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.

Zero is a Real Number

Zero is a real number. Could such a headline possibly be click bait? If so, it is pretty lame. Of course everyone knows zero belongs to the set of real numbers. The problem is the word “real.” A sentence such as “zero is a real number” immediately puts people into mathematics mode wherein they consider the word “real” in only its mathematical sense. Sometimes people recall set theory theory and the curious case of a set containing only one member, zero, as opposed to an empty set with no members. The problem here is that set theory leads people to objectify zero. They think of zero as an object rather than a number.

Zero is a real number. When the truth of this statement dawns, the world changes forever. If you are thinking, “Well, of course zero is a real number. What a stupid waste of time to write about it,” you may be one of those people for whom the realization of this truth in all its depth and beauty has not yet occurred.

My student teacher this year was one of those people in September. 27 years old and she never knew zero was a real number. She thought she knew it, but she betrayed herself when she began teaching first graders to answer the question “how many?.” Although she never explicitly said so, she gave her charges to understand that the minimum answer to the question was “one.” I surprised her by reminded her that “zero” is a legitimate answer to the question, “how many?” She did not quite believe me. “Think,” I said, “Of a time when you may have looked for eggs in the refrigerator and found there were zero eggs.” Her eyes widened. “Oh...yeah!” she said, “I hadn’t really thought about it.” I reminded her that when she set up her counting situations, to let zero often be the answer. Children come to school already preconditioned to disregard zero. Their parents and preschool teachers have given them 6 years of experience ignoring zero. One of the first math tasks at school is to undo that misconception.

Zero is a real number. Tax season provides a perfect example. Consider two taxpayers. One person may complete a tax return and find that his tax liability is zero. Therefore when he pays his taxes, he pays zero dollars. Another person does not even complete the form. One person paid no taxes. However, the other one did pay his taxes, and he paid zero dollars. “Zero” and “nothing” are not the same thing. Set theory was supposed to make this distinction clear, but too often we go into math mode and miss the point.

Zero is a real number. I will never forget the day in November when this realization struck my student teacher. She was in the middle of teaching first grade math when she looked at me sitting in the back of the room and said incredulously, “Zero is a real number,” as if it were her own discovery and not something I had said again and again for more than two months.

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

Nobody Understands Place Value

Parents often ask me to help their children with math homework. My reply is always the same. I am not interested in “helping” with homework. I am very interested in addressing the gaps and misconceptions that give children difficulty with their homework in the first place. Chief among these is a pervasive lack of understanding about place value.

Children's understanding is generally limited to identifying the place value of a given digit or inserting a digit in a given place. I do not blame the children. Most curriculum asks them to do nothing else. Replacing the terms “borrowing” and “carrying” with terms like “exchanging” or “regrouping” represented a tremendous improvement in math education. Even though many elementary math teachers these days play trading games, and the kids appear to know what they are doing, every junior high or high school math teacher has observed that they do not profoundly understand place value, so crucial to understanding the quadratic equation, bases other than base ten, and other topics.

Therefore, I usually start my homework help by first playing simple trading games with the student. The problems I use for the games are ones I know students can calculate correctly, such as 48 + 17. Maybe they can even do it in their heads. No matter.

The first thing I do is dispense with the usual place value names. I use “loose ones” and “packages of ten.” Loose ones is a better term than simply ones. The ones are ones precisely because they are not in a group. They are loose. Children often do not realize that the loose ones' place is fundamentally different from all the other places. Thus they often have trouble with the ones' place in other bases. For example, teachers tell students that when they are working in, say, base five, the largest possible digit in the ones' place is a 4 because “there is a rule that the one's place can be no larger than one less than the base.” Although it is true that the ones' place can be no larger than one less than the base, it is not because of a rule. The reason is much more fundamental than a mere rule. Understanding the ones' place as “loose ones” is key to discovering that fundamental principle.

After we play the trading game for a little while, I have the student do a simple addition problem. When students calculate a problem like 18 + 25, they put a 3 in the ones' place and a 1 above the 1 of 18. Then they add 1 + 1 + 2 and write a 4 in the tens' place, resulting in the answer of 43. Then I ask, “How did you get that answer?”

They usually reply, “I put a 3 in the ones' place and a 1 above the 1 of 18. Then I add 1 + 1 + 2 and write a 4 in the tens' place, so my answer is 43.”

That's fine. Of course they answer in a mechanical, non-mathematical way. They have heard teachers repeatedly explain addition problems to them in much the same way. Then I ask, “Yes, but why did you do that?”

Students invariably reply, “Because that is how the teacher told me to do it.”

Then I ask, “Yes, but why did the teacher tell you to do it that way? Why does it work?” Now they are stymied.

So I show them how to “prove” (not really prove, more like demonstrate) the answer using a picture (similar to this one, but simpler).

I show them how to draw the picture and talk their way through it. “See, you have 15 loose ones. That is enough to make a package of ten. So you gather up a package a ten and put it with all the other packages of ten. You still have 5 loose ones left. Because 5 is not enough to make a package, you leave them loose and show them in the loose ones' column. You add up the packages of ten and put that total in the packages of ten column.”I have them illustrate several problems by drawing the picture.

Very often students realize for the first time that 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.

Instead of the usual place value mat, I like to use a mat that labels the places a little differently. I start with the loose ones, then packages of 10 loose ones, then cartons of 10 packages, then boxes of 10 cartons, then cases of 10 boxes, then pallets of 10 cases, and so on. I usually stop at cargo ship with 10 shipping containers. Kids love it. Even second graders can easily calculate a multi-digit addition or subtraction problem. In fact, after kids master place value as groups of 10, they often ask me to set them problems with any number of digits. I usually refrain from a problem with more than 13 or 14 digits because even though the kids find the problem easy, they also find it tedious and time-consuming. But hey, tedious and time-consuming is a whole sight better than hard when it comes to doing homework.

