Wednesday, January 21, 2015

Conversations on the Rifle Range 22: Tesselations, Border Crossings and Guess and Check

Barry Garelick, who wrote various letters published here under the name Huck Finn, is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number 22:

Tessellations are repeating interlocking designs, made famous by the Dutch artist M.C. Escher. They are also included in many math textbooks –becoming more prevalent over the last two decades. While they are interesting in their own right, I’m not a fan of teaching about tessellations when there are more relevant and useful topics that will prepare students for algebra. But because my school district had dropped California’s standardized STAR test in order to field test the Common Core aligned SBAC exam, the two weeks normally devoted to prepping for STAR were gone. My pre-algebra classes faced a two week gap that I had to fill. I came up with various lessons, assignments and activities. On the first day, I had them construct tessellations.

The tessellation activity involved drawing a design on one side of a square card, cutting it out and taping it to the opposite side of the card to form a template, which was then traced repeatedly onto a piece of paper to make the interlocking design.

I had the students work in groups of their own choosing. Their choices were sometimes surprising. Trevor, the boy who was on the mock trial team, chose to sit with Jacob, a Chilean boy with whom he argued frequently—so much so that Trevor himself had earlier requested that he be seated far from Jacob. Little Esteban, a Mexican boy, who was quite bright but frequently did not do his work, joined them. (He had the habit of singing Mexican songs during tests and quizzes to combat his test anxieties.) When I checked in on them, they were doing the tessellation activity but were engaged in a lively debate. “Listen to this, Mr. G,” Trevor said. “Jacob says the Russian and Chinese military can beat the U.S. in a war.”

“It’s true,” Jacob said.

“It is NOT true,” Trevor said and gave his evidence of troop numbers, weapons and military know-how. “No one can penetrate our borders,” he said.

Little Esteban listened quietly and then chimed in. “OK, tell me this, then! If the U.S. has such great protection at the borders, how do all the Mexicans get in?” As excellent as this question was, Trevor and Jacob ignored it, Little Esteban folded his arms in frustration, and I moved on to other equally interesting conversations too numerous to mention here.

All in all, the tessellation exercise worked out better than I thought it would. Doing such activities also had the advantage of making it look to any border guards of education who happened to be passing through my classroom that I believed in and subscribed to “group work”, “collaboration”, “student-centered learning” and other fads that pass as relevant to education and/or “21st Century skills”.

The last group-based activity I did before returning to the textbook was a “Problem of the Month” activity. I had asked Mrs. Halloran for some ideas, and she sent me some “Problem of the Month” (POM) exercises. These were problems developed by the Silicon Valley Math Initiative (SVMI)—the same folks who constructed the test that was now being given as an extra barrier to taking algebra 1 in seventh or eighth grade. Touted as being “aligned with the Common Core standards”, each POM is a set of five related problems at different levels of difficulty. I decided to use “The Wheel Shop” POM, which involved determining the number of bikes, tandems and tricycles in a shop. The first two levels (A and B) were fairly easy and involved straightforward arithmetic. The explanatory material for teachers stated that the third level (C) “may stretch sixth and seventh grade students” and that “students use algebraic thinking to solve problems involving solving for unknowns, equations, and simultaneous constraints.” Roughly translated, this meant that my students didn’t have the math needed to solve the Level C and above problems efficiently and would have to resort to guess and check among other methods.

To solve the Level C problem efficiently required knowing how to solve systems of linear equations. The problem stated that “There are a total of 135 seats, 118 front handlebars (that steer the bike), and 269 wheels. How many bicycles, tandem bicycles and tricycles are there in the Wheel Shop?”

Indicative of the prevalence of guess and check thinking and instruction in school, most of the students knew immediately to use this technique. Papers filled up with diagrams and tally marks of trials and errors. A very bright boy named Bill called me over. He chose to work alone rather than in a group. “How do I solve this?” he asked

“Most people are using guess and check,” I said.

“I hate guess and check. Can’t you teach me how to solve it using algebra?”

Bill was extremely likable; he had many friends, was on the mock trial team and could argue persuasively about almost anything. He had helped me out of a jam one time when students had finished their work early and I had several minutes dead time before the dismissal bell rang. I wasn’t too good about what to do in such situations. Bill, sensing that I was getting nervous about the rising noise level suddenly stood up and announced “Let’s sing ‘If You’re Happy and You Know It’! ” The class, warming to the spontaneity of what seemed to them a rebellious act, sang along with him; not just once, but two times until the bell rang. “I owe you one,” I told him as the class filed out.

I decided to pay him back by teaching him the algebra necessary to solve the problem. “You know how to solve equations but you haven’t had a lot of what I’m going to show you,” I warned him. “So follow close.”

