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Algebraic Thinking


chat archiveModerator
Good afternoon and welcome to today's chat with NCTM President Cathy Seeley. Today's topic is algebraic thinking.

Here's our first question:

Question from:
Chico, California

One thing that might help develop some algebraic thinking is to present more problems that students have to be able to solve with any variation of the original given constraints. For example, if the question were "The pool manager Jim always keeps 100 gallons of water in his pool. At the end of each day he checks the pool to see how much water is in it. On Monday, only 82 gallons of water was left in the pool. How many gallons of water does Jim need to add?" The kids can figure this out with subtraction fairly easily. However, if there were more parts to the question like, "On Tuesday there was only 52 gallons left. How many gallons does Jim need to add now?" And then a final part to the question, "Write an equation that will tell Jim how many gallons to add at the end of any given day." This type of thinking will help them to develop the concept of variables and equations, and it is done in a fairly simple way.

Cathy Seeley:
This is exactly the kind of thinking that helps students develop the ability to use algebraic thinking to make generalizations. And this is just what we want students to do with algebra—to think beyond specific examples to more general cases. I prefer this idea to just learning recipes to solve problem 'types' (like coin problems, age problems, mixtures, etc.).

Question from
Missoula, Montana

Is algebra more than what most of us learned with x's and y's and many homework problems? You seem to imply that.

Cathy Seeley:
Algebra can be a powerful set of tools for representing situations, analyzing mathematical relationships, making generalizations and solving problems. It can extend well beyond the limited types of problems once filling traditional algebra texts to serving as a set of approaches in a student's mathematical toolkit. Today, algebra can be used to deal with data and make predictions. It can be used to model sophisticated situations from science, social studies or economics, to name a few. If we do our job well to develop algebraic thinking across the grades, American students will never be heard to say that they had no use for algebra. Rather, they will incorporate algebraic techniques into their broader mathematical thinking to deal with everyday life as well as advanced applications in mathematics and science.

Question from
Fayetteville North Carolina

One of the red flags I see in developing algebraic thinking is how we develop students' conceptual understanding of equality, especially in the early years. We must allow students to discuss and explore all aspects of what equality is and is not in order to break away from the idea that math is all about "the answer." Number relationships and balanced equations may not boil down to a nice little number for an answer, and it's important in the developing years for students to understand this.

Cathy Seeley:
I couldn't agree with you more. Equivalence/equality is undoubtedly one of the most important, connecting ideas in school mathematics. As you note, too many students see the equal sign as what precedes the answer. Developing this concept of equivalence calls for lots of experiences with materials as students are developing their conceptual understanding of numbers and operations. More important, it calls for teachers to help students connect their experiences with the mathematical idea(s) they are developing, in this case, equivalence or equality.

Question from
Lanham, Maryland

A major mistake is when we don't tell our first grade students the truth about the numbers. We should tell them the whole story about the numbers. We should tell them about positive and negative numbers both and also tell them that the real numbers are a part of all the numbers, and there are more numbers to learn (complex numbers). I have seen many students who think 2 minus 4 can't be done by asking, "If we have only two things, how can we take four out of them?" This line of thinking is because of not introducing students to negative numbers and algebraic thinking early. I hope we also show the power of algebra to our young students by letting them use it to solve problems. Thanks for asking.

Cathy Seeley:
Thanks for sharing this perspective. I recently visited a second-grade classroom where the (excellent) teacher was leading students in solving problems using subtraction. I heard her tell students to remember that you can't take a bigger number away from a smaller one. Unfortunately, this message, if sent often enough, can plant a future misunderstanding related to negative numbers and to the meaning of subtraction. Better to keep coming back to what the numbers represent and what the problem calls for in terms of a solution based on the particular situation. I'm not sure I'd advocate negative numbers with first-graders, but it's an interesting concept.

Question from
Omaha, Nebraska

I am a community college math teacher. Approximately 90 percent of the students who take our placement test (COMPASS, through ACT) assess into a developmental class in at least one of the areas: math, reading, English.

Yesterday I had a student wanting into my Intermediate Algebra class who assured me he had passed Algebra II in high school this spring. He could not combine like terms (he thought 5x - 5 = x), he could not graph a linear equation, he could not factor x2 - 5x -6, and so on. I don't really mind, since it does mean job security for me, but what is the disconnect between grades and knowledge?

Cathy Seeley:
This is a troubling phenomenon. My opinion is that the disconnect between grades and knowledge is more about how we teach than about how we grade. It is quite likely that this student, like others, passed a test at one time that called for these skills. I am guessing that this learning was temporary and superficial if it was based on lecture and memorization. Unless we connect algebraic skills to meaningful situations, build conceptual understanding and facilitate the use of algebraic thinking to solve problems, we risk perpetuating the study of algebra as a vast set of meaningless abstract rules and procedures that, once learned, may be quickly forgotten, never to return. The key, I believe, lies both in what content we choose to teach, and in how we engage students in their learning; watch for my next two President's Messages on this one-two punch. (By the way, in my recent teaching experience in a French system in West Africa, I noted what I had read elsewhere—that other countries do not use factoring as a general method for solving quadratic equations.) Thanks for bringing up this important and all too common problem.

Question from
Murfreesboro, Tennessee

I feel that students need to practice critical thinking skills in the elementary grades. This can be done throughout the curriculum, not just in the mathematics courses.

Cathy Seeley:
I couldn't have said it better myself. Not just starting earlier, but incorporating critical thinking across the curriculum. I believe that the different ways we teach problem solving and critical thinking in various content areas can help students access learning in different ways, based on their strengths.

Question from
Riverside, California

What is algebraic thinking?

We think of algebra as an art of symbol manipulation (e.g., let "x" be an unknown.) But I sense that teachers of algebra are not offering sufficient training in axiomatic thinking.

To prove the point, not a single algebra teacher in my experience could tell an inquisitive student, why (-a)x(-b) is PLUS (a)x(b).

Sure, there are hokey, plausibility arguments, like, "Why not?" or "Symmetry demands it." In fact, the axioms of distributivity, associativity, etc. come into play. Also, the real explanation may be too subtle for the classroom, which the teacher should acknowledge.

MORAL: Although the result may sound elementary, its justification may not be. The teacher should have the discernment to know how much to teach and how much to omit.

Cathy Seeley:
Your point brings up an important issue and one of NCTM's primary goals: providing high-quality professional development. This year's professional development focus of the year on algebraic thinking supports the kind of teacher understanding that you describe. Some universities and regional educational facilities offer high-quality professional development that gets to the heart of algebraic thinking. Unfortunately, such experiences are not accessible to all teachers. A rich set of resources to help teachers expand their deep understanding of algebraic concepts can be found throughout this year on NCTM's Web site. Look for the magnifying-glass icon on the home page and throughout the site to lead teachers to articles, books, and online resources. Teachers can use these individually, or, ideally, with colleagues as starting points for professional growth.

Question from
University Park, Pennsylvania

I think we need to clarify what we mean by "algebraic thinking." I believe too much emphasis is placed on patterns (which is an important idea). As I look at the Japanese curricular materials, what seems to be missing in the typical US approach is the emphasis on writing equations. We rarely discuss the importance of expressing our mathematical thinking using mathematical expressions and equations, nor trying to interpret mathematical thinking expressed as an equation or expression. Thus, for too many children expressions like 2x(5+3) simply means to calculate and find the answer.

This is just one simple example to illustrate the need for mathematics teachers to critically re-evaluate what we mean by "algebraic thinking." Representing our thinking through mathematical symbolisms must be an important part of algebraic thinking.

