In New Jersey, which is where I grew up, there is a running argument over state funding for education. In the latest salvo, reported recently in the New York Times, a report by Judge **Peter E. Doyne** concludes that the state is not spending enough on the poorer school districts.

The monetary amounts, as too often happens, are reported in an almost meaningless form: as a total for the entire state. For example, the state budget for aid to schools is given as $10 billion (after I round off slightly). Is that amount large or small, or even reasonably likely to be correct? Who knows until it is turned into a human-sized number as spending per pupil. Combining this state funding, the local spending, and the federal contributions, New Jersey’s per-pupil spending turns out to be $17,620 per year.

This spending level is the highest in the country, which ranges from about $6000 (Utah) to $17620 (New Jersey). It is also a good fraction of a private-school tuition! Which started me wondering what we, the public, are getting in return for funding public schooling. I went to public school for all but two years. And I am shocked at what seems to be the main purpose of public schools today: practicing for and taking standardized, high-stakes tests.

In particular, where I now live it is Massachusetts Comprehensive Assessment System (MCAS) season. Therefore, the email newsletters that we receive from a public school in our city talk almost entirely about testing. We are urged to send our children to four Saturdays of test-preparation boot camp, and we are exhorted to write a “positive, heart-felt message giving your child encouragement and support to show us [on the tests] how much he/she has learned this year.”

Maybe this test obsession will be okay if the tests themselves are good. Ever hopeful, I studied an MCAS test’s mathematics section. Alas, the questions do not even resemble mathematics. For me, mathematics is about exploring patterns and using numbers and quantitative relationships to understand how the world is put together. However, the MCAS test is to mathematics as disconnected notes are to a symphony.

For example, here is the first part of a 2010 MCAS question (Question 42 on page 57 of the “March 2010 retest items”):

*A small box of snack mix weighs 18 ounces and costs $4.32.*

a. What is the cost per ounce of the snack mix? Show or explain how you got your answer.

The snack mix is also available in a large box that costs $6.60. The large box has a cost of $0.22 per ounce.

*b. How many ounces does the large box of snack mix weigh? Show or explain how you got your answer.*

This question only seems like mathematics. First, the large box’s information is given in an absurd form. I have never seen a box of snack mix, cereal, or anything that gives the price per ounce but not the number of ounces! Second, as a mathematician, if I had to compare the cost effectiveness of small and large boxes of mix, I would never compute the cost per ounce. Not only is dividing by 18 painful, it is pointless: I would instead compare the costs using proportional reasoning, for that method emphasizes relationships.

To make the comparison, I’ll rewrite the large-box portion of the question to be realistic:

A large box contains 30 ounces of snack mix and costs $6.60 [thus costing $0.22 per ounce, as in the original question]. How much extra am I paying, per small box of snack mix, if I buy my mix in small rather than large boxes?

Time for proportional reasoning. The small box (18 ounces) has 60 percent of the weight of the large box (30 ounces). Thus, if both boxes had the same price per weight, the small box should cost 60 percent of the large box.

Sixty percent of the large-box cost ($6.60) is $3.96: 50 percent would be $3.30, to which I add 10 percent, or $0.66, to get $3.96. Thus, at large-box rates, the small box would cost $3.96. In reality the small box costs $4.32, which is $0.36 more than it would cost at the large-box rate. With that information I could decide whether the small box’s convenience justifies its extra cost (for example, by being less likely to go stale before I finish it).

The mathematical method of proportional reasoning is simple and direct. Most importantly, it emphasizes the relationships (here, in size and price) between the boxes. Building connections and relationships is at the heart of deep understanding. That is why proportional reasoning is crucial in fields as diverse as physics, engineering, and biology (where it explains, for example, why large animals need circulatory systems).

A powerful, general mode of reasoning is just what mathematics can and should offer to the world. In contrast, the high-stakes tests’ offer what American physicist **Richard Feynman** might have called cargo-cult mathematics. They use numbers and mathematical operations, they look like mathematics, but the substance is missing. By teaching to these tests, we are teaching students disconnected notes and calling it a Bach fugue.

Great essay. This is what we mathematics reformers have been saying for years. Please see the National Council of Teachers of Mathematics “Curriculum and Evaluation Standards for School Mathematics” 1990, 2000.also see the movie “Race to Nowhere”. Thanks for backing this up!

