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

The following picture has been making the rounds lately.

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.

The solution of this particular question.
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?
The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
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.

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.

The solution of this particular question.
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?
The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
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.

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.

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.

The solution of this particular question.
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?
The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
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.

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.

How Common Core Standards Kill Creative Teaching

While I have little patience for much of the fear-mongering tactics used by some critics of the Common Core, I do appreciate thoughtful criticism. A recent editorial in U.S. News & World Report (found at http://www.usnews.com/opinion/articles/2014/03/17/how-common-core-standards-kill-creative-teaching) definitely falls under the latter, succinctly summarizing my point of view:

Understandably, proponents of the Common Core say they want greater depth of instruction and lessons that engage students. They say that the standards are only a guide. But reformers betray their cause by over-emphasizing tests and grading teachers with formulas and test scores demanded by both No Child Left Behind and Race to the Top.

To try to live up to the new demands and ensure better test scores, states, districts and schools have purchased resources, materials and scripted curricular modules solely developed for test success. Being lost is the practical wisdom and planned spontaneity necessary to work with 20 to 35 individuals in a classroom. Academic creativity has been drained from degraded and overworked experienced teachers. Uniformity has sucked the life out of teaching and learning.

To me, the operative verb in the above citation is betray, because I certainly feel betrayed. While I personally had no input into the Common Core standards for mathematics, I’ve attended presentations as they were developed over past 8 or 10 years. And the presentations that I heard have little resemblance to the way that mathematics is being assessed in schools right now.

One more thought: I live in a non-Common Core state (Texas), but the same pressure to “follow the book” exists here. So the “follow the book” mentality is not unique to the Common Core.

New Math, by Tom Lehrer

To add a little levity to the recent posts about the Common Core, here’s the song “New Math” by Tom Lehrer that poked fun at the New Math curriculum of the 1960s.

For some context, here’s a succinct summary of the new math fad by Richard Feynman, who won the Nobel Prize in Physics and served on a commission for choosing math textbooks for California in 1964:

I understood what they were trying to do. Many [Americans] thought we were behind the Russians after Sputnik, and some mathematicians were asked to give advice on how to teach math by using some of the rather interesting modern concepts of mathematics. The purpose was to enhance mathematics for the children who found it dull.

I’ll give you an example: They would talk about different bases of numbers — five, six, and so on — to show the possibilities. That would be interesting for a kid who could understand base ten — something to entertain his mind. But what they turned it into, in these books, was that every child had to learn another base! And then the usual horror would come: “Translate these numbers, which are written in base seven, to base five.” Translating from one base to another is an utterly useless thing. If you can do it, maybe it’s entertaining; if you can’t do it, forget it. There’s no point to it.

From the chapter “Judging Books by their Covers” in Surely You’re Joking, Mr. Feynman!

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.

Reflections by a teacher on the Common Core

The implementation of the Common Core has left a lot to be desired, but it’s heartening to see that some teachers have embraced what the Common Core attempts to accomplish. I saw the following first-person person referenced in the Washington Post; the original post can be found at http://www.youngedprofessionals.org/1/post/2014/03/is-the-common-core-working-in-the-classroom.html.

The Common Core State Standards are a reality now for teachers in Maryland and DC, while Virginia is one of six states to omit the standards from their state education approach. YEP-DC asked local educators how the Common Core is playing out in their classroom. Are the standards increasing student understanding or presenting obstacles? What’s changed in pedagogical approach, and how are students are reacting to the shift?

Meredith Rosenberg, fourth-grade teacher

Compare 1/4 and 5/6. This seemingly simple problem is a no-brainer for adults. We know right away that 5/6 is greater than 1/4. But where do you begin with a student who has no conceptual understanding of what a fraction is?

One of the most defining features of the Common Core is how it introduces concepts to students through different modes of comprehension. By the end of a six-week Common Core unit on fractions, my students were talking about, writing about, drawing, and playing with fractions. When they encountered the above problem on a quiz, some students drew a picture, while others found common denominators. A few used a strategy called common numerators, which requires a deep understanding of the denominator of a fraction. One student drew the fractions on a number line. The takeaway: The students in my class were able to compare these fractions in no fewer than five different ways.