Cathy Seeley:
Your comments reinforce the importance of being able to use multiple types of representations for a situation, as called for in Principles and Standards for School Mathematics. The symbolic representation is an important part of a student's algebraic development. I hope you would agree that this representation needs to be well grounded in understanding what lies behind the symbols. By incorporating algebraic opportunities throughout the curriculum and across grades, we can help students develop all types of representational skills.

Question from
Archbald, Pennsylvania

When it comes to middle school math in the United States (specifically grade 8), what percent of the students do you think should be taking an Algebra 1 course as an elective instead of taking the traditional 8th grade math course, which includes some algebra concepts?

Cathy Seeley:
I don't think there is a set percent of students we can identify. Of more importance is the question of what is the nature of the middle school curriculum? A middle school program that addresses central features of Principles and Standards in School Mathematics, including such key ideas as proportionality and the transition to algebra, can clearly be beneficial to many students, much more so than what we might consider 'traditional' eighth-grade mathematics. We are no longer in a time, as we were when I started teaching middle school mathematics 35 years ago, when the middle school curriculum is simply a repeat of K-6 arithmetic. Now that we have so much to offer, there is far less reason to accelerate students into algebra. At the same time, if a student is to be able to take calculus in high school, or another course beyond the level of pre-calculus, they must either start the high school sequence in grade 8 or find a way to double up during high school. In summary, the major reason to accelerate: opening up options for advanced mathematics study to all students, especially those not typically represented in such courses. Cautions against accelerating: make sure the student is motivated to continue mathematics study every year in high school; make sure there are good course offerings for students in 11th and 12th grade; ensure that students have the benefit of the rich mathematics (especially proportionality) that should be part of the middle school program.

Question from
Honolulu, Hawaii

I think the ideas around equality or equivalence in the early grades are very important, as the question from North Carolina implied. In fact, young children can handle much more sophistication in ideas than we might have expected. This causes us to think about early mathematics in different ways.

Cathy Seeley:
Absolutely. As we rethink both priorities and opportunities at the elementary grades, we can likely not only build a foundation for algebraic thinking, but also strengthen students' development of concepts related to number, operations, and other mathematical strands.

Question from
Seattle

I attended the 6–8 and 9–12 Navigations E-workshops, which were very good in showing algebraic thinking across the grades and the development across the grades.

Cathy Seeley:
Great! This model for professional development is one we think holds a lot of promise, and the Council will continue to explore how we can use it to meet the needs of more teachers. In whatever form our professional development takes, this notion of developing this important thread of algebraic thinking across the grades is critical, starting with a student's earliest school experience.

Question from
Turners Falls, Massachusetts

Are we perpetuating an artificial segregation of math content (algebra, geometry, etc.) by emphasizing "algebraic thinking" as a concept? Why not place the emphasis on mathematical thinking?

Cathy Seeley:
Mathematical thinking is very appropriate for our broader attention. However, the term may be too broad. While all teachers might agree that they should help students learn to think, many elementary teachers, in particular, don't see where algebra fits in with their teaching. By focusing on what skills and knowledge build toward algebraic understanding, teachers can see more easily how these might fit with their teaching. For example, a teacher might agree that working with patterns, developing the notion of equivalence, and exploring relationships are important, even though they might not have thought of these as topics related to algebra. By making explicit what we mean by algebraic thinking, we can increase the likelihood that this type of experience can be part of what students do in school. But, as you observe, the bigger picture demands that we not teach algebraic thinking in isolation from other strands, but rather capitalize on our attention to algebraic thinking to find ways to connect it with other important parts of the curriculum toward a student's broader mathematical thinking.

Question from
Rochester, Michigan

I have two favorite problems that can help accomplish your goal. These are mathematical computations that are very down-to-earth and real-life, and they are things that everyone should have been empowered to solve by the time they leave elementary school. They are arithmetic but very much lead to algebraic thinking.

1. You and your best friend went on a weeklong vacation and agreed to split the costs down the middle. Over the week, each of you paid for various joint expenses (like gasoline, meal checks, motel bills) and kept track of what you had paid. At the end of the vacation, it turns out that you had paid A dollars and your friend had paid B dollars. Assume that A is less than B. How much money must you give to your friend at this point to even things up? This can be solved with algebra (in more than one way), by example, verbally, or with a picture, among other approaches. Multiple correct answers are possible.

2. (a) Exactly how many days old are you today? (b) Use your answer to (a) to figure out what day of the week you were born on. (The solution gets into the algebraic ideas of modular arithmetic.)

Problems like these make math be lively, relevant, and interesting, and they start leading students' thought patterns in the right directions.

Cathy Seeley:
Thanks for sharing these ideas. Adapted for appropriate grade levels, problems like these can be engaging tasks as part of a comprehensive approach to incorporating algebraic thinking in the elementary grades.

Question from
Hopewell, Virginia

As a first-year high school algebra teacher I am struggling to find ways to connect with students. Nearly 40 percent of my students are repeaters or students with learning disabilities, and keeping them from giving up on themselves is my main challenge. I have been moving around the room, asking students to stand up and act out parts of equations and mathematic properties, or even simply come up to write an answer on the board and then defend it. Still I hear the kids saying things like "I'm just dumb," or "I'll never learn this," or "Even if I knew how, it wouldn't do me any good." Are there any strategies out there to help engage students with limited motivation?

Cathy Seeley:
I think you have identified a critical factor—engaging students. I think that such engagement can precede motivation. By choosing tasks that allow students to tackle challenging and interesting problems, we can set up a classroom where students work in small groups to come up with solutions, discussing, justifying, and even arguing about approaches and consequences. Some of the National Science Foundation curriculum projects, both for middle school and high school, provide such tasks. This type of activity fits beautifully with the functions-based approach advocated in NCTM's Principles and Standards for School Mathematics.

Question from
Ontario, Canada

At what point do we introduce algebra to students who still have not mastered basic numeracy skills? Students who have troubles with operations using fractions are not likely going to understand or be successful with basic algebra. I have students in my grade 10 class that still use calculators to add and subtract fractions. They cannot even grasp the concept of solving for an unknown.

Cathy Seeley:
It would certainly be ideal to have all students be proficient in arithmetic before progressing to algebra. However, a few years ago I had an experience teaching a ninth-grade algebra class that caused me to re-examine my beliefs about necessary prerequisites for learning algebra. One particular student, whom I call Crystal, could not do fraction operations and asked if she could use a fraction calculator in the algebra class. I quickly discovered that, in spite of her arithmetic deficiency, Crystal was an outstanding algebraic thinker, as long as she had her fraction calculator to help her get answers to fraction problems. To make a long story a little shorter, eventually Crystal was motivated by her success in algebra to go back and learn fractions. She continued through precalculus in high school and went on to graduate from college and graduate school. We need to be careful not to let our own beliefs about how mathematics must be organized get in the way of allowing all students the opportunity to show us what they can do. Even though computational proficiency helps in higher-level mathematics, there is no evidence that students who are weak in some areas of computation cannot succeed in algebra or higher-level mathematics.

Question from
Athens, Georgia

What place does memorization of facts have in developing algebraic thinking? While we want to develop higher-order thinking and understanding of concepts, are there not certain facts students must simply "know?" And how do calculators in the classroom fit in to this?