It goes beyond just math classes. History is in some ways worse … memorize dates, not understand cause and effect. But I couldn’t agree more that teaching to the test is awful, including Math and English. It was true when I graduated high school in 1996 and had the required essay style down pat. Went to college and every professor said to never use that style again.

I hear it is even worse now. Regurgitating facts isn’t learning.

Stupid 5 paragraph essays.

My Sophomore English class consisted 100% of us writing 5 paragraph essays. For “the test”, no less. The first thing my Junior year teacher said was, “If you write in 5 paragraph essay style, you will fail.” Never used that style again. A complete waste of a year.

Worse, that was a year we could have spent reading fundamental American novels that I didn’t get a chance to read until after college. Simple things like One Flew Over the Cuckoo’s Nest and Something Wicked This Way Comes.

Also, that’s why I hated history classes until the one I took my last semester Senior year in college. We were expected to know names and dates, but those were minor points. Most of the class (and homework and tests) was explaining the hows and whys. It was a History of War class, so it mixed in things like technological advances and even cultural diferences, tying those into the way battles were fought.

If I knew history could be so entertaining, I would have picked up a few more history classes as electives.

While I would agree that the way the question is worded is unrealistic, I would disagree that your way of addressing the problem is any easier. I would argue that the relevant question is ‘Which box is a better deal?” For that, I would convert both to a per ounce base and immediately realize the small box is more expensive per ounce. How much more would the large box cost if priced at the small box price doesn’t really weigh into the decision of whether to buy or not.

I agree with you. I always compute cost per ounce when making a purchasing decision. It’s quick and easy, assuming I have my cellphone.

This is exactly why I had such a hard time learning math as a kid and hated standardized tests. It wasn’t until I took math in college and applied it to REAL problems that it all made sense. The public school system doesn’t offer our kids a good return on investment.

And, I have a family full of teachers so I’m the bad sheep for hating the public school system.

I live in Brookline. I tutor in math at a charter school and know people who design math curricula. The MCAS isn’t perfect but the general curriculum for math instruction is better and focuses on everyday manipulation of quantities.

My experience with tutoring really bright kids from poor areas – and often from highly fractured families – is that their environment matters more than school. These kids never thought much about school because no one around them thought much about school. The kids I see are fortunate enough to be only a moderate way behind because intensive instruction catches them up and moves them into high schools – mostly private ones on scholarship. But a lot are so damaged by their environment that they can’t catch up in the time this school has to “fix” things – about 1 year, btw.

That has nothing to do with the MCAS. Expectations are set by family, friends and community. Go to school in Watertown, which is a decent place, and aspirations are set lower than in diverse Brookline and of course lower than in rich Wellesley. Since as we move down the economic ladder, family structure falls apart, we see even more influence by peers – which research says carries a ton of weight – and thus even less intellectual interest.

I see little chance for the general schools to fix this. It is a larger social problem of aspirations and thus commitment to education.

It’s absolutely freaky to deal with these bright kids who are now turned on to school and who are getting so much enjoyment out of learning.

I like Charlie Brown’s True-False test method:

The first answer is true, because they like to start out on a positive note;

The second answer is false to break up the pattern,

The third answer is false to break up the other pattern.

The fourth answer is true. You don’t want three falses in a row.

The fifth answer has to be true.

I could keep this up forever. Hey, what’s so hard about taking tests?

Spot on! I particularly enjoyed the reference to Feynman. PBS did a special on him years ago and in the interview Feynman talked about how he would go on walks with his father and how his father would use proportional reasoning to give him a sense of scale and to relay scientific information in terms that he could comprehend as a child. Feynman said it was one of the greatest gifts his father ever gave to him — the ability to reason.

This essay is a good explanation of why I have so much trouble with the freshmen in my general education astronomy courses. They don’t understand proportional reasoning at all! (They also seem to freak out about doing simple calculations of the plug and chug type, but that’s a whole other level of incompetence.) The students are not prepared to think about math related problems at all. You tell them how gravity works and ask them things like “how would the force of gravity on the earth’s surface change if you doubled the mass of the earth without changing the radius?” and they just completely freeze up.

They’re not really looking to teach reasoning. What teaching to the test does is allows you to pretend you’re measuring success in some hard objective way while allowing bad teachers to teach failed students and still get out the end. Actual teaching is too much work and too fuzzy for school administrators or teacher unions. The student is a product, milked by the system like any other cow.

Note: I’m not saying all teachers are bad here, and I sympathize with the good ones. But ‘teach to the test’ allows the minimum level of functioning.

