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.

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.