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.

Missing Key to Understanding Place Value

I write a lot about place value. Place value (along with zero) may arguably be the most important math concept because it underlies every single calculation we do. Yet teachers often do not teach place value well. Teachers (and most curriculum) are satisfied with a very superficial understanding of this essential concept. If a child can identify the place name of a given digit or put a digit in a given place, most teachers deem the child to have a good understanding of place value. Place value is so much more.

Groups of Ten

Place value is all about making groups of ten. Well, yeah, the reader might say. Tell me something I don’t know. The key to understanding place value is the realization that each succeeding place represents a group of ten of the preceding place. Duh. Stay with me here. The curriculum and instruction alludes to this key, but rarely makes it explicit. Most textbooks have replaced “borrowing and carrying” with “regrouping,” and this was a positive step, but students still take a mechanical view. They still borrow and carry as they move leftwards through an addition or subtraction problem without realizing that they are actually making or breaking a group of ten at each successive place. For example, if they carry a one from the tens place to the hundreds place, they mechanically add that one to the other digits in the place without realizing that the carried one represents making a group of ten. In fact, most students will say, (correctly on a superficial level), that they made a group of 100 because they put the “1” at the top of the column named “100s place.”

Place value is all about making groups of ten. Subtraction is all about breaking groups of ten into loose ones and dumping them with the other loose ones. Every place except the loose ones is a group of ten something. Teachers tell students that each succeeding place is larger by a magnitude of ten, but somehow children fail to grasp the significance of this fact. The reason the standard addition algorithm works is because you are gathering up groups of ten at every place. Likewise, the reason the standard subtraction algorithm works is because you are breaking a group of ten at every place.

Students betray this lack of deeper understanding when they express surprise that given the number 437, that an equally correct answer to the question “How many tens?” is 43. They are also surprised to learn that when we say 2 tens and 5 ones equals 25, what we really mean is 2 tens and 5 ones equals 25 ones.

A better way to express it is “2 groups of ten and 5 loose (not in a group) ones equals 25 loose ones.” Therefore, I spend a lot of time having students expand large numbers in a variety of ways.

Methods of Expansion

Expansion basically means counting numbers of groups. There are several ways to express this accounting. Given the number 47,396:

Standard Methods:

Place Value Names: 4 ten thousands, 7 thousands, 3 hundreds, 9 tens, 6 loose ones

Multiplication: (4 x 10,000) + (7 x 1000) + (3 x 100) + (9 x 10) + (6 x 1)

Exponents: (4 x 10^4) + (7 x 10^3) + (3 x 10^2) + (9 x 10^1) + (6 x 10^0)

Notice that using exponents displays the idea that each succeeding place is a group of ten, however, most teachers do not make this understanding explicit. Most students just view, for example, the number 10000 or 104 as merely another way of expressing the place value name “ten thousands.”

I give my students practice with alternative expansions.

Alternative Expansion

47, 396 = _______ thousands, ________tens, _____ ones

47, 396 = _______ ten thousands, ________hundreds, _____ ones

47, 396 = _______ tens, _____ ones

And of course, we can repeat this exercise with multiplicative expansion and exponential expansion. This sort of practice has the side effect of helping students later understand rounding to a given place. I am also very picky about counting and zeroes. 0 is a real counting number, and I expect students to show that they know that 102 has 0 tens, or (0 x 10) or (0 x 10^1).

Place Value in Later Mathematics

This sort of foundational learning of place value pays dividends in later mathematics. To give just a couple examples:

Bases: Each succeeding place is a group of the given base. This understanding gives logic to “borrowing and carrying” in other bases besides base ten.

Polynomial expressions: Quadratic and other equations of the form Ax^n + Bx^n-1 + …Gx^1 + Hx^0 are essentially equations expressed in base x. Students will find that working in other bases is greatly simplified if they exponentially expand the number and replace the base with x.

Polynomial (and by extension, synthetic division: When students learn to divide equations such as Ax^3 + Cx^1 + D by say, x + 1, they must remember to insert the missing term, 0x^2. Students do learn to replace the missing term in a mechanical way. However, if they have regularly understood zero as a real counting number and included the zero term in their elementary expansions, it seems obvious to them that of course they must have the zero term if they expect to successfully complete the division.

