"Bear in mind that the wonderful things you learn in your schools are the work of many generations, produced by enthusiastic effort and infinite labor in every country of the world. All this is put into your hands as your inheritance in order that you may receive it, honor it, add to it, and one day faithfully hand it to your children. Thus do we mortals achieve immortality in the permanent things which we create in common." - Albert Einstein

Thursday, June 14, 2012

Why So Many Elementary Students Aren’t Mastering Basic Math Facts

by Lynne Diligent
Remedial Math in College
A friend of mine teaches remedial math at the Community College level. We were discussing the problem of a number of students who never seem to have their addition facts mastered (much less their multiplication facts). He wrote:

“I remember as a young math teacher wondering how many hours of flash card drill it takes in the elementary grades to become fluent in the addition and multiplication facts. I could imagine ten minutes a day of actual flash card drill, five days a week, for 45 weeks in one grade, a total of 50 hours if I multiplied correctly, might be a reasonable guess. Surely it has been studied. Well, if it has been studied I have never seen any evidence of it in the last fifty years. I thought of that as probably pretty basic knowledge about the teaching of arithmetic.”

I’d estimate that at least 30 hours of drill, spread over a period of time would be required for an average child to learn addition facts, and an equal amount of time later on to learn multiplication facts. But there is NO classroom time provided for this in the math curriculum.

The problem here, speaking as a third-grade teacher of 8-and-9-year-old students for a decade, is that the elementary math curriculum (in America) is not structured to provide ANY time for drill such as he describes. Even when I was a child in the early 1960′s, we did not have drill of that type in elementary school. My mother worked on flash cards with me 10-15 minutes every day before I was allowed to play. I HATED every moment of it, but saw the value of it when I got into the working world in my 20′s and was using multiplication every day, knowing my multiplication tables by heart thanks to her efforts.

I’ve been discussing math teaching with other math teachers for several years now, and I find there are several trends I highly disagree with.

Trend 1: The amount of math homework has been cut in half from 24-30 problems nightly, to 12-15. I can only guess that this has come about from parents complaining about too much homework over the years. While I am in favor of not giving more homework than necessary, unfortunately, the current lighter homework often does not give sufficient practice in a certain type of problem for the students to be able to understand or master that type of problem. One or two examples of a certain type of problem are just not sufficient.
Houghton Mifflin Grade 3 Math Homework prior to 2007 
Houghton Mifflin NEW Amount of Grade 3 math homework, starting in 2007
(taken from end of a Grade 2 workbook)

Trend 2: Drill practice is considered “old-fashioned.” Never mind that the teacher can make drill practice into a fun lesson, just like any other type of lesson can be made fun by a a dedicated teacher. Without any drill, and without parents practicing or drilling children at home (such as the type of flash card practice my mother did with me as a child), many children are just NEVER mastering even the basic addition facts, let alone multiplication facts.

I no longer teach Grade 3; I am now a private tutor. Unfortunately, I am now running across a number of 14-year-olds who are using calculators to add 5 + 3, or 7 + 6, or 9 + 2. What’s even worse, THEIR TEACHERS LET THEM!!!! I personally think calculators should just be thrown out until about Grade 11, or whenever math involving higher functions on calculators is started. Prior to that time, they shouldn’t be allowed in school at all.

When I taught Grade 3, I made students show all of their work on their homework, including every carry number, and every cross-out for borrowing; I didn’t allow them to say, “I did it in my head.” (See photo above of example homework prior to 2007.) One reason for making students show all of their work (I had several reasons) is that I knew perfectly well many of them had calculators at home. However, even if they did their homework with calculators, they would have to redo it to mark all the carry numbers and borrowing cross-outs. This makes it better to just do it by hand in the first place. I then spent 30 minutes of my teaching time DAILY, going over these homework problems. It’s so satisfying to a teacher to hear, “Oh! Now I see my mistake!” It’s a big mistake for a teacher just to mark answers right or wrong, as students learn nothing from that.

Trend 3: (mostly at the high school level, I haven’t yet seen it appearing in middle schools, although I could be mistaken): Don’t instruct and explain, and then follow up with practice to master the skills. Instead, put students into groups, and let them see if they can “figure out themselves” how to do problems. Don’t give much feedback, but of course, students will have the same test as if you taught them the traditional way. (So the parents who can afford it get math tutors to do at home the job that the teacher should be doing; the parents who cannot afford tutors or understand the math themselves have children who completely fail math).

High school math class, with students working in groups. 
Trend 4: (has been around for at least 25 years): It doesn’t matter if children don’t master a unit. Just move through all the units, and the same units will be covered next year in a little more detail. If they still don’t get it, the same thing will happen the following year, and hopefully they will get it then. This idea has a name, which is called something like “spiraling.”

Even though I’ve never seen it, in the past couple of years I’ve become aware that “Singapore math” requires mastery of each math subject to a certain degree before moving on to the next math subject.

