My Favorite One-Liners: Part 92

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

This is one of my favorite quote from Alice in Wonderland that I’ll use whenever discussing the difference between the ring axioms (integers are closed under addition, subtraction, and multiplication, but not division) and the field axioms (closed under division except for division by zero):

‘I only took the regular course [in school,’ said the Mock Turtle.]

‘What was that?’ inquired Alice.

‘Reeling and Writhing, of course, to begin with,’ the Mock Turtle replied; ‘and then the different branches of Arithmetic — Ambition, Distraction, Uglification, and Derision.’

A nice subtraction trick

A friend of mine recently posted this trick for subtracting any number from any multiple of 10^n. (I discovered this trick when I was a boy and have been using it ever since.)

Pedagogically, I don’t think I’d recommend requiring every elementary school student to learn this trick. But this does make a nice enrichment activity for talented elementary school students, as it requires conceptual understanding of subtraction and not just the ability to follow a procedure.

 

 

Subtraction Trick 1

Source: https://www.facebook.com/australianteachers/photos/a.1409177146069885.1073741828.1409169412737325/1505873516400247/?type=3&theater

Here’s another approach, taken from the comments of the above webpage: consider 5000 as 500 groups of 10 and 0 groups of 1, and then regroup.

 

Subtraction Trick 2

Common Core, subtraction, and the open number line: Part 4

The following picture has been making the rounds lately.

opennumberlinesubtraction

My bedrock position is simply stated: I’m for teaching any technique in elementary school that’s (1) logically correct, whether or not it’s the way it’s (mythically) “always been taught,” (2) encourages students to think mathematically, as opposed to mindlessly following a procedure with no real conceptual understanding, and (3) prepares students for algebra in a few years’ time.

That said, I have a lot of opinions about this picture, which does not necessarily align with our society’s impatient obsession with 10-second sound bites and 140-character tweets. So be it. I will divide my opinions into several categories of increasing scope.

  1. The solution of this particular question.
  2. The pedagogical reasons for using this technique (called an open number line). In other words, do we only want Jack to get the right answer, or do we want Jack to understand something about the logic behind the answer?
  3. The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
  4. The importance of engaging parents with unorthodox ways of teaching mathematics.

Some of my opinions will line up nicely with supporters of the Common Core. Other opinions will align with the Common Core’s thoughtful critics.

green line

This is Part 4 of this series of posts: the importance of engaging parents and caregivers when unorthodox techniques of mathematics are taught.

I’ve only witnessed the implementation of the Common Core from afar, but there’s absolutely no doubt that the professional development of teachers who have been asked to teach math in a new way has left a lot to be desired. Ditto for explaining these new approaches to parents and caregivers who want to help their children.

True story: I personally did not know about the open number line until the “Meet the Teachers” night that was held for parents near the start of the school year. The teachers explained that they would be doing math a little differently and did a couple of examples using the open number line. I could feel many eyes in the room looking back at me (people know I’m a math professor) with facial expressions saying “Is this stuff really going to work for our kids?”

As this was my first exposure to the open number line, I was skeptical (I would have preferred using base-10 kits) but I held my tongue and listened carefully to the presentation. After the presentation, I was convinced that this was a completely legitimate way of teaching the subject and that the teachers had the requisite depth of understanding to teach arithmetic using this technique. After the presentation, I told anyone who’d listen that this technique was mathematically sound and pedagogically sound, even if it was different than the way that “it’s always been taught.”

Parents generally bought into the technique that evening. I’m not sure that they would have bought into it if its rationale had not been carefully explained to them.

And, as a reminder, Texas is not a Common Core state.

The failure to explain to parents and caregivers unorthodox but correct ways of teaching mathematics has been perhaps the greatest failure of the roll-out of the Common Core. It’s unacceptable that children are crying over their math homework and parents feel powerless to help (a common theme that I’ve heard over and over again from my friends).

Teachers and parents ought to be natural allies in wanting children to have a greater depth of understanding of arithmetic that will prepare them for algebra later. However, because the strategies of teaching the “why”s of mathematics have generally not been carefully explained to parents, they naturally feel somewhat helpless when trying to help their children with their homework.