More attention to a deep understanding of place value in the early years would make much of later mathematics less mechanical and more intuitively comprehensible, thus actually saving instruction time and allowing teachers to teach more math.

Is Division Repeated Addition?

One of Stanford mathematician Keith Devlin's pet peeves is the common “division is repeated addition” meme . He despises it so much he has something like a mantra, “Repeat after me. Division is NOT repeated addition.” Naturally, math teachers give him a lot of pushback because division is indeed repeated addition (except when it's not).

It seems there are actually two topics in play here. 1) Multiplication as repeated addition and 2) the skill of elementary math teachers. I completely understand his frustration with prospect of undoing the poor math instruction college students typically receive during their elementary school years. I also experience the same frustration as a secondary and college level instructor.

Multiplication as repeated addition is not a definition of multiplication, even though many elementary math teachers erroneously think the definition of multiplication is precisely repeated addition. Repeated addition is merely another name for the group model of multiplication. There are other models, such as the array model, the area model and the number line model, to name the ones most commonly presented to elementary students. Devlin rightly maintains that it is inaccurate to say that multiplication is repeated addition, period. As a misleading misstatement, it is right up there with “you cannot subtract a bigger number from a smaller number.”

However, as properly taught (a giant qualifier, I know), the group model is merely the first element in a teaching sequence which eventually progresses to the area model, then to the use of the area model to multiply fractions, and beyond. For example, you can definitely model a positive whole number times a negative rational number on the number line where it very much looks like repeated addition of the given negative number. Turn that number line vertically, and it makes even more sense to students because it reminds them of another number line they are very familiar with, the thermometer.

Devlin writes, “Addition and multiplication are different operations on numbers. There are, to be sure, connections. One such is that multiplication does provide a quick way of finding the answer to a repeated addition sum.” Exactly, and this is precisely the way a good teacher presents the group model. Children sometimes ask questions like, “Instead of saying 8 + 8 + 8 + 8 + 8, and then saying the answer, can't we just say “8, five times” and then say the answer?” Of course we can, and that is what we do when we say 5 x 8 = 40. The group model is meant to express this particular connection between addition and multiplication. The group model is not meant to be a definition of multiplication. Nevertheless, I agree there are too many elementary math teachers who fail to make the distinction, or properly progress through the models.

An umbrella idea I like involves the word “of” as an English language expression of multiplication. We can say “5 groups of 3,” or “5 groups of -3,” or “1/2 of 3,” or “1/3 of 4/5,” or “16/100 of 40” or “75% of 200,” etc and neatly cover most examples of multiplication that children are likely to encounter before junior high. Devlin prefers scaling as the dominant meme and argues that children should readily understand scaling because examples of scaling surround them. The problem is most eight-year-olds have difficulty comprehending scaling as a model and effect of multiplication. Even though they can readily see that a scale model is a perfect replica of the original, they do not understand how it is possible that doubling the dimensions of a garden (to take a simple example) results in a garden four times larger. Most of the scaling children see is usually on maps where the scale is for them an unimaginably large (or small, depending on viewpoint) number.

Teachers are better off working their way up to the scalar model of multiplication. I have found this is best done by reminding younger students early and often with the idea that we have not yet exhausted the possible models and applications of multiplication. I have found it useful to show some examples of these applications, and say something like, “Later you will learn how you can use multiplication to produce an exact scale model, or use multiplication to produce a real-life-sized object from a scale model.”

Actually, most students get their first solid grip on scaling when they work with similar figures (typically triangles) during high school geometry. Personally, I have found success with older elementary students by giving them basic practice in scaling on the coordinate plane or increasing recipe yield and other types of problems. Students also enjoy the products of their work whether it be art or good eats. The number line model is also a good introduction to scaling because you are scaling only one dimension, as opposed to the two and three dimensions involved in scaling area and volume, respectively.

Devlin also laments the constant push to make math “real.”

No wonder children arrive at college not only having little or no genuine understanding of elementary arithmetic, they have long ago formed the view that math has nothing to do with the world they live in...many people feel a need to make things concrete. But mathematics is abstract. That is where it gets its strength.

