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Hard Arithmetic Is Not Deep Mathematics



chat archive 

Moderator
Good afternoon and welcome to today’s chat with NCTM President Cathy Seeley. Today’s chat is based on her President’s Message, “Hard Arithmetic Is Not Deep Mathematics,” which was published in the October NCTM News Bulletin. Here’s our first question:

Question from
Medford, Oregon

I think that more one-on-one training by professionals who know how to read a student’s learning style, how to translate math into his or her language will help achieve this goal. I’m pessimistic that any “mass” approach will work.

Cathy Seeley:
I’m not sure I’m convinced that one-on-one training is the only solution. However, it is clear that we need to find ways to help teachers make significant changes in their teaching approach and expand their repertoire of strategies if we are to meet the needs of students who may not have learned everything we wish they had learned before this year. This kind of professional development might be effective in a setting where small groups of teachers get to interact around the ideas, combined with coaching and follow-up.
 

Question from:
Lewes, Delaware

We need to train teachers to know how and when to embed skills into and as a complement to activities that help students conceptualize math operations.

Cathy Seeley:
Yes. Clearly teachers need a broader set of strategies if students are to conceptualize operations, especially if they have a weak background from previous experience.

Question from:
Northridge, California

I agree with the idea. However, I am finding more students who cannot do or will not attempt the “easy arithmetic” that is required to do some of the deep mathematics. I find it difficult to get them to think critically while reviewing multiples of two. This is in an 8th grade Algebra class.

Cathy Seeley:
This is indeed challenging when not all students have the prerequisites we would like them to have. I would like all students to know basic multiplication and other mental facts. However, this isn’t always the case with our incoming students. Nevertheless, I have seen cases (in my own classes and elsewhere) where students without such skills might still be able to do deeper, higher-level mathematics, perhaps relying on a calculator for help with the low-level skills. Often, these students can even learn the low-level calculations after seeing how useful they would be in their work with more challenging mathematics. (As an example, I described the story of my algebra student, Crystal, in the previous chat.)

Question from:
Clayton, Missouri

How can we deepen the definition of “basic skills” for parents?

Cathy Seeley:
For parents, the main question should be: “Is basic arithmetic enough for your student?” I would invite parents to see what we mean by deeper mathematics or higher-level mathematics either by coming to a Family Math event, visiting your classroom, or through a letter home that might provide examples of the kind of mathematics we’re talking about. Helping families and caregivers to expect more of their students is a critical component to improving how and what we teach their students.

Question from:
Union City, Missouri

Maybe part of the problem is in not having the right words to describe what we mean. Given the current political situation it is easy to see how the various parties choose words to portray a specific message. In our case it seems that “depth” is very soft where “rigorous” is hard or strong. I know the message is the right one. It just seems like our words aren’t working. What do you think?

Cathy Seeley:
This is a perceptive observation. I think you are exactly right that our language often gets us in trouble. I have thought that “deep” was widely understood, but I think you may be right that it could appear softer than “rigorous.” If using the word “rigorous” helps get the point across, that may be the right word to use. Unfortunately, this word has a special meaning to mathematicians that may confuse the issue a bit if they are part of the discussion. Sometimes we can come back to the idea of “'raising the bar” or wanting students to do higher-level mathematics, solve complex problems, and learn more mathematics than simple arithmetic.

Question from:
Las Cruces, New Mexico

How can we focus on conceptual understanding, a deeper conceptual understanding, of mathematics in our teaching if students and teachers are still evaluated on skill/drill type standardized tests?

Cathy Seeley:
I am absolutely convinced that if we teach a good, deep mathematics program that balances skill development with understanding concepts and solving problems, our test scores will be fine. And this is true even when the test may focus on low-level skills. Surely students with experience solving problems and thinking mathematically can figure out how to do low-level computation if they forget the rule for the moment.

However, the reverse is not true. That is, if students' mathematics experience is limited to low-level, rote computation, they will certainly not be able to solve complex or higher-level problems on a test. They will never gain the experience they need for these harder items, or, of more importance, for using mathematics outside math class.

Question from:
Las Cruces, New Mexico

I think it would be a mistake to think of deep and rigorous as synonymous. Indeed, rigorous has a specific meaning in mathematics, and depth need not come with rigor. Often a deeper intuitive understanding is what is really wanted.

