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.


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  1. Robbie says:

    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?

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  2. Arthur says:

    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.

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  3. peevee says:

    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.

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  4. Pam says:

    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.

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    • john says:

      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.)

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  5. Randy A MacDonald says:

    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)

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  6. jack4321 says:

    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.

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  7. MRB says:

    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.

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  8. John Theodore says:

    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?

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