My big beef is that many of the current curricula are set up to “teach reasoning” — which is fine. Who doesn’t want logical thinkers? Students who can think and reason?

Unfortunately, in practice this often means that nothing is learned “to mastery,” as they say. When you have kids who cannot do simple computations and who can’t decide if they should add or multiply or maybe subtract and that they can just sort of do a problem by picking a few things they do know how to do, you will never get to reasoning.

There’s a place for drill, ALONGSIDE abstract reasoning. Concrete precedes abstract and curriculum writers know this, but they seem to think that concrete can just be using “manipulatives” to solve a problem, rather than having a collection of concrete facts at the ready in your brain.

While I do not disagree with your discussion of the uselessness of the question, wouldn’t it be more useful to ask the students to compare on a per ounce basis. Your example of adding 10 percent to 50 percent in order to get the equivalent price of the small box because it is 60% of the size simply seems that you are being unnecessarily complicated about the subject. If you are simply trying to decide what the premium is that you’re paying the only thing you need to get to is price per ounce. This, in my mind at least would be a more useful thought process as a life skill, whether be comparing which food product to buy or which office space to rent ($/sq.ft)

No, actually that is exactly his point. Figuring out sixty percent is kind of hard, but fifty percent isn’t (divide by 2) and ten percent isn’t (divide by 10) and knowing that sixty percent is just fifty percent plus ten percent is a more sophisticated approach to the problem. Price per ounce is one way to do this, but doesn’t caputure the risk/reward system of volume shopping, I can buy cheerios at $.12/oz if I buy 2 tons of cheerios but where do I put all those damned cheerios. The better question is if I buy 2 tons of cheerios how much do I save and is it worth having two tons of cheerios.

A fine article but misses the point when it attributes (indirectly) these failings to public schools. It’s the testing that is the problem, not the fact that they are funded by taxpayer dollars. Unfortunately, it seems to be a common theme of this blog that the authors identity a failing in some part of society and then attribute it to government without any actual analysis demonstrating that government in fact is the problem. In the case of this post, no mention was made of how private schools would fare if subject to the same constraints, nor how their performance would change if the only schooling available was private schools subsidized with vouchers.

I take your point and regret having left that issue ambiguous. I believe that the tests would greatly harm private schools too, if they were required to use them. That they are not required to use them is one of the best educational arguments for private school. in contrast, the public schools must not only use the tests, the students must also show up and take them (at least in my state). Here is a frequently asked question from the MCAS site :

7. Can parents refuse their child’s participation in MCAS tests?

Parents may not legally refuse their child’s participation in MCAS tests. Massachusetts General Laws Chapter 76, Sections 2 and 4, establish penalties for truancy as well as for inducing unlawful absence of a minor from school. In addition, school discipline codes generally define local rules for school attendance and penalties for unauthorized absence from school or from a required part of the school day.

(I am not a lawyer, but I wonder about the careful phrasing of the answer. They don’t say what happens if a parent writes a note excusing his/her child from school. Does that count as truancy? In my day, that would have worked. I once got a parental note to skip school and go with many fellow students to opening day of “Return of the Jedi.” But I have heard that schools today often do not accept such parental notes.)

Thus, in this case I think it is fair to attribute the problem to the government for imposing such restrictions. One of the most cost-effective ways to improve education would be to throw these tests in the river or, following venerable local tradition, in the Boston Harbor.

In my state (Texas), passing the standardized test (TAKs) is a requirement for graduation. TAKs (which is a grade-level test) is going away, and being replaced by end-of-course examinations (no better as far as difficulty, but they will at least cover what every student is learning THIS year) which will also be mandatory for graduation. So even if you have a parent note for skipping school, you’ll still have to make up the test or not walk across the stage.

Massachusetts teachers would LOVE to teach “real math” but are required by their administrators to teach to the test …. and teachers are evaluated based on their students’ MCAS scores.

I hear u Joan, all the way in Illinois!!!