Cathy Seeley:
This continues the discussion of the previous question. I absolutely agree that there are facts that students should know. I also recognize that there may be students who are otherwise ready to proceed to algebra or higher-level mathematics, even though they may not know all these facts (or skills). Technology offers us a way for both the teacher and student to see what students can do beyond what they have learned. And, as in the case of students like Crystal, sometimes success in more challenging mathematics can motivate learning of the things we wish they had known before. Engagement and opportunity can absolutely lead to motivation and success.

Question from
Raleigh, North Carolina

There are many well-respected theories out there on multiple intelligences. If a given student tends to think "geometrically" or visually, are we justified in forcing them to repeat algebra classes until they pass? If we are justified (I hope we are), then how do we get them to make this leap?

Cathy Seeley:
You've identified an important issue. In fact, students do think in different ways, and this might be one argument in support of a more integrated high school curriculum where algebra and geometry are not approached in isolation. But regardless of how you organize the curriculum, students should have opportunities to develop their thinking along different lines. When we teach algebraic problem-solving skills, this can increase students' repertoire of approaches beyond what may be their first line of attack. At the same time, a student who thinks geometrically or visually may benefit from a more visual approach to algebra. This is one of the most exciting advances in the teaching of secondary mathematics—that we can represent situations in multiple ways and can approach the solving of problems in different ways. Using the power of graphing technology and a functions-based approach to algebra can allow students to solve problems from either a symbolic or a graphical approach. There is no reason why any person who is reasonably successful in other content areas cannot be successful in mathematics if we offer multiple avenues toward mathematical understanding.

Question from
Newnan, Georgia

So which is better—a continuous mathematics course and integrated math course, or separate courses (Algebra I, Algebra II, etc)? How is the continuous different from integrated?

Cathy Seeley:
You folks in Georgia are certainly dealing with this issue. While many districts in the United States (and essentially all in Canada and elsewhere) have implemented an integrated approach to secondary mathematics, Georgia may be the first state to try to adopt a requirement for integrated high school mathematics in all schools (at least it's the only one that comes to mind). In any case, there are many apparent advantages to such an approach, not the least of which is the opportunity for students to connect otherwise isolated pieces of mathematical knowledge and skills. But the flip side is that this is a huge change for many schools, and the professional development and materials support needs are tremendous. If the state chooses to go this direction (or for schools outside of Georgia considering such a move), this professional development support and finding appropriate instructional materials are needs that must be addressed, as well as the need for ongoing support for teachers in terms of planning time, collaboration time, and so on.

The benefits may be significant and may be worth the effort, but it is important to recognize what is being asked of teachers and to acknowledge that making such a change on such a large scale may be quite challenging. For an individual teacher or school to adopt an integrated approach may be much more doable and may lead to more visible results more quickly. Regardless of whether you teach an integrated program or a course-by-course program, the essential things are what content you choose and how you actively engage students in their learning.

Moderator
Thank you all for your stimulating participation this afternoon. We had far more questions submitted live during the hour (and in advance) than Cathy could answer today. She will review all questions submitted and add several answers for the final chat transcript, which will be posted on the NCTM Web site Monday or Tuesday.

Thank you again.

Cathy Seeley:
Thanks to all of you for your energetic participation. You've kept me typing as fast as I can! I've enjoyed this professional interchange, and I look forward to reflecting on as many other additional questions as I can. Be sure to check the NCTM Web site for resources on Algebraic Thinking, the Professional Development Focus of the Year. Look for the magnifying glass icon.

Thanks for your interest, and I'll see you at our next chat!

Moderator
The following questions are representative of those submitted in advance or during the hour of the online chat. Time restrictions prevented Cathy from answering all the questions submitted for this chat.

Question from
7. Jakarta, Indonesia

What is the best way to teach algebra in grades 1, 2, and 3 in elementary school?

Cathy Seeley:
This question is way too big for a short response. The best source of information NCTM has to offer on our Professional Development Focus of the Year can be found on the NCTM Web site, accessible from many paths. Look for the magnifying glass icon on the home page and throughout the Web site. You will be delighted at the wealth of resources on incorporating algebraic thinking across the grades. One resource you will find there is the algebra standard in Principles and Standards for School Mathematics, as well as the excellent series of Navigations publications, which includes algebra books for each grade band (Go to: http://www.nctm.org/standards/navigations.htm ). We are also seeing an increasing number of online professional development programs related to algebraic thinking, although with so many available, you should choose any such program carefully.

Question from
Columbia, Maryland

Introducing mental math, mathematical properties, number patterns, representation of objects or numbers and generalization of the concept at an early age/grade. 

Cathy Seeley:
These are important components of algebraic thinking. Mental math is tremendously important and helps students predict answers in thinking ahead about problems they are tackling. Patterns help students develop generalizations, which lie at the heart of thinking algebraically. And we now know that being able to represent situations in many ways is not only an important ability, but can help students deepen their understanding of the situation and develop mathematical ideas at the same time.

Question from
San Jose, California

It seems to me that schools are pushing the expectations for traditional Algebra I classes earlier and earlier.  I have noticed children are having a harder time grasping concepts, and I feel that this is due partly to their cognitive immaturity.  At what age is it cognitively appropriate to introduce these concepts?

Cathy Seeley:
This is an important and timely question. The issue of accelerating students into Algebra I earlier and earlier brings several problems, not the least of which is cognitive maturity. For example, what happens to the critical content of the middle grades, in particular proportional reasoning and increasingly sophisticated ways of dealing with data and statistics? I worry that when we accelerate students into algebra too soon, they may miss this. Also, unless we have good options at the eleventh and twelfth grades, why are we accelerating the students? In addition, if the student is not highly motivated to continue high school study through every year in high school, considering calculus, or possibly another advanced course, what is the benefit of accelerating the student? Furthermore, unless prevented by the school or district, some or even many of these students may stop their mathematics study before twelfth grade, which is a disaster for any student going to college. Finally, I think there may be questions of at what age students are developmentally ready to deal with the level of abstraction called for in a formal algebra course. While this may vary from student to student, we need to seriously question the value of pushing algebra ever further down into the middle school and elementary school. Rather, the NCTM focus on algebraic thinking gives us many ways to incorporate age-appropriate ideas that incorporate algebraic thinking from preschool on.

Question from
Dallas, Texas

I have taught math from grade 6 through AP Calculus.  In my opinion, algebraic thinking should be taught at the lower grades with the use of manipulatives, such as use of the hands-on algebra that uses scales and colored pawns. Many of the students in high school still lack a sense of what equality means and hence are quick to violate it. Also, it would help a great deal if properties were introduced in the lower grades with their correct names.  I have been accused of personally inventing the distributive property by a Pre-AP Geometry student who had never heard of it by the 9th grade, and of being a fool for not knowing what the "popcorn rule" is, as in her words, even the 4th grade teachers know what that is.

Further, a student who can spell Mississippi and knows its capital should also be able to state "subtract 5 from both sides" instead of talking about arbitrarily "moving it to the other side." Vertical consistency in notation beginning at the early grades would help a great deal also.  Everyone who has taught Algebra knows the problems that come up with the use of "slashy fractions," that is, when 1/2 evolves into the ambiguous 1/2x or worse 1/2x + 3.   

Another thing that would help is in the area of reduction of fractions and factoring.  If the lower grades learned to factor the GCF from the numerator and denominator then cancel, their students would not be as confused about canceling binomial terms in a rational expression involving polynomials.

I think NCTM can help by promoting vertical team meetings that involve all levels and grades, not just the MS/HS Pre-AP/AP teachers.