The Common Core implementation is not without its challenges. Many standards are vague, and there are only small bits of information coming from the Partnership for the Assessment of Readiness in College and Career (PARCC) on how they are to be tested. The inconsistency with which the standards have been implemented result in the need for highly differentiated classrooms. For example, some of my students came into fourth grade with a solid conceptual understanding of fractions, while others from other schools had no idea what a fraction meant.

However, my school has prioritized Common Core implementation and tackled its challenges with consistent professional development, regular refinement of unit plans, daily lessons and assessments, and an intense focus on the Standards for Mathematical Practice. As a result, my students are thinking critically about numbers every day, and they are becoming accustomed to attacking problems with multiple strategies and assessing the validity of those strategies. The Common Core standards choose depth over breadth, and with appropriate teacher development and support, this leads to much more critical thinking and analysis in the classroom.

The Failure of Test-Based Accountability

From Marc Tucker’s blog on Education Week:

In my last blog, I pointed to the data that shows that, after 10 years of federal education policies based on test-based accountability, there has been no perceptible improvement in student performance among high school students (which, when you get right down to it, is what really matters) as a whole, or when the data are broken down by different groupings of disadvantaged students. There is little doubt—whether test-based accountability is being used to hold schools accountable or individual teachers—that it has failed to improve student performance.

That should be reason enough to abandon it. But it is not. The damage that test-based accountability has done goes far deeper than a missed opportunity to improve student achievement. It is doing untold damage to the profession of teaching…

Test-based accountability and teacher evaluation systems are not neutral in their effect. It is not simply that they fail to improve student performance. Their pernicious effect is to create an environment that could not be better calculated to drive the best practitioners out of teaching and to prevent the most promising young people from entering it. If we want broad improvement in student performance and we want to close the gap between disadvantaged students and the majority of our students, then we will abandon test-based accountability and teacher evaluation as key drivers of our education reform program.

But no one, certainly not me, would argue that we should not hold our professional educators accountable for their performance. The question is, what would accountability look like if we actually regarded our teachers as professionals doing professional work, instead of interchangeable blue-collar workers doing blue-collar work? That is the question I will deal with in my next blog.

I encourage you to read the whole thing: http://blogs.edweek.org/edweek/top_performers/2014/02/the_failure_of_test-based_accountability.html

The following video made the rounds a few months ago and ties in with the above point. It is less about the shortcomings of the Common Core than our leaders’ fixation with quantifying educational output. As the speaker says well, “If everything I learned in high school is a measurable objective, then I have not learned anything.”

Thoughts on the Common Core and its implementation

The following picture appeared on the Facebook page of Daniel Bongino, who is running for Congress in Maryland.

Source: https://scontent-a-dfw.xx.fbcdn.net/hphotos-prn1/t1/1184774_620433314716100_343011500_n.jpg

Here was his commentary on this picture:

Like many of you, I am a parent who is passionate about my child’s education in an increasingly competitive and unforgiving global economy.

Having stated that, I cannot condemn the Common Core in strong enough terms. Look at the picture I have attached to this post. I gave my daughter a relatively easy long-division problem to do today, in an attempt to gauge her progress, and this is what she gave back to me.

This is completely unacceptable. How is it that we are replacing a time-tested, efficient method of long-division with an absurd, multi-step process that not only confuses the students, but the parents too?

Compounding the Common Core disaster is the fact that in my daughter’s last school year she was taught the older, more effective method of long-division and is now completely confused.

Friends, all politics are local and it gets no more local than your kitchen table. Fight back against the Common Core, and do it quickly, by calling and emailing your local, state, and federal elected officials.

This is not a partisan issue. Your child’s education is suffering whether you are a Democrat or a Republican. Every second we lose is another second our kids are being exposed to a third-rate curriculum in a first-world economy. Count on me as an ally in this fight.
-Dan

Source: https://www.facebook.com/dan.bongino/photos/a.517057181720381.1073741827.101043269988443/620433314716100/?type=1&theater

Yesterday, I discussed the mathematical logic behind this unorthodox approach to subtraction. Today, I want to briefly talk about the Common Core standards for mathematics and their implementation, as this is a topic that I’ve been following for several years.