To Sanjoy Mahajan,

I agree with many of the points you posted. I do not agree with the following statement: Which started me wondering what we, the public, are getting in return for funding public schooling. I went to public school for all but two years. And I am shocked at what seems to be the main purpose of public schools today: practicing for and taking standardized, high-stakes tests.~~~~~~~~~~~~~~~~~~~~~~~~~

I have been a teacher for many years (25+). I stay current on best practices and I am constantly learning more about children by taking professional learning classes. I teach problem solving and life-experiences using Math DAILY. In case you are not familiar with Blooms Taxonomy, take a look at that and understand that teachers are constantly using the analysis level as well as application, synthesis and evaluation techniques. We assess using many different methods in our classrooms. Children are taught to use self-monitoring and self correcting strategies. Rubrics are used as a guide to help children and parents learn what is expected and to what degree to be successful and lead to mastery. The tests to which you refer do not measure these important things. It is not the teachers or the school that have the problem you are mentioning, it is the tests themselves!! High stakes testing is required by our state. We would love to generate lists, portfolios and other assessment data for measuring the learning of students. THAT WON’T HAPPEN. The end of the year tests are inevitable. It is a requirement and does give some information that is good to know. Our tests are Criterion Referenced Tests which can only measure how well the students have learned the actual criteria set forth for our grade level (and we do go above that). I certainly DO look at the standards that these tests include. I also make sure that I have introduced the testing format to my students. I do NOT by any means TEACH a test or teach TO A TEST. There is just not enough time in the day to do that. I am hoping that your school experience will improve as you look at the schools all around the United States. I do agree that there is always more that teachers can do for children.

There are several topics here. On the school system, I think public schools have many issues, but the specific example you gave isn’t one of them.

On the math, price/ounce is a relationship between quantities and is also an important way to look at things. In fact, you used this type relationship, quantity per 1 unit of another quantity, to show how public funding per student is almost as big as private school tuition per student.

While I agree that using proportional reasoning as you did seems to be a more appropriate method for the specific example, I don’t see how the example would have a negative effect on learning an also important way of looking at the world, quantity a per unit of b.

Math is about 2 + 2 = 4 You don’t have to know why it = 4 it just does.

Proportional reasoning is NOT math. Just go to Wal Mart. If you bill is $ 7.53 and you give the cashier a

$ 10.00 bill but HE inputs $ 100.00 he will give you back $ 92.47 because that is what the cash register says.

Don’t confuse THINKING and REASONING with Mathmatics.

uh. no.

That’s like saying writing is about spelling correctly.

First and foremost, it is spelled mathematics.

Secondly, real math does not start until you have to being to think and reason. Your confusing arithmetic with the whole of mathematics which is a terrible mistake. Try reading up on some abstract algebra – a very different concept yet essential to the field of mathematics and our sciences today.

The fact that you seem to associate Walmart arithmetic with the whole of Mathematics makes me think you have not seen much math in your past. Go take some new courses, buy some calculus books, or even read up on numerical approximation… You may find a challenge that requires thinking, reasoning, and like every pure science field: sometimes just dumb luck in discovery.

Calculus is useful to less than 1 % of the population. The ability to add, subtract, multiply and divide simple numbers is useful ( and in reality necessary ) to the whole population . Unfortunately today’s educators don’t see it that way. You can function in today’s world and NOT know calculus but you do need to be able to add.

Most people in this world would be far better served by learning statistics than calculus, yet plenty of undergrads are forced through irrelevant freshman calc courses that they’ll never need.

ime for proportional reasoning. The small box (18 ounces) has 60 percent of the weight of the large box (30 ounces). Thus, if both boxes had the same price per weight, the small box should cost 60 percent of the large box.This is what drives me nuts about most math instructors and enthusiasts. That is not math. That is rote pattern matching. HOW do you know that 18 is 60% of 30? Well, I’ll bet your answer, if it is like the other answers I’ve been given is: “well, you just look at it and you can kind of see that it is…” That’s not math. That is some kind of fashion sense and you can see how quickly it breaks down if you say that the boxes are 18.36 ounces and 37.23 ounces. Now what is the percentage? Well, you can calculate them by saying “what percentage of 37.23 is 18.36? Then rephrase that as “X/100 * 37.23 = 18.36″ Then rearrange and divide giving you 49.315…%. That is math. A mathematical approach should work with any numbers any time, not only convenient ones that happen to come out even and can be memorized.

Then you can proceed with your calculations from there without that artificially handy 60%. You are trying to save making mathematical calculations by hiding them behind this sudden assertion. This kind of intellectual dishonesty is the very heart of what is wrong with mathematics instruction in my opinion, or at least it was when I went to school, and seeing it again suggests it hasn’t gone away… or maybe your complaint means it has?

