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Pushing Algebra Down

chat archive


Moderator
Good afternoon and welcome to an online chat with NCTM President Cathy Seeley. Cathy selected today’s topic, “Pushing Algebra Down” with the intent of generating a diverse discussion, and if the number and variety of questions submitted in advance is any indication, she has succeeded. We’ve received an exceptionally high number of questions, and we’ll answer as many as we can during the hour. A complete transcript of the chat will be posted on the NCTM Web site tomorrow.

Our first question is from
Birmingham, Michigan

I respectfully disagree. Many of today’s students are ready for courses that will present a challenge. It is all about the delivery.

Students need to have choices and diversity. Combined with the use of technology; algebra and other topics of mathematics can be explored and studied with a more visual and diverse delivery that enhances the learning styles of today’s students. My major concerns are centered on providing our students with successful experiences at a pace that they are comfortable with and can understand. Most traditional algebra courses are more concerned with completing a book than truly facilitating a successful interaction between the student and math material being covered. Very few students’ learning style is ready for abstract learning at lightning speed at any grade level. This should be the focus: Pacing and Use of Technology! When these two topics are part of the formula for any grade level Algebra course, it can equal success!

The students are excited to go to high school to learn more, and then are shut down by the same assembly-line style/delivery of material—get through the material regardless if the students get it or not. I am willing to question that this is the catalyst behind the problem that you view???

All students are ready and very capable of learning algebraic topics at a comfortable pace with technology. We need to prepare our students for a rapidly changing world that will require quick problem solving, teaming skills, and use of technology that will help build the confidence of our students. Let’s embrace and excite our future mathematicians with choices and challenges at all grade levels:)

Cathy Seeley:
Actually, I agree with much of what you say. When algebra as a course is taught as it often has been—focusing on teacher presentation of abstract concepts—then many students will not be successful. What we need is a more continuous development of algebra from PK through grade 12. If there is continuity, if we give middle school students the opportunity to learn challenging and relevant mathematics, including proportional reasoning, and if students are engaged in working on problems that require a bit of thinking, then many of my concerns are addressed.

Question from:
Redwood City, California

The state of California has made Algebra 1 an 8th grade standard.  That means that in California we have no choice but to try to teach it.  In addition, my school is both a PI school and an IIUSP school, and the Evaluators (paid for by the feds, I think) have recommended that we teach all our 9 th-grade students from one of three state-adopted texts, all three of which are very traditional, and assume knowledge of arithmetic, which is a false assumption for not only our Title 1 students and ELL students, but also for a considerable number of mainstream students.  I cannot see a way out, until someone recognizes that just as you cannot teach a 2-year-old baseball because they are not developmentally ready, it is asinine to try to teach algebra to a student not developmentally ready (check out Piaget’s work) for formal operations and abstract reasoning.  My students are as happy as clams solving equations using algebra tiles.  However, when I push them into the abstract they take leave of their senses and write nonsense!  I believe we have to align the curriculum with the developmental stages of the child as the very first step!

Cathy Seeley:
Alignment of curriculum with what we know about how learning happens should indeed be central to curriculum. Expecting more of students need not be reflected by pushing down arithmetic and algebra procedures into earlier grades. Teachers have to address their state's expectations. Perhaps by incorporating rich activities that address both concepts and skills, and by having students actively engaged in the mathematics, we can make some progress. We might also look at weaving more algebraic thinking into the grades before the course called "Algebra."

Question from
Waukesha, Wisconsin

After many years of experience teaching students at risk, I know that they are able to master basic algebra if we address their learning styles and ‘where they are’ in their skill at abstraction. By taking the TIME—and I emphasize time—to use contexts and to move students from concrete to pictorial to abstract experiences, with a variety of representations (objects or tiles, tables, graphs, number lines, recursive rules, and functions, etc.) students develop deeper conceptual understanding. They develop flexibility in moving from pictorial/graphical representations to symbolic and vice versa. They are able to select preferred ways to reason about integers, variables, first- and second-degree equations, patterns, etc. The teacher also gains tools that facilitate instruction and is able to refer to prior lessons that involved contexts or models by saying, “Do you remember, (context)..?” or “What ways did you use ... to represent...?” “Which representation do you prefer and why?” “How is ...representation related to ...representation?”