To the mindless critics who think that America is headed to communism because of the Common Core: there’s no point having a rational discussion about this. Michael Gerson is one of many conservative commentators who is not ideologically opposed to the Common Core; see http://www.washingtonpost.com/opinions/michael-gerson-gop-fear-of-common-core-education-standards-unfounded/2013/05/20/9db19a94-c177-11e2-8bd8-2788030e6b44_story.html.
Also to the mindless critics: while Texas (where I live) is not a Common Core state, the standards for mathematics that we’ve had for the past 10 years or so align fairly well with the Common Core. And Texas is about as far away from a blue state as any of the 50.
To the thoughtful critics who are worried about the appropriateness of the Common Core standards: as I said, while not in perfect alignment, for the last few years Texas has had content and process standards for mathematics education that are decently close to those stipulated by the Common Core. I’m more than happy to declare that the implementation of the Common Core has been thoroughly botched from sea to shining sea. Still, I believe that a good implementation is possible, and I hope that you don’t throw out the baby with the bath water when critiquing the potential of the Common Core standards.
To the supporters of the Common Core standards: you better read Diane Ravitch’s thoughtful critique of how the standards have been rolled out: http://dianeravitch.net/2013/02/26/why-i-cannot-support-the-common-core-standards/. It seems to me that textbook publishers are driving the rollout of the Common Core, and educators are desperately trying to shift from the previous standards to the new standards while also trying to figure how they are being required to teach because of the textbook… and not because of the standards themselves.
Also to the supporters of the Common Core standards: voters — and, more importantly, parents — will not tolerate these standards if a rationale for these standards are not carefully explained. I do think that most parents do care about the mathematical education of their children and will rationally discuss cutting-edge ways of teaching mathematics, but they have to be convinced that these cutting edge methods actually make sense. The rollout of the Common Core will be studied in public-relation circles for years to come for how *not* to make drastic changes.
And though they are not specifically required by the Common Core, don’t get me started on the hours we’re wasting high-stakes testing, an intellectually lazy and ineffective way of measuring teacher quality.

Thoughts on unorthodox ways of teaching long division

The following picture appeared on the Facebook page of Daniel Bongino, who is running for Congress in Maryland.

Source: https://scontent-a-dfw.xx.fbcdn.net/hphotos-prn1/t1/1184774_620433314716100_343011500_n.jpg

Here was his commentary on this picture:

Like many of you, I am a parent who is passionate about my child’s education in an increasingly competitive and unforgiving global economy.

Having stated that, I cannot condemn the Common Core in strong enough terms. Look at the picture I have attached to this post. I gave my daughter a relatively easy long-division problem to do today, in an attempt to gauge her progress, and this is what she gave back to me.

This is completely unacceptable. How is it that we are replacing a time-tested, efficient method of long-division with an absurd, multi-step process that not only confuses the students, but the parents too?

Compounding the Common Core disaster is the fact that in my daughter’s last school year she was taught the older, more effective method of long-division and is now completely confused.

Friends, all politics are local and it gets no more local than your kitchen table. Fight back against the Common Core, and do it quickly, by calling and emailing your local, state, and federal elected officials.

This is not a partisan issue. Your child’s education is suffering whether you are a Democrat or a Republican. Every second we lose is another second our kids are being exposed to a third-rate curriculum in a first-world economy. Count on me as an ally in this fight.
-Dan

Source: https://www.facebook.com/dan.bongino/photos/a.517057181720381.1073741827.101043269988443/620433314716100/?type=1&theater

This picture was shared by a friend on Facebook; the resulting discussion follows. I’m sharing this because I think the following reactions are typical of parents when their children are taught mathematics using non-traditional methods.

While I don’t think that any of the commentators said anything personally embarrassing, I’m withholding the actual names of the correspondents for the sake of anonymity.

Anonymous #1: What in the world is this?

Me: In the worst case scenario, it’s a waste of time for children who already know how to divide.

In the best case scenario, it’s an effective and pedagogically reasonable first step — for children who don’t yet know how to divide. (FYI, this technique has been used long before the advent of the Common Core.

Here’s the justification: Young children often have a hard time coming up with the “best” first step that 43 divided by 8 is 5 with remainder 3. However, they often can come up with a reasonable first step, whether it’s subtracting off 10 groups of 8 or 40 groups of 8. The important thing is that they’re reducing 432 by a multiple of 80, not necessarily the “best” or “optimal” multiple of 80. With practice, children hopefully get better at guessing the optimal multiple of 80, thus leading to the traditional method of long division.

The idea is that the children can, with time, figure out the reason why long division works, rather than mindlessly following an algorithm that leads them to an answer that they don’t understand.

