Most American Math Teachers Cannot Teach Math

...because they studied non-math in school, not math. (And most of the rest of us have the same problem.) https://schoolcrossing.blogspot.com/2007/11/are-you-good-at-non-math.html

I read Dr. Nancy Pine's book, Educating Young Giants, with great interest. The book is about her observation of classes in China, her discussions with Chinese teachers and parents (mostly through interpreters), and the comparisons she makes to American education. She admits to being ethnocentric at the time of her first visit to China in 1989, but while she could sometimes recognize her own egocentricity, she was not able to fully overcome it.

She noticed that in Chinese literature classes, teachers emphasized close reading and digging for the author's meaning. She felt that Chinese teachers denied students the opportunity to create personal meaning from the literature they read. Although her research in China centered on elementary literacy development, as a former math major (page 41), she became interested in observing elementary math classes. As everyone who has ever observed Chinese math classes has reported (see, for a few of many examples, Harold Stevenson, James Stigler, Liping Ma), she, too, witnessed superior teaching skill.

I have been teaching math in China for the last several years, and I taught in Japan for nearly two decades. I speak both Japanese and Mandarin. My conversations with people from mainland China, Hong Kong and Taiwan, as well as written descriptions such as Educating Young Giants has led me to conclude that the actual education systems as well as the cultural foundations of both China and Japan are very similar.

Nancy Pine came to appreciate that Chinese teachers teach mathematics, but “most U.S. teachers merely teach arithmetic” (page 45). Dr. Pine is being generous. U.S. teachers teach non-math, specifically routines, tricks and shortcuts, but call it math on the misconception that if numbers are running around, it must be math.

Chinese teachers spend a significant amount of time considering a relatively simple math problem from every conceivable angle. The students probably already know the “answer” and that is precisely the advantage of using an easy problem. Because they already know the outcome, they can concentrate on the process, the concept-building. Once the concept is solid, their homework includes problems that American students eventually spiral to. China thereby reduces the need for the endless review so common in America.*

Dr. Pine herself admitted “that even with my strong interest in math, I would not have known enough about the underlying mathematical concepts to think through the best ways to present the initial problem that would enable students to correctly solve more complex ones” (page 45). See what she is saying? She is admitting that she was great at non-math, but weak at mathematics itself. Not only that, she says she knows “that most American grade-school teachers, who teach five or more subjects, do not have the depth of knowledge to walk children through mathematical concepts to prevent misunderstandings” (page 50).

She believes it is because American teachers are generalists who must teach every subject, while Chinese teachers are specialists who teach only one subject. I would like to suggest that being a generalist or a specialist has nothing to do with it. Chinese teachers could be generalists and their ability to teach math would still “far surpass ours” (page 46) because nearly all Chinese teachers, regardless of their particular specialty, acquired a profound understanding of fundamental mathematics (PUFM, a term coined by Liping Ma) beginning in the primary grades. If our own children acquired PUFM, they would also be much more effective math teachers, even as generalists.

You see, regardless of professional training or subject matter courses, teachers tend to teach the way they were taught. The strident calls for teachers to take more subject matter courses is misplaced. Simply learning more and more non-math will not improve teaching ability. Okay, how about we reteach math at the university level? I tried to do exactly that, only to meet with terrible resistance. “We don't want to know why the math works,” my students complained, “Just tell us how to get the answer.” Fine, let's at least teach those students who aspire to become elementary teachers. Guess what? Most universities require all elementary teaching candidates to pass a series of courses entitled something like “math for elementary teachers.” My students complained that the classes were a waste of their time, since they “had learned all that stuff in elementary school.” Most elementary teachers, even though compelled to take a real math class, most of them for the first time in their lives, end up graduating from college without learning much math due to their resistance. They subsequently teach math the way they were taught in elementary school.

The main reason that Chinese students do so well in international math tests is because they actually learn math in school. American do not. Therefore, the reasons critics cite (specially selected students, lower poverty rates, rote learning, etc) miss the point. What critics are saying is that due to circumstances beyond our control, American students can never compete with Chinese students. I call baloney. If we would actually teach math in our schools, our students could compete just fine.

