Is your math curriculum coherent?

You probably know that in international comparisons, US students don't do real well in math. According to TIMSS studies, US fourth graders are just above average, eighth graders are just below average, and twelfth graders are well below average.

There are many reasons for this. 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.

An alternative

I have created a series of math books that totally do away with the incoherencies, because they are written to be per topic books, instead of by grade.

Math Mammoth Blue Series comprises 22 elementary-level books, ranging in topics from place value, addition, subtraction, multiplication, and division, to fractions, geometry and decimals.

When your child or student studies these books, he will get a comprehensive, in-depth learning experience of a few connected topics.

And it pays off. Read some of the testimonials here!. Children will learn in a coherent, logical manner, and understand the subject matter easily.

And you will not be paying too much either: prices for these e-books range from $2 to $5 each. You can also get all of them for less than $2.50 each if you buy the whole series as a package.

Learn more:
=>Math Mammoth Blue Series

Sources

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.

See also:

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, pp. 13-17.

Comments

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!

patricia greene

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!!!!

STEPHANIE RAINES-SIMPSON

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.

Beverley

Teaching Tips and Math Lessons menu