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
Other Math Posts by Lynne Diligent:
Why So Many Elementary Students Aren’t Mastering Basic Math Facts — Part 1 (of 2)
Tags: Are Chinese and Indian students using a spiral math curriculum, Are we not cooking anymore in American society?, Are we not doing any home repairs or improvements ourselves in µAmerican society?, Does no one sew or do woodworking for pleasure anymore in American society?, In these days of cash registers is there never a need to count back change?, problems with the spiral math curriculum, Why American students are not mastering basic life skills in math classes, why students are not becoming competent in fractions or measurement, why students are not developing estimation skillls, why students aren't learning addition or multiplication tables by heart, why students aren't learning how to make change for customers, why students cannot do math anymore without a calculator, why students cannot recognize a wrong answer when a wrong button has been pushed on a calculator, Young elementary students are mostly concrete learners, younger elementary students are concrete learners
Dilemmas of an Expat Tutor 
July 16, 2011 at 2:30 pm |
One of the things that initially impressed me about Everyday Math was its connectedness to things at home (find things that are circles, triangles and squares that fit in a brown bag. Bring to school and share in the “shape museum.”)
However, especially in older grades, the connection to the language curriculum makes it impossible for more math-minded students to succeed. Students answer questions such as how they FEEL about the math curriculum or, if they could be a number, what would it be and why?
Ordinarily I would say that unschooling or homeschooling would help alleviate these problems, but in some states, homeschoolers must pass the same standardized tests the children in public school do. IMO this decreases the effectiveness of the homeschool, although it may make bureaucrats and society at large feel better because then they “know” that the child is learning at home instead of watching Spongebob all day.
Thankfully, I live in Missouri (no tests, no reporting). I am very pleased to say that not only did we do a LOT of cooking in our homeschool, but that having a larger family meant that the homeschoolers had to learn to add fractions (3/4 cup plus 3/4 cup to double the recipe) and the like as a matter of course. Looking at a lesson plan, however, I could see where this might “look” to an outsider as though Mom just wanted 50 chocolate brownies and made the kids cook for her. 🙂
PS I use tangrams for fun with the littler children. At what point do you think a study in line symmetry and the like is beneficial? Will you follow up with a general outline of what ought be covered and in which grades?
July 16, 2011 at 5:44 pm |
The original Singapore Math did include some of the geometry in the 2nd semester books of the primary grades. What had to be moved from the higher grades into the lower grades to align with the California standards were algebra, probability/statistics, and negative numbers. I’m with you in saying that these are less important to master in elementary school than the basic 4 operations.
July 16, 2011 at 8:40 pm |
Thank you, Crimson wife. Algebra and probability/statistics start in the Grade 3 new (post-2007) books. I totally agree with you that those should be saved until later, perhaps Grade 6.
July 16, 2011 at 10:34 pm |
If I may I’d like to add a few thoughts on spiraling.
I don’t have experience teaching arithmetic, but I do have a few memories, vague and incomplete though they may be, of learning arithmetic as a child in the 1950’s. I remember that we would start each year in math with some review. Indeed if I remember correctly this review would go back to the addition of whole numbers. Is this sensible? Is it spiraling? If it is spiraling, is it good spiraling? I don’t remember how long this beginning-of-year review would last, perhaps a day or so, perhaps several weeks. I didn’t question it, and I didn’t have the term “spiraling” to apply to it. It’s just what was done.
I can imagine elementary teachers would have some disagreement on how much review is sensible at the beginning of the school year. Would it make sense for a fifth grade teacher to flatly declare, “I’m not going to waste a minute in reviewing what the students learned in math last year!”? I don’t know if that is a defensible statement or not. I don’t even know what students learn in four or fifth grade math. I would presume in fourth grade students learn at least something about fractions and decimals. I would guess that most fifth grade teachers would see value in at least some review on fractions and decimals before forging ahead with the fifth grade math topics. Should not this be interpreted as spiraling? It seems to me that the word applies, and therefore the question is not whether to spiral or not, but what is the optimum amount and nature of such spiraling? Should a fifth grade teacher spend zero time reviewing fourth grade math? Or a week? Or a month? I would guess that at least some review would be sensible, and it seems to me that it is properly called spiraling
And let’s take it down to a lower level. Should any of today’s lesson be a review of yesterday’s lesson? When a teacher says, “Remember yesterday we learned that . . . ” is that spiraling? What about when the teacher goes over problems of homework with the class? Is that spiraling? Are there times when a teacher knows a concept is difficult for students, and therefore intends that today’s work is essentially going to be another attempt to get that difficult concept across to the students. Can’t that also be called spiraling? If we say no to these questions, then it seems to me that we are saying that elementary math should be taught pretty much like college math. I doubt if that is a good idea, but again I have no experience to draw on.
So my perspective is that a lot of spiraling is just built into what teachers do everyday. “Spiraling” should not be a bad word. But we should try to be aware of whether we are doing spiraling well or poorly.