Cathy Seeley:
These are great ideas. I think they can be effective as part of a strong middle school mathematics program, regardless of whether a course called Algebra I is part of it.

Question from
Winston Salem, North Carolina

The way to teach students mathematics is to put mathematics teachers at all levels. The ONLY way students will be prepared for mathematics earlier is if they have a strong conceptual understanding of mathematics. The only way to do that is to put teachers with mathematics specialties in the elementary grades!!!

Cathy Seeley:
There is no question that a committed, caring, well-prepared teacher is the most important factor in mathematics learning--regardless of the curriculum, school structure or textbook.

Question from
Kalamazoo, Michigan

Most of my curriculum involves the use of patterns. I take my students to places where they can relate to patterns to life, art, even music. When I present a new concept, it involves patterns and the kids approach it from a pattern point of view versus the new concept. Sort of sneak it in on them and then one day all of a sudden they know the concept. Patterns solve anything!

Cathy Seeley:
This is a nice example of incorporating algebraic thinking across the grades. Thanks for sharing!

Question from
Minnesota

My 8th grade algebra class stimulated me in ways that a typical 8th grade math class could not. It's unfair and immoral to hold back students based on age. Minds develop at different paces, and it's important to remain in the ZPD. To ask teachers to differentiate to the extreme is unreasonable and impossible.

Moderator
ZPD is zone of proximal development.

Cathy Seeley:
There are undoubtedly students who benefit from moving into the study of algebra in 8th grade, and maybe even some who could benefit from it earlier. I think we can do a much better job of facilitating student development by the way we teach mathematics, particularly at middle school, and by choosing mathematics content that is important and engaging (like proportional reasoning). Perhaps we would find more students who could be successful at algebra in grade 8. I would still suggest that we should be very careful before instituting programs that put all students into algebra sooner. We need to be careful not to miss rich middle school content and also ensure that we are supporting students in continuing their study of mathematics through grade 12.

Question from
Brooklyn, New York

Although I have been very much in tune with your essays in the News Bulletin since the “Embracing Accountability” one last summer, something seems amiss between your current one (“Pushing Algebra Down”) and the way you were quoted in yesterday's New York Times article about the state’s returning to Algebra I as a well-defined first course in high-school mathematics after an approximately 20-year hiatus. According to the Times article, one could get the impression you did not favor a return to an exposure to algebra, the language of mathematics, as the exclusive domain of a first high-school math course. Thus, please clarify your comments to the reporter.

Cathy Seeley:
Thanks for allowing this clarification of my quote. My comment was related to the fact that nearly all other countries, especially those that outperform the United States in mathematics, have a continuous mathematics program from elementary through secondary school. The course we call 'Algebra I' is unique to the United States. In my recent teaching in a French system in West Africa, I taught 'Mathematics' not courses called algebra, geometry, trigonometry, even though these were the topics studied. So I made the observation that a few states were taking their lead from Japan, Singapore, etc. to move toward a more integrated high school program. This is consistent with NCTM's Focus of the Year calling for the integration of algebraic thinking from PreK through 12. The ongoing development of topics from middle school through high school is likely to yield much stronger results than separating PreK-8 mathematics from a course-based system of Algebra I, Geometry, Algebra II, etc. That said, I still contend (and I told the New York Times) that WHAT is taught and HOW it is taught is far more important than how it is organized.

Question from
Tallahassee, Florida

I am a graduate student of mathematics education at Florida State University. I want to answer for your first question.

To make the development of algebra a continuous part of the preK-12 curriculum, the contents of new material in the students' textbook should be increased smoothly. Several researchers (Usiskin, 1987 and Flander, 1987) show that from 5th to 7th grade new contents added are 25 to 40 percent and a sudden 8 th-grade algebra book includes almost 90 percent new contents. In this situation, to pass the course students may be imposed to memorize a certain formula instead of understanding concept.

Cathy Seeley:
This is an important observation. First, the studies you refer to are for textbooks from the early 80s, when middle school math was almost entirely a rehash/review of computation from the elementary grades. Today’s innovative middle school programs include much richer mathematics, and their emphasis on proportional reasoning and algebraic thinking allows for this smooth transition to algebra that you call for. This is one of the reasons I wrote in my President’s Message that we need to be very careful before compacting middle school mathematics (and potentially missing this development) in a rush to get students into an algebra course.

