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.

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.