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An innocent recent post, Two SAT-Type Percent Problems Appropriate for Middle School as well..., generated some intense and passionate opinions about the current state of US education, math in particular, from one of our readers, affectionately known as Curmudgeon (I'll abbreviate his name as CM).

Unless you subscribe to my Comments feed (see link in sidebar), you might not have followed this dialog so I've decided to post a link to these comments and to copy some of the more provocative ones in this post.

Based on his comments I find CM to be highly informed, articulate and not intimidated by anyone. He eloquently challenged my difficulty rating system for the SAT-type problems I have posted and he made some excellent points. Re curriculum/instruction, his perspective is somewhat similar to mine although I lean a bit more toward a balanced view. A direct quote describing his blog summarizes his beliefs very well.

"I believe that mathematics should be taught, not collaboratively explored; algebra and geometry are better than a vague course of Integrated Math; spiraling doesn't work nearly as well as learning it properly the first time; "I don't DO math" should be an incentive rather than an excuse. "I don't DO English" should be treated the same way."

I encourage you to visit his blog, Curmudgeon.

Below is CM's long response to my remarks about the teaching of the part:part construct which, IMO, receives far less emphasis in classrooms than part:whole approaches.

Rarely taught? Probably, for three reasons. The first is that teachers are under pressure to "finish" things, to check off the standard and move on, to "get through" the material. As soon as the kids "know" the material, it's time to move on. There is rarely time for the extensive exploration that I seem to recall from my own days. Standards-based educational theory says that you need to pick and choose your topics until you get each kid to the understanding point but says nothing about total mastery. "Drill and kill" is an epithet. "Drill and Practice" is unknown. Many people have also fallen for the "spiraling" fad and never quite complete a thought before they're on to the next one. "We'll spiral around to this again in that module in next year's course" is probably the dumbest thing ever to come out of edumacation colleges. I can't tell you how many times (because I've lost count) I and the other teachers have been told that we needed to get our "bubble" kids over the line into the passing zone. "To hell with knowing math, just know enough for the test" seems to be the rallying cry. The upshot is that you can mention these things to the better students who will get it easily and gain an even better understanding. The weaker students just trundle along. The second reason is that weaker students are resistant to trying a second or third idea. They want to understand THIS one. A second approach is confusing. It takes a while before they get comfortable with multiple approaches and some never get much beyond "I'm not a good math student. I'll learn this but only to a point." It takes a determined teacher to ease them into this, but she can't have anyone breathing down her neck to do test-prep. The worst reason is that a fairly large percentage of our teachers don't really understand math to the level required. They've either bought into failed and worthless education theory or they simply are stupid. I've related the story of the fourth grade teacher in the in-service this year ... Me: "You say you want to be a guide on the side not a sage on the stage, yet you're teaching 4th graders. You still need to teach them things. They have to memorize 4*3 for example." Her: "No. We should be teaching them how to look that up on Google. Did you know that Google will give you the answer to that? It's true." She leaned back, satisfied that she had put one over on me. Is it any wonder that her kids arrive with no understanding of fractions, decimals, percents, operations? "Where's the fraction key?" "Sorry, that calculator doesn't have one, the problem you're doing doesn't need one and you wasted more time than if you just looked at it and solved it." And some elementary and middle-school teachers "just don't DO math, tee-hee-hee." They SEEM to do math - they have lots of test-prep bubbling exercises from the publisher of the math book, but they fundamentally don't understand the nuances of the material. "Let's see what the calculator says." In a different in-service this year, the presenter was showing how pre-schoolers and elementary kids learn math. She had lots of visuals and was "teaching" the teachers as if they were students. The aides were getting questions wrong and the elementary teachers weren't doing too much better. This explains a lot.

To read my response (aka, "rant'), click on Read more...

To read all the comments to this post and/or post a comment, look here.

May I call you C.M.??

I'm not going to say something inane like "I feel your pain." Most of the angst you expressed is a reflection of what I was feeling when I retired. You're describing a system that is in dire need of systemic change, not tweaking.

I will not apologize however for my advocacy of national standards in math. It is unconscionable that students across the country are not learning the same content - concepts, skills, procedures, terms, definitions,... This is truly inequitable.

However, I have also been preaching "LESS IS MORE" for the same period of time. Finally, NCTM has taken this position with their Curriculum Focal Points document. William Schmidt (of TIMSS renown) stated this obvious fact 15 years ago when he described our math curriculum as "one inch deep, one mile wide."