Cathy Seeley:
You are correct that these are not synonymous. However, sometimes this kind of distinction is not helpful in dealing with people who may not be mathematics educators. Nevertheless, you have identified what I was describing in the President's Message as the most important thing we need to help our students achieve. Perhaps “deep” mathematics can help students develop the understanding they need to succeed in more “rigorous” mathematics. In any case, it’s the depth that will open the door to future study, in my opinion. Thanks for helping push this discussion a bit deeper.

Question from:
South Bend, Indiana

I do not see any effort to switch to deep mathematics because the simple arithmetic has not been mastered. It is difficult to get kids to think mathematically when they have trouble with basic addition and multiplication facts. I mean BASIC! Like sophomores and juniors in high school who need to think about what 2:7 or 3 x 6 might equal.
 

Cathy Seeley:
There are many reasons why, unfortunately, some high school students do not have mastery of basic arithmetic. And certainly it makes it challenging to teach students algebra, for example, if they do not have a strong foundation of basic skills. However, to continue to repeat this kind of arithmetic with them at the expense of letting them deal with challenging mathematics is likely to lead to their never progressing much further. We have all seen too many examples of students who get such a negative attitude about mathematics and their ability to do mathematics that they leave it (and sometimes school) at the earliest possible moment.

On the other hand, why not let these students work in groups on good, hard problems that require them to think and discuss their thinking, using a calculator as a tool if necessary? Often we discover that students who have difficulty with low-level computation might be successful on more complex problems. There is no research that I know that says that mastery of computational procedures is related to the ability to think mathematically or solve problems.

Question from:
St. Francis, Minnesota

Our past state standards required us to do “assessment packages” with students to demonstrate their learning. The problem with the packages was the rigor. The writers of packages made the tasks complex, but not necessarily mathematically rigorous. The complexity gave students problems, not the mathematics.

Cathy Seeley:
Solving complex problems is very compatible with teaching deep mathematics. As students develop their understanding of more sophisticated mathematical ideas, they should be wrestling with problems that challenge them to think, even in ways they may not have thought before. When students have this kind of experience in class, they are more likely to be able to deal with complex problems on a test.
 

Question from:
Clayton, Missouri

Sometimes I think we need to do a little more pre-assessing at the onset of a unit to determine which students need to review which concepts and skills.

Cathy Seeley:
Pre-assessing to some extent is fine, but I think we can sometimes limit a student’s progress if we spend too much time going back over things they have learned (and forgotten) before. I continue to believe that the best way to review is in the new context, allowing students to tackle challenging problems and ideas while refreshing their forgotten skills.

Question from:
Tucson, Arizona

I am a community college mathematics teacher, and when we say “depth” is not the same as difficult arithmetic, many of my colleagues feel that students are not getting the rigor they need. They point out that placement test scores have dropped in recent years due to this fact. I counter with the fact that our current placement tests no longer test the curriculum being taught in the high schools. We as an institution are not in line with the progress that is being led by NCTM, even though the American Mathematical Association of Two-Year Colleges (AMATYC) has a crossroads document that closely resembles the NCTM standard initiative. Is NCTM reaching out to community colleges or AMATYC to help bridge this gap?

Cathy Seeley:
Connecting to two-year college faculty is an important aspect of what the Council does. NCTM has participated in various ways around the Crossroads document and in other activities with AMATYC. You have identified an important need at this level—to bring into alignment placement procedures with the high-quality and high-level mathematics we are coming to expect from high schools implementing programs based on NCTM’s Principles and Standards.

Question from:
Coquitlam, British Columbia, Canada

This topic is very timely in view of the probable changes that are going to happen with the Western and Northern Canadian Protocol (WNCP) Curriculum framework here in Western Canada. There is always a debate about breadth versus depth, and it is obvious that the breadth form of curriculum has not given what was hoped for or intended with curriculum.

Cathy Seeley:
This kind of shallow treatment of too many topics seems not to have worked in either the United States or Canada. The vision of Principles and Standards for School Mathematics paints a picture of the depth that we can achieve with all students, and gives us hope for the future that our students can indeed learn high-level, high-quality mathematics.