My own field of research is not mathematics education. So, at professional conferences, I’ve asked friends and colleagues the same question over the years:

Letting children use their own natural curiosity to get at they “why”s of mathematics is good. Letting children use their own natural curiosity and also having the support of parents at home is better. So what research has been done on strategies on successfully engaging parents with how mathematics is currently taught versus how it was taught a generation ago (or, more accurately, what parents remember of their own experiences from elementary school)?

To my surprise, people that I greatly respect did not have an immediate answer to my question. So I’m guessing that while there’s been a lot of research into successful strategies for teaching mathematics in the classroom, there hasn’t been a lot of research into how these strategies can be supported when children are away from the classroom and asking their parents for help on their homework.

I’ll repeat the close of yesterday’s post: I won’t defend the indefensible way that the Common Core has been rolled out. Voters will be more than justified in voting out anyone who supports the Common Core if its implementation isn’t fixed in the very near future.

 

Common Core, subtraction, and the open number line: Part 3

The following picture has been making the rounds lately.

opennumberlinesubtraction

My bedrock position is simply stated: I’m for teaching any technique in elementary school that’s (1) logically correct, whether or not it’s the way it’s (mythically) “always been taught,” (2) encourages students to think mathematically, as opposed to mindlessly following a procedure with no real conceptual understanding, and (3) prepares students for algebra in a few years’ time.

That said, I have a lot of opinions about this picture, which does not necessarily align with our society’s impatient obsession with 10-second sound bites and 140-character tweets. So be it. I will divide my opinions into several categories of increasing scope.

  1. The solution of this particular question.
  2. The pedagogical reasons for using this technique (called an open number line). In other words, do we only want Jack to get the right answer, or do we want Jack to understand something about the logic behind the answer?
  3. The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
  4. The importance of engaging parents with unorthodox ways of teaching mathematics.

Some of my opinions will line up nicely with supporters of the Common Core. Other opinions will align with the Common Core’s thoughtful critics.

green line

This is Part 3 of this series of posts: assessing the depth of a student’s knowledge of mathematics in ways that are developmentally appropriate. To be blunt, this has been one of the great weaknesses of the roll-out of the Common Core in the early years of its implementation.

Where I agree with most critics of the Common Core is that, even though I want students to understand why (for instance) subtraction works, 2nd and 3rd graders are still learning how to write complete sentences. So of course they will get easily frustrated with explaining their reasoning in paragraph form. I think there are better ways of assessing whether young children really understand subtraction that is more developmentally appropriate. (Here in Texas, students in this age group are instead asked to explain their reasoning by drawing pictures.)

I’m happy to defend the standards of the Common Core themselves, but it’s patently obvious that the implementation of the Common Core standards were driven by textbook publishers and not educators. I don’t doubt that much of what’s assigned for homework under the guise of the Common Core is developmentally inappropriate.

I’m perfectly happen to defend the logic behind a specific pedagogical technique like the open number line. But I can still be genuinely annoyed that the first years of the Common Core has been so badly botched. And I’m especially annoyed that most public defenders of the Common Core seem deaf to the legitimate and thoughtful complaints of its critics (as opposed to the braying expressed in the original picture).

Here’s another example that made the rounds in recent months, for which I have a considerable amount of sympathy.

The problem in question that forms the basis of her argument:

Mr. Yumata’s class has 18 students. If the class counts around by a number and ends with 90, what number did they count by?

The “correct” answer requires students to divide 90 tick marks into equal groups of 18, counting the number of tick marks in each group. Again, the textbook publisher missed the mark. The phrasing of the problem asks for just the answer (5). If the problem wants to know a rationale for the answer, then it should have been phrased in a different way.

There absolutely is a place in the curriculum for the method described in the video — dividing X objects into piles of Y to see how many objects are in each pile. This works well when children are first exposed to division and X and Y are reasonably small integers. However, by the time students get to X =90 and Y =18, another method should be used. By this point, dividing objects into piles is logically correct but pedagogically questionable, as it requires precisely 108 steps (as noted in the video). Any little careless mistake in counting will lead to an incorrect answer.

Flatly, I won’t defend the indefensible way that the Common Core has been rolled out. The textbook publishers have clearly missed the mark on how to assess the depth of a student’s knowledge. Voters will be more than justified in voting out anyone who supports the Common Core if its implementation isn’t fixed in the very near future.