Rote Does A Lot of People A Lot of Good

Recently I read a forum comment somewhere to the effect that “rote does not do anyone much good.” Ironically, this comment was part of a comment whose main idea was that educators should question buzzwords and dubious tenets of pop-education. In spite of the current popularity of this sentiment, rote actually does a lot of people a lot of good. I agree that sometimes educators have abused rote. I remember a fourth grade science book from around thirty years ago that included a whole unit on state birds and flowers as if such information had even the remotest relevance to science. Naturally, since most schools consider the textbook to be THE curriculum, many teachers felt compelled to “teach” children the state birds and flowers. Inquiry or constructed learning will not work for this kind of arbitrary knowledge. Rote is really the only way. State birds and flowers should never have been part of the science book in the first place.

Today we have Common Core. Just as in the past, of course publishers are scrambling to roll out new textbooks that reflect Common Core. Thus yet again, the textbook will become the default curriculum. Yet, even within Common Core and even within a commitment to teaching concepts over rote, there are still a number of topics for which rote is still the best method. The order of the alphabet, sight words, broad overviews of history, and arithmetic facts are just a few. People forget the intense amount of repetition and memorization involved learning a first language, let alone subsequent languages. Churchgoers know memorizing the books of the Bible greatly aids in finding the preacher's text. If you want to pass your first driver's license test, it would behoove you to memorize the rules of the road. Rote memorization of poems provides an avenue of future pleasure. If you live in a character-based literary system such as Chinese, you will need to memorize several thousand characters to simply be a literate person.

Another related canard holds that the purpose of education is not knowledge itself, but the ability to find knowledge. However, students who lack a substantial reserve of memorized knowledge have a great of difficulty even figuring out what search terms to use. Personally, as much as I hated memorization when I was young, I have come to appreciate the easy access to information, internet or no internet.

Finally, there is an important reason to refuse an ideological stance against rote learning. Sometimes it is all a student has left. Math is a subject area well-suited to discovery methods. All mathematical procedures are based on the real and predictable behavior of numbers. Very little math knowledge is actually arbitrary. The best math teachers who consistently use the best discovery methods to help students acquire mathematical concepts still sometimes encounter students who simply cannot get it. If they cannot acquire the concept, and we also also deny them rote learning, we leave them with nothing. Although rote should never be the first resort in a non-arbitrary domain such as mathematics, rote still remains the best last resort to ensure that all students acquire the basic skills they need for their adult lives. As educators, we need be at the forefront of confronting ideological statements wherever found

Nothing Wrong with Rote

A popular view among educators is that rote learning is bad, bad, bad. Point out how well Asians do in international tests compared to Americans, and defenders will likely counter that maybe so, but Asian education depends on that bad rote learning, but we Americans, no matter how poorly we compare, are superior because we emphasize concepts and creativity.

To be honest, as a math teacher, for a long time I believed that rote learning, while undeniably effective, was merely second rate. Kids must need lots of memorization of mathematical recipes and homework practice to pound poorly understood mathematical procedures into their brains. As a math teacher, I believed that if skilled math teachers developed a strong conceptual foundation within children, the logic and elegance inherent in math would minimize the need for tedious, time-consuming homework.

As a junior high and high school math teacher, my ideas about foundation building were only theoretical. It was easy to look at my students, the products of elementary school instruction and conclude that elementary math teachers were doing a terrible job of building foundations. After all, there is plenty of documentation for the inadequate math teaching skills of elementary teachers. I could be charitable. I did not blame elementary teachers too harshly, because they themselves did not acquire mathematical foundations when they were elementary students. They cannot teach what they do not know.

However, until I came to China to work with elementary students, I had never had a chance to test my hypothesis that all students really need is a great foundation. Actually, for most students my hypothesis worked. They could demonstrate a deep and thorough understanding of the concepts I taught. I often assigned homework of only five to ten problems, and after this little bit of practice, they could reliably get the right answers.

However, there were a few students for whom concepts were not enough. One day they would demonstrate terrific understanding. The next day we had to start almost from scratch. I tried everything, every approach I could think of, including a lot more practice. What worked when nothing else did was lots of practice---yes, tedious, time-consuming practice designed to activate rote memory.

I finally concluded that if a child can successfully utilize the concept to solve problems, that's great. However, if the only way they can master the procedure is to repeatedly execute it until they reliably get right answers (my standard was 80% or more), then so be it. Better that then sending them on their way with nothing.

Sadly, as comparative studies show, too many American children lack essential conceptual foundations because the documented lack of teaching skill means teachers fail to actually effectively teach the concepts. Students also fail to do adequate procedural practice because of the American educational aversion to “boring” homework. So I say let's teach concepts and teach them well, AND let the students practice until they can get the right answers. Some students may need more practice than others. So be it.

Interestingly, recent research on children's and teen brains explains why children have such great memories, and how practice, even in an environment of complete concept understanding, is necessary to build brain pathways.

the whole process of learning and memory is thought to be a process of building stronger connections between your brain cells. Your brain cells create new networks when you learn new tasks and new skills and new memories. And where brain cells connect are called synapses. And the synapse actually gets strengthened the more you use it. And especially if you use it in a patterned way, like with practice, it gets even stronger, such that after the practice, you don't need much effort to remember something.

When we dismiss rote learning, we forfeit a valuable tool for building neural pathways in the brain.

See related posts:

Patient vs Impatient Problem Solving

Common Cart---Cart Before Horse

I Love Manipulatives...But

Cultural Sacred Cows of American Education

MacDuff: The New Math

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.