Next: examples of non-math teaching I encountered in a child's algebra class.

After seeming to correctly solve a number of simplification problems of the form -(ax-b) or -(ax+b), a child complained she could not simplify this one: +(ax+b). “What do I do with all those plus signs?” she wailed. What did you do with the other ones? I flipped them (referring to a mat-and-tile manipulative she is using in class). Even after all that flipping, she still had no idea what was going on. As long as she flips correctly, she can get the right answer without ever understanding how the flipping was supposed to communicate the concept. (Here is another topic: how American teachers routine take great resources like manipulatives and use them ineffectively  https://schoolcrossing.blogspot.com/2010/11/i-love-math-manipulativesbut.html.

In another example, the child needed to solve for x by first combining +2 + ¼, easy—the answer is 2¼. But the next problem was +3 –  -½. She wrote 3-½ as her answer, then complained because the problem was coming out “all weird.” I straightened that one out with her, only to have her evaluate the next problem, -2+ 2/3, as -2 2/3.

Dr. Pine realized that the depth of Chinese math learning far surpassed ours. Yet she seemed unable to perceive that the digging for meaning she observed in literacy classes was precisely the same digging for meaning evident in math classes*. She lauded it in math but lamented it in literacy, saying that teachers denied Chinese children the expression of their own personal opinions.

* Everywhere I wrote an asterisk I am referring to the Chinese philosophy of math education as evidenced by the textbook presentation of concepts and the implementation I and others have observed  in many Chinese classrooms.  HOWEVER, honesty compels me to relate that there are a number of  Chinese math teachers whose delivery of math concepts is at best cursory.  These teachers also assign an overwhelming amount of homework and "practice tests."  These teachers have been know to "steal" time for "unimportant" subjects like art and music for more practice tests in their never-ending quest to maximize test scores regardless of understanding. This approach is absolutely murderous to the spirit and curiosity of Chinese students.

When Are We Ever Gonna Use This?

Raise your hand if you have ever heard this question, “When are we ever gonna use this?” When I was a young teacher, I tried hard to answer. I used to give my students (junior high and high school) examples of math problems from various occupational fields. I bought a large poster that listed many occupations along the top and many mathematics topics along the side with black dots showing exactly which occupations use which topics.

Years passed. Film projectors gave way to Youtube videos. Mimeograph machines gave way laser printers. Whole new field of occupations emerged. I metaphorically threw up my hands in exasperation. When the inevitable question arose, I answered that I had no idea how they were going to use this information. I had no idea how their interests would develop, or which occupations they would pursue, or what the jobs of the future would be. All I could do was teach them a little bit of what had taken thousands of years for people to discover about math. My students were not always satisfied.

Then Paul Lockhart came along and wrote “A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form.” https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

Now I had an answer that captured their imaginations:

“In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide -- a relief from daily life, an anodyne to the practical workaday world….People don’t do mathematics because it’s useful. They do it because it’s interesting … The point of a measurement problem is not what the measurement is; it’s how to figure out what it is.”

The question of the usefulness of any particular subject stems from the mutual internalization of both the teacher and students of a questionable, yet unexamined assumption.

“To say that math is important because it is useful is like saying that children are important because we can train them to do spiritually meaningless labor in order to increase corporate profits. Or is that in fact what we are saying?”

Thus instead of teaching real mathematics, we teaching “pseudo-mathematics,” or what I have often called non-math, and worse, we use math class to accomplish this miseducation (See https://schoolcrossing.blogspot.com/2012/11/tricks-and-shortcuts-vs-mathematics.html and others). According to Lockhart, we teach math as if we think “Paint by Number” teaches art.

“Worse, the perpetuation of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of symbols, creates its own culture and its own set of values….Why don't we want our children to learn to do mathematics? Is it that we don't trust them, that we think it's too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon. Why not about triangles?