We established that B, T and R equal the number of bicycles, tandems and tricycles. Knowing there is one set of front handlebars on each type of bike, one seat on bicycles and tricycles and two on a tandem, and two wheels on bicycles and tandems and three on a tricycle, I coached him through setting up the following equations:

B + 2T + R = 135
B + T + R = 118
2B + 2T +3R= 269

I led Bill through the elimination method to solve for T by subtracting the second equation from the first, eliminating the variables B and R. Then, substituting the value of T into the second and third equations, I led him step-by-step, in solving for B and R. In the end, he solved it: 68 bicycles, 17 tandems and 33 tricycles.

I knew I was at risk of criticism for “telling” rather than “facilitating” and not letting Bill discover the solution by himself. But there were neither recriminations nor accolades of praise. I passed through all borders unobserved, which suited me fine.


lgm said...

Your students do not need algebra to solve this, nor do they need bar models. They need logical thinking, a key insight, and the ability to do a multistep solution.

First, notice that 118 front handlebars means there are 118 total bikes, tandems, and trikes.

The bikes and trikes have one seat each, while the trikes have two. There are 135 seats. Key insight: if we give one seat to each vehicle, there are 135-118=17 seats leftover. They cant belong to the bikes or trikes, but can belong to the tandems, so there must be 17 tandems.

There are 269 wheels. Key insight:all 118 vehicles have 2 wheels. Trikes have more. So 269-2 (118)= 33 wheels left after passing out two to each vehicle. That is the number of trikes, since each needs one more wheel.

You have reasoned that there are 17 tandems and 33 trikes. That means of the 118 vehicles, 118-17-33=68 are bicyles.

No algebra required, despite that red herring teacher manual statement. Just plain old algebraic thinking, of the type found in elementary school, extended to more steps.

My kids had the two step version in 5th grade with Houghton Mifflin 10 years ago. Use the 'draw it out strategy', not ' guess and check'.
The point of my kids lesson was to stop g and c, and use Polya's first step...understand the problem.

lgm said...

Please change typo...2nd trike in sentence 3 should be tandem

SteveH said...

Let the math do the thinking as I like to say. However, many educational pedagogues seem to believe that the goal of math is a thinking process. Yes. You think about the governing equations that apply and whether you have mn. Math is not about thinking your way to a solution when you don't have a clue. Besides, they can't have all kids discover all things. Even in group learning, only one or two "discover" anything, and it could be wrong. Teachers in most classes, student-centered or otherwise, do a lot of "telling."

R. Craigen said...

You're quite right, Igm, and a gold star for you.

Nevertheless, the student did not ask our hero to show him how to reason through the problem -- he knew that they were in prep for algebra, correctly fathomed that this was probably an algebra problem, and requested to be shown how algebra may be used to solve it. He wanted the door opened a crack, to see what was on the other side.

Now the student could have been shown your seat-of-the-pants method (which I don't mean to denigrate -- it's perfectly good!) and come away with some idea of how to solve similarly-posed problems that are elementary enough to solve in a series of very easy one-step deductions like this.

But instead, the student -- evidently already highly motivated for this -- came away with the rudiments of understanding how algebra is done, how to use it to model a large class of problems, and some of the first techniques therein. A valuable hour of learning for that one child, I'll say, and a lesson that will pay back much in later educational dividends.

While in class the student generally ought to be focussed on the problem in front of them, the teacher should always have an ulterior motive: getting the most educational bang for his or her chronological buck. You have only so many hours with those children, and it is your job to give them as much academic shove as you can, so their educational bikes will roll as far as possible after they walk out one day and never return.

A child would have a perfectly good educational outcome by solving this problem with your method -- I dare say a much better outcome than he would gain by an hour of guess-and-check that finally hits on the correct answer. But for that one student, we saw a supercharged lesson, the sort that good teachers lie in wait for, hoping to spring on students who show readiness. I think that's the point of this chapter. Or it's one of the points. Creative breaking of the rules, I suppose, is another point.

I thought I'd also point out (something I'm sure you noticed) that your solution is the same as our hero's except for the lack of symbols and equations. The steps can be mapped from one solution to the other -- because what you are doing is a classic elimination-of-variables as is taught in starting algebra, only informally in words, without the variables and equations. It is a nice observation that the extra clothes are unnecessary. However, those clothes are where the power of the method lies, and it is good to learn how to put them on.

lgm said...

Your last paragraph states the reason that the student should not have been shown the algebra. He does not have the preliminary understanding . Just like a first grader who memorizes the multiplication tables, he will never come to understand. It will be plug and crank for him, all the way.

lgm said...