Anonymous #2: It’s the longest division problem ever. Lol

Anonymous #3: OMG John, that answer was more confusing than the picture!! LOL just kidding! What I want to see from that picture is, did she eventually get the answer right? Did she give up? If the kids learn how to get a right answer, I’m hard pressed to find a valid argument against any teaching method. If it frustrates them to the point that they give up, well then that is a problem. That picture he posted doesn’t give us any real information. It just makes us old farts think “what the hell??” Because it’s so different from what we learned.

Maybe it isn’t pulling up right, but I don’t see an answer in that picture. Is because she couldn’t do it or because he just wanted to post the weird method to promote fear of something new?

My daughter was taught the “lattice” way to do 3 digit multiplication. I wanted to cry trying to figure that out. But it made sense to her and she got the answers right.

But, I will admit that it looks crazy to me, too!

Me: I agree that the person who posted the picture did not (deliberately?) show if the student ultimately got the right answer. I can say that the partial steps that are shown are correct.

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.

I’ll also say this: unorthodox teaching practices usually go over better when both the practices and the rationale for the practices are clearly explained to parents. Sadly, while a lot of thought has gone into improving mathematics education, not much thought has gone into justifying these new practices to parents, and that’s a shame.

Anonymous #2: The problem isn’t teaching the method. I’m all for showing kids multiple ways to do things. The problem is forcing all kids to use this method. We are all different and therefore we all think differently. If it makes sense this way to you great however if it doesn’t make sense then why not let kids use the way that works for them. Yes teaching different methods is great but forcing kids to use methods they don’t understand is foolish.

Me: No argument from me.

Anonymous #4: I am troubled by this and other styles of math that no longer require children to learn and memorize simple mathematical tables of simple addition, subtraction, multiplication, and division. It disappoints me to no end that people allow children to avoid learning thoroughly these tables, as though they are not necessary in life. I am appalled here that kids are encouraged as early as 3rd grade to start using a calculator for basic math!

I appreciate different styles of doing math here, Subtraction and Division are quite different in (European Country) than in America. But sometimes it just seems that so many new methods are obscure attempts to help an overly super small subset of kids which are then exposed to them, and at times, forced on them; much to the chagrin of parents.

Me: (Anonymous #4), I agree about the importance of children memorizing mathematical tables at a young age. I disagree that this particular algorithm — unorthodox long division — necessarily tells children that such memorization isn’t particular useful.

My own daughter struggled with long division when she first learned it. She already knew that 36 divided by 4 was 9 and hence knew the “right” step when computing 368 divided by 4. However, when the problem changed to something like dividing 384 by 4, she had difficult with the first step, as she didn’t have anything memorized for “38 divided by 4.”

My friends who are elementary teachers tell me that this particular conceptual barrier is fairly common when children first learn long division.

For 384 divided by 4, the best first step is subtracting 90 groups of 4 from 384, but she was having trouble immediately coming up with the largest multiple of 10 that would work. However, subtracting *any* multiple makes progress toward the solution, even if it isn’t necessarily the “best” step for solving the problem as fast as possible.

In those early stages of her learning, she computed 384 divided by 4 using suboptimal steps. I can’t remember exactly how she did it, but a reconstruction from memory is shown in the attached picture. She knew that 50 times 4 was less than 384, so it was “safe” to subtract 200. When she did this, I didn’t correct her by telling her that she should have subtracted 90 groups of 4. Instead, I let her make this step (emphasis, step — and not mistake) and let her proceed.

The step that always surprised me was when she’d occasionally subtract 12 groups of 4… she had memorized her multiplication table up to 12 and instinctively knew that subtracting 12 groups of 4 brought her closer to the correct answer than subtracting 10 groups of 4.

Obviously, as she got better at long division, she made fewer and fewer suboptimal steps when dividing. That’s the beauty of this unorthodox method… children don’t have to stress so much about making the best next move, as any next move will bring them closer to the answer. Hopefully, with practice, children get better at making the best moves quicker, but that’s a skill that they develop as they get used to long-division algorithm.

Me: One more thought: (Anonymous #1), I’m sorry if I’ve completely commandeered your original post! 🙂

Anonymous #1: John you crack me up! I have never had such lengthy discussion about anything I have ever posted! I still have NO idea how to do all these extra steps-but I know who I will be asking for help when the time comes for me to deviate from my old school method of math!