Cathy Seeley:
Your comments address a range of issues. I'll start with the last: I think meeting across grade levels is one of the most important things that can happen within a school system. Such meetings are most constructive when all involved both share with and learn from each other. Often, I have found that secondary teachers benefit from seeing some of the powerful mathematical ideas addressed before they see students. And all teachers benefit from seeing what content their students may later deal with in school. I think the most important content directions we can give elementary teachers is to teach for a balance of understanding, skills, and applications and to do whatever is necessary to actively engage students in their own learning. I also think that we can do more toward maintaining mathematical precision in terms of definitions and language, but only if terms are attached to a sound understanding of what they represent. And by the way, I have never heard of the "popcorn rule" either.

Question from
Oakland, California

I wonder if you agree with trying to level the playing field by publicizing the resources available at zero or nominal cost over the Internet. 

I am speaking of Web sites like AOL@School, Hotmath.com, Quickmath.com, etc.  There are many others.  These can relieve students of math anxiety when it comes to absorbing new concepts in algebra. 

If teachers incorporate them into homework assignments, then more students might be able to keep up in class and "get it" more easily. Especially those without math help at home or tutors.

Thank you!

Cathy Seeley:
I think the best way to level the playing field is to provide all students with the opportunity to be actively engaged in learning mathematics that will serve them in the future. In looking at a couple of such Web sites, it appears that they provide help on doing homework exercises from common textbooks or answering questions. This kind of help may be quite useful for some purposes, especially if there is limited assistance at home. However, I also hope our mathematics teaching extends beyond this kind of assistance toward the rich classroom experiences that help students learn mathematics deeply in a way that will stay with them through their future mathematics study.

Question from
Auckland, New Zealand

New Zealand has had a strand of Algebra in the mathematics curriculum since 1992, for students from age 5 up. This strand emphasizes algebra as relationships. For the first few years the equal sign and > and < are part of that focus. There is a continuing strand on patterning.

More recently a National Numeracy Project has been introduced for ages 5–14. This includes dealing with part-whole relationships in numbers for ease in mental calculation. For example 19 + 7 is done more easily as 20 + 6. We hold that the thinking behind this, which differs for different numbers and different operations, constitutes algebraic thinking.

My colleague Murray Britt and I have been doing research on the extent to which students can generalize from such examples to the correct use of algebraic notation to express this relationship as a variable.

Anyone interested in learning more about this can see Chapter 5 in my report available on
http://www.tki.org.nz/r/literacy_numeracy/
professional/2003Y7_9NPReport.pdf.
For a description of the algebraic thinking involved, write to me and I will send you a copy of a paper in press with Educational Studies in Mathematics.

k.irwin@auckland.ac.nz

Cathy Seeley:
Thanks for sharing this resource. I think the American mathematics curriculum can benefit a lot when we examine what is done outside the United States. I particularly like your early emphasis on the important ideas of equivalence and relationships. These are indeed central ideas to the development of algebraic thinking.

Question from
Archbald, Pennsylvania

Do you feel that calculators should be used on a daily basis in middle school math classes, specifically grades 7 and 8?

Cathy Seeley:
I think calculators should be available for students to use at these grades, understanding that the teacher helps students decide when and how to use them. It is critical that students learn to make these decisions. Teachers can help by identifying when calculators should not be used ("Put your calculators away; we're going to do some mental math.") and when they can be helpful (as in solving complex problems). We also need to make sure that we capitalize on the availability of calculators by giving students challenging problems that go beyond the limitations of what we can do with students when they do not have such access. Simply giving students long lists of computational exercises and then giving them calculators to do them defeats the purpose of having this tool available.

Question from
Oak Park, Illinois

This is not terribly profound, but it is a part of the puzzle of developing algebraic thinking preK–12:  Students need lots of experiences in varied real-world contexts of creating data tables (especially 2-variable T-charts or T-tables) and thoroughly examining the patterns in the data.  Then they need to graph the data and again thoroughly analyze those patterns.  Then they should relate the patterns in the graph to the patterns they saw in the table. 

I have done many such activities with students of varying backgrounds in grades 3 through 8.  With those students who are ready, we examine the patterns again and use the two variables to create expressions and then equations.

Cathy Seeley:
This is a wonderful summary of the power of learning how to represent situations in multiple ways. It provides students with the opportunity to see algebra as the study of patterns and relationships, which sets the stage beautifully for their increasingly sophisticated development of algebraic thinking and, eventually, symbolic procedures.

Question from
Chicago

There is a challenge with students coming into high school and understanding very few concepts of algebra.  There is a program in CPS to have students who scored lower than 50 percent on the Iowa Basic Skills test to take a double Algebra course as 9th graders.  Part of this is Algebra Problem Solving (IMP or Mathscape).  These programs do not seem to emphasize any real algebra equation work that the future tests call for.  Are problem solving curricula really beneficial if the goal is to pass tests that requires students to do equations quickly?

Cathy Seeley:
Problem-solving curricula are absolutely essential to prepare our students well for success in algebra and the courses that follow it. However, the flip side is not true. Teaching only equation-solving skills without adequate attention to understanding and problem solving is short-sighted and likely to backfire on students as they move deeper into the secondary mathematics program. Of course, balance is the key, and an ideal algebra program will include not only problem solving, but developing conceptual understanding and skills development as well.

Question from
Ft. Lauderdale, Florida

The first order of business is to ensure that the K–5 teachers have enough content knowledge to prepare the youngsters. Content knowledge is lacking in most K–5 teachers. Mathematics has evolved over the last decade, however, some teachers are still teaching for the industrial era. They believe that computation is the basis of elementary education. Some elementary teachers are not comfortable teaching mathematics. NCTM could offer online courses to help elementary teachers acquire the needed content knowledge with which to help the K–5 students develop a strong solid base on which middle and high school teachers can build.

Cathy Seeley:
Teaching algebraic thinking beginning at the elementary level presents increased demands on elementary teachers' understanding of mathematics at a deep level. Professional development is a priority for NCTM. Online courses offer great potential for delivering professional development to teachers who might not otherwise have access to it.

Question from
Fairburn, Georgia

Many teachers don't know what "algebraic thinking" looks like in the elementary classroom. If they did, they would feel more comfortable when they are told to include it in their curriculum.

Cathy Seeley:
This is another reason why professional development plays such an important role. Teachers of mathematics at all levels, not just elementary, need to make a lifelong commitment to their professional growth. For elementary teachers, experiencing the kind of algebra that is conceptual and relevant can be a liberating experience, not to mention the benefits for guiding their students' learning.

Question from
Marion, New York

We can help elementary students look for patterns through the combined creation of tables, graphs, and equations.  Even young children can extend patterns to create tables, learn to graph the results, and describe the pattern of the table and the graph in words.  Students can think and question critically in an algebraic context when they are given the opportunity to extend a problem through patterns and see it through a visual model. For example: My third graders create tables and graph the story of the Gingerbread Man.  If the Gingerbread Man begins to run as soon as the oven is opened, and runs 2 feet per second, the wife begins to run 2 seconds after the oven is opened and she runs 2 feet per second, the wolf begins to run 5 seconds after the oven is opened and he runs 4 ft per second does the Gingerbread Man ever get eaten?  They pull amazing stuff off the tables and graphs.

Cathy Seeley:
This is a nice way to develop informal ideas of algebra. Thanks for sharing your example!

Question from
Tempe, Arizona

People who share these views are usually "burned at the stake," but several years of my 39 years of teaching math I taught with the Saxon series.  Students liked it and did better on college entrance exams! It was an incremental development and geometry was integrated throughout. Students could go at their own pace. Independent study was much easier. It de-emphasizes the sacred role of the teacher. Students wanted their own copy of the text to use in college. (Most students would like to burn their own copy of the math text they had to use!)