And, R. Craigen, let me also state that I am opposed to, the current practice of omitting problem solving techniques from the instructional, offerings. Throwing a POD or PaoW out and expecting more than guess and check is the equivalent of throwing the kid in the deep end of the pool and hoping someone else discovers how to tread water or swim or demonstrates what they learned at home and the kid can copy before he drowns...or the bell rings and the water level is lowered.
Providing an activity without an instructional objective is baby sitting, not teaching.

SteveH said...

"Just like a first grader who memorizes the multiplication tables, he will never come to understand."

No, it's not just like memorizing the times table. There is no magic connection between trying to figure something out with no help and with understanding the beauty and power of defining variables and equations, and making sure that you have M variables equal to N independent equations.

Besides, students able to tackle this kind of problem should already have experience with creating simple equations from words, like "Mary is three years older than Jane and Jane is 12 years old. How old is Mary?" Students start to write equations when they can still do it easily in their heads. You run a big risk to wait until problems get too hard before you learn the basic skills of algebra. But even at three equations and three unknowns, you think that there is some math understanding that will never happen because they can't first do this without math? At what point does this stop?

Understanding in math is understanding the tools, not understanding some vague thinking process that can leave you high and dry if you don't see the missing pieces. Algebra, however, gives you the tools for organized thinking.

I used to tell my students to start defining variables and finding equations that they know are correct. They don't have to "think" to find the best or minimal set of equations. It's usually easy to find at least one equation. Then that will help you find the others because you can more easily see missing relationships. If you create too many variables, then it's easier to see when some combine or drop out. This is not a mechanical times table rote process.

Just thinking with no tools is anti-math.

Guess and check-based discovery is neither necessary or sufficient. Besides, one cannot do that for everything and none of the discovery educational pedagogues do that anyway. It's just an excuse to push student centered learning in class.

R. Craigen said...

Hi IGM. You say "Providing an activity without an instructional objective is baby sitting, not teaching"

I almost agree with this. I would say providing an activity without HAVING an instructional objective is ... not teaching. The difference is that I do not think that one must "provide" the instructional objective (i.e. to the students in advance). It is often good for teachers to have a hidden agenda and students to believe they are doing something as a lark or getting away with wasting time. This notion that teachers must explicitly tell students the objectives of every lesson is pedantic nonsense. The teacher is a craftsman who crafts learning experiences. Sometimes good teaching is stage magic, where in the end the rabbit comes out of the hat and everyone gasps. Students should often be pleasantly surprised at how much they have learned over time.

You say "Your last paragraph states the reason that the student should not have been shown the algebra. He does not have the preliminary understanding . Just like a first grader who memorizes the multiplication tables, he will never come to understand. It will be plug and crank for him, all the way."

This is absurd. You are talking about knowing "why" without knowing "what". Understanding without knowing? I encountered this years ago as the ministry folk here were pressing this understanding-first business. It is essentially a doctrine saying that you can build a house from the roof down. No, you can't. There's a reason why basic skills and knowledge are always put at the bottom of Bloom's taxonomy -- they are foundational.

Mathematics is relentlessly hierarchical. Every lesson has a foundation, has precursor skills and knowledge, which ought to be in place, quite often before the lesson begins. And understanding quite often (not a firm rule -- I only say "often") is far up the ladder. Higher skills generally are built upon lower ones, not the other way around.

Your last statement that someone giving students basic skills dooms them to "...never understand. It will be plug and crank for him all the way" is the old false dichotomy between understanding and skills. I've got news for you -- far from getting in the way, facts and skills REINFORCE understanding. Ask any cognitive scientist.

Further, you want to know what "plug and crank" looks like? Guess and check! -- it's almost exactly that. Guess and check only barely registers in my view as a "problem solving skill" -- for many it's a problems-solving-skill-AVOIDANCE procedure. Here's a problem ... rather than approaching it systematically and analytically, let's ... uh ... try 17. Nope. How about 23. Nope, -5? No good... That's not problem solving! And it's an impoverishment where a good teacher could and should be helping students to apply systematic reasoning and organized methods.

You are probably a very good teacher -- I liked your analytical approach to the bicycle problem and believe you would prefer to see students doing this than guess and check. So I know you "get" this. But you seriously need to divest yourself of this mythology about the relationship between facts/skills and understanding. They are not enemies.

Recommended reading: Daisy Christodoulou's "Seven Myths about Education". In particular, Myth #1: Facts prevent understanding.

Daisy has posted an excellent summary of the seven myths on her website, here:

R. Craigen said...

Hi Steve. You say

"Just thinking with no tools is anti-math."

As a career mathematician who lives and breathes the subject I agree 100% with this statement. Very well put. With this statement you've nailed one of the most disturbing trends in contemporary "math education" -- the stripping out of actual math in favour of generic and poorly defined "thinking skills" at the expense of systematic development of generalizable, established tools of the trade.