Phrases which should be banned from mathematics instruction and discussion:

“You can just kind of tell”

“It’s obvious”

“It’s intuitively obvious”

“Well you look at this one and you look at that one, and then you can just see it”

and so forth.Computation and Mathematical Reasoning are different things. The point the author is trying to make is, if you are trying to test reasoning, do not make the students waste time on computation. This is a view, which has become popular very recently as computers make it much easier to compute difficult calculations. The creator of Wolfram Alpha strongly advocates for this shift in mathematics education (see his TED Talk). I am not saying whether his argument is right or not. I merely think that your response is a little strayed from the main issue at hand.

Perhaps it is a little strayed, but I’m not sure that it is. I agree that computation and reasoning are different things. (I’ve seen Wolfram’s Ted talk and I disagree with him too, but as you say, whether he is right or not is a side issue.)

I *do* think my point is very much on topic if you consider that the topic is more effective mathematics education.

Learning mathematical reasoning requires seeing and practicing the application of step by step mathematical principles. Reasoning requires *reasons*. What the author of this piece has done is to argue for better reasoning by dropping steps and the reasons for them claiming that makes things simpler. As a student it makes them much more difficult. Furthermore it leaves the student unprepared to reason through more complex versions of the same class of problem.

As far as testing goes, I’m not sure that any multiple choice test can work for reasoning, but if it did, it would have to be much more specific. I think in the end those question would look something like proofs.

Perhaps the question could say: I assert that the small box has 60% of the contents by weight of the large box. Which of the following reasons support my assertion?

In regard to the value of calculation, I can see the reasons for wanting to segregate it from the reasoning, this would allow you to leave an expression unreduced, a fraction not in simplest terms, etc. For purposes of understanding and practicing mathematical reasoning this would be very helpful and I can’t argue with Wolfram there. OTOH, as the author of the piece clearly demonstrates there is an intuitive side to these things. In fact, I think what Mahajan has done, and many mathematics instructors routinely do, is to apply intuition mistakenly believing they are applying reasoning. Practicing calculations like (18/30=.6) until they are second nature contributes to that intuitive grasp of mathematics.

Honing intuition can be very valuable in any field, it is part of what we mean by “experience” but let’s not confuse it with reasoning.

I understand your desire for math instruction to have more depth but I think this is a poor example.

Dividing $4.32 by 18 to find the price per ounce then comparing the answer to $.22 is a lot easier than your proportional reasoning. How is finding 60% of $6.60 “simple and direct” ( 66×6 = 396, then subtracting $3.96 from $4.32) while dividing by 18 is “painful” The problem is set up to divide easily by 18. A student who notices patterns, connections and relationships in test questions could see that.

I think you are being naive in overestimating children’s ability to reason at this level and age. Although I agree that the tests given to public school children are poorly made, I also believe that this test is made to keep track of the laggards, not those who excel. It is unfortunate that in order to get everybody to fulfill the already low the bar we are also lowering the ceiling, as we pay more attention to those who are falling behind.

“get everybody to fulfill the already low the bar we are also lowering the ceiling, as we pay more attention to those who are falling behind.”

Early differentiation is the key. Special Ed, regular program, GT. Even more would help. Smaller class sizes, with each class for similarly-advanced students, would REALLY help.

Now my version of a test problem.

Instruction in a X elementary school starts at 9:15 am and ends at 1:25 pm on Monday and 3:55 pm Tues-Fri.

Breaks and recess take on average about 1 hour 10 minutes (less on Monday, more on Tues-Fri).

Average number of students per FTE (full time equivalent) teacher is 14. If you visit any call on any time of day, you will see classes of 26-30 being taught by one teacher.

The question is: how many hours an average teacher actually teaches?

Additional questions for GT students:

1) how many additional time an average teacher spends for class preparation if a material covered is the same year after year, is provided by school system according standardized curriculum, and work sheets (at about 3 per student per day) are produced by copying at a Xerox machine producing 60 pages per minute?

2) how many time a day an average teacher spends grading home work, if the homework consists on average from 3 pages total (on all subjects), all questions for all students are the same and thus are graded at rate of at least 3 pages a minute, and there is no homework on Friday? Disregard the fact that some grading happens during class when pupils are writing tests or doing other classwork.

3) how many time a day an average teacher spends grading tests, if there are 3 tests a week (on all subjects) 3 pages each?

PhD question:

How the class sizes can be reduced in half without hiring a single extra teacher, given the fact that minimal full time salaried work day in other occupations is 8 hours?