Math is like playing a game. As with any game, it has rules to be sure. However, it is more fun and more elegant than all other games because it is literally limitless.

Physical reality is a disaster. It’s way too complicated, and nothing is at all what it appears to be. Objects expand and contract with temperature, atoms fly on and off. In particular, nothing can truly be measured. A blade of grass has no actual length. Any measurement made in the universe is necessarily a rough approximation. It’s not bad; it’s just the nature of the place. The smallest speck is not a point, and the thinnest wire is not a line. Mathematical reality, on the other hand, is imaginary. It can be as simple and pretty as I want it to be. I get to have all those perfect things I can’t have in real life. I can never hold a circle in my hand, but I can hold one in my mind. […] The point is I get to have them both — physical reality and mathematical reality. Both are beautiful and interesting… The former is important to me because I am in it, the latter because it is in me.

Mathematics offers infinite possibilities for storytelling. I tell many stories as I teach math. My students are positively enchanted and remember them forever. One of my favorites is the kimono story.

I tell my students how in old Japan, servants helped geisha to put on the multiple layers of kimono. Each layer has to arranged and offset just so in order to reveal the colors of each layer. I tell them we are going to start with a geisha like 1/3. First we put on the 2/2 layer. 1/3 x 2/2 = 2/6. Notice that the geisha looks a little different, but underneath it is the same geisha. How about another layer, maybe 3/3. Okay 2/6 x 3/3 = 6/18. How about another 2/2 layer. 6/18 x 2/2 = 12/36. We can take off the layers one-by-one as well. This is called “simplifying a fraction.” Simplifying a fraction is simply a process of finding out which geisha is at the bottom of all those layers. If we are in a hurry, we can remove all the layers at once. How would we do that? In the case of our geisha, dividing by 12/12. The students love it.

The most elegant math story is the proof.

A proof is simply a story. The characters are the elements of the problem, and the plot is up to you. The goal, as in any literary fiction, is to write a story that is compelling as a narrative. In the case of mathematics, this means that the plot not only has to make logical sense but also be simple and elegant. No one likes a meandering, complicated quagmire of a proof. We want to follow along rationally to be sure, but we also want to be charmed and swept off our feet aesthetically. A proof should be lovely as well as logical.

How to Evaluate a Math Textbook

Regardless of Common Core, everybody knows that practically speaking, the textbook IS the curriculum. Therefore, it behooves textbook adoption committees to choose carefully. First, ignore the beautiful graphics. The beauty may truly be only skin deep. Reject books that teach tricks, procedures and shortcuts. Choose books that teach the profound understanding of fundamental mathematics. You do not have to read the entire book. Look especially for how the book handles the following topics:

Place value---Place value is arguably the most essential foundation stone of all future math understanding. Yet most textbooks provide only a rudimentary presentation of place value. Students are expected to do no more than name the place of a given digit or write a certain digit in a given place. The understanding of place value actually begins with counting. Make sure children name what they are counting and start with zero, “0 frogs, 1 frog, 2 frogs, 3 frogs...there are 7 frogs altogether.” Remember, place value depends on fully knowing the name of what is being counted, and not simply as part of a memorized pattern. 203 means you have counted 2 hundreds, 0 tens, 3 ones. 203 can also mean you have counted 20 tens, 3 ones. Which version is more useful depends on the context of the real life math. An early emphasis on place value helps students with later concepts such as fractions (203 thousandths), volume and area (203 cubes vs 203 squares), or the difference between like and unlike terms (3a + 2b). There are many more math concepts that depend first on naming what is being counted and understanding the significance of the name to place value.

The equal sign—An equal sign means everything around the equal sign is equal to everything else. Therefore an expression like 2 + 3 = 5 x 4 = 20 is not allowed because 2 + 3 does not equal 5 x 4. However the separator bar within the vertical format is allowed, because the separator bar does not mean equal; it is a separator bar.