We cannot expect students to learn math well if we fill our textbooks each year with every topic under the sun. If, for example, our teachers could focus on the essential ideas of ratios and ALL students were required to solve a range of problems from the basic to the more challenging, then most students would eventually learn ratios and be able to handle fractions. BUT facility with ratios and fractions requires facility with division which requires mastery of multiplication, etc.

"Jack, you will learn the times tables. The facts you got wrong in class today, you will have to write five times each for homework tonight. Tomorrow, I will ask you to answer just those." I know there are some teachers out there who are doing this. Is everyone?

While base ten blocks, unifix cubes and learning software have their place, we all know that nothing replaces repetition. Some students need far less than others but all students need some.

Yes, C.M., there are serious teacher preparation issues out there. Read my comments at the bottom about Finland. Yes, C.M., I share your feelings about spiraling, although there are aspects of spiraling which make sense.

I am also concerned as you are that standards-based learning has devolved into learning only for the state assessment. While we are moving toward national standards, we have to rethink how we will evaluate student learning and the bottom line is: "WHAT IS THE REAL PURPOSE OF TESTING?" What you and I and millions of other teachers see is that testing has little to do with helping children improve. It has everything to do with POLITICS:

"MY DISTRICT IS BETTER THAN YOURS; MY STATE IS BETTER THAN YOUR STATE; LET'S HOLD THOSE D*** TEACHERS ACCOUNTABLE FOR THOSE 'HIGH' SALARIES THEY'RE GETTING PAID"; "WE HAVE TO JUSTIFY ALL THAT TAX MONEY WE PAY FOR EDUCATION."

Remember that quote:

"In other countries, education is seen as an investment; in the US, it is seen as an expense."

Perhaps this administration will "see" it differently. I truly hope so. In Finland, teachers, have to take additional years of training beyond college before they officially are certified. This additional 2-3 years culminates in a year of working in a laboratory school with real students. A true internship. And, by the way, THE GOVERNMENT PAYS EVERY PENNY FOR ALL THIS ADDITIONAL TRAINING. This is how Finland has turned around its system in the past 20 years.

INVEST IN EDUCATION, INVEST IN OUR CHILDREN, INVEST IN OUR FUTURE. Don't look for short-cuts, folks. There are none. Expedient solutions lead to students who only care about the bottom line, the grade, not about learning. This "get results without really earning it" mentality is pervasive in our society. In the worst case, this mentality produces the AIGS, the Enrons and the Madoffs of the world.

Ok, now you got me to rant too. I guess I needed that catharsis. otherwise it sounds like I'm just pontificating about challenging our best and brightest with all these problems I'm writing. But there's so much more to it, C.M...

P.S. I have a feeling that your comments and mine are not being read by most of my readers. I'm thinking I should copy them into a separate post and really incite a riot! I think I will do that unless you state an objection! THANK YOU!

## Tuesday, April 28, 2009

### Strong Opinions from Curmudgeon Re Math Education...

Posted by Dave Marain at 7:54 AM

Labels: curriculum, issues in education, more

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## 11 comments:

Dave, thanks for copying these comments to a separate post. You’re right, I would not have seen them otherwise. CM raises a critical point.

Curmudgeon said: ”The worst reason is that a fairly large percentage of our teachers don't really understand math to the level required.”

There is no doubt in my mind that this is the main problem with math education, and we are not addressing it. Although we talk about it, somehow it is pushed aside. I would like to quote from your interview with Professor Steen. “Enthusiastic and imaginative teachers who are both mathematically and pedagogically competent are more important by far than anything else in the educational system.

… It is cheaper by several orders of magnitude to convene a consensus process to write standards than to attract, educate, and retain people with the interests and skills needed to teach mathematics well to all our nation's students. When you don't have enough teachers with the required competence, then the way politicians "make do" is to lay out specific standards and assessments for everyone to follow. I don't think we have much evidence that this strategy will work.”

Here is a quote from the Frontline interview with Dr. Schmidt that you referred to in your introduction of your dialog with him. William Schmidt: “My focus, nationally, has been trying to get the dialogue going to develop the standards and get them in place. But once in place, the most serious issue would be the quality -- and I don't mean the quality in a generic sense -- but the knowledge of the teachers, how we would get teachers that would have the deep subject matter knowledge that's required to teach those kind of standards.”