I’d like to see school unions wiggling around those.

Hi Team,

An interesting article… even from New Zealand, where mathematics is taught very differently from US methods.

Oddly enough, in many supermarkets here, many products are displayed with ‘grams/cent’ values, so folk can compare the relative value of different size packages, jars, etc.

It may well be done so the average non-mathematical shopper can see at a glance which is the ‘best buy’. I’d say it is inherently more sensible than the method you propose.

What I find depressingly familiar about your explanation/methodology is that it assumes that people being tested with this type of question actually have your mathematical grasp of the benefit and usefulness of proportional reasoning, and have accordingly developed what can only be called higher order thinking to apply to such questions. They have not, and intellectual snobbery will turn them off learning it even faster.

Forgive me for being blunt, but I think you have the cart before the horse. The end-point of such mathematics is that they arrive at an understanding of its power and value, they don’t start with it.

Secondly, I think you may never have taught this method to students with a view to having them understand it. Sure, they might manipulate figures quickly, as you do to show off how ‘simple’ and ‘obvious’ it all is. They might be able to spin tricks with algorithms that yield good answers. And they might be able to sit one of the stupid mutiple-choice tests to prove they can ‘do it’. But where in all that is the test to show that they ‘get it’?

You know, if there was an oral test where they explain and describe and communicate their understanding, which is is a fundamentally different test than some rote-rule-rubbish, with possible correct answers but possible wrong understanding, is it not? Such a test fails to deliver understanding, and, I’d suggest, so does your explanation to the average student sitting that test.

Yes, the test asks for reasons for student’s methods, but your answer fails to deliver that to them.

I’d suggest an academic and intellectual explanation/answer is no use to students (age 10-12who want understanding. It will appear to be just another smart-alec trick to show off: they will learn to do the trick, and feel a little superior that they can make the rabbit appear, or they’ll learn that math is void of understanding and meaning.

Your statement “as a mathematician” is the giveaway here. Your students are not mathematicians, and have not spent years becoming one, as you have. So, maybe ask them what their understanding is beofre you tell them what yours is – you might be surprised.

Best regards

Stephen Barrett

PS Just so you don’t think I’m not qualified to comment

BSc (Econ/Math/Phys)

High scholl math/econ/phys teacher for 16 years.

I never normally mention this to anyone, there’s no point.

While I do the same reasoning as you, I see 3 benefits to the test question. First, nearly everyone has calculators on them (if you have a cell phone you almost surely do!), so calculating things on a price per ounce basis is a reasonable approach. Second, they are trying to show how math can help in the real world. Third, sometimes we get information in messy formats and we have to use what we get. Sure, we don’t get information this way at grocery stores, but this is supposed to be a quick problem.

As someone who writes math content (Mathalicious) focused on real-world topics, I agree entirely with not only the article but also most of the comments. Clearly there’s a problem when students see mathematics as wholly divorced from reality, and it’s no surprise when a recent Raytheon study finds that 50% of middle schoolers would rather take out the garbage than do their math homework.

At the same time, though, I find the specifics of the author’s argument to be fairly anemic and half-baked. Instead of addressing the mathematics head-on, though, let me offer a different scenario:

Imagine you take a wood shop class. If on Monday you learned how to use a screwdriver, Tuesday a hammer, Wednesday sandpaper, but never actually built anything, that’d be a pretty pointless class, right? But just as pointless would be a class where you’re told, “Go build something,” with no training on how to use the tools.

The author thinks unit prices are unintuitive, and thus shouldn’t be taught. Fine. That’s a perfectly reasonable perspective. But I can point to any number of people who find them perfectly intuitive…indeed very graceful. Per capita. Per diem. Percent. All of these are unit rates. Is there anyone here who *wouldn’t* ever use them?

The point, of course, is that skills and context are both important, and teaching one without the other is to teach nothing at all. Sadly, in math education today–just as in politics–there are two camps: those who advocate fuzzy, explorational math; and those who want a more skills-based, Singapore-like approach. While such shouting matches may make for entertaining sport, the solution is almost certainly somewhere in the middle.

Yes, you need to know why you’re doing what you’re doing. You need to understand what slope, unit rates, exponential notation, etc. really mean. But at the end of the day, you also need to know how to use them. Because frankly, it shouldn’t take fifteen minutes to figure out which bag of peanuts to buy. There’s a reason humans came up with “per ounce.” It’s useful…you just have to know how to use it.

