My Back Pages: Math for 21st Century Citizens

BackpagesPleas for reform are sometimes justified with exaggerated indictments of the past or present, so it’s always refreshing—and often more convincing—when advocates for change present a clear-headed assessment of how things are and make realistic suggestions for improvement. In the December 1983/January 1984 issue of Educational Leadership, National Council of Teachers of Mathematics President Stephen Willoughby takes a sober look at the state of mathematics in the early 1980’s and makes practical recommendations for a successful future.

“Today the schools of our nation are doing more for more children with fewer resources than has ever been the case for any nation in the history of the world,” concedes Willoughby. Despite this achievement, however, he cites a number of factors—poor teacher preparation and students’ disinterest in mathematics, to name a couple—that point toward a potentially problematic future.

After reviewing the successes and failures of past reform efforts, Willoughby lays out a number of concrete suggestions to ensure that schools effectively prepare students for the future. Many of his recommendations, like smart hiring practices and professional development for staff, transcend mathematics instruction and easily apply to other disciplines. Others, like ensuring all students have calculators as early as possible and redefining computer science as a fundamental part of the curriculum, reflect his sense that changing times demand new ways of thinking from math educators.

More than 20 years later, Willoughby’s measured approach and thoughtful advice provides a great model for modern educators looking to the future.


  1. I believe that many students and teachers have a fear of math. Teachers fear it because they were never confident in their teachings of it. Students pick up on this and can not get passed the lack of excitement form their teacher. Teachers have to believe in their subject in order to gain the excitement and will to learn the material. Most schools do have a lck of materials to work with but we are learning to compensate for this.

  2. As middle school teacher of mathematics I have to agree and disagree with the comment above. I personally have a love and passion for math. My goal is not only to provide a thorough education for my students, but also pass on some of the love for the subject that I have. I agree with the initial post that districts need to regulate who they hire and how that teacher will be an asset to the students, department, and school. I think that if teachers do fear math they should use that fear to build both their confidence and that of their students. They can be a role model for their students who fear math.
    Personally, I feel there is a struggle with students at the middle level when they do not know the basic skills necessary to continue on to higher-level math in high school. Students are memorizing the information and not learning or practicing it. We all know math is a skill that needs to be practiced in order to succeed at it. Time and resources aside, I think we as educators need to get to the root of our subject and how students learn math.

  3. It is wonderful to hear from another math teacher who loves mathematics. I too teach middle school math and find that many students lack the basic number sense they need before they can move onto algebra. At the elementary level, we need to make sure all students get the basics they need, and not allow them to use a calculator or help sheet to do arithmetic. If my students can’t multiply or understand fractions how can I expect them to understand linear change or slope of a line? Accountability for math instruction needs to start at the lower grade levels so students feel confident to explore algebra and geometry in middle school.

  4. I agree with Beth who states that teachers with a fear of math can be role models for students. I also agree with Elaine about students not being prepared for the grade level. I teach 5th/6th grade math and my 5th grade students did not know place value let alone multiplication facts.
    I feel that math is a language and if students miss the basics it will be very difficult to gain new concepts.

  5. As an educator who has taught at the high school, middle school, and elementary school levels, I would first like to say that I have witnessed the pecking order, in which those who teach the upper grades blame the failures of students on those who teach the grades below them. I currently teach 4th grade math and I have a big job (and a job that I enjoy) of instilling the love of math in my students. I would absolutely agree that students need to have a solid background in basic skills and understanding of number sense to be successful in upper grades. However, in defense of the elementary teachers, there are more and more demands on teaching concepts at younger grades. The third grade teachers are required to teach multiplication facts, even though the students have not mastered addition facts. Therefore, when they come to me, I have to insist on mastery of all facts in a short amount of time. I give a weekly timed test on addition and multiplication facts and I do assign grades for these tests. An important thing to understand is that parents do not always see the need for their child to study these facts at home and students continually “fall behind” when new concepts are introduced.

  6. I agree that some teachers are intimidated by math and would rather not teach it. In turn, students adhere to these same anxieties about math and never learn to like it. This is not healthy for students to do! We as educators need to instill a sense of enjoyment within our students when it comes to mathematical concepts. We will all be much happier with the results!