I think most of your audience have read or heard about Liping Ma’s book comparing the mathematical knowledge of American teachers and Chinese teachers. The American teachers knew how to compute (I am being generous here), while the Chinese teachers could compute and explain concepts. The Chinese teachers showed a different attitude toward mathematics than the American teachers. The Chinese teachers expected their students to understand mathematics conceptually, and to be able to justify any actions they took. The Chinese teachers justified steps using mathematical principles and logical reasoning; while the Americans often said that’s what they were told to do; they relied on authority rather than their own reasoning. They did not have the conceptual knowledge to justify the procedural steps that they had memorized.

While Liping Ma worked with a non-random sample of 23 American teachers, Stigler and Hiebert, in The Teaching Gap, wrote about a video study using random samples of classrooms from three countries, the United States, Japan, and Germany. This study was part of the Third International Mathematics and Science Study (TIMSS). What struck the team of researchers was that each country had a theme, or cultural approach to teaching. The image of teaching in Germany was “developing advanced procedures;” in Japan it was “structured problem solving;” in the United States it was “learning terms and practicing procedures.” One math educator on the team said that in Japan, the students engage in mathematics and the teacher mediates; in Germany, the teacher owns the mathematics and parcels it out to students, giving facts and explanations at the right time; in the United States, there is interaction between teacher and students, but he couldn’t find the mathematics. He didn’t see any real math in memorizing terms and procedures.

How do we break the cycle of children not learning math becoming the next generation of parents and teachers, and passing on their attitudes and (lack of) knowledge to the following generation?

Burt,

Thank you for that highly informed and reasoned comment. Also, thank you for being such a faithful reader!

How do we improve teacher preparation? I wish I had the 'magic bullet'. I do think it makes sense to explore and perhaps model our teacher prep programs after those of highly successful countries. I personally like the Japanese model but surely we can design our own model which incorporates the best features of several of these.

What are the implications of more intensive teacher preparation? A redesign of current ed departments in our colleges? Far more dollars being invested into preservice education? A longer internship for teachers than a semester of student teaching? Raising the professionalism of the entire teaching profession? Absolutely!

I would love to see more teacher exchanges with China, Japan and other nations. How enlightening that would be!

I have much more to say about this but I need more time than I have at this moment. Besides, I'm sure you or CM or others have something more to say!

Dave, I’d like to comment about the part-part ratio that you labeled as conceptual. Here again is the problem and the solution:

------------------

With a special promotion, Al received a 60% discount on a new stereo system and paid $x. Sylvia bought the same system (same original price) but only received a 20% discount. In terms of x, how much (in dollars), did Sylvia pay? Assume x > 0 and disregard sales tax.

Method II (conceptual)

Al paid 40% of the original price, Sylvia paid 80%, therefore Sylvia paid twice as much as Al.

-------------------

Dave, you remarked that the part-part ratio approach is not emphasized. The problem asked to compare what Sylvia paid to what Al paid, so the part-part comparison is really the most direct and simplest, if you understand the concepts. In my earlier comment, my main point was that in this country, we don’t learn about concepts. What passes for conceptual learning is memorizing definitions. The American mindset is that mathematics is all about computational algorithms and applying formulas. We mistake the procedure for the concept. In this case, the main procedure is part-whole, so that is how most Americans of any age would approach these problems.

With all of the rational numbers, it really goes back to how we learned multiplication. Remember the article that Keith Devlin wrote that multiplication is not repeated addition, and the strong pushback that he got from teachers. We compute the product by repeated addition, at least at first with whole numbers, so that procedure is the concept. It gives the right answer. This shows again that we think in terms of procedures instead of real concepts. But at least we don’t say that addition is “repeated counting.” How did we escape that one?

If we learned multiplication as using an intermediate unit, or of changing the scale, then we would learn to focus on the reference quantity, what is one whole or 100%? We would become flexible in changing our point of reference, in using different scales (different units as the whole) and this skill is needed when working with any multiplicative relationships such as ratios, proportions, fractions, decimals, and percents. In your problem, the original price is the reference quantity for both Al’s and Sylvia’s cost, so they are on the same scale so there is no problem in directly comparing the 80% to the 40%. If they were on different scales, then we would need to convert one to the other scale, or both to a common scale, whichever is easiest.