I think students would be far better served by having HALF the number of math topics (eliminating topics in Grade 2 such as Data, Graphing and Probablility; Congruent Shapes and Symmetry; etc.) and making sure they have mastered basic addition facts (by heart), addition and subtraction of two-digit numbers, and multiplication tables up to 5 (by heart) before moving into Grade 3. If parents don’t have time to drill children at home on these facts, then some time for it should be allowed in the school curriculum.

One of the major problems with the “spiral” math curriculum is that in every grade, limited and precious classroom math time is being wasted on unnecessary math concepts, given the age of the students. Those who have put the spiral curriculum together have moved math education from practical, daily skills to incorporating many advanced and unnecessary skills (for the age of the students). Many of these topics could be saved for higher grades (6-8) and students would arrive better prepared, and intellectually ready.

Some important topics, which are covered briefly in the curriculum, but to which little or no time is devoted to practice or mastery of these important life skills: making change for customers, knowing addition and multiplication tables by heart, knowing how to do the simplest operations without a calculator, being able to recognize a wrong answer when a wrong button has been pushed on a calculator, developing estimation skills, becoming competent in measurement and fractions (useful to every housewife in halving or doubling recipes on a daily basis).

Consider: Are we not cooking anymore in American society? Are we not hanging picture frames? Are we not doing any home repairs or improvements ourselves? Is there never a need to count back change? Does no one sew or do woodworking for pleasure anymore?

There is also a great disconnect in many classrooms between the material students are working on, and on knowing the reason for learning it. Instead of letting students feel that they are learning skills which can be useful to them NOW, so much time is wasted on learning concepts where the only use is for passing a test which seems useless to the child. Younger elementary children are mostly concrete learners, and they love and appreciate fun concrete tasks to work on.

Here are five examples of the types of things I feel should be eliminated from the Grade 2 curriculum (for seven-year-old students):

Grade 2 spending time learning to differentiate between types of quadrilaterals.
Grade 2 spends time working on composition of shapes viewed in 3D.
Grade 2 spends time learning about congruent shapes.
Grade 2 spends time on learning about lines of symmetry. 
Grade 2 learns about slides, flips, and turns of geometrical figures.
Are the Chinese or Indian students spending time on these things at age 7? I doubt it.

In my opinion, more time needs to be spent on mastery of basic life skills in the early elementary grades.

One last point about the spiral curriculum. Math educator Brian Rude feels that the spiral curriculum should not be thrown out entirely, but that the problems are caused from barely touching on subjects each time, instead of cutting a bit deeper, so that information is retained. He points out, however, that if cuts are too deep, that there is a danger of never having time to return to that subject, and students will also forget. He feels a balance between the two extremes is best.

-Lynne Diligent


Singapore Mathematics Curriculum: An Example of Spiral Approach

"Singapore mathematics curriculum emphasizes the spiral approach based on Jerome Bruner's explanation on spiral curriculum. The idea of the spiral curriculum, according to Bruner - 'A curriculum as it develops should revisit this basic ideas repeatedly, building upon them until the student has grasped the full formal apparatus that goes with them.' Most people will not miss the idea of 'repeatedly' but may miss the subtle notion of 'building upon them' and 'until the student has grasped the full formal apparatus' of the target concept. In Singapore curriculum, addition is taught four times in Grade 1 (this is a core idea and they are new to it) - addition within 10, within 20, within 40 and within 100. Students get to revisit the idea of addition repeatedly but each time building on the strategies that they already had. When they add with 10, they count all and count on, perhaps with the use of concrete objects and drawings. Later, in addition within 20, they learn to make ten before adding, effectively acquiring the notion of place value. Later they progress to more formal approaches such as adding ones and adding tens in the formal algorithm. Thus, it is not mere a review of materials. It involves extension. In a similar way, multiplication of whole numbers is taught in grades one through four; addition and subtraction of fractions is taught in grades two through five; area of plane figures is taught in grades three through seven; solving equations is taught in grades seven through nine. As a result, in Singapore, Algebra is taught across grade levels in high school (grades seven through twelve). Thus, we do not have the practice of teaching Algebra, Geometry etc separately. They are all under the subject of mathematics. There is geometry in all grade levels."

-Dr. Yeap Ban Har, Director of curriculum and professional development at Pathlight School, an autism-oriented K-10 school in Singapore, and the principal of Marshall Cavendish Institute, a global teacher professional development institute.


  1. Students generally learn mathematics as a part of syllabus but don't take it as a research topic or subject.This is the reason the math researchers are declining.

  2. Wow, really well researched and nicely written article. I too believe that practice is the key when it comes to Math. Recently a nice website for Math practice www.edugain.com/ph is launched for students in Philippines.

  3. hi can you please recommend some research titles for science education? thank u.