  7. Hi Beth,
    This is the first time I have seen a fellow student from our masters class on one of these blogs. I hope you find it more beneficial than me.
    I get the impression that you don’t think that we at the elementary level have high expectations for our students in math. I can’t speak for anyone but me, but I can tell you we do. We are held accountable for our students’ math performance on state assessments just like those teacher who teach junior high or high school.

  8. As an educator who was a Chapter Math teacher at one time teaching 1st – 5th graders know exactly how you feel when students have not been taught their basic skills. There are different skills that must be taught at every level. As an educator, we must also build on what the student has been taught the previous years. I agree that the district should hire teachers that are knowledgeable in subject areas especially in middle and high school. Also, the teacher must be excited and show interest in what they are teaching.
    The elementary teachers need to also make sure they are doing their part. As a first grade teacher now, I know how important it is for me to make sure all my students have been taught their basic skills before they are promoted to 2nd grade. Also, I let my students use calculators in 1st grade. Exposing them to calculator at an early age will allow them to be familiar with them. All elementary teachers at all grade levels must prepare their students in math in order for them to succeed in middle and high school. Dr. Nieto talks about how important it is for teachers to be knowledgeable in their subject area in the “DVD”, “Teaching as a Professional.”

  9. I think it is safe to say that over time, expectations can and should change. We need to keep up with the times and demands that we are inevitable. I agree that there is no one to blame when students come without previous knowledge of basic math facts. It is our job to make sure that they enjoy math, feel confident doing it and from there they will thrive. It might sound elementary but I give my students math raps for multiplication facts in the beginning of the year to prepare for the rest of the year. If we set another new goal for ourselves as the president of the NCTM did in the 80’s maybe we will get to where we need to be as well. Having a vision is the first step!

  10. I teach math at the high school level. I think that at times students will struggle with math because of the different things that they will hear at home. I feel that at times the parents are feeding negative thoughts into their students like saying that they are to young to be learning this information, or that they will tell their child that they weren’t good at math so they won’t be good at math either. The students will respond to the information that they recieve whether it is positive or negative.

  11. I disagree with giving students calculators at a fourth grade level let alone a first grade level. They should not be able to use a calculator until they have mastered addition, subtraction, multiplication and division facts. I teach 6th grade math and still have students who do not know their facts. They too are allowed to use calculators (because the state has said they are allowed to use them on the State Proficiency tests). They rely so much on their calculators that they have forgotten and do not care to learn their facts. Why should we set our students up for failure by giving them calculators to solve everything? We need to give our students responsibility and a calculator is not the way to do it. Parents may not be involved in our student’s education lives so we must instill in our children what we want them to know, not with a calculator.

  12. The portion of the article that I struggle with as a high school science teacher is the emphasis on calculators in instruction. I see the value in getting students to see that they do not have to be completely reliant on a calculator. I think the earlier that calculators are introduced as essential components to a math curriculum as opposed to useful tools, the more students are reluctant to use their brains instead.
    I can remember many times in an Algebra course struggling to decide when it was time to continue with the arithmetic that I thought was essential and when to just give in to calculator use. I still go back and forth on the pros and cons involved with calculators.

  13. I have to agree that the biggest obstacle students currently face in mastering algebraic concepts is a lack of basic number sense. I also believe that using calculators too early clearly contributes to this gap in mathematical knowledge. I cannot count the number of times a student has said to me: “if you’ll just tell me what to put in the calculator, then I can find the answer.” If a student does not understand what the calculator is doing, then he or she will not be able to take the current lesson and apply it to a new situation.
    Therefore, I must respectfully disagree with the comment that kids need to be “… exposed to calculators at an early age to be familiar with them.” In my mind, that is equivalent to saying ‘kids should be allowed to get behind the wheel of car at a young age so that they will be able to drive when they are sixteen.’ —When it is time to learn about calculators, the kids will be able to learn how to use them in a very short amount of time. Personally, I never used a calculator until I was in college, at which time I not only used one, but I learned to program complex equations into it in less than a week.
    My experience (15 years as an engineer and 9 years as a teacher) has taught me that the ‘cons’ associated with using a calculator too early far outweigh any potential benefits.

