Good afternoon and welcome to today’s online chat with NCTM President Cathy Seeley. We've received an overwhelming response to this month’s President's Message on mental math. We'll answer as many questions as we can during the hour. A full transcript of the chat will be posted on the NCTM Web site tomorrow.
Our first question is from
Sewell, New Jersey
Students need to focus on answers that make sense. Strengthening their mental math skills can help them determine if the answer makes sense. We as teachers need to help students know why it is important to be able to do mental math. What are some of the best ways to help students know the importance of learning mental math?
Mental math can absolutely help students decide whether their answers to complex problems make sense. Perhaps a balance of making the mental math like a game, trick, or puzzle can help with young children initially. Eventually, the usefulness of mental math techniques helps students realize how much time they can save. This in itself can be a powerful motivator.
Pawtucket, Rhode Island
Over the past 20+ years, I have found that those students most in need of support in developing conceptual understanding of number, children with an identified learning disability, are, in fact, usually prescribed only computational intervention in an IEP. Shouldn’t it be that we instead target the development of mental math and flexibility with numbers, providing these children with basic understandings that they can use and develop in solving problems and thinking mathematically? Even those computational goals could be better achieved if the kids understand what these processes are all about, and find a meaningful strategy to figure it out. Don’t these misdirected practices only serve to further handicap these kids?
I absolutely agree with your take on the importance of understanding and mental flexibility, especially with respect to the development of computational skills. It’s a mistake to limit some students to superficial memorization without understanding, and it’s inefficient. To do so misses the opportunity to help students both understand and learn computation. Memorization can be one dimension of a broader base of numerical work.
NCTM has a new initiative to develop a small number of curriculum focal points at each grade level. In this way we can make clear the importance of deep conceptual understanding of addition and subtraction as well as proficiency in performing the operations.
Great Bend, Kansas
It is important in some areas for students to “know” math. Unfortunately, that is not something you can teach a student. The major problem with the math standards is that they are not written as measurable, concrete objectives that can be taught. Some people are born with more math sense than others, and I do not understand why we as educators are afraid to admit that. It doesn’t bother anyone to say that some students are born with the ability to run fast or jump high. Sure we have to help students reach their potential but I am tired of being asked to do the impossible with some students.
Your comments echo the frustrations of teachers across the land. However, I would argue that indeed mathematical knowledge is something we can teach students. It is probably true that some students may have a particular affinity or talent in mathematics, or in some aspect of mathematics. Nevertheless, it is clear that many students have talents that the system or their particular situation has not allowed to develop fully. I think there is a subtle but important difference between blaming teachers when some students don’t learn everything we might hope and asking teachers to try to do whatever they can to help students learn. There are many reasons why some students do not learn challenging mathematics, and as teachers, we might not succeed with every one. However, if we constantly remind ourselves that each student has untapped potential, it helps us shift our thinking to continue to learn and try different approaches that might tap that potential for different students. This can help us address the responsibilities we have for helping every student achieve as much mathematics as possible.
In terms of math standards, I agree that expectations for what students learn at some point must be measurable and at all points should be clear. Sometimes I worry that we have created such specific lists of isolated items that we limit mathematics teaching and learning to small superficial items. Isn’t this one of the criticisms of the American math curriculum when people label it as “a mile wide and an inch deep”?
We’re going to combine three related questions, which Cathy will address together. The questions are from
New York City
Learn by heart the time tables
Knowing prime numbers under 100
General knowledge about prime factors (Be able to break down the numbers to prime factors quickly.)
I think that knowing the squares up to 256 and the cubes up to 125 are also important for mental math. They come in handy for factoring in algebra. The difference of squares crops up in many problems. For instance 22*18=20*20-2*2=400-4=396.
and Newberry, Michigan:
I believe that students should be able to add and subtract double digit numbers using mental math. They should also know their times tables through 12. Having this base to build on makes mathematics easier and much more enjoyable.
These are all important topics and skills to bring to the table. We might argue for or against any of these on the “must have” list. Certainly knowing basic addition and multiplication tables is important. Many other skills will come in handy for students who learn them. I think teachers need to weigh the cost (in terms of instructional time) and benefit (in terms of long-term helpfulness) in identifying which particular mental math skills to emphasize. I also think that some students will latch onto particular skills that might be fun or easy for them or that might make sense to them. The key, in my opinion, is to help all students develop a mental math toolkit that grows over time. This is an important part of students’ mathematical learning as they increasingly understand the power of mathematics to make hard problems easier.