I know I didn’t explain the last paragraph in enough detail. Let me just say that with multiplicative relationships, the scale changes all the time. If I compare 2 with 8, 2 is 25% of 8, and 8 is the whole. But if I compare 8 with 2, 8 is 400% of 2, and now 2 is the whole. Multiplicative comparisons are relative. The reference quantity, what is the whole or 100%, changes. In additive comparisons, the difference of 2 and 8 is 6. I can say 2 is 6 less than 8 and 8 is 6 more than 2, but the scale is always the standard unit 1, and the difference is always 6 units. Even with signed numbers, the scale is still the standard unit 1. (If I multiply with the standard unit 1, then I don’t change the scale. Hence, 1 is the identity element for multiplication.)

One last point, we use different units all the time. It is not a strange notion. Take the number 325. We have three digits and therefore three units: hundreds, tens, and ones. The 3 is the count of the number of hundreds (3 x 100); the 2 is the count of the number of tens (2 x 10), and the 5 is the count of the number of ones (5 x 1).

Excellent analysis, Burt.

You have that capacity for recognizing the essential underlying concepts and articulating them clearly. You also recognize that the clearest explanation cannot replace a concrete example - as in "one picture is worth..."

I hope you are in a position to promote conceptual understanding in our classrooms? Have you presented these ideas at NCTM conferences or with local or state education groups? I tried to do this for 20 years at all levels and my message fell mostly on deaf ears. That's why I started this blog:

Using challenging problems as the vehicle to provide models for teaching conceptually.

Now I do believe we need to balance conceptual and procedural instruction. One cannot live without the other and both rest on a foundation of basic knowledge.

I am very reluctant to make general statements about the quality of instruction in this country. The majority of teachers are wonderful dedicated professionals who are doing the best they can with the tools they have.

In math, most of us still teach the way we were taught, which, of course, emphasized procedures over conceptual understanding. We need to address these issues at the college level as the next crop of math teachers go through their preservice training. I could go on forever here but...

Dave, I agree with your comments about teachers. I appreciate your balanced views. I noticed in other posts that you try to be objective, and fair to different sides of an issue.

I want to stress that I am not criticizing teachers or suggesting in any way that they are foolish or stupid or bad. I am saying that everyone who grew up in this country, myself included, is subject to a cultural mindset that thinks of math in terms of procedures and calculating an answer and not in understanding mathematical concepts and relationships. As Stigler and Hiebert point out in The Teaching Gap, we aren’t even aware of this mindset. It was in studying classroom teaching in different countries that they realized this cultural tendency existed.

The TIMSS video study also showed how difficult it is to break the pattern. 70% of the teachers said they were implementing reforms such as those published by the NCTM, and they pointed to places in the video where they were doing so. When the researchers looked at the video, they found only surface changes; the lessons were still consistent with the image of memorizing definitions and practicing procedures.

We can’t break this mindset until we recognize that we have it. We need to recognize that what we think are concepts may not be concepts—for example, that repeated addition is a computational procedure, not a conceptual definition of multiplication. When you were promoting conceptual understanding before you started this blog, could it be that your audience fully agreed with you and thought they were already doing what you were promoting, or that they were adopting your ideas when they were not really doing so?

I didn’t even think of what multiplication was until I stumbled upon a discussion of Devlin’s article. And I only recently read The Teaching Gap. But these ideas have captured my attention and I am becoming more aware of how true they are.

I agree that we need to address these issues at the preservice level, but why not at all levels? It is a cultural mindset; it is pervasive. I don’t think it is possible to change it unless it is addressed at all levels.

Burt,

Thank you for recognizing my attempts to be objective. I recognized the same fairness in you from your first comment. My comment about criticism of teachers was not directed in any way towards you -- rather it was a general comment about the negativism in our society toward teachers in general. That too is pervasive.

Whenever I gave presentations to teachers locally or around the country at NCTM Conferences, I felt there was genuine support for my ideas

at that moment. BUT there's a huge gap between recognizing the need to develop conceptual reasoning and actually implementing it in the classroom.(a) Breaking the cultural mindset to which you referred

(b) Learning HOW to develop lessons which are conceptually-based while maintaining algorithmic and skill proficiency. I didn't become adept at this overnight, if, in fact, I am adept at it! However, it was my credo from day 1 of my commitment to my profession and I never abandoned it. This mindset definitely came from my dad who continually asked me probing questions to get me think more deeply about what appeared obvious at first.

(c) Learning the ART of questioning. Many teachers I've spoken to feel there is such poor response to any higher-order questions they ask, that it's just not worth the effort the "sacrifice of time" in the lesson. My reaction is that, if students do not respond to our questions, then we need to

change the questionsor create a more questioning environment. Learning how to seize that teachable moment when a student asks a question or makes a comment without "sacrificing the lesson" requires much practice and skill. Taking the first step here is the hardest!(c) You're right that change needs to come at all levels. My point was that it is of paramount importance to reach this next generation of teachers as soon as possible.