Question from:
Los Angeles California

The math teaching situation in Los Angeles is in a complete mess because of the recent requirement that algebra and geometry are required for graduation. The high school general math class has been eliminated, and students without basic math skills are forced to sit in algebra over and over and over and over again. (For an analogy, just imagine if playing in the school orchestra were to become a graduation requirement and scores of children would be forced to hold an instrument and sit in orchestra for years without passing a beginning instrument class first.) Our administrators have forbidden us to fail them, so we give gifts of passing grades. As a result, of my current Math Analysis class of 40 students, fully two-thirds do not have the necessary skills for success in calculus. And, I’m not allowed to fail two-thirds of the class. So I give them credit for a test question if any part has something correct in it. These students, for example, need a calculator to figure out what 4-cubed is. They have been given open book and open notes tests for a decade and have memorized nothing at all.

These students have been deluded by the system into believing they are qualified in higher math and can succeed in college. My assistant principal’s comment to me last year, when I failed three-fourths of my Algebra 1A students at the 5-week report card was, “None of their other teachers failed them, therefore, you are not meeting their needs.” So, I met their main need, which was to have a passing grade. It is that, or lose my job.

So, the really qualified students are not getting challenged in class because class time is spent on elementary math stuff only. We have to “dumb down” our teaching. And we are using a really awful, “dumbed down” Algebra 1 textbook. Isn’t this sad?

High school math teachers are incredibly discouraged and are leaving the profession in droves, or going back to middle school where actual teaching still is allowed.

Cathy Seeley:
Teachers in many states are facing a dilemma as new policies call for teaching algebra (and sometimes geometry) to all students. Without a lot of teacher professional development and support for both teachers and students, the natural tendency is either for teachers to water down their content so that students will pass or else to maintain rigor and have many students fail. I would like to suggest that there is another alternative (one I'll discuss a bit more in next month's President's Message), and that is to change our teaching style to engage more students. This is possible (but challenging) even for students who are lacking some computational background, provided we use a tool like a calculator in appropriate ways to help get past computational barriers. When we get students hooked into problems calling for different levels and types of thinking, we really can discover new stars. Learning to teach differently calls for a lot of professional development and support, but it is definitely worth it. We'll pursue this a bit more in next month's chat. Hang in there!

Question from:
Las Cruces, New Mexico

I think it is important when thinking about how to teach deeper mathematics, that we realize we can do this at any level. Teaching first graders what addition really is, rather than just drilling on addition facts, invites them to participate in the intellectual interchange that underlies mathematics.

Cathy Seeley:
This is a great observation! Deep mathematics need not be reserved for the big kids. How great it would be to have students really “get” what addition means and what numbers mean as they develop their ability to perform addition. Children with this kind of foundation are so much better armed to succeed in future mathematics than children who go straight to rules.

Question from:
Morehead, Kentucky

I’m interested in the professional development of teachers in relation to their personal philosophies of teaching. For some, the “broader set of strategies” and problem solving approach do not fit their personal philosophy. Can we expect those teachers to embrace teaching with more depth? What if they don’t? Can there be teachers of differing philosophies teaching across the hall from each other? What effect, if any, will this have on the students?

Cathy Seeley:
These are important questions. I think that the maximum benefit for students comes when teachers can work together. However, every teacher who improves his or her practice helps students.

I think that good teachers, even if they come from different philosophies, can learn how to teach differently if they see the benefit to students. Perhaps professional development needs to include both a realization of the limitations of our current practice and visits to places where teaching is happening differently. If we are to truly meet the needs of an expanded audience of students—all students—we simply must re-examine whether our philosophy as teachers may be limiting some of our students.

Question from:
Greenville, South Carolina

Connected Mathematics and Middle Grades Math Thematics provide deep mathematics to middle-school students without making them do hard arithmetic. Does anyone know approximately what percentage of the schools are using these Standards-based curricula nationwide?

Cathy Seeley:
These are great programs that provide good examples of deep mathematics. I have been in some of these classrooms where the level of mathematics being done by students blew me away. I don’t know the exact figure for the extent of implementation of these programs, but one place to find out more about both the programs themselves and the extent of their use is at the Show Me Center at the University of Missouri: http://showmecenter.missouri.edu.