Question from
Terre Haute, Indiana

Do you think Geometry should be taught between Algebra I and Algebra II, or after both?

Cathy Seeley:
This has been a discussion for years, and there has never been a clear answer. The problem is, again, the uniquely American idea of separating out these courses. What I'd like to see is more geometry in our algebra courses and more algebra in our geometry course. Or perhaps an integrated secondary program that is part of a continuous PK-12 program. But I'll come back to saying that WHAT we teach and HOW we teach it are more important than the arbitrary structure we use.

Question from
Vienna, Austria

I think that the earlier students are led to develop symbolic sense, the better they will understand underlying algebraic structures, hence analytical computation. Traditionally, we attempt to instill number sense at a very early age, even before students understand number structures. With the advent of computer algebra systems we now have powerful tools, which if used appropriately, can be a veritable algebracy tutor, and can be far more effective at instilling this symbolic sense than “chalk and board” methods have been. In the international schools’ curricula that we teach, we introduce Algebra in the 6th grade, and as with any language, the earlier the students are exposed to the symbols in the language, the more intuitive they approach it. I am witnessing incredible progress in my classes by using computer algebra systems as an aide and ally in instilling understanding for basic algebraic structures.

Cathy Seeley:
Thanks for being with us from Austria! Your system seems very wise. It’s clear that you are developing ideas continuously from elementary through high school. If we have a structure like this, we can ensure that students are getting the content from a good middle school program that includes algebraic ideas, proportionality, geometry, etc. I strongly believe in the power of computer algebra systems to help students visualize algebra and to allow them access to much higher-level problems than they can otherwise do.

In my President’s Message, I urged caution when accelerating students. We don’t want them to miss out on important mathematics, and we do want them to continue their study of mathematics through high school. I’m guessing that you are paying attention to these two ends of the “accelerating-into-algebra” question.

Question from
Falls Church, Virginia

I am a fourth-grade teacher, with a curriculum with no algebra per se. Other than patterns, what are some other pre-algebraic concepts that I can infuse into my lessons?

Cathy Seeley:
One of the most important things you can do at this level is have students represent mathematical situations in many ways—graphically, with numbers/symbols, in tables, etc. You can also have students investigate situations with related quantities, like tickets sold for a raffle and the amount of money earned, and so on. Principles and Standards for School Mathematics, as well as the Navigations series, have other examples of how to do this. The main thing is to build this awareness of generalization and representation that comes from exploring patterns.

Question from
Aurora, Illinois

I do NOT think that algebra is appropriate for all 8th graders. We have confused more and faster with better. The result is that kids are mastering less and less. Also, with our emphasis on algebra sooner and sooner, kids are missing out on basic life skills math such as percentages and more consumer applications. Not every child is ready for algebra at that level. They are still in the concrete. We need to have more differentiation at the middle school level so more kids can learn at a pace that is comfortable to them.

Cathy Seeley:
I agree with much of what you say. Middle school math should be rich with challenging math problems and should have a significant emphasis on proportionality (well beyond that unit on ratio, proportions and percent that I used to teach as a junior high teacher).

I’m not sure tracking students equips them to do mathematics well. I’d rather see students engaged in problems, really struggling with mathematical ideas and having to think about how to apply their skills. Many will make significant growth in this environment.

Question from
Boston

All math teachers, including elementary teachers, should see arithmetic as the sole basis for all future mathematics. A strong emphasis on knowing basic math facts and then understanding the many ways they can be applied is necessary without the use of calculators. An emphasis on discussing math use math terms, such as, distribution, reflexive, inverse, etc. will do wonders for students when they are introduced to pre-algebra and algebra.

Cathy Seeley:
I respectfully see things a bit differently. I’d love it if all students knew all their arithmetic procedures. But they had better know a lot more than that, at all grade levels. They need to build their arithmetic on a solid understanding of numbers and what they mean and on what the operations mean, and they need experience with geometric concepts. They also need a lot of opportunities to refine that understanding and hone their skills by solving a range of problems that push their understanding. And they need to apply their mathematical thinking to real data from real situations.