  14. I am a teacher of mathematics and one who loves the subject. As a first grade teacher, I tried desperately to get my children to love the subject.
    If I must be honest, I was not always happy about the way that some of my coleagues taught the subject as the children progressed. In my opinion instead of teaching steps for them to memorize, we should be laying the foundation by teaching the concepts. Once a child understands the material, they will be able to apply it.
    It’s true that some teachers pass on their fear of the subject to the children while some use their fear as a challenge to themselves and their students to do better.
    Two things that I don’t like the use of, too early in the subject are, calculators and multiple choice testing. Currently my fourth graders and I are having a tussle every time we review a multiple choice test. Based on their responses, I am of the view that they are more inclined to just pick a response randomly instead of solving the problem.
    As with multiple choice assessment, calculators encourage the young child o be lazy. They stop using their brain for the simpliest problem.

  15. “Today the schools of our nation are doing more for more children with fewer resources than has ever been the case for any nation in the history of the world,” I agree and disagree with this statement by Willoughby. I agree because teachers do seem to have larger class size averages than 20 years ago. Also, teachers seem to have greater responsibilities, many times taking on other roles such as parent, friend, guidance counselor, secretary, “paper pusher”, etc. However, I disagree with Willougby’s opinion that teachers have fewer resources. With access to the internet, in your school or public library, teachers have an unlimited supply of resources from podcasts, You Tube, Teacher Tube, SMART Notebook, online manipulatives (, publishers online resource books ( or or blogs; a teacher only needs to be resourceful. The biggest challenge is time; time to explore the vast supply of resources and find something that works for your students.
    In previous posts, colleagues stated “many students lack the basic number sense they need before they can move onto algebra. At the elementary level, we need to make sure all students get the basics they need, and not allow them to use a calculator” and “how important it is for me to make sure all my students have been taught their basic skills before they are promoted.” I was working with my niece Destinee (a fifth grader) on her math homework over the Thanksgiving holiday; she was doing mean. I discovered she has poor basic skills but an understanding of many higher thinking/reasoning skills. Destinee has difficulty recalling her multiplication facts and sometimes adds on her fingers, but she understands how to find a missing number given the average and the other numbers; a concept even some secondary students can not understand. So, I wonder would it be better for her to remain in the fifth grade because she has difficulty with multiplication or promote her because she has an understanding of concepts that are more difficult. Where is the promote versus retain line? Should students know all their basic facts/skills with easy recall before leaving elementary school? Also, districts have rules about retention; a child can fail one subject and still be promoted. Is this a good policy? A student can fail math every year from K-8 and never be retained or attend summer school.
    A previous post stated, “I have witnessed the pecking order, in which those who teach the upper grades blame the failures of students on those who teach the grades below them.” Another post mentioned, “I agree that there is no one to blame when students come without previous knowledge of basic math facts.” These comments relate to the previous discussion of promotion versus retention. Sixth grade teachers will have a difficult (not impossible) chore of teaching fractions if students do not understand divisibility, LCM and GCF. Fifth grade teachers will have a difficult (not impossible) chore of teaching divisibility, LCM and GCF if students do not understand multiplication and division. Third and forth grade teachers will have a difficult (not impossible) chore of teaching multiplication and division if students do not understand addition and subtraction. It is a spiral effect; and, this is just one example. When do these students get retained? Districts give the students extra assistance (AIS, IEP, 504, after school tutoring), but the students lack of basic skills can hinder them from learning/understanding other skills.
    I agree that introducing the calculator too early may be disadvantageous. However, how early is too early? Ask yourself: at what point in the curriculum does a basic 4-function calculator stop “giving” a student the answer? If a student does not have some understanding of how to find a common denominator, change a fraction to a decimal or percent, find equivalent fractions, solve proportions or solve equations, a basic 4-function calculator will not help. Is it better for that student to use a basic 4-function calculator to promote understanding of new concepts? I believe there is no perfect answer to this question. Each student should be examined independently; it is a balance of paper-pencil and calculator activity.

  16. I have read the article and each of the blogs. As a teacher for 30+ years at the Middle School Level and now an administrator, I can tell you one of the areas of math I find most frustrating is when teachers see a problem in the classroom; and then, as a school, or a district do not take the time to find out where the curriculum is lacking. It is not a grade level or certain materials; but usually the concepts of mathematics that are lacking. One of the things I did each year after the standardized tests came back was to do an item analysis to see if the students consistently missed certain concepts in math. I then worked to teach the students the skills they were lacking. Did all students succeed? No, but when that light went on and the Mathematics language cound finally be understood, there was no stopping the success rate in the classroom.

