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:
“….,based on my observations of many math classrooms throughout New York City, I have seen that there are many early teenaged students who are not yet sufficiently cognitively developed to think about complex mathematical ideas, and they are being left behind, unable to integrate the abstraction of algebraic ideas at this point in their lives. I value the idea of developing deep conceptual understanding and believe it is the only means for someone to develop the ability to work with ideas in higher mathematics. But what is appropriate conceptual understanding for a student in ninth grade? Fourteen year olds will now be expected to engage with linear, quadratic, exponential, absolute value, step, radical and polynomial functions, while developing an understanding of linear and exponential regression. Even most adults have no understanding of this level of mathematics. I would love to believe that students are well-prepared, but I have sat in over 50 different ninth grade math classes this year and have witnessed that what is being asked of our students is “disproportionate to their knowledge.” Too many students have come into ninth grade with limited understanding of basic important ideas like the variable, equality, and solution. Students lack an understanding of the relationship between arithmetic and algebra.”
Katz writes:
Facts about the CCSS and the New Common Core Algebra Regents
-Jonathan Katz, Ed. D.-
Mathematics is a wonderful discipline. All people should have the chance to see and feel some of its beauty and magnificence. I have spent the last 33 years in the world of mathematics education. I taught students from grades 6-12 for 24 years and have coached mathematics teachers for the last nine years. When the Common Core was presented five years ago—specifically, the 8 Standards of Mathematical Practice—there was hope among high school teachers that they would have the support needed to make math come alive for students. They wanted to open up to students the excitement of really grappling with problem solving and mathematical thinking, as opposed to merely asking them to follow standardized solutions closely tied to procedural goals rather than mathematical thinking. But with this year’s introduction of the Common Core assessment in algebra, it’s clear that this is not what the State of New York is expecting teachers to do.
In June 2014 NY students will be taking a new exam in algebra created by the New York State Department of Education that is “aligned” to the Common Core Standards. Only recently, sample questions were published to give teachers a sense of what their students will be asked to do on this exam. I have looked closely at the sample problems and have had many discussions with teachers about these questions. I have come to see that we have created a situation in New York that is causing tremendous harm to its students and that there needs to be an immediate moratorium placed on the dissemination of the new Common Core examination in algebra.
Why do I make this statement?
George Polya, who has had tremendous impact in math education in the United States, stated,
Thus, a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking. (Boaler, 2008, p. 26)
Two questions arise from Polya’s statement.
• What is a mathematics “problem”?
• What does it mean to challenge students with “problems proportionate to their knowledge”?
The first Common Core Standard of Mathematical Practice can help us to understand the meaning of a problem.
MP. 1 – Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
This Common Core standard seems 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.
The second question I posed from Polya’s statement was,
What does it mean to challenge students with “problems proportionate to their knowledge”?
The Common Core Standards is asking students to think deeply about algebraic concepts at an earlier age. Students in 7th grade are being asked to understand linear relationships and are introduced to y = mx + b. Students in 8th grade are asked to make sense of systems of linear equations. All this to prepare students for high school. But based on my observations of many math classrooms throughout New York City, I have seen that there are many early teenaged students who are not yet sufficiently cognitively developed to think about complex mathematical ideas, and they are being left behind, unable to integrate the abstraction of algebraic ideas at this point in their lives. I value the idea of developing deep conceptual understanding and believe it is the only means for someone to develop the ability to work with ideas in higher mathematics. But what is appropriate conceptual understanding for a student in ninth grade? Fourteen year olds will now be expected to engage with linear, quadratic, exponential, absolute value, step, radical and polynomial functions, while developing an understanding of linear and exponential regression. Even most adults have no understanding of this level of mathematics. I would love to believe that students are well-prepared, but I have sat in over 50 different ninth grade math classes this year and have witnessed that what is being asked of our students is “disproportionate to their knowledge.” Too many students have come into ninth grade with limited understanding of basic important ideas like the variable, equality, and solution. Students lack an understanding of the relationship between arithmetic and algebra. Ninth grade teachers have needed to develop the basic ideas of algebra as they attempt to get students to develop a strong understanding of functions. It has put students and teachers in a very difficult position. Teachers have had to ask, “What is fair for my students? What should I be doing to make sure I help them to grow and develop an appreciation of mathematics?”
Many teachers have been doing an incredible job, and my respect for them is enormous. One of those teachers, who is working in a school where most students come from struggling situations, was shocked when he saw the sample questions for the new Regents exam. He knew immediately that his students would not be able to answer most. He saw that many of the questions would have previously been on an Algebra 2 exam. Students will have to answer questions about an exponential regression, graph the residuals of a linear regression and describe its meaning, graph a cube root function, find the zeroes in a quadratic function, graph an absolute value equation and state the domain over which the function is increasing.
I remember in my early years of teaching I gave my students a test and most students did poorly. Instead of looking at why this happened, I blamed my students and simply gave them a harder test next time, as if that was a solution. I’ve learned a lot since then. I learned to redirect my teaching from what I hoped “to cover” to better understanding the thinking process that my students were experiencing—how they were making sense of the mathematics we were engaged in. In NYS we have decided that since too many students who graduate high school are not prepared for college, we will simply make things harder, as if exposing them to more and more complex mathematics at younger and younger ages will solve the problem of college readiness. We should be asking why students struggle to learn how to think mathematically and what needs to change so that math can begin to make sense to them?
New York State education officials are not totally oblivious to what is going on. They are concerned about what is going to happen when the algebra exam is administered for the first time this June. But they “jumped into a solution” rather than grappling with all the “givens, constraints, relationships, and goals.” Their solution has been to require that students take the CCSS Algebra Regents in early June and then have the option to take the old Regents exam three weeks later. Students can choose the highest result as their final score. It is a no-brainer that teachers will let students take the old Regents since we already know it is considered the easier exam, but this creates a new set of problems. The two curricula are very different. What is a teacher to do? Try to cover material from both curricula? Stop teaching the required CCSS curriculum and teach the old curriculum only since students would have a better chance of doing well on that exam? What is fair for our students? To what extent are we feeding into students’ already negative attitudes about mathematics?
I see only one solution at this time: a moratorium on the testing of students in ninth grade algebra. Then we need a concerted and informed effort to bring together teachers, math educators, students and parents to grapple with the question, “What is mathematics and why do we teach it?” Why do we ask students to spend 12 years in school studying mathematics? Since true mathematics is not a rigid subject, when will we recognize that all students are not the same and the way they express mathematical understanding can take different forms? Do we need to recognize that standards can be very detrimental if we treat them as if etched in stone but very useful if they are approached with more openness and flexibility? We must continue to ask questions so that we can truly meet the needs of our students.
References
Boaler, J. (2008). What’s math got to do with it? New York: Penguin Group.
Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.
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