Is your math curriculum coherent?
In international comparisons, US students have not been doing exceptionally well in math. According to the 2003 TIMSS study, US fourth graders were just above average, eighth graders were just below average, and twelfth graders were well below average. Thankfully, these results have improved in the 2007 TIMSS study, where US fourth graders and eighth graders scored above average.
There are many reasons for this lagging behind. William Schmidt, Richard Houang, and Leland Cogan have done research into the curricula in the best performing countries versus US, giving us one clue as to why it happens. The following article of mine is based on their report A Coherent Curriculum: The Case of Mathematics, which appeared in Summer 2002 in American Educator.
Schmidth et al. compared the national curricula of the best performing countries versus US state standards and district standards for math (since US does not have a national curriculum). Several differences emerged:
US math curricula tend to be
- not focused. No country in the world covers as many topics as US in their mathematics textbooks. For example, in Japan, eighth-grade textbooks have about 10 topics whereas US books have over 30 topics.
- highly repetitive. The average duration of a topic in US is almost 6 years (!) versus about 3 years in the best-performing countries. Lots of spiraling and reviewing is done. Like Schmidt says, "We introduce topics early and then repeat them year after year. To make matters worse, very little depth is added each time the topic is addressed because each year we devote much of the time to reviewing the topic."
- not very demanding by international standards, especially in the middle-school. In the USA, students keep studying basic arithmetic till 7th and 8th grade, whereas other coutries change to beginning concepts in algebra and geometry.
- incoherent. The math books are like a collection of arbitrary topics. Like Schmidt et al. say, "...in the United States, mathematics standards are long laundry lists of seemingly un- related, separate topics."
What this means is that typically in the US, a math topic is studied for a short time, and then the next one, and then the next one, on and on. A good part of this short time is spent reviewing previous year's knowledge. It follows that any particular math topic is NEVER studied very deeply in any given school year.
Also, during the school year, many topics are covered but not in a coherent and logical order. Instead the topics tend to jump here and there in somewhat of an arbitrary fashion.
So, the end result of following a curriculum that is like hodgepodge and "inch deep and mile wide", by the end of eighth grade US students are about two years behind their counterparts in the best performing countries.
In America, kids are often still learning basic arithmetic (such as fractions and decimals) on 7th and 8th grade, whereas in those other countries students have moved on to algebra and geometry and trigonometry topics.
See for yourself! The following two charts show which topics are typically covered on which grade, either in the States, or in the best performing countries.
Notice the upper triangular structure in the first chart. It means more difficult topics are studied on later grades.
The difference between the two is striking. In the US, several topics are studied over all 8 grades. On each grade, the US has much more topics per grade. The topics do not seem to flow coherently and logically from easy to difficult.
Click to enlarge
Click to enlarge
Check your math books
Just check your own math books for several grades: do you find lots of overlapping? And you might not find too much if you're using a homeschooling curricula, because this problem is especially present in math books used in public school.
Check 1st and 2nd grade books together, then 3rd and 4th grade books, then 5th and 6th grade books, and lastly 7th and 8th grade books. How much overlap is there? Reviewing of previous year's stuff?
Do you find for example the topic of fractions on each of the books from 1st till 7th or 8th grade, with little bits each year? Or, does your chosen math curriculum teach the concept of perimeter or octagon on many many grades?
How about decimals? How many school years does it take to learn to add, subtract, multiply, and divide decimals?
What to do?
If the books have lots of overlapping, you can probably easily eliminate some of that just by skipping some pages. You can also consider teaching the same topics from two grades one right after the other, on one grade, skipping the overlaps.
For example, you could teach the geometry sections of 3rd and 4th grade books one right after the other on one grade only. Or, combine sections on decimals on two different grades into one longer run, eliminating some o.f the review you will undoubtedly find. Remember, there is no need to study decimals on 4th, 5th, 6th, 7th, and 8th grade... kids surely can learn them quicker than that.
You can do similarly for other topics.
Now, this doesn't mean that you wouldn't need to review topics. I am sure your student needs review every now and then. But maybe longer periods spent on one topic will help fix things in his/her mind better so that less review is needed later.
And like I said, your particular curriculum may not be guilty of this constant spiraling from year to year without adding much depth. The research done was based on state standards - not on any particular math book. But obviously math books do follow the standards.
I have created a series of math books (Math Mammoth Blue Series) that totally do away with the incoherencies, because they are written to be per topic books, instead of by grade, and a complete grade-level curriculum (Math Mammoth LightBlue Series).
Both of these series are built on the mastery principle. The grade-level series does include some review and repetition of topics over a few grades - but no topic is dragged on beyond 2-3 different grade levels.
The Role of Curriculum - short article by William Schmidt.
A Coherent Curriculum: The Case of Mathematics, Dr. Schmidt’s Summer 2002 article for American Educator (PDF file).