In order to become proficient at mental math, students need to become excellent at visualization! This is a crucial skill for math “in the head,” but it also is valuable for nearly every other academic or life skill, such as reading comprehension or goal-setting. I teach visualization daily in many contexts. Anything you can clearly “see” you can do or become!
Visual representations can greatly expand any math student’s understanding and proficiency. We are also beginning to see the power of visual representations in helping students succeed in mathematics who might not be the typical “math types.”
After teaching mathematics since 1962, at every level, I am convinced that the best thing we can do for students is be sure they can quickly recall their multiplication facts before they leave grade 4. This does not mean they must spend great amounts of time—ten minutes a day, three days a week should suffice. What is missing from most classrooms I have seen is an emphasis on oral drills coupled with the teacher emphasizing the importance of quick recall. We must let students know up front that this—quick recall of multiplication facts—is an important part of mental arithmetic. Of course, the students should spend a lot of time engaged in activities that demand they understand the concept of multiplication.
Quick recall is important, but we also need to provide varied ways to build the recall—oral, written, looking for patterns, using things like divisibility to check facts for accuracy, etc. And, yes, students need to conceptually understand how, for instance, multiplication works, so that they can use the distributive property to help them build some facts (for example, 8 x 7 is 8 x 5 + 16 [8 x 2]). Understanding makes practice more efficient and more lasting. Some teachers find that just a few minutes a day (perhaps even less than 10) for a few days a week helps keep mental math skills fresh.
My question deals with maintaining students’ facility with number sense and mental math abilities. While the majority of states require 4 years of English in high school, only five states require the same for mathematics and only three states require Algebra II for graduation. As a result, many students opt not to take math late in high school, which is counterproductive for those who are college bound. This is evidenced by the high remediation rate for mathematics in colleges, which is nearly double that of reading or writing. Should states require 4 years of rigorous math in high school for college-bound students (as Arkansas currently does) in order to maintain students’ mental math skills?
In my opinion, states should probably require 4 years of appropriate and challenging math for all students, but especially for those going on to higher education (and we can never be sure who those students are). We have a serious problem in many states when students stop their mathematics short of their senior year. This is not helpful for maintaining mental math skills, mathematical thinking skills, or college-preparatory math content.
I teach preservice elementary education majors. I agree that kids should learn more mental math. I have two comments: To have kids get good at mental math requires: (1) that they practice it often, and (2) that they understand at a deep level what the properties are (commutative, associative, etc.) and what the definitions are (x% = x/100). And doing mental math will help drill those in.
I agree that developing math calls for both understanding and practice. My personal bias is that the more of the latter you have, the more efficient the former can be. This means that more time on practice is not always the answer, but taking the time to develop understanding can be a powerful investment in order to get the most out of practice, as well as to increase the likelihood that students will be able to use what they learn to solve a range of problems.
Mt. Laurel, New Jersey
Assessment of mental math capability is difficult. Multiple choice tests such as those available with our math program leave much to be desired. How can students be evaluated for growth in this area other one on one?
This is an important question. I agree that many of our current test models, especially for large-scale assessments, do not capture whether students can do math in their heads. I know that the National Assessment of Educational Progress (NAEP) several years ago piloted an audiotaped block of questions intended to assess mental math and estimation. It was fairly successful, although they no longer conduct this type of assessment as part of the main NAEP test to my knowledge. Similar techniques can be used in classroom assessment with teachers either saying questions out loud or flashing slides for mental math skills.
It is important to note that some students might have good mental math skills even though it may take them a moment to process a problem. In other words, it may not be the fastest students who have the strongest set of mental math tools at their disposal.
University Park, Pennsylvania
I agree with you completely that mental computation should be emphasized much more. When people criticize about calculators by saying, “What would students do if the batteries die?” I often wonder “What would they do if they don’t have paper and pencil?”
Anyway, one important thing to keep in mind is that all basic facts (single-digit addition, multiplication, their reverses for subtraction and division) are mental computation.
One interesting observation from my analysis of the Japanese elementary school textbooks is that they reserve the vertical notation as the ‘paper-and-pencil computation format,’ and they would not use that notation until that is the focus of the study. So, there is no vertical notation for addition & subtraction in the Grade 1 textbook at all. They would study the sum of multiples of ten (e.g., 30+40) or a multiple of ten and a single digit (e.g. 40+5) and their reverses, but they are all written horizontally.
Some might say this is just an issue of format, but I beg to differ. I think if we want to emphasize the importance of mental computation, we need to communicate that idea to our students. By clearly distinguishing the notation is a way to communicate that importance. Just as Principles and Standards for School Mathematics recognized representation as an important mathematical process, we should not take this distinction too lightly.