 

Common Core, subtraction, and the open number line: Part 2

The following picture has been making the rounds lately.

opennumberlinesubtraction

My bedrock position is simply stated: I’m for teaching any technique in elementary school that’s (1) logically correct, whether or not it’s the way it’s (mythically) “always been taught,” (2) encourages students to think mathematically, as opposed to mindlessly following a procedure with no real conceptual understanding, and (3) prepares students for algebra in a few years’ time.

That said, I have a lot of opinions about this picture, which does not necessarily align with our society’s impatient obsession with 10-second sound bites and 140-character tweets. So be it. I will divide my opinions into several categories of increasing scope.

  1. The solution of this particular question.
  2. The pedagogical reasons for using this technique (called an open number line). In other words, do we only want Jack to get the right answer, or do we want Jack to understand something about the logic behind the answer?
  3. The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
  4. The importance of engaging parents with unorthodox ways of teaching mathematics.

Some of my opinions will line up nicely with supporters of the Common Core. Other opinions will align with the Common Core’s thoughtful critics.

green line

This is Part 2 of this series of posts: the pedagogical reasons for using an open number line.

I firmly reject the premise, implied by the picture, that the only goal of teaching mathematics is just getting the right answer. If that’s the goal, then we can eliminate much of the K-5 curriculum by handing out 19-cent solar calculators to every student. That will pretty much guarantee that the students will solve elementary arithmetic problems like the one shown. It will also fail to develop their higher-order reasoning skills and leave them hopelessly unprepared for algebra.

A great example of a teacher using children’s natural curiosity to foster mathematical thinking can be seen in this video: https://www.teachingchannel.org/videos/common-core-teaching-division.

I absolutely agree with Common Core critics that students in elementary school should, at the end of the day, be able to quickly compute 427-316 using the standard algorithm in 5 seconds or less. I would also argue that it’s also important for students to learn why is algorithm works and not just how to use the algorithm. This deeper level of conceptual understanding begins to lay the foundation for the more abstract thinking required in later years.

The open number line predates the Common Core by about 25 years; for more information, see http://www.k-5mathteachingresources.com/empty-number-line.html. When used properly (a big “if”), it will lead students to the standard algorithms for addition and subtraction. I’ve personally seen the open number line successfully used in the elementary schools where I live in the hands of skilled teachers. Bottom line: the students had no problem with the technique. (By the way, Texas is not a Common Core state.)

So I take great issue with the quote “The process used is ridiculous and would result in termination if used.” Of course the process is ridiculous for people who already know how to subtract. For students who are first learning how to subtract multi-digit numbers, however, it is a completely appropriate way to introduce students to the topic which will lead them to the standard algorithm for subtraction. Once learned, then there will be no need to revert back to the previous method.

green line

I have heard some critics say that teaching the “why”s of mathematics, using the open number line and other techniques, merely favors the stronger math students and the expense of the weaker students. Perhaps this happens in some classrooms. However, this has not been my personal experience. I have supervisory authority over aspiring teachers who have taught math to literally thousands of third, fourth, and fifth graders — all in a non-Common Core state. With careful supervision, they have used the children’s natural curiosity and inductive reasoning (but not deductive logic — that’s an important distinction) to get at the “why”s of mathematics as well as the “how”s. And I’ve not personally witnessed anything like the frustrations that I’ve heard as the Common Core has been implemented. Hence my “don’t throw out the baby with the bathwater” approach when I hear and appreciate legitimate concerns and frustrations with how the Common Core has been rolled out.

One more thing: a great motivation of the Common Core has been to improve student achievement in algebra by fostering mathematical habits of thinking in the lower grade levels. It’s certainly debatable about whether or not this ideal has been achieved. But the motivation for the Common Core and improving conceptual understanding at the lower grade levels is to increase access to higher-level mathematics, not to favor the high achievers.

Though I’ve seen it used effectively, I’m personally not an advocate for the open number line. Given the choice, I would use a base-10 kit (shown above) for students who are first learning subtraction. I would start by collecting 4 plates of 100, 2 rods of 10, and 7 cubes of 1. I would then remove 3 plates of 100, 1 rod of 10, and 6 cubes of 1. The remaining pieces (1 plate of 100, 1 rod of 10, 1 cube of 1) represents the answer of 111.(This particular problem didn’t require trading a plate of 100 for 10 rods of 10 or else a rod of 10 for 10 cubes of 1; this would be handled later in the curriculum.)