Moderator
Thank you all for your participation this afternoon and for the questions submitted in advance. The next chat with President Cathy Seeley will be at 3:00 p.m. EST on Tuesday, November 16. The topic will be “Engagement As a Tool for Equity,” the President's Message in the November NCTM News Bulletin.

Cathy Seeley:
Thanks to all for your questions, comments, and suggestions. These chats always cause me to push my own thinking just a bit. I appreciate your participation, and look forward to our November chat.

Happy Halloween!


Moderator
The following are additional questions received before or during the chat that could not be answered during the chat:

Question from:
Pine City, Minnesota

How do we retrain ourselves and our colleagues (K–12) when staff development money is at a premium?

Cathy Seeley:
This is a serious problem. Investing in professional development is one of the most, if not the most, important investments a school or district can make. Even though many teachers end up attending workshops and taking classes on their own, this should not be their burden alone. Professional development should be a shared responsibility of school systems and the teachers who commit their time and energy to lifelong learning that serves students.

Question from:
San Bernardino, California

Yes! All students CAN learn/master depth mathematics. I am reeling to read research about the myriad effects of parenting dynamics both maternal and paternal. There are so many types of strengths and weaknesses in parenting styles that kids benefit/suffer from, and so many types of effects these dynamics have on attitudes about learning, aptitudes for learning, and self image, that a vast catalog could be compiled to help with alternative instructional techniques for the INDIVIDUAL student. So, you could have 20 high-achieving students, all with different reasons for their achievement, and 20 very difficult students with correspondingly different reasons for their poor attitudes/lack of achievement. All 40 students may require a significantly different approach to instruction!!! This is how I interpret research given my experience in the classroom. I wonder what the experts have to say about my opinion?

Cathy Seeley:
I agree! All students CAN learn deep mathematics and the higher-level skills it leads to.

There are many factors that affect student learning, and certainly parenting is one of these. Even though 40 students may be in 40 places, some instructional strategies are effective with a wider range of students than others. For example, it is becoming increasingly obvious to more and more teachers that simply presenting a lesson by telling or lecturing is not the best way for many students to learn. When they shift to letting students do more of the talking on structured tasks than teachers, they discover that more students can become engaged in their learning and can learn mathematics.

Question from:
Ketchikan, Alaska

I teach Special Education at the high school.  We have a state test required for graduation.  It requires Algebra/basic geometry.  Many of my students still struggle with fractions/decimals/percents.  We recently convinced the state to allow students the accommodation of a calculator.  Last year all of our seniors except two passed the math portion.  I have had many students get through Algebra with this accommodation.  I think they need to get past the “computational block” to be successful with application problems.  Only a few have successfully completed geometry.  I know my students have difficulty with abstraction (including estimation).  They can learn algorithms and with a calculator find success in the math needed for post high school experiences.  I believe deep math should include calculators for all students to get past difficult arithmetic to deep mathematics.

Cathy Seeley:
I agree with you wholeheartedly. When a student has trouble with computation at the secondary level, over and over we have found that spending more time on more computation does not often help. Allowing students to have access to algebra and even geometry can let students show us that they can indeed think. You have already found this to be true for algebra. I am betting that, over time, as students have more experiences to be successful before high school, they can come to learn geometry as well, especially if we use engaging strategies for teaching geometry.

Question from:
Riverside, California

California is definitely NOT shifting to deeper standards.  However, we have tons of room for going deeper than we do.

Some immediate changes can be made by having better examples of “best practice” and instruction in ideas.  We need a multi-point strategy.  Teachers should be having conversations with board members, principals, parents, and community stakeholders.  We must speak in a language they can understand.  We should point to evidence of success. We must continually build teacher content knowledge—starting with better K–16 instruction, continuing through preservice, then throughout the education career.

Cathy Seeley:
California made many strides toward deep mathematics in the 1980s. Unfortunately, today’s state standards may have replaced some of the richness, rigor and depth with hard arithmetic that may not help students advance their understanding or their ability to solve problems. All your suggestions make a lot of sense. Teachers are the key—in terms of what they know, what they do in their classrooms, and also how they reach out to work with administrators and the community.