His comments seem contradictory, but they are not. One of the most enjoyable aspects of teaching math is showing students the leap from concrete to abstract. For example, I love showing students a cube and showing them the edge e, the face e x e, then showing them cube e x e x e. I usually let the idea hang, and love when someone asks if I can show e x e x e x e on the cube. No, I answer, I have nothing to show them that. And therein lies the power of math. Math can help us express ideas we can understand, but for which we have no physical representation. I am thrilled when someone asks if e x e x e x e can be time. I ask what would e x e x e x e then express. The children are really products of this century, and one of them is likely to answer, “the coordinates of a time-traveling spaceship.” So much fun. One time, a child said, “Maybe someday we will have a real meaning for more es.”

To the extent that teachers present repeated addition as a property, which in some applications, connects the operations of addition and multiplication, no harm done. However, considering repeated addition as THE definition of multiplication is a serious problem, and Devlin is right to be concerned about it. As Denise Gaskins pointed out, “...if (a particular) model doesn’t work universally, then (the model) certainly cannot be used to define the operation.”

Devlin asks an interesting, and in today's pedagogical environment, nearly taboo question:

The "learn the technique first and understand later" approach is very definitely the only way to learn chess, and millions of children around the world manage that each year, so we know it is a viable approach. Why not accept that math has to be learned the same way?

I would say that technique over understanding has been the preferred approach for centuries, but by all accounts, many adults have never made it to the “understand later” stage. In my experience teaching concept concurrently with technique works the best. The problem I am seeing is that these days too many elementary teachers attempt to teach concepts they barely understand, and then give short shrift to technique because the kids have calculators for that. The result is legions of kids who not only have faulty understanding of the concept, but also lack the ability to perform the technique quickly and accurately.

Additional Discussion:

https://www.quora.com/Why-is-it-incorrect-to-define-multiplication-as-repeated-addition

https://denisegaskins.com/2008/07/01/if-it-aint-repeated-addition/

http://scienceblogs.com/goodmath/2008/07/25/teaching-multiplication-is-it/

https://numberwarrior.wordpress.com/2009/05/22/the-multiplication-is-not-repeated-addition-research/

http://www.quickanddirtytips.com/education/math/is-multiplication-repeated-addition

http://billkerr2.blogspot.com/2009/01/multiplication-is-not-repeated-addition.html

http://rationalmathed.blogspot.com/2008/07/devlin-on-multiplication-or-what-is.html

http://rationalmathed.blogspot.com/2010/02/keith-devlin-extended.html

http://homeschoolmath.blogspot.com/2008/07/isnt-multiplication-repeated-addition.html

http://www.textsavvyblog.net/2008/07/devlins-right-angle-part-i.html

http://mathforum.org/kb/thread.jspa?threadID=2045768

http://www.qedcat.com/archive_cleaned/114.html

Slaying the Calculus Dragon

No doubt about it. Many students consider calculus scary, right up there with monsters under the bed. Calculus is the Minotaur or St George's dragon of math at school. Sadly, schools have done little to undermine its almost mythological reputation, what with “derivatives” and “integrals” and those frightening numberless equations recognizable by the initial elongated “∫.” There is like, what? 100 equations, that, according to most teachers, need to be memorized.

It is a pity, because calculus is really the Wizard of Oz, terrifying to behold, but quite tame behind the curtain. Did you know that most people do calculus in their heads all the time? In fact, because the numbers associated with calculus are ever-changing, moment by moment, doing calculus with numbers is a bit pointless. A mother filling the bathtub very often does not want to stand around watching the water. She knows that the water is coming out of the faucet at a certain rate. She knows the bathtub is filling at a certain rate. Every moment the volume of water is changing. Yet, she reliably comes back to check the tub before it overflows.

The high school quarterback and his wide receiver communicate an even more difficult calculus on the field, seemingly by telepathy. The quarterback never aims the ball at the place where the receiver is standing. That mental math is too easy, more like algebra or even arithmetic. No, he aims the ball toward the place he hopes the receiver will be. In his head, he calculates the trajectory of the ball, the amount of force necessary (oh my gosh, not physics, too!), the speed of the receiver, and every other factor. And most of the time he gets it right, and the pass is completed. The fun thing about calculus is that the numbers are ever-changing. It is like hitting a moving target, whereas algebraic numbers thoughtfully stand still.