I do recognize that teaching children multiplication and especially division with a base-10 kit is very cumbersome and is significantly easier using an open number line. So I see the rationale for using an open number line for addition and subtraction if it will be used in later years to teach multiplication and division.

I don’t support requiring children to learn subtraction or any other concept in mathematics by any one specific technique. The big issues are (1) getting the right answer and, of equal importance, (2) understanding why the answer works. I take issue with the original picture because it favors (1) but not also (2), and a surface level of understanding will not serve students well when they hit algebra.

Bottom line: I have absolutely no problem with schools using a technique like the open number line, as it is a mathematically correct way of explaining to students why the answer works.

 

Common Core, subtraction, and the open number line: Part 1

The following picture has been making the rounds lately.

opennumberlinesubtraction

My bedrock position is simply stated: I’m for teaching any technique in elementary school that’s (1) logically correct, whether or not it’s the way it’s (mythically) “always been taught,” (2) encourages students to think mathematically, as opposed to mindlessly following a procedure with no real conceptual understanding, and (3) prepares students for algebra in a few years’ time.

That said, I have a lot of opinions about this picture, which does not necessarily align with our society’s impatient obsession with 10-second sound bites and 140-character tweets. So be it. I will divide my opinions into several categories of increasing scope.

  1. The solution of this particular question.
  2. The pedagogical reasons for using this technique (called an open number line). In other words, do we only want Jack to get the right answer, or do we want Jack to understand something about the logic behind the answer?
  3. The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
  4. The importance of engaging parents with unorthodox ways of teaching mathematics.

Some of my opinions will line up nicely with supporters of the Common Core. Other opinions will align with the Common Core’s thoughtful critics.

green line

This is Part 1 of this series of posts: the solution of this particular question. Here it is:

Jack correctly started at 427 on the number line. He then correctly understood that 316 consists of 3 groups of 100, 1 group of 10, and 6 groups of 1. He then correctly subtracted 3 groups of 100 (for an interim answer of 127) and then correctly subtracted 6. However, he forgot to subtract 10. That’s why he got a wrong answer (121) that was 10 more than the correct answer (111).

Just to make sure I wasn’t completely missing the mark on this, I rewrote the problem (without the handwritten commentary) and showed it individually to a few elementary school students. They all saw Jack’s mistake within 15 seconds. They may not have been able to explain what Jack did right and what Jack did wrong in the form of a letter (more on that in a later post), but they certainly identified the core problem quickly.

I understand a parent’s frustration with knowing how to subtraction but seeing a child learning subtraction in a different way. (More on that in a later post). I also understand that some may argue with this technique of teaching children how to subtract. (More on that in a later post.) But there’s no way to sugarcoat this: an engineer who took differential equations and read this problem but couldn’t figure out that Jack forgot to subtract by 10 has little conceptual understanding of mathematics.

So let offer some free advice to critics of the Common Core who want to share this picture to vent their complaints. I am totally sympathetic with frustrations expressed in this picture. Sharing this picture with your fellow critics may feel good, perhaps with the self-justification “If an engineer can’t figure this stuff out, then how can I?!?!” However, sharing this picture is not going to persuade anyone who disagrees with you to your cause. Remember: some children can solve this problem in 15 seconds or less. If anything, sharing this picture only communicates to those who disagree with you that the critics of the Common Core are the people who have little conceptual understanding of elementary school mathematics. Once again, I am sympathetic to the emotions expressed in this picture, but there are better ways of criticizing the Common Core and persuading its unabashed supporters to your cause.

In the posts that follow, I will provide plenty of criticism of how the Common Core has been implemented in its initial years.

 

Teaching for understanding and teaching procedures

Many critics of the current state of mathematics education take issue with asking students to explain their reasoning. They’d rather students just apply an algorithm and get the answer.

The following is quoted from QED: The Strange Theory of Light and Matter, where Richard Feynman describes how he’s going to explain for a lay audience the techniques behind quantum mechanics that earned him a Nobel Prize. (By the way, I highly recommend this book.)

How am I going to explain to you the things I don’t explain to my students until they are third-year graduate students? Let me explain it by analogy.

The Maya Indians were interested in the rising and setting of Venus as a morning “star” and as an evening “star” – they were very interested in when it would appear. After some years of observation, they noted that five cycles of Venus were very nearly equal to eight of their “nominal years” of 365 days (they were aware that the true year of seasons was different and they made calculations of that also). To make calculations, the Maya had invented a system of bars and dots to represent numbers (including zero), and had rules by which to calculate and predict not only the risings and settings of Venus, but other celestial phenomena, such as lunar eclipses.