What we don’t want is to withhold problem solving and complex mathematics from students who may have a gap in arithmetic. Many such students are great mathematical thinkers if given the chance.

Question from
San Jose

What a timely discussion. We recently found out that the district will require all 8th-grade students to take high school equivalent Algebra I. That means Algebra I will supposedly be remedial if the students must retake Algebra I in high school. I would like to know how this can be admirable if it’s not the best solution.

What is there to admire if the students are not being serviced? Why should I care that my students are exposed to quadratic equations if they cannot factor, multiply, divide, calculate a 15% tip, figure out if they are being duped out of a 20% discount, make change for someone on the fly at a register, understand instructions for daily dosages on a prescription bottle, or read a table in the paper?

My district claims this is to give all the students the opportunity to be “college prep.” Well, what’s the point of being pseudo-college prep if they are not “life prep?”

Cathy Seeley:
Good points. Also, how motivating or useful is it for students to fail algebra in grade 8 or to repeat it in grade 9? We don't know enough yet about what it takes to succeed in algebra. I have seen students who could not do fraction computation but who could do sophisticated algebra. But I hope we don't jump at a quick fix like moving algebra down for all students without attending to deeper issues of what we are teaching and how we are engaging students in learning.

Question from
Evanston, Illinois

What criteria should be used to accelerate students?

Cathy Seeley:
The most important criteria should be the student’s interest (not necessarily the parents’ interest) in pursuing mathematics through a level of about calculus in high school. What I think is not a good criteria is to look at mastery of arithmetic.

My most recent experience teaching an Algebra I course showed me that a student who had no mastery of fractions was a fantastic algebra student. This is why I’d rather see us look at what a student learns and how his or her thinking develops at every level, and then moving naturally into algebra content, rather than thinking of algebra as a threshold into higher math.

Question from
Tempe, Arizona

I am currently in the middle of finishing my last semester before graduating with my masters in Math Ed.  My goal is to teach at the middle school level, beginning next year.  I have been debating often about the struggle of integrating algebra into 6 th and 7th grade math, and I feel I have come up with a few ideas that could help the transition.  First off we should introduce the idea of a variable at a very young age.  Students often complete problems with empty squares or blanks to fill in the missing solution.  This could be replaced with a variable, to begin the discussion at an early age.  Second, in your message you mentioned the use of proportional situations to make the bridge.  I feel this is the best way to work with this in sixth or seventh grade.  What I would like to ask is do you agree with what I have proposed, and if so, how is a first year teacher supposed to design an all-encompassing curriculum when the one they are given is full of holes?  Why isn’t there a curriculum developed already that everyone can use?

Cathy Seeley:
I like the idea of introducing algebraic ideas in the elementary grades and in middle school. I think it is worth investigating how students do in making the transition from open spaces or boxes to variables. I think our students are far more capable than we have expected, and certainly in grades 6 and 7, many students should be able to do this given the right kind of experiences. And you already know how I feel about proportionality! So I think this is a great goal.

Now, finding a program to help you do this is tricky. But there are some out there. Look at the NSF-funded curricula (you can e-mail me if you need a specific link to a site for more information). Your challenge will be whether the school you are assigned to will be using a program that fits your philosophy. This is really challenging, especially given how many other challenges a first-year teacher faces. Perhaps choosing your school is the place to put a lot of energy, so that you are more likely to be somewhere that can nurture this philosophy. Welcome to the profession, and we look forward to having you be part of this professional community!!!

Moderator
Thank you all for your participation in an exceptionally stimulating exchange over the past hour. Please see the NCTM home page for the topic and time of next month’s chat with President Cathy Seeley.

Cathy Seeley:
Thanks so much for this energetic exchange of ideas. I don’t think there is a simple answer to when we should teach algebra. I’d like to see a continuous PreK–12 mathematics development, regardless of how a school or district organizes the pieces within. But more important is taking into account these many issues you have raised, and making the best professional decision we can for students at each step.

Let’s continue this kind of discussion in all the faculty lounges across the land. We can learn from each other and continue to refine our thinking about how best to help every student learn high-level mathematics.

 

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