At its core, calculus is nothing but slope. Remember humble slope, change in y over change in x. Slope is an expression of rate, such as, change in miles over change in time, most commonly called “miles per hour” or mph. The graph is a straight line, so you can pick any two points to find the slope. But what if the graph is curved? If you were to magnify each point and extend its line, every point has a different slope. That is because straight lines illustrate rates like speed (velocity), while curves illustrate rates like acceleration (getting faster and faster each moment), just like the football getting slower and slower until it reaches the top of its path, comes to dead stop (but only for a moment) and then gets faster and faster.

A graph has three basic pieces of information, the x data set, the y data set, and slope. “Derivatives” are used to find the slope of a curve at any point when the x and y data are known. When you know one of the data sets and the slope, you can use “integrals” to find the other data set. Techniques like finding the area under the curve are used to get as close as possible to the exact answer. First you divide the area under the curve into rectangles all having the same “x” length. Then you add up all the area. Obviously, some of the rectangles are a little too small for the curve and some are too big, so your answer is only an approximation. If you shorten the “x” length to make narrower rectangles, your approximation will be closer. Integration allows you to find an exact answer instead of an approximation, however close the approximation may be. But there are “limits” (you have no doubt heard of limits). Your “x” length may be very small indeed, but it can never be zero, because then the sum of the rectangles would illogically be zero.

If you can find a skilled dragon slayer, that is, a teacher who can demystify it, studying calculus can be great fun.

How Should Students Show Their Math Work?

In this post, I am pinging off Maria Miller of Math Mammoth. I recommend Math Mammoth for its concept-based lesson development and worksheets.

Many students resist showing their work. They feel they are demonstrating their smartness by not showing their work, as in “See, Ma. No work.” However, when you ask these students how they got the answer, they cannot remember what they did. Sometimes they say they used a calculator. OK, I say, but what numbers did you put into the calculator? They cannot tell me. I explain that since we cannot record thoughts the way we can record voices, students need to make a record of their thoughts when they solve a problem. Dispensing with the work is not actually smart at all.

Now, here is where we see the real difference between strong students and weak students. Strong students respond to my words, and start showing work ever after. Weak students respond (eventually) only to action. I make them do their homework again, and I mark right answers wrong if there is no work.

As Maria says:

The purpose of writing down the work allows someone else to follow the person's thought processes. This is of course important for students to learn no matter what their future occupation: they need to be able to explain to others how they solve a problem, whether a math problem or a problem in some other field of life!

As strong as Chinese math teachers tend to be, they do not encourage students to show their work. Chinese teachers expect “clean” papers, with only answers. Chinese teachers check whether answers are right or wrong. They are completely unconcerned with why the student got a wrong answer, or if the answer is coincidentally right for the wrong reason. Retraining my students has been quite a challenge. Today they appreciate the need to show work, and they work hard to demonstrate that their work flows in a logical manner. Today, they show off their work instead of showing off the lack of work.

Even though Chinese teachers do not want to see work in the final product, they actually have high standards for the format of work. They train students from first grade in this format, and one reason they do not care to see the work in, say, fifth grade is they trust the student followed the format to get the answer the teacher does see, a dubious assumption at best.

Maria says she would ask primary student to verbally explain how they got an answer. Chinese teachers expect students to translate verbal (or written) math problem to mathematical expressions. Students learn to write “number sentences” from the very beginning. Perhaps there is a picture of a tree branch with three birds and two more birds landing. The child translates this picture in the number sentence “3 + 2 = “, and then writes “5 birds”.

I modify this approach a little. I expect children to write “3 birds + 2 birds = The idea of ignoring the units and then plugging them back in at the end leads to all kinds of confusion in later grades. Leaving the units out of the work is a major reason students persistently forget to square the unit when finding area. The math sentence should be 4 cm x 5 cm =

When students first begin studying area and perimeter, I make them write intermediate steps. In the case of area the intermediate step is: (4 x 5) x (cm x cm) = 20 cm2. In the case of perimeter, the intermediate steps might be (2 x 3) cm + (2 x 5) cm = 2(3 +5) cm = (2 x 8) cm = 16 cm. There are many types of problems where keeping track of the unit is vital. An early example is division, especially division with remainders. Often the unit for the quotient is different from the unit for the remainder. Knowing the difference is the key to understanding the solution.