And yet it was banned from use in some areas because it didn't have colored pictures and it supposedly did follow the standards.  If students do better in college because of it and like it much better, then maybe the standards need to be revised or replaced.

Cathy Seeley:
Thanks for your honest and direct comments. You are correct that the Saxon text has not been adopted in many states and systems. Teachers tend to have strong feelings either for or against the program, and, frankly, results are rather mixed. I think that if students are to take responsibility for their own learning and become independent algebraic/mathematical thinkers, they need to come to rely more on themselves than either the textbook or the teacher. I continue to believe in the importance of the teacher, not for telling students things, but for structuring a classroom where learning is likely to happen. A textbook is only a tool, and it cannot address the wide range of problems students will encounter. It may help students deal with certain types of problems, and that's great. My preferred model of teaching would have students engaged in a variety of activities, including a good dose of small-group work on solving engaging problems that draw students into figuring out how they will use mathematics to find solutions. The role of the teacher is critical not only in structuring and facilitating their work, but in asking good questions that push students' thinking farther than they thought they could go. One of the strengths of NCTM as an organization is the diversity of views of its members. When we openly discuss our different points of view, we can all become better educators.

Question from
San Francisco

I am a Math Learning Specialist.  I help learning-disabled students really understand math.  Why do so few math teachers understand the ways in which various learning disabilities hamper students' abilities to master math?  

Cathy Seeley:
Unfortunately, many teacher education programs are limited in the amount of time they can spend learning about special needs students of all kinds. One of the roles you can play is to connect classroom teachers with resources, including yourself, on how they can better meet the needs of their students. It's great to have someone in a resource role who understands both mathematics (at a very deep level) and also special needs students.

Question from
Centreville, Virginia

I love this topic!  I am constantly trying to find ways to incorporate algebraic thinking in my classroom both within formal mathematics discussions and in the thinking that is essential when students are solving practical applications of algebraic concepts to real life.

I think that weaving the strands throughout mathematics is very important in grade school and middle school.  I do, however, support the structure of separate courses that exist in many of the schools throughout the United States. This approach allows students to concentrate their attention on one strand so that the topic can be studied in depth.  I taught mathematics for 15 years in California, and the schools in which I taught had this form of "traditional mathematics."  However, I tutored students who were attending schools that had adopted "integrated mathematics."  Their understanding of mathematics was shallow and they felt scattered.  One student in particular, who was actually naturally very talented in mathematics, was very frustrated.  Her conclusion was that she was just dumb and unable to understand math.  She gave up somewhere in her sophomore year and couldn't wait to unload that course.  I kept trying to assure her that she was actually very talented in mathematics, but her test and quiz scores were lower than she wanted and only furthered her self-assessment in this field.  I was very sad for her.  Her brother, on the other hand, went to a different school that had a traditional approach.  He was also talented in mathematics but was lazier than his sister.  He still excelled in the subject and continues to be confident that he is good in mathematics.  What was sad for me was that both students were extremely capable, but only one believed it.  I attribute that, in part, to the way they experienced mathematics in the classroom.  The girl's understanding was a mile wide but an inch deep.  The boy's was narrower, but he had a deep understanding of algebra and algebraic thinking that allowed him to solve complex problems and feel successful in mathematics.  The breadth of his knowledge would undoubtedly grow as he took more courses. 

These are not the only students with whom I have had this experience.  I have actually tutored many students in both integrated mathematics courses and in traditional courses and have seen the same results across the board.  It is, to me, striking. 

I actually had many students come to our high school having been in an integrated program with a similar experience.  They had very little exposure to in-depth algebraic thinking and a smattering of knowledge about different strands of mathematics.  They generally demonstrated an inadequate knowledge of Algebra when tested for placement in Geometry and would be enrolled in Algebra I for the school year.

Perhaps with an excellent textbook and an excellent instructor, integrated mathematics would be able to allow ALL students to be successful. I don't know. Certainly there will always be those brilliant students who could learn mathematics on their own without regard to the methodology implemented.  What I have seen generally, though, is that otherwise talented students feel frustrated and inadequate.  It seems like a well-intentioned program is failing them in this country.  

Cathy Seeley:
Thanks for your participation in this chat. You raise an important issue. I honestly think that we don't have adequate information to determine whether an integrated program works better than a traditional one at the secondary level. The rest of the world uses such an approach. Perhaps one of the issues is getting clarity about what outcomes we want to see. If we are evaluating students on traditional equation-solving skills, then it may well be that students coming from a non-traditional program might not perform the skills at the same level at the same time as more traditionally prepared students. However, if we evaluate the ability to use algebra, for example, to solve diverse problems, I think we would all agree that many of our students over the years have not understood how to apply the skills they learned to the problems they encountered. I found this over and over again in teaching and tutoring students at this level. And this was the case when algebra was taught only to the more successful students. If we are now to help more/all students learn algebra and higher-level mathematics, I think we must be open to different ways of teaching and even different priorities in terms of the content we address and the organization of the curriculum.

Question from
West Liberty, Ohio

I've long been a believer in an integrated approach to mathematics, combining algebra, geometry, statistics, and data analysis.  We have not taught a formal geometry class at my high school for at least the past 25 years.  The state of Ohio seems to be moving in this direction also with the state standards, indicators, and the Ohio Graduation Test.

Cathy Seeley:
There continue to be success stories where school systems, and now states, may be moving in this direction which is widely used outside of the United States.

Question from
Oakland, Maine

I think there is a misunderstanding of what "algebra" is with teachers, students, and parents. Many teachers are reluctant to use the term algebra in the early development of mathematical thinking, and children begin to think of algebra as a monster that is too scary to conquer.  Parents are always willing to jump on the "My child needs to be taking algebra" bandwagon, without looking at the development of the child's mathematical thinking and realizing the background is rich with algebra.

I feel teachers need to use terminology and make connections to algebra at an earlier stage of a student's mathematical development.  If this happens kids will begin to work with concepts more fluidly, and parents will realize that algebra is not a separate subject but part of understanding of what we call "math."

Cathy Seeley:
It would be great if students didn't think of their mathematics experience as a set of isolated topics. That's just the goal of incorporating algebraic thinking as a strand within a balanced mathematics program PK-12.

Question from
Towson, Maryland

What can we do about the biggest textbook publishing houses that continue to produce and peddle those books? How do we "push" school districts to consider alternative programs such as IMP or CORE or Connected Mathematics?

Cathy Seeley:
I think we may be focusing on the wrong battle. If we pay attention to what mathematics we want to teach and help teachers grow through professional development that lets them learn how to engage students in learning, then I trust teachers to demand tools that support that learning. Publishing is all about supply and demand.

Question from
Culpeper, Virginia

The current math standards of many of our states create barriers to offering integrated secondary math courses.  For example, in Virginia, we were beginning to offer math this way, but in most cases have backed away, since our state standards are fairly traditional and require tests in Algebra I, Geometry, and Algebra II, each separately.

Cathy Seeley:
Our accountability systems do influence what we teach. But I would argue that if we teach a rich, balanced problem-solving-focused program that includes skills, concepts and applications, our students can do fine, even on low-level skills-based tests. (See our chat from last month on accountability)

Question from
Jefferson, Ohio

Our guidance counselor says that the colleges and universities are not on the same page.  They want to know whether a student has had Algebra II, Trig, Calculus, etc.  How can you change everyone's thinking?

