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.

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.

The Issue is NOT Algebra 2—The Issue is First-Grade Math

A novelist writing for Harpers believes students should not be required to study Algebra 2. The fact that students "are forced, repeatedly, to stare at hairy, square-rooted, polynomialed horseradish clumps of mute symbology that irritates them, that stop them in their tracks, that they can't understand." is not an argument against Algebra 2. It is an argument against the ineffective math foundations instruction occurring in the primary grades.

I mentor two first-year teachers. One just gave her second graders a page of double-digit subtraction with regrouping, and all of them scored 100%. She concluded her students already "understand" subtraction. I replied, "The only thing you learned from that is they all memorized the step-by-step blind procedure and can execute it. You have no idea if they actually understand the math." We often teach non-math shortcuts and call it math. As an article from the New Yorker points out, "These shortcuts aren’t a faster way of doing the math; they’re a way of skipping the math altogether."

Our school system continually tells students who are successful with non-math that they are good at math, and then wonder why these same students struggle with real math. The situation is even worse for students who never mastered the blind procedures in the first place.

Our biggest problem is that our elementary math teachers understand only non-math themselves, as Liping Ma documented in her now famous book, Knowing and Teaching Elementary Mathematics. A review of the literature shows that our elementary teachers lack what Ms. Ma calls “a profound understanding of mathematical foundations.” The first step needs to be the development of skilled math teachers at the critical elementary level.

True, Authentic, Real Life Math Problem of an Eighth Grader

A certain student had recently missed much of the first quarter, so her band teacher did not count those weeks when figuring the credit for the weekly practice logs. Therefore, the student had only five weeks worth of practice log grades for the quarter whereas her classmates had ten weeks worth. She got 100% for each of the first three weeks she was back. When her report card came, she was shocked that she had a C. She thought she had an A in the bag.

Looking back at her practice log grades, she saw 100, 100, 100, 70, 0. “I forgot to turn in last week's practice log,” she explained. BUT, she knew how to figure averages, and when she did, she got 74%. “Ah, so there's the C,” she said.

Then she asked me, “How many minutes will I have to practice to bring my grade back to an A by next week?” True to form, I irritated her by telling her to figure it out. A real life math problem was staring her in the face. She had been wondering if there was such a thing as a real life math problem. “How do I do that?” she wailed. I told her to think about what she already knows and what she needs to know.

Her train of reasoning: One more week means I will have six total weeks of practice log grades. To average 100%, I will need 600 total minutes. I have 370 minutes. So I need 230 more minutes. I will need to practice 230 minutes this week. (An aside: I wonder if my teacher will give credit for so many minutes in one week). My practice log is due on Friday, so I have six days to practice 230 minutes.

So far, so good, but then her reasoning began to go awry. She divided 230 by 60, and got 5.5 on a calculator. She did not question the result. We need to teach students to determine the neighborhood of the result before doing any actual computation. I do not like to call this process “estimation,” because almost all kids have reduced estimation to mere rounding, and nothing more. Most kids tolerate estimation lessons at school, but basically tune them out because they have been socialized to value answer-getting techniques. Estimation does not, in their minds, yield “answers.”

(Now I have to explain that during this whole process, I was busy with my own work, so I was only seeing pieces intermittently, as she showed them to me. She showed me the calculator with the 5.5 in the display, which at this point was all I knew. I reconstructed her train of reasoning later from her comments).

I asked, “What does 5.5 mean?” She said, “5 hours and 50 minutes.” Remember, this student has all As in math, but as I have explained before, much of math in schools is misnamed. It is really non-math, but since schools call it math, students believe it is math, and if they get good grades in non-math, they believe they are good at math.

I probed, “How did you get that?” She looked at me like, well duh, isn't it obvious and said a little too loudly, “5.5 is 5 hours and 50 minutes.” Then turning away, she poked something into her calculator.

“How do I round this?” she asked. The display showed 0.9166666.

“You have asked the question wrong. No one can answer your question the way you asked it. You need to specify what place you want to round it to.”

“The thousandth's place. So 0.917.”

“That's right. But what are you counting?”

She pondered a moment and wrote 0.92.