If you don’t like the tests, propose a better solution to measure the effectiveness of the education system. The test may not be perfect, but they are the best near-term measurements for school and teacher effectiveness we have.

On another note, are the name and email field no longer saving for some of you?

Maybe we don’t need the tests at all? Kids in Finland are not required to take tests, and yet they have been outperforming US and most of other countries around the world (not only in maths). Look up Pasi Sahlberg’s book or youtube presentation “Finnish Lessons”.

All school systems should insist that students learn the 6 Functions of a $1. It would allow those students to make better decisions about their money. When they grow up, most won’t be doing math problems but they will be making basic finance decisions.

I agree with you that mathematical reasoning is not being taught (very well) at our schools.

I don’t think you’re right about proportional reasoning. In my experience (as a public school math teacher) proportional reasoning is actually over-taught to the point that many students A) use it where it’s not appropriate, B) use it when another method is more appropriate (or easier) and C) can’t really use anything but proportional reasoning. When the only tool in your chest is a hammer, everything starts to look like a nail…

I also don’t think you’re right about private schools being immune from this problem. Sure, they don’t have to take the state-mandated minimum competency exams. But to get into college, every student has to take the SAT (or ACT) and most will take one or more AP exams. Let’s limit ourselves to the math portion of the SAT and the AP Calculus test. Now, these are certainly more difficult than, say the TAKs test (the SMMCE in Texas). But they’re still tests, and they can still be taught to. I happen to teach at a very successful high school in Houston (competitive with private education, but free, and largely minority rather than white/asian). We do very little to teach to the TAKs test (and we still do very well). We have been very successful on both the SAT (average score on math across this year’s entire graduating class: 635) and the AP test (all of our students take AP Calculus, and about 90% of last year’s class passed the test). This success is not due (solely) to us working with kids who are good at mathematical reasoning. In fact, a significant portion of our student body are, frankly, not very good at real mathematical reasoning (although, they ARE very good at proportions). This happens because we spend all four years preparing them for the AP calculus test. The test can be taught to, and I guarantee you that a significant number of private schools (maybe not all, but a large number of them) are teaching to the test as well, or at least providing a significant amount of test-prep near the administration of the test. (To be fair, the AP test does a much better job of measuring reasoning, but the questions tend to follow a certain pattern, and if you don’t expose students to that pattern, then most won’t do well, no matter where they go to school).

Now, you can complain about standardized testing all you want and how it’s foisted on us by governmental bureaucrats or whatever. But the reality is that the SAT and the AP tests are actually foisted on us by you (ie academia). You wanted an objective way to measure the quality of students coming out of very different high schools (both public and private). And so we have these tests that are based on a faulty bit of reasoning of your own. It’s true (to a large extent) that if you are good at mathematical reasoning, then you’ll do well on these two tests. However, it’s not true (and this is the fallacy that you in academia, and we as a country have fallen into) that if you do well on these tests you are good at mathematical reasoning. It may have been true to a certain extent when the tests were first given. But it has long since ceased to be true now. Now, what’s the solution?

Well, you need a way to assess student’s mathematical reasoning skills that is:

A) Transparent – you have to show us how you’re assessing MR skills.

B) Scalable – it has to be administered to the millions of students who are going to apply to college every year, and then graded efficiently (by contrast, the AP exam is graded by 500 or so teachers over the course of a week, and the number of students who take it is at least 2 orders of magnitude less than the number of students who will be taking this hypothetical assessment)

C) Truly tests MR – a student will pass the test if AND ONLY IF they can reason mathematically

D) Can’t be taught to.

If you can do that, you’re the smartest person on the planet. Frankly, I think A and D together make this task impossible. You can argue that C is the most important part of this list, but in reality, if you don’t have A, B, and D the test is useless.

The alternative is to get rid of standardized tests altogether. From a political point of view, that’s a bit pie-in-the-sky. But even if you could swing it, would you want to? They may be imperfect, but if you get rid of them, then standards, as a whole, will fall, not rise. Sure, some students will be able to excel further because they will spend less time worrying about these stupid tests. But I would argue that, to a large extent, those students don’t spend nearly as much time worrying about these tests as you think they do. The rest of the students, including some good ones, are going to end up not achieving to the same level because they aren’t going to be pushed to the same extent.