Long Division---Although the idea that division is nothing but repeated subtraction is a bit oversimplified, the long division algorithm exactly depends on repeated subtraction because when you multiply within the algorithm, you are multiplying negative numbers. That is why you subtract the result of the multiplication. Look for a text that presents long division as more than memorizing the steps of the algorithm.

Multiplication and Division of Fractions---½ x 2/3 means one-half of two thirds. This example highlights the value of word problems. Word problems put math where it belongs and from where it arises, that is, math is the solving of real life problems. All math problems have a story. A page of naked problems has simply lost the stories. Suppose I have a ribbon 60 cm long. 2/3 of the ribbon is 40 cm, and half of that is 20 cm. 20 cm is 1/3 of 60 cm. Through examples like this, students can see that ½ x 2/3 = 2/6 = 1/3.

Division works the same way. Say I need to measure ¾ cup sugar and all I have is a 1/8-cup measuring cup. How many times do I need to fill my measuring cup to get ¾ cup sugar? ¾ cup divided by 1/8 cup therefore equals 6 times. (Notice again usefulness of knowing what you are counting. In this example, the answer is counting “times,” not “cups”). Texts should require kids to solve math problems by drawing pictures. When the student can reliably use a diagram to solve a problem, they are ready for the algorithm. Only at the end of the learning process should we teach the shortcuts. Math first, then shortcuts. Pictures are also the first step to proofs.

Absolute Value---Make sure absolute value is presented as distance from zero, NOT as simply a negative number turning into a positive number. A football analogy may help. If the quarterback is sacked, the ball may be 5 yards from the scrimmage line, but from the quarterback's point of view, it is still a negative 5.

Canceling---I loathe this word. Students are not “canceling.” They are simplifying a fraction. Simplifying a fraction means finding “1.” It does NOT mean crossing off numbers. Canceling leads students to lose track of the difference between “0” and “1.”

Multiplying and Dividing Decimals---Multiplying and dividing decimals has nothing to do with moving decimal points. It has everything to do with multiplying or dividing by powers of ten. 12 x 1.4 means 12 times 14 tenths. 14 tenths means 14 divided by ten, so 12 x 1.4 means [(12 times 14) divided by 10], which means 168 divided by 10, which equals 16.8. Students can tell where the decimal point goes, not by counting decimals places but by realizing the answer must be a number close to the product of the whole numbers. 12 x 1 = 12, so the answer must be close to 12. 1.68 is too small. 168 is too big. Therefore the answer is 16.8.

It is easy to confuse students by changing the problem slightly to 12 x 1.40. They will likely say they need to count 2 decimal places so the answer is 1.68. Giving them a new rule about ignoring zeroes does NOT build math understanding. Shortcuts are just that: shortcuts---and should be taught only when the student knows the actual road, not to replace the actual road.

Ignore the glitzy graphics and choose textbooks that handle all these topics well.

I Love Math Manipulatives...But

I love math manipulatives. I really do. Manipulatives allow students to physically model mathematics concepts. But manipulatives are no panacea. Manipulatives have significant, often overlooked, limitations.

Mistaken Modeling

Many teachers view math instruction as teaching standard algorithms, that is, teaching students the conventional step-by step recipe for computing an answer. Thus teachers use manipulatives to model algorithms. However, teaching algorithms is not the same as teaching math. For example, the most common explanation for dividing fractions is to multiply by the reciprocal. Multiplying by the reciprocal works because something mathematical is going on. However, we usually teach the superficial procedure and ignore the mathematics. The purpose of manipulatives is to model the mathematics, not the algorithm. The difference is subtle, but crucial.

Manipulatives Cannot Model Everything

Math is far more powerful than physical manipulatives. Manipulatives are merely a bridge to that power. Manipulatives cannot model beyond three dimensions, but manipulatives can lead students to math beyond the three dimensions. Some Montessori schools have a manipulative that physically models a quadratic equation, Ax^2 + Bx + C. If the factors of the quadratic equation are equal to each other, the quadratic equation models a square. If the factors are unequal, the quadratic models a rectangle.