SteveH said...

I think the biggest myth is the idea of rote knowledge. Rather than see it as partial or incomplete understanding, many mischaracterize it as lacking any understanding and use that to justify flipping the education process around and approaching it from the top-down, where they hope that some vague understanding will drive the development of "rote" skills. This in turn justifies student led discovery in class.

Real mathematical discovery happens individually when one consistently, day after day, year after year, works hard on homework sets - problems carefully crafted to increase in difficulty and explore a range of variations in the material in the current unit. Proper mathematical development requires a carefully controlled feedback loop that tries to ensure mastery of the material. Detailed understanding comes from this work on skills. As annoying as they might seem. weekly quizzes work well.

My son suffered through MathLand and Everyday Math in K-6, but when he got to pre-algebra onward starting in 6th grade (with proper textbooks), life became much better. His high school math teachers did not do discovery-driven math. They all directly taught the class, but some left time at the end of class to allow them to start doing their homework. They could collaborate, so one could claim that they were doing group "discovery" in class. What this did do, however, was to help the students get over the initial hump generally encountered when doing homework. There might not be enough time to do all of the homework in class, but all of the students were off and going in the right direction. This required less time going over homework in class the next day, and it was more effective.

Talking about understanding never works as well as doing understanding individually. When I tutor students in math, I can see that they "understand" when we go over a problem step-by-step. However, I always tell them that they have to prove it by doing the exact same problems themselves - alone. Too many students breeze through homework sets and end up with all sorts of partial and incomplete understandings. It's not rote, just incomplete. If you want better math students, you have to enforce a proper homework feedback loop year after year.

One cannot be successful with rote skills in math unless the teachers do not know how to create proper tests. Students would love to see the exact same homework problems on the tests, but just with changed numbers. It never works like that. These are variations that won't be solved by general logic or understanding. They are solved by the understanding that comes from time spend on homework sets.

R. Craigen said...

Well-said Steve. I'll only add this, that I think the very word "rote" has been abused and tormented into something that it never was before, by modern discourse.

Today when one says "rote learning" they generally mean "without understanding". But that is not what the word means, although some dictionaries now include that meaning. "Rote" means, essentially, "by repetition". That is, rote learning is that which is learned through repetition of some task. Think of words sharing a common root: "rotation", "rota", "rotary". It means to repeat something over and over.

Repetition ... is the basis of most memory-building. And memory is the seat of learning. So where there is no rote, there can hardly be much learning. An actor learns his or her lines by rote. Does this mean they do not understand those lines? You'll find few in the business who would agree with that conclusion. Athletes learn their competitive skills by rote ... repetition. Does this mean they have no understanding of the game? No, in fact most would tell you that "understanding", in a vacuum of those skills, is of little value. Same in mathematics.

Rote is not "understanding", or certainly not "complete understanding". But understanding is not complete even after years of excellent learning. Even as a professional mathematician I am continually learning my subject matter at a deeper level, even the elementary topics. I rankle at the suggestion that there is some royal road and down which teachers should supposedly lead students directly to understanding, prior to mastery of requisite skills. Plato understood well that there was no such road, and the situation is no different today.

Anonymous said...

Before there was writing, the traditional cultures that we all are descended from accomplished most of their learning by demonstration, repetition and practice. Writing added the ability to catalogue learnings and have them for reference, but it did not invalidate the older forms of learning.

Anonymous said...

“Just like a first grader who memorizes the multiplication tables, he will never come to understand. It will be plug and crank for him, all the way.”

I take issue with this statement. It seems as though educators have a view of memorized knowledge that is only negative. It is as though they imagine all children who produce such knowledge to be pale and forced to sit inside by parents who make them do repetition for hours on end. Their performance of this knowledge being only a parlor trick with no depth. First, why will the child “never understand”? It seems inevitable that in fact even if he is just chanting or singing the facts, in the coming months or years he will continue to apply this knowledge to math problems and develop the same understanding as his peers. Likely he will go faster through the material as he can do problems without struggling to access the facts. Second it seems as though as much as this is denigrated, teachers know that this must happen at some point. A first grader who has this memorized is a victim of his parents distorted vision, but a fourth grader who has this memorized is just an ordinary child. Certainly as an adult when I think of 3x5 I rely on my memorized knowledge and never consider the concept of how that is achieved.

Also I don’t think it is true in all cases that a child could not understand. Multiplication is not that difficult to grasp. 2 groups of 3, equals 3 groups of 2, equals 2+2+2 or 3+3, all of this can be shown and explained fairly easily. We are abroad this year in a new school system and I noticed looking ahead in the first grade textbook, that multiplication is introduced toward the end of the year.