Curriculum Focal Points is a 2006 report by NCTM (National Council of Teachers of Mathematics). In it, they advice everyone to do the same thing as I wrote in the article above: to have focal points on each grade so that the curriculum is focused and coherent! I recommend you to read the focal points by grade. I hope these new NCTM recommendations will influence the state standard writers and textbook makers.
What Are Science & Math Test Scores Really Telling U.S.? - by Brown, A.S. & L.L. Brown. 2007. Published in The Bent of Tau Beta Pi, Winter 2007, pp. 13-17.
While I disagree with comparing our America with foreign nations, I do agree that our school children are being taught most subjects on a superficial level. This is one of the many reasons that we, my husband and I, chose to home school our children.
My state's Department of Education has published standards on its website. I do have to conform to those standards if I wish to home school our children. However, I don't like to teach "a mile wide and an inch deep," and our children don't like to be constantly switching topics. Fortunately, our state has heard the call from mathematics educators and has changed its standards beginning next school year. It's similar to the "Figure 1" page.
I got my confirmation by reading your article. I am foreign born, and could not understand why the American children could not grasp math as any other foreigner. I can see it from both sides of the coin, and realized that one major problem is they are not versatile in the times table, because their brain is the calculator. It tried my utmost best to have my kids learn and and know it, because in some exams you are not allowed to use a calculator. In the earlier grades, the children are not pushed enough to grasp this nice subject "math". I agreed it's a long drawn, waste of time subject which confuses the students. I think it's time for the Educational System in this country to get the message, and do as the foreigners. I did some college courses, and when I see what is being taught, I smiled to myself, because I already did it in my country. I was way advanced in the class. Another area, the children are crammed with too much homework in grades 5-8. If a different pattern occurs, maybe the children could relax, and get on with an easier flow. It's time to get up America.
I am a retired public school teacher. I taught first, third, and 1-5 self contained LD and resource classes 1-5. As I walked the halls of my school and heard the lessons being taught, I was so enraged! I knew students sitting in those classes that had learned those skills I still heard being taught and retaught. The work on display outside the classrooms showed very little improvement beyond what the same children had done in the earlier years. We spin our wheels over and over teaching the same thing year after year and wonder why they can't do better on testing. They are bored to death. This is not just in math. It is in phonics, language skills, creative writing, basic logic skills. As a nation we teach a swallow and spit back out curriculum that is mandated by the state standards. If a child is capable of going beyond, teachers are not allowed to take them higher. "It's not on the grade level standards". Likewise, if a student needs to review last year's standards, it is not allowed. We must stay on grade level standards. Now do you see why Johhny can't read, spell, write, solve problems, of think. He never had to. He just memorized information for a test and never saw it again until the next year when he rememorized the same facts and passed with flying colors. Homeschool teachers, please don't teach the memorize and spit back out method. Make you children reason, explain why or why not, defend a position with research data. Don't put out robots. Public school has already handled that job very well.
I whole heartly agree with you. Just this week I was reading about America's educational status compared to the international community (www.oecd.org/pisa). While I'm not home schooling (unfortunately don't have the time or patience) we are doing alot to supplement school and encourage logical thinking.
I see your data is referencing 2001. There is a newer study by PISA which shows our lack of progress. If you are interested, the 2006 US summary is here. Thanks for the great product and info,
I really appreciate your blog. Math is not and never has been my strongest area, so I am no genius, but at times I've been frustrated by the sheer number of concepts to be taught at each grade level. I have texts for 3-5th grade and I see lots of overlapping, but thanks to your article, I know how to remedy the problem. Thank you so much for this timely article!
Now I see what the problem stems from as far as our students not grasping concepts. Its not the curriculum but overlapping and not sticking to the basic concepts until the students have mastered the topic being taught. I don't see how our sudents will ever catch up with other countries in math if they never learn or master any of the concepts before moving on to any of the other areas of math. Thank you so much for this article. I am glad that I am not alone in thinking about what the problems stem from and how I can better my teaching strategies in mastery alone!!!!
I am a 8th grade mathematics teacher that homeschools his four kids. There is no way in the world I am putting my kids back in public school for the exact reason I just read in your article. As a matter of fact, I am going into my 3rd year of teaching (I am 27) and I have been ranting and raving about precisely the same argument that is being posed within this article. There is too much stuff in a math curriculum over here (in the states) and we spend entirely too much time reviewing. I should not have 16 year olds in my 8th grade class who have no idea how to change a improper fraction into a mixed number, that's 3rd, 4th grade stuff. It's absolutely ridiculous and I commend you for this article.
Marcus D. Sanders
thanks for this article. Having always saw math as a threat and recently attempting to face it head on, realized that it is not an impossible task, but have still been wondering why so many of my classmates (including myself) have difficulty grasping the concepts. A professor recently mentioned to our class that the U.S. rates lower in math than any other country. I know the standards of teaching in other countries is much higher but could not understand why the entire US population did so poorly in math. I now have the answer!!!!! I appreciate knowing that I am not alone.