I know that this use of the horizontal form of computation has been shown to be quite powerful for students as they learn basic facts and develop their understanding of the operations. I also know that even the way problems are structured on a page can help students see patterns within the facts that help them develop mental math skills and learn their facts. This is done very well in some of the Asian programs. The use of appropriate representations in appropriate ways is one way that master teachers and expert curriculum developers can help students. Thanks for sharing this insight.
We’re going to take two similar questions, which Cathy will answer together. They are from West Palm Beach, Florida
I think that using the distributive property to multiply by whole numbers or fractions is a time-saving skill that, in fact, emphasizes the conceptual understanding of the operation.
and East Meadow, New York:
The distributive property is only touched upon in many textbooks, but it remains a powerful tool for increasing speed and accuracy in multiplication. For instance, 14 x 3 = (10 x 3) + (4 x 3) = 30 + 12 = 30 + 10 + 2 = 42. While many students begin tentatively, they soon report that this method helps tremendously.
The distributive property is a great example of the importance of developing the number system as part of a broad, balanced, comprehensive mathematics program. The properties of numbers, especially the distributive property, can significantly help students at in their mathematics development at the elementary, middle, and high school level. When students understand this way of taking numbers apart to make problems easier to do in their heads, they advance their mathematical understanding and they also expand their set of mental math tools.
How do we incorporate mental math practice on a daily (or weekly) model into our typically full 50-minute periods in middle school?
The greater challenge is when we see class periods even shorter than 50 minutes. I think that to allocate only 40 or 45 minutes a day for mathematics for students at any grade level shortchanges students and stretches teachers too thin.
Nevertheless, regardless of the time allocated to mathematics, developing mental math skills can be incorporated into appropriate places in the mathematics program, such as emphasizing shortcuts in multiplying by 10 when developing multiplication skills. In terms of keeping these skills fresh, a few minutes daily as part of ‘sponge’ or warm-up activities can be an important way to start the day’s lesson without too much intrusion. Even a few times a week can help. I visited a great fourth-grade classroom where students spent just 2 to 4 minutes at the beginning of most class periods doing rote, oral practice, and then followed that with rich, in-depth activities that developed understanding and application. Just a few minutes on a regular basis, either in writing or out loud, can make a big difference over time.
I believe that students need to know how to find 10% of a number in their head. This skill allows them to find almost any percent in their head by doubling, halving, etc. I teach mentally finding 10% of a number in my class on a regular basis.
Well, you identified one of my favorites! This can be a relatively easy one to learn with understanding and it pays off over and over again for the rest of your life, both in school and outside school. The familiar 15% tip becomes a piece of cake when it is built on this mental skill of knowing how to find 10%.
The more mental math ability a student has, the more control that student has over the problem. More control over a problem builds greater confidence. Greater confidence nourishes a healthier attitude towards mathematics. A healthier attitude generates a life-long love for learning mathematics. Sounds simple, doesn’t it?
Divisibility rules are an important skill in middle school because of the great emphasis placed on the understanding of fractions at these grade levels.
Ironically, the calculator is a great tool to improve mental math skills. The calculator makes it easy for students to challenge each other to get the closest answer to any computation problem and then use their calculator to quickly determine a “winner.” Time would be well spent on student discussion about different mental math strategies used.
You raise important points. I should mention that we will be dealing with the use of calculators and other technology in a forthcoming President’s Message (scheduled for March 2006) and accompanying chat. Meanwhile, I would agree that there are positive ways that this tool can be used in support of students developing computational skills, including mental math skills.
How do you see the role of estimation in the context of mental math?
Estimation and mental math go hand in hand. Each supports the development and use of the other. Estimation is a mental ability that calls for understanding something about numbers (including 'about how big'), and often involves understanding operations when we estimate a computational result. A strong set of mental math skills enables students to make good estimates by trying out approximate answers in their heads. Wouldn’t we love to see all our students able to estimate answers to problems and compute in their heads?
Silver Spring, Maryland
I am a ninth grade Algebra 1 teacher in a large urban high school in the Washington, D.C. suburbs. The low-level students in this class have failed Algebra 1 at least once, some as many as three times before. So many of them cannot do basic computations in their head, and I believe that slows them down in their ability to work our problems and frustrates them so much they become disengaged. Yet some of the students do know their addition and multiplication tables; they generally moved to this area from other states that felt it was still important to know these tables. 8 + -9 should not be too hard for someone in ninth grade to do in their head, yet many have to use their calculator to find the answer. Shouldn’t all students be required to know these tables, as well as have good number sense?