This information, although provided by the government does not jive with the government’s other published data. Here, if you perform the calculation yourself, you will see that it shows that $3,146.78/pupil was spent in New Jersey. What gives?

http://nces.ed.gov/programs/stateprofiles/sresult.asp?mode=short&s1=34&s2=49

I am in general agreement about this illness of teaching to tests, and largely about your plea to teach concepts. It’s certainly what I would want. Yet, not all students will get that, and there may well be a place in a test for all for simple drills. “May well be” — I’m raising this as a question: this is a test for all students, not the top 50%, &c.

Sanjoy, you write: “For example, here is the first part of a 2010 MCAS question… The snack mix is also available in a large box that costs $6.60. The large box has a cost of $0.22 per ounce.

b. How many ounces does the large box of snack mix weigh? To make the comparison, I’ll rewrite the large-box portion of the question…

A large box contains 30 ounces of snack mix and costs $6.60 [thus costing $0.22 per ounce, as in the original question]. How much extra am I paying, per small box of snack mix, if I buy my mix in small rather than large boxes?”

The problem is that you did not rewrite the question. You wrote totally new question. And that new question is of substantially higher complexity. If the original problem required only one arithmetic operation, your question requires three and some connecting logic (and if solved by your “estimethod”, even more). Thus the original question is at approximately second grade level, while yours at third or fourth (I am not a teacher, just estimating from my daughter’s experience at school).

Now, if the test (not being the test to look for potential GT students) is given to 2nd-graders, the original question (or another one with similar complexity level) is totally fine and yours is out of place. If it is given to 8th-graders, both the original question’s complexity, your question’s level complexity and much higher complexities should be present to satisfactory attest education levels of different students (except the very far tails of the distribution for whom special education/GT programs should be available with different test questions).

If THIS test in its present form is given to 8th-graders it is certainly a tragedy and epitomizes dumbing-down of American education. The 2nd page of the document can help to explain WHY.

Thank you – great example of what I call foolish math. If I want to find out this answer, I’d look on the pricing info labeling in the grocery store where the math already exists typically. I’d like students looking at multiple boxes of the same size with the pricing data and making a decision which one to buy based upon nutritional info (math), amount of money in my checking account budgeted for groceries this week, and number of people in my household which has something to do with the longevity of the box of snack mix. BTW, I haven’t seen any evidence that Finland has parents writing their children selected response letters of love or adding jukus on Saturday for test prep. Even the POTUS says these tests are boring and punish children and schools unlike his children’s Sidwell Friends experience. Our children need authentic mathematical experiences to build conceptual understanding, problem-solve authentic problems they actually might have to do some day in the real world, and learn the mathematical fluency needed as a means to the end rather than being the endgame. Thank you.

You say: I’d look on the pricing info labeling in the grocery store where the math already exists typically

Because my habit of calculating price per (ounce/pound/gallon, etc.) long predates those printed labels, I still do it reflexively. In so doing I’ve discovered and reported several errors to the manager of our local grocery store. I think she thinks I’m a little crazy, but she sighs, runs the numbers through her calculator ( a process which by itself seems to mystify her younger employees), sighs again, and sends someone out to take the erroneous label down.

I wouldn’t count too heavily on those labels if I were you. (Then again, expanding the discussion to probabilities, I’ve found three incorrect labels among thousands, over the course of years… maybe it isn’t really a concern.)

The question looked mathematical to me, since arithmetic _is_ mathematics. I guess the poster was put off by his more advanced perspective being ignored at the expense of testing the fundamentals. Introducing the concept of spoilage because the larger volume would take a longer time to eat is a red herring, since I doubt that cereals have a short shelf life. The ‘explain your answer’ part got no attention in the post, (Apply the formula: Total Cost = Weight x per Weight Cost)

There is being a technique of mathematical concept to teach. It should clear the student base as per there syllabus it carries general formula for the specific method.

http://www.biblehealth.com/ear-infections/ear-infection-a-deep-look-at-it.html

For years, we heard [especially in relationship to low-income communities where kids were less likely to go to college] that the math taught in schools was too theoretical and not related to the types of real-world knowledge most kids would need to make simple decisions to get through life. These types of problems address exactly the problems that someone shopping on a limited budget would make.

This doesn’t seem to be a condemnation of public schools as much as it is a condemnation of how poor standardized tests are at evaluating the effectiveness of schools and their teaching methods.

Also, could you please present some statistical evidence to support your claim that private school tuition is a fraction of what we spend per student on public education?