Question from:
Mineola, New York

Math A is not for physically challenged special ed students.  The concepts are not important to them.  The negative signs, exponents, and other “codes” are far too confusing for them.  When they try to explain it to others it is impossible.  These students need some algebra. Most importantly they need to have math apply to their lives.
Get with it New York, not everyone has the same needs.

Cathy Seeley:
Certainly, some special education students with particular needs may have difficulty learning some concepts. We know that certain teaching strategies are more likely to be successful with all students than other strategies—teaching in ways that engage students in tackling engaging mathematical problems is likely to help many more students learn mathematics than more teacher-centered approaches. Nevertheless, we also have to recognize that we may need to make accommodations for some students in extreme cases if they are to learn high-level mathematics. The challenge for us as educators is to recognize that a student’s ability to do high-level mathematics may not be related to the student’s past success or failure with low-level computation.

Question from:
Las Cruces, New Mexico

How do we raise the bar in light of No Child Left Behind?

Cathy Seeley:
On its simplest level, No Child Left Behind calls for standards and achievement for all students. Deepening mathematical understanding only helps improve student achievement, including for students who have not previously been successful. What I have found is that when groups of students show a disparity in achievement levels, it is usually not based on computational skills. Rather, differences tend to show up on the more complex problem-solving parts of tests that call for students to go beyond rote computation to actually using computation to solve problems. If we spend too much time teaching low-level skills to some students, those students have no chance whatsoever to do well on these more challenging parts of their tests and no chance whatsoever of showing adequate yearly progress on challenging goals. What this means is that we have no chance whatsoever of closing the achievement gap unless all students have access to deep, rich, high-level mathematics.

Question from:
Walla Walla, Washington

What does it mean to do “rigorous” mathematics?   I don’t see students mimicking back a series of steps that are meaningless to them to produce a product that also has no meaning as rigorous.  What would you say are characteristics of rigorous mathematics?  And what can the Council do to help the public see the difference?

Cathy Seeley:
I would refer back to the question about “deep” versus “rigorous.” I think that understanding mathematics deeply and being able to use it to solve challenging problems is the main goal if we are to raise student achievement. When students have this deep understanding, they are far more likely to be able to learn mathematics in ways that help them use it, and they are well prepared to study continually higher levels of mathematics.

When mathematicians use the word rigorous, they often have particular meanings in mind, usually involving proof or particular methods of solution. When the public talks of ‘rigorous mathematics,’ they tend to mean that they want their students to do challenging mathematics and move to advanced courses. In communicating broadly then, I think we should know our audience, and use the word “rigorous” with care. Perhaps better words for the public are “challenging” or “high-level.”

Question from:
Las Cruces, New Mexico

I think we are asking a lot of our students to expect ALL of them to want to engage with the material conceptually.  It is up to us to pull them in, to give them a reason to “think critically while reviewing multiples of two” and to do it with enthusiasm ourselves.

Cathy Seeley:
It is definitely up to teachers to draw students into the mathematics. I learned a long time ago that motivation isn’t necessarily inherent in people. Rather, sometimes action precedes motivation. Creating engaging situations and hooking students into mathematical activities can often generate from students motivation to learn more.

Question from:
Clayton, Missouri

I’d like to piggyback on the Greenville question and ask about CorePlus at the high school level.

Cathy Seeley:
The URL for the secondary dissemination center with information about CorePlus is: http://www.ithaca.edu/compass. For information on any of the K-12 curriculum projects funded through the National Science Foundation is: http://www2.edc.org/mcc.

Question from:
Clayton, Missouri

Can you recommend any research that can help to convince parents (that may keep our shift from happening) that greater depth and active student engagement are the kinds of mathematics our students need for the future?  (I lifted many of your phrases from your message!)

Cathy Seeley:
These topics are becoming increasingly of interest to researchers. One good source of reader-friendly research summaries related to teaching and learning mathematics is EDThoughts: What We Know About Mathematics Teaching and Learning, published by McREL (Mid-Continent Research for Education and Learning; http://mcrel.org). However, some of the research in this area occurs in educational psychology, rather than in mathematics. For example, James Greeno of Stanford has done some work in this field.

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