“And what is that?” “Minutes,” she said, and wrote 92.

“How did you get that?”

“I need minutes, so I moved the decimal point.”

“'I moved the decimal point' is never a mathematical explanation for anything. You need to give a mathematical reason for the math you do. What did you do to get 0.92 in the first place?”

“I divided 5.5 by 6 to get the number of minutes I need to practice everyday. 0.92 minutes doesn't make sense so I need to move the decimal to get a number that makes sense.” (With this kind of reasoning, is it any wonder our students are so poor at math? And if they use the same faulty reasoning for any of life's other problems, no wonder decision-making ability is also poor. When they become adults, they are easily scammed by poor reasoning that sounds good to them).

She has three main problems:

1. Using disembodied numbers

Teachers have allowed her and her classmates to disembody numbers since first grade. What I mean is students have been trained to compute with only the numbers and attach the units to the result later. When students do that, they attach the unit they want, not the unit their computation produces. What she should have done is written 5.5 hours = 0.92 hours/day. Her unit was “hours/day.” However, since she was looking for minutes, she did the math the way so many students (and adults) do: 5.5/6 = 0.92 minutes.

2. Mixing bases

She did not realize that decimals numbers are base 10, and clock numbers are NOT base 10. I set up some place value columns for decimal numbers, and another set of columns for clock numbers. Then we did some counting so that she could see how numbers end up in the columns they do. First, we counted decimally, that is, in base ten. Then we counted time. As our paper time clock ticked over 59 in the minutes column to 1 in the hour column and 0 in the minutes column, she exclaimed, “Oh, base 60, like the Incas.” She could tell me that 0.5 = 50/100 = 50%, but still insisted that 5.5 hours = 5 hours and 50 minutes. She realized that she was looking for 50% of 60 minutes, but insisted she should divide 50% by 60. Eventually, understanding dawned. She realized that since 50% means half, then half of an hour is 30 minutes, so 5.5 hours means 5 hours 30 minutes. (My own work had come to a complete standstill long before). “So 'of' means multiply, right?”

3. Misunderstanding “Decimal Number”

She thinks, like so many kids do, that a decimal number is a number with a decimal point. Just take out the decimal point and presto, changeo, it is not a decimal number anymore. What else do we expect when we teach kids tricks,shortcuts and blind procedures,and call this strange conglomerate "math?"

In quite East Asian style, we had spent over an hour on this one problem. Eventually, she determined that (leaving aside the original calculator error), she actually had gotten her answer way back at 0.92 hours/day. She realized that the math had “spoken” to her if she had only thought about it correctly. What the math said was that she would need to practice a little less than an hour a day. She never noticed the calculator error, and I did not point it out.

Epilogue:

She practiced 60 minutes (in 30 minute increments) three days in a row. Then it occurred to her that if she practiced 60 minutes per day for 6 days, her total would be 360 minutes, not the 240 minutes she was expecting. She has not practiced for two days, but plans to practice 60 minutes on the sixth day. She got a real-life lesson in checking the math by plugging the solution back into the original problem, a step her teacher requires, but she resents as a time waster. We talked about that maybe her teacher really does have some wisdom in her requirements. She also admitted that her goal is to do the minimum necessary to secure an A. Excellence and doing one's best is just adult yadayada. At least her bar is set at A.

Patient vs. Impatient Problem Solving

According to Dan Meyer, the problem with a steady diet of TV sitcoms is students learn to expect easy problems resolved in twenty-two minutes “with a laugh track.” We have now raised several generations of “impatient” problem solvers, and typical math textbooks pander to the syndrome instead of challenging it.

Mr. Meyer has a prescription for what ails our math teaching.



According to Mr. Meyer, there are two kinds of mathematics: computation, or “the step you forgot” and math reasoning. Within computation, there are a lot of tricks and gimmicks, like counting decimal places. The tricks work because of the underlying math reasoning. We teach the tricks, the non-math, and call it math. Good grades for non-math amount to “congratulating students for following the smooth path and stepping over the cracks.” No wonder our students display symptoms of impatient problem solving syndrome:

Lack of Initiative,
Lack of Perseverance,
Lack of Retention,
Aversion to Word Problems, and
Eagerness for Formulas.