I first saw the intriguing quadratic equation model in a Montessori school in Japan where preschoolers were enthusiastically absorbing the geometry of the quadratic equation without resorting to pencil and paper. FOIL? Who needs it? The factors were perfectly obvious to them. Add a “height” factor to model three dimensions. If the height is “x,” we have a model of a third-degree equation. We have an “x-cubed.” Cubed! How cool is that? Can we build a model in of an equation in the fourth degree? Well, now we have bumped up against a limitation. Mathematical representations can express math much more powerfully than physical models.

The Training Curve

It can sometimes require substantial training in the symbolism and design of the manipulative before the child can use the manipulative. For some children, imagining that one thing stands for another can create an obstacle to the mathematics itself. It is an adult myth that children have superior imaginations. Children represent, pretend, or re-enact what they already know. They have trouble with pretending something they do not already know. Adults can manage with the incomplete sets of manipulatives often found in classrooms. Children may be stymied. Children especially have trouble with strings of representations. Dr. Kamii says manipulatives can end up being “abstractions of abstractions” rather than the concrete models usually intended. For example, a teacher might say “We do not have enough hundred-flats for every group to make their number. You can use a teddy bear to stand for a hundred-flat if you need to.” Such instructions only make things more perplexing for the kids.

Impractical for Problem Solving

If manipulatives are used as algorithm aids, students may not be able to solve problems when they have no manipulatives, like during a test. Constance Kamii, who researches the ways children learn math, found that when young children were given a problem for which they had received no instruction and free access to a variety of manipulatives, writing instruments and paper, children preferred their own constructions over those imposed by others. Children preferred to think their way through problems with pictures they draw themselves rather than with manipulatives.

Broken analogies

Math manipulatives are analogies. Every analogy breaks down at some point. Math manipulatives are no exception. Manipulatives have lots of features which may or may not be salient to the math. Children may have difficulty understanding which features to pay attention to and which to ignore. For example, Cuisenaire rods are different lengths. Each length is a different color, but the color is arbitrary and has nothing to do with the math. However, the colors are sure convenient because kids can use them to express math without numerals.

Too Much Fun

Perhaps the most dangerous limitation of manipulatives is the fun. Student teachers have often reported to me that their math methods courses were little more than a term's worth of “playing” with manipulatives. They loved their methods course, but when they got into a real classroom with real kids, they found to their chagrin that they were woefully ill-prepared to actually facilitate the acquisition of mathematics concepts. I have often observed teachers use manipulatives as a fun diversion without ever getting to the point of the mathematics involved. I have seen educators demonstrate the use of manipulatives without ever building the bridge to the concept.

Manipulatives cannot substitute for the teacher's own profound understanding of the fundamentals of mathematics (PUFM). Sadly, nearly every college of education has a version of the course “Principles of Mathematics for Elementary Teachers” because so many elementary education students lack PUFM.

The over exuberant adoption of manipulatives is yet one more instance of educational pendulum swinging. Good ideas get over-used and misapplied all the time, often turning what could have been promising strategies into just another education fad.

Stand and Deliver? No, Sit Down and Shut Up

The movie, Stand and Deliver, told the inspirational story of one teacher's success in using Advanced Placement (AP) calculus with his demoralized students. The students complained, worked hard, fought back, bought in, and eventually passed the AP calculus test. Test administrators thought the students had cheated and canceled their scores. The students retook—and passed---the test. Garfield High in Los Angeles would never be the same. Or would it?

Texas hopes to replicate Jaime Escalante's resounding success. More and more schools are offering more and more AP courses to more and more students. But Texas school officials do not like the results. At least they do not like the statistics. More and more students are failing.

But the latest data show Texas high school students fail more than half of the college-level exams, and their performance trails national averages.

School officials wring their hands and wonder what could be going wrong. The students who are expected to fail are failing, and surprise, students from elite schools, the top tier, are failing in increasing numbers, too.