Cathy Seeley:
The state of New York, for one, has offered integrated mathematics for many years with no negative impact on students entering college. I have found guidance counselors generally quite open to meeting the needs of students, but often uninformed about how to do that in mathematics. As mathematics educators, we have a responsibility to work together with counselors, providing accurate information from the universities and colleges most often selected by your students. It may be appropriate to engage in conversation with mathematics faculty from these post-secondary institutions to get a clear message of what will work. If there are, in fact, barriers at that level, solutions will start by connecting and communicating.

Question from
Princeton, New Jersey

How can assessment be modified to encourage the integration of topics and early attention to algebraic thinking? Do assessment strands such as Numbers, Measurement, Geometry, Data, and Algebra help or hinder?

Cathy Seeley:
I think the strands do not in and of themselves hinder learning. I will return to the notion that if we teach a good, rich, balanced mathematics program, our students can do fine on pretty much any test they encounter.

Question from:
36. Taichung, Taiwan

I'm currently teaching in an American school in Taiwan and we still use the Algebra I, Geometry, and Algebra II courses.  My own children, however, went to school in Arizona where those courses were integrated.  My son seemed to pick up on the various concepts great and had a great score on his SAT while my daughter, on the other hand, struggles to understand math and had a very hard time with the SAT test.  She reviewed for the test quite a bit and was unable to have any concrete knowledge or comprehension of the algebra concepts that were still Algebra I concepts but the more advanced ones.

My first year here, I taught the middle school math where the math is integrated, streaming up from elementary as well as the 8th graders that were placed in Algebra I.  Having done both at the same time, I did note that it was very beneficial for my 8th graders to learn the algebra concepts—although they had a hard time understanding them—and then see them all the way throughout the book for the rest of the year.  I continually heard comments such as, "Oh, I get it now. I never understood that before."  It was easier to teach Algebra as a coherent class and much less time consuming in preparation.  I also found that the more advanced 8th graders who were in the Algebra I class did fine on the formula memorizing and plugging in numbers, but they had a difficult time with problem solving and the pace we needed to keep in that class. And our block program didn't allow us to delve into the problem solving very much.  I've moved into the 5th grade classroom this year, and they were just introduced to the concept o =
"n."  You should have heard my room; you'd have thought Martians had landed.  Once they calmed down, they could do the math, they just had to learn to rearrange the equation so they'd be ready later on.

Previously, I taught kindergarten.  They were quite open to problem solving because they didn't have any preconceived notions of how things had to work.  I had my kids really doing multiplication and division before the year was over as well as carrying and borrowing, but they weren't afraid of it because it was encased in games.

I've noticed that we tend to want the kids to know more per grade level as the years progress so we can do more with them in the later years, but if they don't spend enough time with some very concrete things at a young age, they have no real concept of mathematics for the more abstract things.  Unfortunately, we are in a time crunch that makes it very difficult to spend the time with the concrete manipulative—the way I would like to at least.

Cathy Seeley:
Students learn in different ways, and there are many variables that affect their learning. As teachers, we must continue to offer a variety of learning experiences and give students opportunities to deal with the mathematics themselves, not just watching the teacher present the "rule du jour." I agree with you that our curriculum is still too crowded. We need to take the time to talk with teachers above and below our grade level to identify where we should spend more time and where we can spend less. We certainly need to free ourselves from the belief that we have to teach every page, or even every chapter in a textbook. Developing lasting understanding, as you observe, takes time. And I would argue that this time is not only worth it, but necessary.

Question from
Vilas, North Carolina

What kinds of staff development is offered to help teachers gain a deeper understanding of the content in algebra? Where can teachers find extensions/projects to differentiate instruction in the algebra strands in all grades?

Cathy Seeley:
I would refer you to your local and regional educational centers, as well as the universities in your state. North Carolina has done some nice work in assessment and mathematics over the years. In terms of NCTM, check out the Algebraic Thinking Focus on the Web site. Look for the magnifying glass icon, and you will discover a wealth of resources. The Navigations and Illuminations examples are among many others.

Question from
Naperville, Illinois

There is increasing pressure for all students at the eighth grade to take algebra. I put the emphasis on all, because there appears to be little or no concern on ability or prep for the course. It is just implied that students do better on tests when they have had algebra. The concern tends to be with math test scores and not with the content or subject matter that should be presented to the students at any level. What are your thoughts on this aspect of the mathematics curriculum?

Cathy Seeley:
I think we must be careful if we move all students into a formal algebra course. Whether we do this or offer a rich middle school curriculum that develops a strong base of algebraic thinking, we should not shortcut the development that leads to symbolic understanding. For example, one of the most important concepts that leads to success in algebra, in my opinion, is a solid understanding of proportionality well beyond the simple study of ratios, proportions, and percent. This critical connecting concept needs to be well developed, and often, when the middle school curriculum is compressed, students do not develop this understanding. I have shared in other responses my concerns about what happens to these students in high school. The main thing is to prepare all students well for success in four years of academic mathematics so that they can succeed in college.

Question from
Ridgefield, New Jersey

What are your thoughts on how much algebra should be taught at the middle school level and also being from an urban area where many students are low level? How do we catch them up?

Cathy Seeley:
There is increasing evidence that many students thought to be low-level, especially when they come from a background of poverty, are in fact victims of lack of access to educational opportunities. The best thing we can do is to hook them into engaging tasks where they get to show us what they can do. I have found in situations like this that we can often discover new stars among students that nobody thought could succeed.

Question from
Mobile, Alabama

I have children that are very weak in fundamental skills such as facts and procedures.  As we all are aware, high-stakes state tests are trying to push me to objectives that I feel won't be fully understood if that groundwork is not laid down properly.  When administration comes in and sees a lot of broad framework and understanding being done, they get nervous.  They do not see the DIRECT OBJECTIVES being covered.  An incentive bonus at the end of the year is at stake for test grades for all the teachers here.  After the long-winded introduction—Is there a point at which you can feel safe that the students will be motivated to go back and work on facts and operations while you're pushing ahead to stuff—like simplifying fractions for instance—or do you feel that something that low level has to be tackled before moving on?   I read the Crystal story, but this is more elementary. 

Cathy Seeley:
This is a real and challenging problem. First, I continue to believe we must teach a mathematics program that has integrity. We cannot be pulled down to teaching in ways that we think do not serve students well for their future. If we teach a sound mathematics program, grounded in understanding, driven by challenging and engaging problems and inclusive of computational development, our students will do fine on the tests, even low-level tests. And do not underestimate the potential of a calculator to allow students access to problems that would otherwise be beyond their skill level. If motivated, students can learn these higher-level skills. A comment Crystal made could be made by an elementary student with this kind of success as well—when the opportunity arose for her to learn fraction operations AFTER succeeding in algebra, she observed: "The time has come; I'm wasting too much time using my calculator to do fractions." I'm not suggesting giving up and handing out calculators for all computation. But we must allow students the opportunity to solve problems. The biggest difference in test performance among groups of high-performing and low-performing students is not found in low-level facts and procedures. Rather, the differences show up in the more complex problem-solving situations that many students have never experienced.