Sometimes, Burt, I feel like the "prophet in his own land" who goes unheeded. I'm not being egotistical here but it is so frustrating when I'm paid lip service by my own mathematical community and I am basically left out of the mainstream. That's my choice however and I will not concede defeat. I am grateful to my faithful readers like you for reassuring me that I am not really alone.

I could go on interminably here but then my point would be "blunted!" Let's keep this dialog going, Burt, and hopefully others will join in!

Yes, maybe others will comment if we look at concepts and procedures in context of a non-routine problem. Dave, would you like to propose a problem? I don’t want to put you on the spot, so here is a backup problem, in case you would rather use this, from Krutestskii, The Psychology of Mathematical Abilities in Schoolchildren, page 106. He does not give a solution.

For Dave’s problem or this one, what are the concepts and relationships that come into play here? What are the computational steps and how do you justify them?I went shopping. I pay for my purchase in 3-ruble notes. I must give out 8 more notes than if I pay in 5-ruble notes. How much does the purchase cost?We tend to think in terms of the tools that we learned to use. For most of us that is setting up algebraic equations. For the Singapore Math people, it would be setting up block diagrams. And then we would get the answer and move on. Yes, in setting up equations we need to understand some concepts, but I don’t think we see the full mathematical structure of the problem, and we wouldn’t increase our understanding of the concepts or our ability to think conceptually in future problem solving. The block diagrams would give us a better chance of seeing the deeper math structure than algebra, but we would still need to employ Polya’s last step in problem solving: looking back. Even the block diagrams can become a procedure that we employ, like solving equations, if we focus only on getting the answer and not on understanding the mathematical relationships. Since getting the answer validates what we did, then we think we are done. I think most of us miss the last step of looking back because improving our conceptual understanding is not a real goal. It’s not part of our culture. In practice, we say we don’t have time for it. Learning procedures to get the answer is the real goal. It is not balanced by conceptual understanding.

So I am asking for an explicit description of the mathematical structure of the problem, the computational steps, as well as the answer. Is there more than one way to conceptualize the problem?

Awesome, Burt! This apparently simple exercise in algebra is an ideal problem to "look back" on. Your "assignment" would be a great discussion exercise for prospective elementary, middle and secondary teachers.

One way I've addressed your objectives both in the classroom and during teacher training is to give the directions:

"

You need to come up with at least 3 methods in your group."This forces the algebraic person to think more intuitively and the quick intuitive thinker to work through the algebraic procedure. Algebra is a wonderful tool but, as in any procedural approach, it "hides the information"!

Another approach I've used is to give the numerical answer immediately. At that point, they have to focus on methods!

After solving your problem algebraically, I rethought the problem in terms of "exchanges", i.e., 5 threes for every 3 fives. This led to another solution. But the most conceptual approach, IMO, occurred when I really thought about the extra 8 three-ruble notes and the difference between the value of the notes. At that point the problem reminded me of the classic rate-time-distance problem in which one walker is walking at 3 miles per hour and the second walker starts out 8 hours later going 5 mi/hr. When will the 2nd walker catch up?? Developing reasoning is all about making connections...

I've already given too much away (if anyone is reading these comments!).

I would like to publish your problem as a separate post (I think that's what you were asking). We may make a good team, Burt.

Algebra and concept knowledge.

Algebra requires some conceptual understanding. In this problem it requires just an understanding of the problem and how to translate that into mathematical expressions. The cost is calculated using the basic concept of multiplication: the number of units times the size of the unit. This is the only knowledge that must be brought to the problem. The rest are given: the size of the units are given as 3 rubles and 5 rubles, and it is given that the number of 3-ruble notes is 8 more than the number of 5-ruble notes.

Let x = the number of 5 ruble notes,

then the cost is 5x

then x + 8 = the number of 3-ruble notes,

and the cost is 3(x+8)

The cost using 3-ruble notes is the same as the cost using 5-ruble notes so the equation is:

5x = 3(x + 8)

The rest of the solution is procedural. The concepts needed are related to solving equations, not to the specifics of the given problem. Algebra is very useful, especially when the problem situation is very complex. Once the basic relations in the problem are understood and equations are defined, we don’t need to think through the problem situation. Solving the equations are usually easier than thinking through the actual relationships in the problem situation. That’s the power of algebra.