But high failure rates from some of the Dallas area's elite campuses raise questions about whether our most advantaged high school students are prepared for college work.

What is the problem?

For one, you can not just “helicopter-drop” AP courses into a school and expect instant education reform.

Because, two, the teachers may not be qualified to teach AP courses.

So, three, the teachers tend to fail to cover the material and properly prepare the students.

Besides, four, too many students enroll without adequate academic foundation for the courses.

The problem with looking to a movie for direction in education reform is that Garfield High's AP calculus program was just a bit little different than the movie version. Mr. Escalante spent years preparing the students, requiring them to take summer courses and come to school from 7:00 am- noon on Saturdays.

Even Garfield High did not sustain their own success. Please read that link. Mr. Escalante's experience is emblematic in terms of reform obstructionism, professional jealousy, and society's lack of respect for teachers.

Place Value Part 2: Base Ten for Young Students

One of the most fundamental mathematical concepts, yet one of the most poorly understood, is place value. The typical primary school lesson presents only a superficial, nominal understanding of place value. Students learn merely to correctly name the place-value columns, or identify the digit in a given column, but they often do not understand the significance of the column names.


In Part 1, The Chocolate Factory, I introduced a middle school activity for rebuilding often weak base ten foundational concepts. The activity extends understanding to place value in other bases. In Part 2, I will introduce activities suitable for much younger children. Young children can construct the meaning of base ten place value through many activities and games.

There is some evidence from Jean Piaget's work as illustrated in the video, that base ten is conceptually out of reach for very young children. If there is demand, I will present some activities that help young children explore “Two Land” and “Three Land.” Years ago I field tested a unit called “The Land of Hand” which of course would be “Five Land” in the terminology of the video.




Today I am going to concentrate on base ten, or “Ten Land.”
1. Morning Circle
Many kindergarten and first grade teachers have a regular morning circle time when they gather the children and go through a structured routine of talking about the calendar, the season, birthdays and other topics using a set of visual materials that are permanently on display. The two main math components are the calendar and the base ten pocket chart. The periodicity of the calendar lends itself to a number of activities for building number sense. The base ten pocket chart is decribed below.

The teacher prepares a display of three horizontal pockets with transparent envelopes on the front of each pocket. On the side is a cup full of Popsicle sticks and a stack of cards numbered with the digits from 0 to 9. Pocket charts can also be purchased from various vendors. Every morning the teacher takes one Popsicle stick and places it in the far right pocket (as you face the display). Each day the teacher replaces the card in the envelope to reflect the number of sticks in the pocket.

On the tenth day, the teacher places the tenth stick in the pocket and then makes a show of pointing out there are ten sticks. The teacher then bundles up the ten sticks with a rubber band and places the bundle in the middle pocket. The pocket envelopes should now show (empty, 1, 0) representing 1 bundle of ten sticks and 0 single sticks. The teacher goes through the Popsicle stick routine every day.

On the hundredth day, a celebration day in many schools, the teacher gathers the 10 bundles, ties them together with a piece of yarn and places the whole bundle in the far left pocket and changes the display to show (1,0,0) representing 1 packet of 10 bundles, 0 bundles of 10 sticks, and 0 single sticks. The teacher continues the routine until the last day of school at which point the display should show something like (1, 8, 5).



2. Trading Activities and Games

Playing games is a natural way for children to acquire all sorts of different aspects of number sense. Years ago I checked a book out of the library that was chock full of wonderful tutoring games. The book has long since gone out of print but no matter. I found the author, Peggy Kaye's website. Here is my version of a game she calls "Fifty Wins."

The teacher creates two boards on heavy card stock, one for each player. Each player also has a die. I recommend using extra large die if you can find them. Each player also has a collection of 50+ beans, pennies, or other counters. My own modification involves using the board at first, then doing away with the board and playing with pennies and dimes.