Question from
North Chicago area, Illinois

I think one of the major issues we are faced with as a 'Nation at Risk' is how to present mathematics problems in a way that encourages students (not discourages them) to think and interpret what is being asked for themselves—and then validate their non-linear thinking.  Too often, teachers approach a problem with a particular agenda and do not encourage creative solutions to problems.  Here's an example of what I mean:

In a 3rd grade lesson about place value, the opening question for individual reflection looks like this "With the follow numbers, what is the largest number that can be made and what is the smallest?  (given  8,  5,  3,  0,  1,  9)   What teachers sometimes do is put so many constraints on problems that students are encouraged not to think creatively? (The teacher WANTS 985,310 and 103,589 so she/he adds statements to the original – carefully crafted – question, like, since there are 6 digits you need a 6-digit number in your answer … whereas students might come up with all sorts of other creative notions (exponents or decimals or mathematical symbols). 

Or

Talking about patterns 2,000  1,200   800  600  500  ___  ___  and having a student describe how we divide the difference by 2 each time (which is clever but since the teacher has a particular agenda, the answer is not accepted and dismissed in search of the 'correct' answer that was being sought (1/2 of the difference is being subtracted).  Both are correct and furthermore making the connection between should be celebrated and emphasized!!!

I think this does a disservice to students' development of mathematical thinking.  I don't think there is ever a moment where a teacher should not take every opportunity to point out connections between concepts.  I am keenly aware that not all teachers of math are acutely aware of all these connections but shouldn't that be a priority of ours to help make meaningful connections and/or set up meaningful connections in the years to come?  Do you have any thoughts on the relative importance of this notion?

Cathy Seeley:
Increasingly, I believe that the role of the teacher is to ask good questions, not tell students answers (or even hints). Far better than minute coaching are questions like: How do you know? Why do you think so? Would that still be true if you had twice as many? And so on. I think you are correct that students need opportunities to express their developing algebraic/mathematical thinking, even if that goes in a different direction than the teacher expected. The key is for the teacher to help the student connect what the student has done with the appropriate mathematics being used so that the student has the possibility of using what he or she has learned in another situation.

Question from
Savannah, Georgia

I teach honors Algebra II and a couple different levels of precalculus.  Though I have the students use their calculator often in class to enhance their understanding of many different concepts, I still mix in non-calculator quizzes and portions of tests that require good computational skills and a command of operations on fractions.  Is this overkill or a necessity for students who should be preparing for calculus courses and need a strong algebraic foundation?

Cathy Seeley:
I think it is helpful to reinforce skills previously taught, and I definitely think some non-calculator work is appropriate, especially for mental math. However, I also think we need to put in perspective the level of computational proficiency that is actually necessary for success in higher-level mathematics. It may not be a useful way to spend time in algebra II to have students do pencil-and-paper long division, for example, when they are unlikely to need this algorithm in the future, and when there are more important concepts they need to practice. But the teacher can choose how to balance the program so that students' needs are best addressed.

Question from
Hopkinsville, Kentucky

Has the NCTM studied the types of courses and curriculum used in England, Spain, or even South American countries.  I often had foreign students, and their problem solving skills left my students in the dust.

Cathy Seeley:
There is mixed evidence about problem-solving expertise in other countries. One of the complaints of some international mathematics educators is that their students are better at skills than at creatively solving problems. However, there are definitely some interesting programs in problem solving, especially from England and the Netherlands. (I'll have to check out Spain and South America a bit more…) In the Unite States, our most traditional skills-based secondary programs probably did leave many students ill equipped to handle more complex problems. Consequently, many of the newest mathematics curriculum programs have incorporated the more innovative aspects of some of these programs in their focus on problem solving.

Question from
Eldersburg, Maryland

I am struggling with helping teachers understand the importance of teaching alternative algorithms as a way of facilitating an understanding of algebra. In order to be able to manipulate an algebraic equation, very young children can think about and discover many ways to solve a regrouping with addition problem or subtraction, multiplication, and division.  They learn to think about pure computation in a multitude of ways.  How can we help elementary teachers see this as a valuable part of their instruction?

Cathy Seeley:
I think that seeing other approaches to solving problems is valuable, as is experiencing other ways to perform computational skills. Other countries often use different computational algorithms than we do. Among other benefits, students can see that mathematics is not magic, but rather, efficient ways to deal with numbers and problem situations. Professional development can help teachers themselves experience alternative algorithms, which can provide them with insights into student thinking.

Question from
Saint George, South Carolina

How do you get students out of a find-the-answer mode and into understanding that problems can have multiple solutions?

Cathy Seeley:
We have conditioned students through our teaching techniques to look for quick answers. The best way to combat this kind of student thinking is by teaching differently. When we engage students in interactive solving of problems that may not easily be solved and/or that may have more than one possible correct solution, over time they can learn to trust their own thinking, rather than trying to guess which rule to use.

Question from
Cleveland

How should algebraic thinking best be developed for middle school students?  Many people believe that a traditional algebra course, focused on rules and manipulations, should simply be taught earlier.  Shouldn't middle school algebra look different than that?

Cathy Seeley:
YES!!! Middle school mathematics should be a time of rich exploration of new ways of thinking. It should also be a time of powerful connections between elementary work with numbers and operations leading to symbolic awareness and the ability to make generalizations. The most important connecting idea at this level, in my opinion, is the development of proportional reasoning. If this is done well, students can see how quantities can grow proportionally, leading naturally into the study of linear relationships. Whether we offer algebra as a course in middle school, or whether we develop algebraic thinking that sets students up for success in an algebra course later, middle school mathematics needs to be a rich, balanced program, with an emphasis on representing and solving problems.

Question from
Macomb, Illinois

Can you say some more about "early algebra," especially the need to think in abstractions and understand the varied meaning of the equal sign?  For example, in the 70s the program Developing Mathematical Processes had young children working with equations such as W + D = B + C.  Perhaps we should try to work with these ideas even before the push to use numbers and counting to verify answers.

Cathy Seeley:
There are many ways for students to explore patterns at an early age, especially patterns dealing with equivalence, such as the one you suggest. Students can explore equivalence with balance scales (I believe these were an important tool in the DMP program; they used a nifty two-piece blue plastic balance that was nearly indestructible), representing what they find with pictures and, possibly, basic symbols. The key idea is that this kind of mathematical exploration can plant important seeds that can develop into algebraic understanding if nurtured well.

Question from
Reston, Virginia

Robert Gagne offered a hierarchy of learning that progressed from verbal skills through problem solving - with intermediate steps such as concept learning, analysis, synthesis and principle learning in between. Similarly, wouldn't we do well to pursue a generally acceptable definition of algebraic thinking? The more precisely we can define our objectives the better our chances of developing strategies for achieving them.

Cathy Seeley:
It is often helpful to rely on a theory of learning that lays out developmental or instructional stages. In the case of algebraic thinking, there have been several articles and other publications that address this development. One excellent source is Principles and Standards for School Mathematics. Other NCTM resources can be found through the NCTM Web site, especially through the articles and links included as part of the Algebraic Thinking Focus (look for the magnifying-glass icon).

Question from
Fairfield, Connecticut

At what grade level can you start to involve students in critical thinking?

Cathy Seeley:

If you ask my friend, Denise, an outstanding kindergarten teacher, she would tell you that you can involve students of any age in appropriate critical thinking. Asking students to justify their thinking and make predictions are just a couple of ways we can do that.

Question from
Lexington, Kentucky

You wrote, "The rest of the world, including our colleagues in Canada, teach mathematics, not as separate courses, but as a continuous program from elementary through secondary school. In the United States, some schools offer an alternative, such as an integrated program that incorporates algebra as a strand blended with geometry and other advanced topics."  I would be interested in knowing about comparisons of the "successes" of such approaches in developing algebraic reasoning and conceptual understanding.