There was a time when spiraling was the latest educational fad. It was seen as the wave of the future, of course. I don’t remember just when this was, but I do remember somewhere picking up a book titled, “Physics, A Spiral Approach” or something very close to that. I didn’t look closely at that particular book, but I have my doubts about a spiral approach to physics, of all subjects. But that book illustrates the general idea of spiraling at that time. Spiraling was not seen as just a regular part of teaching and learning. It was seen as something new. That fad went away in due time, of course, as I think it should.
Making spiraling into an end unto itself, and then bending the subject to fit the fad, is definitely not a good thing to do. I presume it is in this sense that people have a low opinion of spiraling. And I presume it is this sort of thing that is happening in elementary school math.
Why should second graders learn about shapes, symmetry, and geometric transformations, as in the examples that Lynn gives? There are several possible answers to this.
One answer is that any knowledge is good to know. But at what cost? If this is taking time away from learning arithmetic fluently, then the cost is probably way too high.
Another answer, probably more relevant, is that students will learn it later, and if they are introduced to it now, then they will gain the benefits of spiraling. At this point I think the fad of spiraling has run amok. This is not beneficial use of spiraling. Surely it’s a waste of time. Shapes, symmetry, and geometric transformations, if indeed they are worth being somewhere in the math curriculum, need a thorough “straight through” treatment. These topics need isolation and concentration at an appropriate level where students are able to deal with them and learn them.
The NCTM in their 2000 “Standards” has the ideal of having a thread of every major topic in every year of the math curriculum. Thus they think there should be an algebra component to first grade math. Where in the world did this idea come from?
I can’t answer that very well. Maybe the best answer is that educational theorists (me and thee excepted, of course) tend to be shallow thinkers, and ideological thinkers. There is a quote that came to mind when thinking about this, and I was able to find it. In 1960 Jerome Bruner wrote a book called “The Process of education”. It is a report of a meeting of scientists (Woods Hole, 1959) about science education. Chapter Three starts with this:
“We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development. It is a bold hypothesis and an essential one in thinking about the nature of a curriculum. No evidence exists to contradict it; considerable evidence is being amassed that supports it.”
It is a short step from this idea to NCTM’s idea that every major topic of math education will be present in some form in every year of the math curriculum. I don’t know if NCTM was thinking of Bruner when they compiled their 2000 “Standards”. Maybe it’s just a coincidence. The important thing, it seems to me, is that this hypothesis is basically not true. It is not true that any subject can be taught effectively in some intellectually honest form to any child at any stage of development. One might argue that there is some truth in it, but on a practical level it is simply a false statement. Bruner says “No evidence exists to contradict it: considerable evidence is being amassed that supports it.” I read this as the usual “research shows” which can usually be read “my opinion is . . . . . ., and I’m sticking with it”.
So while I’ll stick to the defense of spiraling as a regular part of teaching, I am very aware that the shallow, trendy, conception of spiraling can be unproductive, and ought to be challenged. Perhaps we should call it gratuitous spiraling.
August 4, 2011 at 2:37 pm |
When speaking about spiraling, it refers to Jerome Bruner’s definition, which is that content is revisited and reviewed after the student has mastered. However, curricula such as EDM and Saxon Math have morphed this theory into something other than what Bruner originally intended – hence, a sound theory of learning is now anathema. These curricula “touch on” content without ensuring that the student master under the guise that it will be touched upon again in a deeper way through the spiral. The National Math Advisory Panel advises against this practice.
The Singapore Math curriculum spirals in the way Bruner had originally intended. The concept takes that which was learned in previous lesson to deepen understanding. Old ideas are continually used in the service of new ideas. This is a true spiral. It is the reason why students in those countries are better prepared in math than those in our own country.
July 17, 2011 at 1:33 am |
Brian, I think you’ve made me not feel so opposed to spiraling, if it’s done in a reasonable manner. I appreciate the extremely good and thorough information you’ve provided here.
August 18, 2011 at 3:59 pm |
I understand why, as math concepts, you would be opposed to teaching the distinction between various quadrilaterals and 3D shape formations. While I agree with you from a mathematical standpoint, perhaps it would be helpful to to consider these as visual logic preparations. They encourage the student to observe and use their minds to further think about what they are seeing. You also mentioned children in Indian schools. I am a graduate of the Indian schooling system and we did, in fact, spend quite a bit of time in geometry on quadrilaterals in primary school 🙂
If by spiraling, we are talking about reviewing, I think that depending on the material covered, it can be beneficial. Some basic concepts automatically are required to deal with advanced concepts but some things do fall by the wayside. Part of learning new material is to ensure that the base knowledge is mastered, intact and functioning.
I am now a homeschooling mother who uses the Math-U-See curriculum. We spend as much time as needed on a topic until it has been mastered before moving on. I think that this allows us to review less as the topic has been thoroughly ingrained in the student’s mind. Getting through a topic based on a schedule, without requiring mastery of the topic would likely result in having to spend more time on the same topic later.
January 9, 2012 at 2:03 am |
Thnk you for this great dialog. When I first started teaching elementary school back in the dark ages…1967, we were just moving toward use of the spiral approach. In the mid nineties I was teaching high school and felt that many students’ puzzlement that they were supposed to have retained something of their previous learning was directly related to the lack of mastery fostered by spiralling. They felt no reponsibility for mastering ANY subject.