Each child casts their die in turn, and draws the number of counters that matches the number of dots on their die, placing one counter in each of the small squares of which there are nine. Upon accumulating the tenth counter, they transfer ten counters to one of the five big squares. The first person to get fifty counters wins. Children learn there can never be more than nine in the one's place, and that the ten's place is precisely groups of ten. If three big squares are filled and none of the little squares, they can see very clearly 3 (groups of ten) 0 or 30.

A modification I have made is to use poker chips for counters. I change the design of the board so that the nine little squares become a long rectangle outlined in one color (say blue) and the big squares are outlined in another color (say red). Then as the child accumulates 10 blue chips, the child exchanges the 10 blue chips for one red chip and places it in one of the red squares. The poker chip modification leads quite naturally in the penny-dime modification I mentioned earlier. I have also used the same poker chips with the same color signification for "The Chocolate Factory" activity, blue for leftovers, red for boxes, white for cases.

Another modification of mine which may be considered a weakening of the game is the use of a die to generate numbers. The original game uses a spinner where some of the fields say “Win 10.” At the beginning the child will dutifully count out ten beans and place them one by one in the small squares, only to have to transfer the entire group of ten to a big square. Very soon the child counts out the ten beans and straightway places them in a big square. The opportunity to realize a group of ten in one turn is lost when die are used, but I suppose you could use a set of two dice. I like the die because the child does not have to read words or numerals. With die, the child has only to match, by one-to-one correspondence, the beans to the die spots. There is no need to reference numerals at all, so the game stays squarely focused on number and avoids number/numeral conflation.


“Make Fifty” is just one example of what is known as a “trading activity.” Cuisenaire rods also work well for trading activities. Every ten cubes makes one rod. Any base-ten block set goes one step further where every ten rods makes one flat, and every ten flats makes one cube. Many base ten block worksheets can be adapted to active lessons.

All manipulatives have limitations and some researchers are concerned about the limitations of base ten blocks. Nevertheless, with a good mix of activities, the teacher can address the differing learning styles of each student.

Stuff to Avoid
Worksheets
Generally speaking, worksheets should be avoided. Nevertheless, I like to design special worksheets as data recording instruments for math labs utilizing base-ten blocks and Cuisenaire rods. Students can learn a lot of math without writing numerals. In fact, a foundation of math reasoning skills without reliance on numerals helps children acquire the concept of the difference between numbers and culturally-determined symbols for numbers such as Arabic numerals. Schools “accidentally on purpose” teach children to confuse number and symbol. Cuisenaire has a few such worksheets along this idea, but I have some problems with the worksheet design. Maybe I'll collect my math lab worksheets into some kind of cohesive with comprehensive directions for using them with children and make them available.

Computer-Based Materials

Too many of the computer-based materials, animated mathematics and virtual manipulatives, though so appealing to adults, often have a magical quality to young children. Regrouping happens before their very eyes but they do not understand the mathematical concept and mechanism. They do not get from the computer what I call the psychology of numbers, or how numbers behave. It is just a lot of cool special effects without specific mathematical concept acquisition benefit.

Calculators

Despite the National Council of Teachers of Mathematics (NCTM) claims to the contrary, calculator studies with the youngest students show no advantage in the development of children's number sense. In 2002, I conducted a major survey of research, research critiques, case studies, and editorials. I periodically asked NCTM to provide me a list of what they characterized as supporting research, but they never did. I found no basis for NCTM's assertion that research backed their recommendation for calculator use in the earliest grades. I found that calculator usage need not hinder the development of math reasoning skills, but it may in fact do so. Teachers report that children become overly dependent on the calculator and have difficulty learning to evaluate the reasonableness of their answers. They trust the calculator more than themselves.

Links

The following is a list of links of base ten lessons. They are presented as is. Many exemplify what I believe are the main weaknesses of most base ten teaching.
Lesson plans reviewed by teachers:
Crayola Tally Sticks:
Applet: but better off using concrete manipulatives.
An Unreviewed Collection of various resources:
A favorite resource for getting teaching ideas:
Vendor: