Public School Math Doesn't Teach Students How to Reason

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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.

Peter Smyth

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!

James D

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.


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.


Eric M. Jones

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?

Johnny Mudshark

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)


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.

Sanjoy Mahajan

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.



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!!!

Viki Bozeman

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.