I also require the box. The box makes the number sentence a complete sentence. Later, we will replace the box with a variable, and later still the variable may appear somewhere besides the end. Take this problem for example: There are 5 birds in the tree. After a certain number fly away, there are 2 birds left. How many birds flew away? I expect children to translate this sentence to math as written, without doing any preliminary math in their heads. Thus “5 birds - = 2 birds”.

Most teachers have the children write this math sentence as 5 birds - 2 birds = 3 birds. Doing so requires the students to do some math in their head first. The purpose of the number sentence is to accurately translate the problem to math terms. The number sentence must follow the story. The number sentence for a multi-part story should incorporate all parts into one number sentence. When problems become more difficult, the ability to translate the story to math as written becomes essential. The crucial part of solving a math problem is the number sentence. When the number sentence is correct, absent any silly mistakes in the work, the solution will most certainly be correct.

Finally, I require the students to answer the question with a complete sentence. The purpose of answering the question is to help student differentiate the solution from the answer. For example the solution to the question, how many cars do we need for the field trip might correctly be 5.2 cars, but the answer is 6 cars.

Summary

The work for a word problem needs to have three parts.

1.  A translation of the word problem into a complete math expression that includes the units and follows the story.

2. The arithmetic which tracks the units all the way through to the solution and may include intermediate steps for as long as necessary for mastery.

3. The complete answer to the question.

Sample

Math Expression: 10 x [$10.50 – (2/5 x $10.50)] = n

Work: 10 x {$10.50 – [($10.50 ÷ 5) x 2]} = n

10 x [$10.50 – ($2.10 x 2)] = n

10 x ($10.50 – $4.20) = n

10 x $6.30 = $63.00

Answer: Annie's total bill is $63.00 or Annie paid a total of $63.00 for the shirts.

Well-trained fifth graders have no trouble displaying their work as in the sample. This vertical work format, started in first grade, gives the students excellent preparation for mathematics involved in algebra, chemistry, physics and calculus. In fact, starting in third grade, I often have students format their work in two vertical columns, the second column for the math property used, as in this simple sample:

Problem: There are 5 birds in the tree. After a certain number fly away, there are 2 birds left. How many birds flew away?

Number Sentence: 5 birds – n = 2 birds

Work:

---

Arithmetic	Property
5 birds – n = 2 birds	given
+ n              + n	both sides rule
5 birds + 0 = 2 birds + n	additive inverse (opposites rule)
5 birds = 2 birds + n	additive identity
-2 birds = -2 birds + n	both sides rule
3 birds = 0 birds + n	math fact/additive inverse
3 birds = n	additive identity

Answer: Three birds flew away.

Do Not Use Baby Talk to Teach Math

Number sense is like a mighty oak rooted in the subconscious. Beginning in infancy, it is little more than a humble acorn. Misconceptions are weeds that also root in the subconscious and stunt the acorn's growth. The language we use to express number sense can nurture the acorn or plant the seeds of misconceptions. The resulting weeds are pulled only with great difficulty.

The baby talk some teachers use to teach addition can plant misconceptions that prevent students from properly developing the concept of mixed numbers. We should never, ever say, “2 and 3 makes 5.” Even a good quality text like Singapore Math talks baby talk, but that is because something was lost in the translation to English. We should say properly, “2 plus 3 equals 5.” Children are perfectly capable of learning correct language, and it saves them the trouble of unlearning it later. After all, we do not expect them to say “2 and 3 makes 5” forever. We expect them to transition to adult math expressions sooner or later.

So what is wrong with “and” anyway?

AND means something mathematically, and it is not “plus.” For example, 2½ does not mean 2 + ½. You do not believe me? How about -2½? Does that mean -2 + ½? Of course not, but that is not obvious to kids. The mixed number -2½ means “minus 2 and ½,” not “minus 2 plus ½.” more technically, it means “minus 2 and minus 1/2” or -(2 + ½). Subtracting a mixed number is often the child's first exposure to the distributive property, however I have never seen a textbook make it clear. Instead, we routinely plant misconceptions and then wonder why kids sometimes have so much difficulty with math.

