Author: John Quintanilla

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.

Jonathan Katz on Some Problems of Common Core Mathematics

Courtesy of Diane Ravitch:

Jonathan Katz taught mathematics in grades 6-12 for 24 years and has coached math teachers for the past nine years.

He prepared this essay for the New York Performance Standards Consortium, a group of high schools that evaluates students by exhibitions, portfolios, and other examples of student work. The Consortium takes a full array of students and has demonstrated superior results as compared to schools judged solely by test scores.

What is of special concern is his description of the mismatch between the Common Core’s expectations for ninth-grade Algebra and students’ readiness for those expectations.

Here is a key excerpt…

[The Common Core standards seem] to honor the idea of problem solving and the many ways a student might engage with a problem. It seems to value the process of problem solving, the ins and outs one goes through as one tries to solve a problem and that different students will engage in different processes.

To implement such a standard, a teacher would need to present students with problems that allow for and encourage different approaches and different ways to think about a solution—what we call “open-ended problems.” Yet, when you look at the sample questions from the Fall 2013 NY State document you would be hard pressed to find an example of a real open-ended problem. Here is one example in which a situation is presented and three questions are then posed.

Max purchased a box of green tea mints. The nutrition label on the box stated that a serving of three mints contains a total of 10 Calories.

a) On the axes below, graph the function, C, where C (x) represents the number of Calories in x mints.

b) Write an equation that represents C (x).

c) A full box of mints contains 180 Calories. Use the equation to determine the total number of mints in the box.

A situation is presented to the students but then they are told how to solve it and via a method that in reality few people would even employ (who would create a graph then a function to find out the number of full mints in the box?). If you are told what to do, how can we call this solving a problem? (This would have been a very easy problem for most students if they were able to solve it any way they chose which is what we do in real life.) In fact, all eight problems in the same of Regents questions follow the same pattern. Students are told they have to create the equation (or inequality or system of inequalities or graph) to answer the question. Thus there is no real problem solving going on—merely the following of a particular procedure or the answering of a bunch of questions. Why don’t we use problems where there is a real need for an algebraic approach? Why would we ask students to look at a simple situation then force them to use an algebraic approach, which complicates the situation? We should be helping students to see that the power of algebra is that is gives us the means of solving problems that we would have great difficulty solving arithmetically.

If we were truly trying to find out if our students are developing the ability to problem solve, we would never create questions of this nature. They would be more open-ended so students had the chance to show how they think and approach a problematic situation. But that can’t happen on a test where everyone is instructed to do the same thing so we can “measure” each student’s understanding of a particular standard. This is not real mathematics and a contradiction of the Common Core Standards of Mathematical Practice!

Why does this matter? The consequences are huge, and not just for students. Consider the message we are sending to teachers. Since students will be assessed on following given procedures rather than how they strategize and reason through a problem, then teachers’ lessons will become all about following procedures to prepare their students for an exam they must pass in order to graduate. This will simply perpetuate the same failing math teaching practices we had in the past, will compound the dislike that students already have for math class, and will not in any way help our students to develop mathematical thinking.

Source: http://dianeravitch.net/2014/04/02/jonathan-katz-on-some-problems-of-common-core-mathematics/

How the Texas Testing Bubble Popped

The Dallas Morning News recently ran a three-part long-form article on the passing of HB 5, which significantly rolled back the number of high-stakes exams that are administered in Texas. From the concluding paragraphs:

So in a relatively short time, a Legislature that had been the most all-in in the nation about high-stakes testing as the key tool for accountability became almost as all-out as federal law would allow.

As inevitable as it may look in retrospect, however, the shift was anything but at the time. Politics, policy and more than 30 years of history pushed hard against the change in course. As House Speaker Straus put it recently:

“We got as close as we could to something not happening, but it happened.”

HB 5 did not have my unequivocal support, as it removed the requirement that all high school students take Algebra 2 before graduating from high school. But, on balance, I think HB 5 definitely helps more than it harms.

Part 1: http://res.dallasnews.com/interactives/2014_March/standardized_tests/part1/

Part 2: http://res.dallasnews.com/interactives/2014_March/standardized_tests/part2/

Part 3: http://res.dallasnews.com/interactives/2014_March/standardized_tests/part3/

Statistics Done Wrong

I happily provide the following link to Statistics Done Wrong, a free e-book illustrating pitfalls when using statistical inference. From its description:

If you’re a practicing scientist, you probably use statistics to analyze your data. From basic t tests and standard error calculations to Cox proportional hazards models and geospatial kriging systems, we rely on statistics to give answers to scientific problems.

This is unfortunate, because most of us don’t know how to do statistics.

Statistics Done Wrong is a guide to the most popular statistical errors and slip-ups committed by scientists every day, in the lab and in peer-reviewed journals. Many of the errors are prevalent in vast swathes of the published literature, casting doubt on the findings of thousands of papers. Statistics Done Wrong assumes no prior knowledge of statistics, so you can read it before your first statistics course or after thirty years of scientific practice.

http://www.refsmmat.com/statistics/index.html

Statistical errors and their tendency to mislead

As a follow-up to yesterday’s post, here’s a recent article in the scientific journal Nature about the slippery nature of P-values, including a history about how reliance on P-values has evolved in the past 100 years or so: http://www.nature.com/news/scientific-method-statistical-errors-1.14700

While I’m personally familiar with many of the pitfalls mentioned this article, I have to admit that a couple of the issues raised are brand new to me. So I’ll refrain from editorializing until I’ve had some time to reflect more deeply on this article.

Fun with Bases

Something I noticed last month:

10 in base ten…

is 101 in base three…

and 1010 in base two.

Notice that the representation changed by just adding an extra digit. Also, this is the maximum number of times that this could happen for 10; base two is the smallest base, while a two-digit representation is the shortest that could have a pattern like this (since 10 is represented as A for any base higher than ten).

Naturally, I’m curious if there’s another number like this. 64 comes close:

10 in base sixty-four…

100 in base eight…

1000 in base four…

1 000 000 in base two.

But the pattern breaks for base three.

Flipping to Offer Low-Enrollment Courses

I just read a very interesting article about how an instructor at Ohio Dominican University is simultaneously teaching several upper-level mathematics courses with low enrollment by flipping the classroom (using the catchy title One-Room Schoolhouse). Here’s the article: http://digitaleditions.walsworthprintgroup.com/display_article.php?id=1639570