The older your students, the more likely you can be teaching math reasoning well and still encounter not only the symptoms, but also resistance to the cure. Your students have been so conditioned by previous experience, that like chemical tolerance, they do not believe they can function mathematically any other way. It might be a good idea to show this video the first day of class to shock their systems into even entertaining the idea that math could be different.

His description of his presentation of the water tank problem is very like the way Japanese elementary teachers have been teaching math for decades (that I know about). They can easily spend a whole period on a single problem, but they actually save time, because they are not wasting it practicing a forgettable blind procedure on twenty problems. They invest the time it requires to think about math, for as Mr. Meyer says, “Math is the vocabulary for your own intuition.”

Mr. Meyers suggests a five-part prescription:

Use Multimedia,
Encourage Student Intuition,
Ask the Shortest Possible Question,
Let Students Build the Problem, and
Be Less Helpful.

Teachers ignore many features of a problem as irrelevant without discussion as if we expect students to figure it out on their own. Many do, some do not. Asking what matters, says Mr. Meyer, is probably the most underrepresented question in math curriculum.

After, and only after, students have acquired the math reasoning should we give them shortcuts, tricks and mnemonics.This video is an excellent example of a math teacher receiving accolades for teaching non-math.




And finally, just for fun.

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.

No Surprise Algebra-For-All Fails

“Algebra-for-All Policy Found to Raise Rates Of Failure in Chicago”

Math educators, with good reasons, have long recommended that students be required to study algebra. Many districts mandate algebra in the ninth grade. California, one-upping everyone else, currently requires eighth graders to take algebra. Japanese children begin studying algebra in the fifth grade. So how's it working out?

Findings from a study involving 160,000 Chicago high school students offer a cautionary tale of what can happen, in practice, when school systems require students to take algebra at a particular grade level.

160,000 is a lot of students, and normally the bigger the sample from the population, the more reliable the conclusions. Researchers studied eleven “waves” of students entering ninth grade from 1994 to 2005.

(Researchers) compared changes within schools from cohort to cohort during a period before the policy took effect with a period several years afterward. They also compared schools that underwent the changes with those that already had an “algebra for all” policy in place.

What did the researchers find?

The policy change may have yielded unintended effects, according to researchers from the Consortium on Chicago School Research, based at the University of Chicago. While algebra enrollment increased across the district, the percentages of students failing math in 9th grade also rose after the new policy took effect.

By the same token, the researchers say, the change did not seem to lead to any significant test-score gains for students in math or in sizeable increases in the percentages of students who went on to take higher-level math courses later on in high school.

Not much upside. More students failed, test scores were flat, and the percentage of students motivated to take advanced math course did not rise, but, gee, “algebra enrollment increased.” The district says more students will fail when required to take harder courses without supports in place. Yet the district made attempts to include supports over the last seven or eight years.

Steps include developing curricular materials introducing students to algebra concepts in grades K-8, requiring struggling 9th graders to take double periods of algebra, and providing more professional development in math to middle and high school teachers..

One of the researchers thinks that test scores did not improve because teachers may have “watered down” the content since “math classes included children with a wider range of ability levels following the change.”

But Japanese elementary schools are not tracked. All children study exactly the same material with such predictability that some observers have quipped that every child in Japan is on the same page of the textbook on any given day. I have successfully taught Algebra 1 to high school special education students, or to give due credit, special education students have successfully learned Algebra 1 under my guidance.

The problem is with issuing mandates without a coherent, integrated societal commitment to the foundations of education, mathematics in particular. I have seen Montessori preschool students exploring algebra with manipulatives. I have often said that lots of profound math can be learned without any resort to pencil and paper. Children do not necessarily need numerals to understand number.

There is one other thing. Japanese children from kindergarten age regularly take abacus lessons the way American children take piano or ballet. Generating a sum with the abacus is different than generating a sum using the written algorithm. The very process of thinking about number and computation in more than one way leads to greater mathematical flexibility. Japanese students can therefore more readily absorb and manifest algebraic thinking. That's my hypothesis anyway and maybe the Gates Foundation or somebody else will provide me a grant to test it.

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.