It is not all that hard to teach either, especially if using money to illustrate. “If I have three dollars, and I spend two and a half dollars, how much do I left left?” I spent 2 dollars AND I spent ½ dollar. A seventh grade teacher mentioned in this blog the difficulty his own students were having.

I saw this post about a week after it appeared, and so I was prepared to prove MY 7th grade pre-algebra students would not make such mistakes. Equation-solving did them in, with this as a solution: -5¼ + 2½ = -3¾. I had previously showed them how illogical such a thing was, and how it didn't make "number sense", yet the method error persisted.

Break it out the way students do, and the thinking error emerges: -5 + ¼ + 2 + ½ = -3 + ¾ = -3¾. Our long custom of misrepresenting “plus” as “and” has led them to the idea that all you have to do is take out the plus sign and shove the fraction up against the whole number. If it is already shoved together, pull it apart, put the plus sign back in, and voila! The problem is solvable.

Because the root of the misconception is in the subconscious, even if they get some number sense training and even understand the training, they will fail to see the error of their thinking, and so the error persists. The teacher will probably have to name this misconception directly and explain to students how they were mis-taught in the past. They may then be able to pull it into their conscious mind and deal with it.

Decimal numbers might help. 37.2 is not “thirty-seven point two.” It is “thirty-seven AND two- tenths.” The function of the decimal point and the meaning of “and” is to differentiate the wholes from the part, whether in decimal numbers or mixed numbers (which brings me to another pet peeve. It is not that we are “converting” from decimal numbers to mixed numbers. Both forms are essentially the same: a whole number with a fraction). The decimal point does NOT mean “perform the operation of addition.”

AND is a mathematical operation called “union.” The performance of AND yields a result similar to addition only when the sets contain entirely discreet members. Otherwise, the result of the AND operation is a smaller number than the result of ADD. It used to be that AND (and OR) could be tough to teach. Nowadays, with Internet searches, lots of kids readily understand that search terms with OR between them will get you a bazillion, mostly useless hits, while search terms with AND between them will get you a smaller number of hits than each search term alone. Set theory using sets of hits makes sense, and a great way to exploit technology such that technology actually increases learning, instead of being the usual monumental distraction.

Tricks and Shortcuts vs. Mathematics

The issue is not whether algebra should be taught in the eighth grade or later. The issue is not whether local schools should be able to make their own textbook adoption decisions. The issue is about how easily states make big changes based on flimsy research which asks the wrong questions, only to backtrack later because solutions that solve the wrong problem do not work. California reverted to phonics in 1995 after abandoning it for a faulty implementation of whole language based on research that answered some questions, but not the questions that matter.

The emphasis on algebra in the eighth grade is misplaced when even students with good math grades enter algebra weak in math concepts. I am working with an A student now who is solving for x in problems involving mixed numbers. She wrote these "computations:" 2 + ¼ = 2¼, 3 + (- ¾) = 3-¾, and -2 + ½ = -2½. Do you see the pattern? In her mind, numbers are disembodied entities with no real meaning. She thinks all she has to do is take out the plus sign and push the fraction up against the whole number.

These silly errors happen in an education system where children have been taught tricks and shortcuts since first grade. The problem is teachers call tricks and shortcuts "math," and when children do well on a test of tricks and shortcuts, they learn their good grade is proof they understand math. Actually the grade proves only that they can reliably implement tricks and shortcuts.

I have worked with children who have terrible math anxiety because they do not do well with the tricks and shortcuts. Some part of their mind has rejected the tricks and shortcuts as not making sense, so "math" does not make sense. If they ever get a chance to acquire true number sense, then they find out they are good at math after all.

Sometimes we reward unthinking compliance (as when kids memorize the tricks and shortcuts) and punish the thinkers for whom the tricks and shortcuts do not make mathematical sense.

Algebra in 2nd Grade?

In February, 2009 a teacher in Montana made EdWeek headlines because she was teaching algebra to second graders and had been doing so for five years. Why all the oohs and aahs?


Elementary math is supposed to prepare students for high-level math classes in middle and high school. Students should not need a dedicated pre-algebra class. When I was a kid, pre-algebra did not exist. Now it is part of every school's math course line-up.