In those days, only a few Maya priests could do such elaborate calculations. Now, suppose we were to ask one of them how to do just one step in the process of predicting when Venus will next rise as a morning star – subtracting two numbers. And let’s assume that, unlike today, we had not gone to school and did not know how to subtract. How would the priest explain to us what subtraction is?

He could either teach us the numbers represented by the bars and dots and the rules for “subtracting” them, or he could tell us what he was really doing: “Suppose we want to subtract 236 from 584. First, count out 584 beans and put them in a pot. Then take out 236 beans and put them to one side. Finally, count the beans left in the pot. That number is the result of subtracting 236 from 584.”

You might say, “My Quetzalcoatl! What tedium, counting beans, putting them in, taking them out – what a job!”

To which the priest would reply, “That’s why we have the rules for the bars and dots. The rules are tricky, but they are a much more efficient way of getting the answer than by counting beans. The important thing is, it makes no difference as far as the answer is concerned: we can predict the appearance of Venus by counting beans (which is slow, but easy to understand) or by using the tricky rules (which is much faster, but you must spend years in school to learn them).”

To understand how subtraction works – as long as you don’t have to actually carry it out – is really not so difficult.

That’s my position: I’m going to explain to you what the physicists are doing when they are predicting how Nature will behave, but I’m not going to teach you any tricks so you can do it efficiently. You will discover that in order to make any reasonable predictions with this new scheme of quantum electrodynamics, you would have to make an awful lot of little arrows on a piece of paper. It takes seven years – four undergraduate and three graduate to train our physics students to do that in a tricky, efficient way. That’s where we are going to skip seven years of education in physics: By explaining quantum electrodynamics to you in terms of what we are really doing, I hope you will be able to understand it better than do some of the students!

In the same way, I want students in 2nd and 3rd grades to understand what they are really doing when they subtract, and not just mindlessly follow a procedure to get an answer that they do not really understand.

Where I tend to agree with most critics of the Common Core is that students are asked to write miniature essays to explain their reasoning, and that’s probably a bad idea. Even though I want students to understand why subtraction works, 2nd and 3rd graders are still learning how to write complete sentences and can get easily frustrated with explaining their reasoning in paragraph form. I think there are better ways (like drawing pictures) of assessing whether young children really understand subtraction that is more developmentally appropriate.

An analysis of subtraction algorithms from the 18th and 19th centuries

Today I happily link to this wonderful article about how elementary school students “should” subtract two numbers, as it challenges the commonly held notion that there is only one way that subtraction should be implemented.

The common algorithm taught in schools today is the Decomposition Algorithm.

http://www.youtube.com/watch?v=3itmfsP6HoM

But there’s also the Equal Additions Algorithm.

http://www.youtube.com/watch?v=AN8XN_MSucI

And the Complement Algorithm.

http://www.youtube.com/watch?v=krNVuaIwi-o

And the Austrian Algorithm.

The author concludes:

The teaching and learning of subtraction is just as important today as it was in the past. Innovations in technology and mathematics curriculum have certainly occurred since the 1700s and 1800s, but the need for the teaching and learning of subtraction has not changed. Today, in many classrooms, subtraction is often taught through student-invented algorithms. Looking to the past may give teachers insight into invented algorithms or other algorithms students may use. Additionally, many teachers who do not encourage students to invent strategies teach only the “standard subtraction algorithm” presented in nearly every textbook across the United States, the decomposition algorithm. This research and analysis provides the modern teacher with an opportunity to reflect on the algorithms being taught in his or her classroom and allows the teacher to begin to think about why decomposition became the dominant algorithm in the United States. Teachers can ask their students to reflect on whether they agree with this historical turn of events. Incorporating the history of subtraction algorithms into modern elementary school mathematics invites robust mathematical discussion of subtraction and also of how, for many mathematical operations, there isn’t just one algorithm, but rather many algorithms from which to choose.

Exploring the history of subtraction in past school mathematics may provide us with insight into students’ mathematical struggles as they attempt to conceptualize not only subtraction, but also negative numbers and other notoriously challenging mathematical concepts. As educators and researchers, we need to devote more attention to issues in mathematics education such as the development of specific algorithms in elementary mathematics.