Certainly knowing facts and having number sense are appropriate goals throughout the mathematics program. We run into challenges, however, when a student hits a stumbling block for whatever reason (school absence, missing prerequisite, distraction, or simply having difficulty learning). The particular challenge is whether we need to hold that student back until he or she learns the facts. The first part of the challenge is to identify which facts are essential for success at the next level. The next part of the challenge is to determine whether the student is likely to master the problem skill/fact(s) by spending more time right now working on it. Finally, the important question must be asked as to whether the student might be able to tackle a higher-level topic or problem and revisit the problem spot at a future time. To appropriately address these issues calls for teacher expertise both in mathematics and in the learning process. It also calls for excellent articulation from teacher to teacher as a student advances. In many cases, as a student learns more challenging mathematics, the student might become motivated to learn the lower-level skill or might see what he or she has been doing wrong within the context of the problem. Teaching is a complex process calling for attention to each student and the particular path the student takes.
Fargo, North Dakota
Being able to multiply numbers in your head using the distributive property. Reliance on calculators really scares me.
I think students can learn mental multiplication using the distributive property, as well as using other properties and tricks that arise from noticing patterns. Using a calculator appropriately need not interfere with this learning if we make effective decisions (and help students learn to make decisions) about when to use a calculator and when not to.
Bethpage, New York
What are some ways to help students develop their mental math facility?
Students value what we test, so if we believe mental math is important (and it is!), we need to assess it somehow. I recently gave an oral quiz to my 10th graders. They had one minute per question and no calculators were allowed.
What is log(1000)?
What is log, base 4, of 1/2?
And so on.
If I really wanted to test their mental arithmetic, I would have restricted their use of paper and given less time. But it was their first oral quiz, so I took it easy on them. Although the experience was a little stressful for my calculator-dependant class, they did well and most of them can do these problems in their head. The same approach could apply to more elementary mathematics.
You can also validate mental math by asking your students to talk through a problem without a calculator. When I taught precalculus, I was able to get them to think in radians this way. Students would verbalize the thought process to evaluate sin(pi/2).
“What is the sine of pi over 2?...John?”
“Let's see, pi over 2 is 90...”
“OK, but just working in radians, where is it on the unit circle?”
This type of classroom dialogue validates and promotes mental math.
Thanks for sharing these examples. Mental math can continue throughout a student’s study of mathematics.
This is not so much a question as a comment. For my pre-service elementary teachers I have a no-calculator section on every test. They have to describe mental strategies that reflect number sense. There was an article called “Decomposition and All That Rot” written by Stuart Plunkett in 1979—Judy Sowder mentioned it in an article she wrote in Teaching Children Mathematics. The article makes a strong case for emphasizing mental math.
As someone who is a strong advocate of appropriate use of technology in the classroom (see the forthcoming President’s Message and chat in March 2006), I absolutely see the value in a no-calculator part of any test for either students or pre-service teachers. Students need to develop all their tools—the use of their heads as the most important tool, as well as the use of a calculator.
Asbury Park, New Jersey
I believe more important is to know the patterns of the hundreds board. In addition there is a multitude of fun activities that children can play at all levels. I know my daughter can do the math in her head because of the hundreds board.
The hundreds board can be a great learning tool. It is a strong visual representation that can stay with students in their heads forever. As students become stronger with their mental math skills, they may not even realize the role the hundreds board has played, but they will carry the model with them as part of their mental understanding of the base-ten number system.
Manhasset, New York
I totally agree with your ideas. In order for mental math to be internalized for students you do need parent involvement. I find that parents are quick to read with their children and help them with reading but many of the parents are math phobic and tend to not engage their children as much as they should. As educators how do we make the parents less math phobic?
Mental math is one place where parents can really help. Most of the math we are trying to teach students to do in their heads is familiar to parents. Providing concrete shortcuts may even help some parents who do not like mathematics. Perhaps the same ways we use to engage children can be used to design family-friendly letters that describe how to do the tips/facts/skills you are emphasizing at the time. Family math events can also include an opportunity to identify and describe the mental math skills you are working on for that particular grade/school year.
Thank you all for your participation this afternoon and for the many questions submitted in advance. Unfortunately, with those and the questions submitted during the hour we had more questions than we could address today.
The next chat with President Cathy Seeley will be at 4:00 p.m. EST on Thursday, February 2 on “Teaching to the Test.”
Thanks to everyone for your participation in this energetic exchange today. Happy holidays, and we'll see you in 2006 for the next chat!
“Do the Math in Your Head” (December 2005)
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