days, and so then 30 days for the 3 pounds of fudge and the half pound of fudge would be 5 days, since 5 is half of 10.

We switched to decimals at this point, but scaffolded this newer, more unfamilar territory by sticking with tenths. "An animal at the zoo eats .1 of a pound of food each day. If the zookeeper has 36.8 pounds of food for this animal, how many days can she feed the animal before the food runs out?" Chris spent considerable quiet think time before writing anything on his paper. Then he wrote:
10 X 30 = 300
10 X 6 = 60
360 + 8 = 368 Days

We asked him why it made sense to multiply ten. He replied that the animal eats .1 per day and it takes ten to make a whole. " So like the 6 in 36.8, it would take ten tenths to make a whole and then you would have to do that 6 times. We asked him how he knew to add 8 days. He again pointed to the .1 per day, saying. "Obviously, you would have to have 8 days to eat .8.

At this point we decided to take the plunge into equations with multiple variables. We began with a fairly easy one (we hoped), given his place value understanding. I should point out that the use of parentheses was a familiar one to this student.

(m x 10) + c = 65

I asked Chris if he had seen anything like this before. He said, "Yes, but I can't remember what m is." I told him that this was what these problems were all about; he was to suggest a number combination that would work for m and c. He did not have to think very long at all before suggesting that m be 6 and that c be 5. So we decided to try a more challenging problem for him, particularly in light of his earlier struggle with the number of tens in 1000. This was where we were really able to see the outcome of his previous cognitive processing.

6345 = m + (b x 100) + (y X 10)

You might stop at this point and try to think through for yourself what you would suggest as feasible number combinations for m, b, and y. Chris thought about this one longer than the first one. After a period of time, he wrote 3 under b, then 4 under y, and then 6005 under m. He told us in the explanation phase that he thought of m as "the leftovers after I have b and y". We told him that with these kinds of equations which have multiple variables, usually we can think of several different number combinations that would work, and challenged him to try to find another. He wowed us with his flexibility in place value understanding when he came up with this option:

6345 = 45 + (60 x 100) + (30 x 10)

His understanding that 60 hundreds would be 6000, given the struggle he experienced earlier in the interview with the number of tens in a thousand, indicated to us that the cognitive struggle he had experienced had paid huge dividends in his understanding. We asked him to try one more number combination. He then wrote:

6345 = 345 + (30 x 100) + (300 x 10)

He explained that he had decomposed (yes, this is part of the mathematical language in his school) the 6000 into a two 3000's and expressed one of them as 30 hundreds and one as 300 tens. He was able to explain the sense in this using language similar to what has been expressed earlier in this entry, which is rapidly becoming lengthy. We still had 15 minutes remaining to our scheduled one-hour interview, so we gave him the "finale"--an equation with multiple variables AND decimal numbers.

576.25 = (100 x b) + c + (.01 x p)

He had never seen one one-hundredth represented as a decimal before, so we had to tell him what it stood for. We pointed to the number 576.25 and asked him to show us how he would say this. Given that decimals were still fairly new to him, we wanted to ensure that he knew what the number was. He stated the number correctly and immediately recognized that p could be 25, "because I have 25 hundredths". He then wrote the following:

576.25 = (100 x 5) + 76 + (.01 x 25)

I was thrilled with this interview experience. I don't think it would work with all 5th graders, but I could see how children who have built strong place value understanding AND developed their sense-making skills could experience a highly scaffolded interview, which starts with ideas that clearly part of their schema, then creates cognitive conflict, and then extends their new understandings into gradually less familiar territory, could be used to help students use their own thinking and understandings to make sense of algebraic understandings.

What am I doing writing a blog?

I am sitting here shaking my head at my audacity in setting up a blog. I am not a technophobe, but I am also by no means a "pioneer." In the case of blogs, in particular, I have not really seen the point. It seemed like an ideal opportunity to exhibit self-promotion, and I didn't really see the purpose in that. However, I have recently visited the blogs of two other math educators who focus on children's learning with meaning and I actually really enjoyed reading and exploring the blog. I thought it would be a nice opportunity to continue a dialogue, particularly with my former students. This is all quite new to me. I'm not quite sure how it will turn out. I imagine it to be a place for pondering intriguing ideas and pointing people to links that may be of interest. My hope is that it might be one of several sources available to current and former students. Initially, my focus will be on mathematics. If all goes well, I will extend the focus to include science instruction as well.