The author of a pre-algebra text wants students to build math reasoning skills. However math reasoning often does not happen. Many teachers treat pre-algebra as a last chance for students to get those blind elementary math procedures down pat. Problem is, a student can be A+ in procedures and still not understand algebra. In fact, students competent with procedure often believe they are good at math. It is not their fault. Our education system has been telling them for years that grades equal understanding. So if they get a good grade in math, naturally they conclude they are good at math.

Math has been misnamed. What passes for math in schools is often non-math. “Carry the one” is not a mathematical explanation for what happens in addition. It is a blind procedure. Students get good grades in non-math believing it is math. No wonder algebra is such a shock. Math reasoning skills actually matter in algebra.

Still a student with a good memory can get by, at least until they meet a new math monster, calculus. However, since middle and high school math also fail to teach math reasoning, now students take pre-calculus, another relatively recent addition to course offerings. Without a major change of emphasis, pre-calculus prepares students no better for calculus than pre-algebra prepared them for algebra.

By now pre-calculus students have so internalized non-math that they complain to the instructor, “Just tell us how to get the answer. We don't want to know why.” Just give us some more blind procedures.

The Vital Place of Place Value

Perhaps one of the most important foundational concepts in mathematics is place value. As the Massachusetts Department of Education rightly observes, “The subtly powerful invention known as place value enables all (my emphasis) of modern mathematics, science, and engineering. A thorough understanding removes the mystery from computational algorithms, decimals, estimation, scientific notation, and—later—polynomials” (Massachusetts Department of Education (2007). In fact, it is when students first meet polynomials in algebra, that the lack of a proper grounding in place value becomes painfully apparent. Most likely a significant number of the difficulties that students experience with math may be traced to place value.

I reviewed the state standards of various states with regard to place value. I looked for an explicit reference to “regrouping,” the current term for what we used to call “borrowing” and “carrying.” My survey of state standards resulted in a mixed bag. Some states require students to do little more than name the place value of a particular digit. Other states expect students to use various means to model place value. Alaska asks students to not only perform the operations of addition and subtraction, but to explain those operations.

State standards have their utility, but apparently whatever the specific state standard, students are able to follow the regrouping recipe without having any real understanding of why the recipe works. In fact, adults of all ages add and subtract by mindlessly following the recipe. Most adults, and of course, all children could do with a solid grounding in place value.

I have a number of activities I use to make place value explicit. Tomorrow I will tell you about an activity I like to call “The Chocolate Factory.”

Are You Good at Non-Math?

One of the most persistent issues in math education has been the reliance on non-mathematical explanations of mathematical principles. For example, we tell students that when multiplying positive and negative numbers “two negatives make a positive.” Such an explanation clarifies nothing about how the numbers behave or why an ostensibly English grammar rule should apply to math.

What is worse, we tell students who successfully master such non-math explanations that they understand math, or that they are good at math, when really what they are good at is the blind procedures of non-math. Young children have no way to distinguish non-math from math. They believe, because we have told them, that they are learning math, when in fact they are learning non-math. If it does not catch up to them earlier, it often catches up to them in algebra class where historically “A” students may find themselves inexplicably failing to understand the subject material.

Children rely on adult teachers to initiate them into the joys and delights of math, but often teachers make math a difficult subject, usually because they themselves understand non-math rather than math. After all, if numbers are running around, it must be math, right? Even sadder are the number of elementary teachers who lack an interest in acquiring what math education researcher Liping Ma called “the profound understand of fundamental mathematics” even while believing that they “know” math.

Many colleges of education and community colleges have sought to address the serious weaknesses in the mathematical understanding of elementary teachers by either requiring, or at least offering, coursework in mathematics for elementary teachers. I am quite sure a survey of professors teaching such required courses would report remarkable levels of student resentment at being forced to take a class in something they think they already know, to “jump hoops” as they say . Some of these students may wake up and get motivated to learn the math concepts. Some seethe inwardly as they pass the class. However, most students will pass the class and eventually be certified to teach regardless of their poor attitude toward or lack of understanding of the vital core subject of mathematics.

Only later, once they are in the classroom, will they be likely to regret the squandered opportunity to finally get math. Perhaps they may grow to appreciate the professor who tried to give them the gift of mathematical understanding, a gift they resisted at the time.