But conceptual knowledge is more powerful. It allows flexibility and creativity. This problem is from Krutetski’s Arithmetic test, not his Algebra test. This problem only requires basic math concepts. I think we would almost automatically think in terms of algebra, but we should be able to solve it without algebra.

If we solved the problem without algebra, one conceptual insight can lead to two solutions. It is the idea of balancing—the smaller sized note needs more notes; the larger sized note needs less notes. We can explore this balancing of size and number in additive relations or multiplicative relations. Dave, I liked the connection you made to the rate-time-distance problem. Since you focused on the additive relations (differences) there, I’ll expand on the multiplicative relations (ratios).

The ratio of the size of the notes is 3:5. We know that there must be more 3-ruble notes than 5-ruble notes, but how much more? It must be just enough to balance the difference in the sizes so that the cost is the same with both notes. The ratio of the number of notes must be the inverse of the ratio of the sizes. It must be 5:3. Then the products will be equal. 5 x 3 = 3 x 5. But the cost is not necessarily 15. The ratio doesn’t mean that the actual numbers are 5 and 3. There is a constant scaling factor or proportionality factor involved. Call that constant c, then the number of 3-ruble notes is 5 x c and the number of 5-ruble notes is 3 x c. The cost is (5 x c) x 3, or (3 x c) x 5. The problem is now like a routine ratio problem: “The ratio of two numbers is 5:3, the difference is 8. What are the numbers?”

We don’t need algebra to solve this. We can think in terms of determining the value of the scaling constant c. The difference between 5 and 3 is 2, and we know that the difference between the numbers that we want is 8, so the scaling factor must be 4 since 2 x 4 = 8. The number of notes then are 5 x 4 = 20 and 3 x 4 = 12. The cost is 20 x 3 rubles = 60 rubles, or 12 x 5 rubles = 60 rubles.

To realize that the ratio of the number of notes must be the inverse of the ratio of the size of the notes in this type of situation is something we can readily do with simpler ratios. For example, if I paid with $5 bills and $10 bills, I think people would readily see that since 5 is half of 10, we would need twice as many $5 bills as $10 bills. That it is harder to see with numbers like 3 and 5 is largely because we don’t work with the concepts much; we work with computational procedures. We need more of a balance.

There is something to be said for the beauty of math. I can remember as a young student being amazed by the fact that there were different ways of solving a problem, that they gave the same answer, and that they were connected and not so different after all. Maybe it was being surprised by the connection, maybe it was the way they all fit together. I liked it. I didn’t like the drill. I liked the ideas. I liked the way they were connected and how they always seemed to work out, agreeing with each other. Maybe that is why I am passionate about balancing procedures with conceptual understanding. It is in understanding the concepts that we can see the connections and appreciate the beauty of math. The added bonus is that math becomes easier. There aren’t so many different things to memorize. Knowing one concept makes it easier to remember others that are related. The pieces in a network of concepts support each other. And the concepts make remembering procedures easier, too.

Let me finish the thought of balancing number and size of notes in this problem, this time looking at differences instead of ratios. (I promise—this is my last post on this thread.) Here again is the problem:

I went shopping. I pay for my purchase in 3-ruble notes. I must give out 8 more notes than if I pay in 5-ruble notes. How much does the purchase cost?The 5-ruble note is 2 rubles more than the 3-ruble note. For the price to be equal, there must be more 3-ruble notes, just enough to make up the cost difference due to the 2 rubles. We know there are 8 more 3-ruble notes, which is 8 x 3 rubles = 24 rubles. For the 2-ruble difference to balance the 8 note difference, the number of 5-ruble notes must be 12, since 12 x 2 = 24. The number of 3-ruble notes must then be 12 + 8 = 20. The cost is 20 x 3 rubles = 60 rubles, or 12 x 5 rubles = 60 rubles.

Note that the intermediate equation in the algebra solution is 2x = 24. This can be interpreted as, “the part of the cost due to the 2-ruble difference in note size (2x) is equal to the part of the cost due to the 8 note difference in number of notes (8 x 3 = 24).”

Burt,

It's a bit scary when I was skimming through these comments quickly and I lost track of whether you or I was writing the comment! That's how closely we are aligned in our thoughts and beliefs.

Interesting that Krutestskii thought of that problem as an arithmetic-conceptual question. That's precisely how it would be viewed in Singapore Math!

BTW, I particularly liked your "scaling" approach to ratios...

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