Cathy Seeley:
In terms of international comparisons, the TIMSS studies are the best source of information. For the integrated secondary programs in the United States supported by the National Science Foundation, you can visit the Web site for the dissemination of information about these five programs at: http://www.ithaca.edu/compass. The biggest challenge in answering your question is that we don't have widely accepted measures for algebraic reasoning and conceptual understanding to use in comparing various approaches.

Question from
Acton, Indiana

As we strive to achieve the goals of "No Child Left Behind" and raising standards/expectations of our students in the middle school environment, it seems that these two ideas do not always mesh very well.  Many schools have either increased the number of students enrolled in Algebra I courses in the middle grades and/or have begun teaching Algebra I and geometry courses. Unfortunately, some of these courses have led to algorithmic teaching of basic algebra skills instead of an understanding of a variety of mathematical concepts such as those outlined in the Principles and Standards for School Mathematics. 

Would many students in the middle grades be better served if an emphasis were placed on developing understanding a wide range of concepts and standards instead of offering a traditional algebra course?

There area several factors that influence the necessity to offer an algebra course and/or a geometry course in the middle grades, including demands from the local school board and parents as well as restrictions of coursework due to limitations of textbooks.  Should there be another course offering developed for the middle grades, which would include the development of student understanding in the areas of algebra, geometry, data analysis, number sense, and measurement?

Cathy Seeley:
This seems like an important balance for all students to experience during the middle school years. Even if we teach an algebra course, I would hope that these elements would precede the course, and also would be incorporated appropriately in the algebra course itself.

Acton, Indiana
What is the best practice to identify students that would be best served by a traditional algebra course?  

Cathy Seeley:
Increasingly, I am coming to question whether any students are well served by a traditional algebra course, if by that you mean a course focused on learning abstract rules and procedures. Even students who have been successful in such courses in the past could well have benefited from experiences with graphical and other representations, as well as focusing on using algebra to solve complex problems, rather than primarily solving certain types of story problems.

Question from
Besancon, France

What are the main weaknesses that have been identified in student achievement in the algebra field, and at which levels? What does NCTM suggest to try to overcome these problems?

Cathy Seeley:
For many years, students who study algebra have had difficulty applying what they learn to solve problems. Teachers have searched for ways to 'teach story problems.' As we try to teach algebra to more students, this problem is exacerbated, especially in traditional algebra classrooms focused on abstract rules and procedures. Principles and Standards for School Mathematics calls for a much richer vision of algebra, building on the use of functions to represent and analyze relationships. In such a program, many or all students can learn a range of problem-solving approaches that serve them well regardless of the type of problem.

Question from
Sigourney, Iowa

I use an integrated curriculum and use a lot of explorations. The students do very well with it, but I am having difficulties getting that knowledge to transfer to textbook problems. Any ideas about how to ease this transfer?

Cathy Seeley:
If your students are successful in a good integrated program, they are likely learning important knowledge and skills. In terms of transferring to textbook problems, I'm not sure which types of problems you mean. If you mean routine equations to solve, this is an important skill that may call for more practice than they have had, connected to what they know about how equations help in solving problems. If you mean typical types of word problems (coins, age, etc.), you will need to decide how much time to spend on such problems based on your curriculum and whatever test is used in your accountability system. Otherwise, it may be reasonable to question to what extent traditional textbook problems are useful. But I continue to believe that a strong program that balances skills, conceptual understanding, and problem solving is the best preparation we can give students for whatever types of tests they face.

Question from
Accra, Ghana

What will be Algebra III, Algebra IV, Algebra V, etc.?  I think algebraic concepts pervade the mathematics curriculum from K to university.  So segmentation might not be useful.  Rather, we must find ways to teach algebraic concepts appropriately to different groups of students and help them to make sense of those concepts as they relate to the students' daily lives.  For example, commutative law remains commutative at all levels K to university.  How can we teach it to K students so that they can recognize it in terms of a + b = b + a at the middle school level?  Can they also apply the concept in real-life situations?  I think these are some of the fundamental issues to be concerned with.

Greetings from Ghana.

Cathy Seeley:
Hello to Ghana. I agree with you that segmenting the curriculum in this way does not serve students well. There are many ways to connect mathematics, including algebra, to students' lives both in and outside of school. Many elementary teachers give students experiences dealing with commutativity as a concept, even if they may not give it a name. Developing many properties and characteristics of equivalence is definitely worth the investment at the elementary level.

Question from
Springfield, Missouri

When parents ask, how do we respond to the question:  What is so important about algebra anyway?

Cathy Seeley:
Algebra has long been a gatekeeper, and the evidence is solid that it is critical as a first step toward college. Since many students don't know themselves whether they are likely to attend college, we need to start them along this path. But this in itself is not enough to drive us to call for algebra for all students. Algebra, as described by today's vision in Principles and Standards for School Mathematics, is a powerful set of tools that helps students represent relationships, make generalizations, analyze situations, make predictions, and, most importantly, solve many kinds of problems in mathematics, other disciplines, and life outside of school. Today we know how to connect algebra to engaging and useful situations. Invite parents to experience this kind of problem experience in your classroom (ideally) or through materials you send home.

Question from
Kingfield, Maine

As a high school mathematics teacher, I see many students that come in to my classroom without basic number sense.  Would you think that before we can work on algebraic thinking, we need to work on number sense first, or do you think they can (or even should), be emphasized simultaneously?

Along those lines, it would seem that part of the answer to that question depends on the resources of the school and district itself.  If developing algebraic thinking at the elementary and middle grades requires the purchase of manipulatives and other items not normally in a math class budget, then it would seem to be prohibitive for smaller and more rural districts.

Thoughts?

Cathy Seeley:
Absolutely, number sense is critical as a primary goal of elementary mathematics. I believe that incorporating algebraic thinking can support this number sense. For example, as we develop the idea of equivalence, students can learn multiple ways to represent a number. As students explore patterns with numbers, they are laying the foundation for more sophisticated algebraic patterns later. These goals can and should be connected. I might argue a bit with your statement that manipulatives and other tools might not normally be in a math class budget. These are as essential to the teaching of mathematics as maps and globes to the teaching of social studies or science equipment to the teaching of science. There are ways to economize, but basic counting, place-value and geometric manipulatives, to name a few, should be a priority even in small schools.

Question from
Indianapolis, Indiana

Do you have some data indicating how parents can support their children in fostering algebraic thinking?

Cathy Seeley:
There are many ways for parents to help their students developing algebraic skills. Asking students to notice patterns, formulate generalizations and make predictions is a great way to do this throughout elementary and middle school. This can start with simple situations like the sequence of lights in a traffic light and can extend to noticing patterns of house numbers. Asking students to give many names for a number and talking about equivalence can also help. Having students justify their thinking any time they are working with a mathematical situation is helpful. Questions like "How do you know?" "Why do you think so?" "What else can you say about the situation?" can also be useful.

Question from
Naperville, Illinois

Mathematics programs: Where does one find an unbiased evaluation of mathematic programs? That is, one that looks at the aspects of the curriculum and gives an honest refection of how the programs meet the goals and objectives of NCTM.

Cathy Seeley:
This is the question of the hour (and day and year). Gathering and packaging this type of information is a priority of many efforts right now, both within and outside of NCTM. Recent efforts by the U.S. Department of Education and the Mathematical Sciences Education Board have told us that there simply is not enough information to make such judgments at this point. Individual programs are often accompanied by related data, but it is important to look at studies across districts and schools. Instead, we can look at recommendations such as those presented in Principles and Standards for School Mathematics that are based on more focused studies of effective practices.

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