  17. The period of the ’90’s was proclaimed the decade of the brain. We are now in the global knowledge age bringing the promise of challenges far greater than experienced by previous generations. I believe the broader problem that subsumes much of what has been said above is most students aren’t too interested in disciplined thinking. Thinking with criteria, posing logical arguments and defending them, guided by stringent rubrics that can be posted in the classroom, more naturally segues into using numbers to validate, or generate, a point.
    Teachers can make this happen by conducting inquiry rich lessons in any subject and modeling/facilitating question-asking behavior to lead children and youth through thoughtful processing of ideas. This should be part of the intellectual fabric of classrooms at any grade level. Mathematics rises naturally like a leavened bread within rich philosophical contexts. When I entered education in 1982, I searched for programs that fostered this approach. IMO, the best of them–Philosophy for Chilren–is still out there. Brainchild of Mathew Lippman and others at Mont Claire University in New York, this approach will take your students to new heights in using logical analysis and examining ethical issues within a rigorous, age appropriate, framework. Even if you don’t conduct the program–and there’s certainly enough rich material to do so if you wish–you will learn to use language to provoke better evaluation of ideas. By moving your students along the path toward formal thinking, you truly empower them to handle information well. Until math programs can generate real visceral interest, students in general will not invest the time, effort and concentration needed to become masterful–at any grade.
    This is the path to social efficacy, and the road to leadership and intellectual prowess. Number sense, concepts and applications are all a part of it. Habits of mind that incline kids in this direction start early–as do those that incline kids toward mental apathy. Sadly, philosophical inquiry is left out of the reform dialog and believe as long as we continue to ignore intellectual development students of all ages will meet “rigorous” classes with the all too prevalent collective yawns.

  18. In the International Journal of Mathematical Education in Science and Technology (1994, 25, 549-561) I pointed out a form of mathematics teaching to children that should be stopped. I refer to the addition, multiplication, subtraction, and division of fractions. Students do not need to know that information until they take Algebra 1 in high school.
    Patrick Groff, Professor of Education Emeritus, San Diego State University

  19. Jessica,
    This is also for all the other colleagues who disagreed with me about using calculators in 1st grade. My disrtict requires this and I have to follow my curriculum. I do not let them do their work on the calculator. I show them how to add by touching the numbers and using the plus and equal signs. I do not use it for subtraction. Cheryl, my sister is a successful chemical engineer that is a superintendant at a large refinery however, she thinks it is a great ideal to use calculators in 1st gade.

  20. Interesting that none of the postings mention the value of multisensory math; that is, presenting math with multiple avenues to the brain by including visual, autditory, and kinesthetic input simultaneously. Many of our dyslexic or atypical learners need this. If it is true, as I’ve heard, that 80% of learners have a visual learning style, then it is particularly important not to jump to abstract symbol representations of math concepts prior to permitting the student to “SEE” the concept and have a visual reference. “Oh, I see it!!” isn’t a common statement for no reason.
    2nd point, WHY don’t schools permit individualized pacing of math learning? If teachers were placed in rooms with access for ANY student during the “math period” then I suspect we’d find some 2nd graders “bumping up” to fourth grade while some fourth graders would revert to third grade. Allow students to stay until they have mastery and perhaps we’d alleviate some of the blame game, and true suffering that is engendered by marching kids onward without the basic grasp they need to be successful. This sad process extends all the way into college and closes doors all along the way. We can do better. It should not have to require “failing” a student and retaining them for all subjects. Address the math singly and get it right.

  21. I was a student who refused to rote learn my mutiplication ‘facts’ and hated to do time test. I failed arithmetic in 4th and 5th grade. I met a pre Algebra teacher in 7th grade that helped me understand the concepts of Mathamatics. I learned my basic arithmetic facts over time thru usage. I ended up with a college degree in Mathametics. I like the concepts of solving a problem using an unknown symbol and am very good at it. After years in the world of business, I am now a High School Math teacher. I am so glad someone did not insist I stay stuck until I could quote all my multiplication tables by rote. What upsets me the most are all the students who feel they can memorize every answer, and do not know how to approach a new problem. they can not apply learned concepts to a slighty different situation. In high school we are giving them tools to use to solve all types of real life math problems, that might arise, where not all the numbers are ‘known’.


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