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.
1
Diane Ravitch's blog 
https://twitter.com/PARCCPlace
The PARCC Twitter feed.
It’s 100% positive responses on the testing from what I can tell from this Twitter feed, which is sort of miraculously high….
Everyone loves this test and it’s working perfectly 🙂
LOL
Jacques Barzun had a great word for this sort of curriculum (and yes, it’s a curriculum): Preposterous; literally putting the end at the beginning. We keep trying to make 40-year-olds out of 18-year-old high school graduates.
A bit more …
This preposterousness is result of the bait-and-switch game behind the CCSS. By making the standards that are in reality ridiculously inappropriate, Gates & Co. hope to sell the public and politicians on the pipe dream that just “setting higher standards” will magically fix our economic problems by providing high school graduates with educations that look like those of the students entering the best colleges (i.e., their kids).
Of course this can’t work. Standards at best are just hopes and wishes. Without providing funding, training, and dealing with childhood poverty, without setting standards that are developmentally appropriate, this can’t work. Of course, while we’re learning this, the supporters of CCSS will be raking in money from “aligned” materials, tests, big data, and every other “service” they can think of. But these people don’t care. If this works, they’ll claim victory and get rich. If this fails, they’ll dismantle “failed” public education and get rich.
In the end, it’s all about getting rich(er). It’s not about the children.
Weird contradiction: you cite Polya, whose work has been operationalized in education by Alan Schoenfeld, and you cite Jo Boaler. Both Schoenfeld and Boaler have publicly come out in favor of CCSSM. Having worked with both of them I suspect they would say you grossly underestimate kids’ ability to grapple with complex mathematical idea. Your developmental argument, frankly, smacks of elitism.
I think that perhaps you are missing the point of teachmathculture. I am curious. Do you teach k-12 math?
Jonathan Katz did not say that he was opposed to the CCSS in math. He talked about meeting kids where they are and actually allowing them to begin to explore and think mathematically. I would summize that you are the elitest here if we are going to start throwing around labels. Are we supposed to be immpressed that you worked with Boaloer and Schoenfeld?
Strike the word ‘of’ from the 1st sentence.
Is that a sincere question or a jab? I will give you the benefit of the doubt and assume the former.
I taught high school in urban settings and now am an educational researcher. I am not seeking to impress but pointing out what I read as specious reasoning. I believe in meeting children where they are, but in my view, which is informed by my own teaching and my work as a researcher, this author is underestimating what children are capable of. I am concerned that some anti-CCSSM pieces focus on what children can’t do.
bmarshall: I agree.
Perhaps a review of the recent discussion on “grit” and “determination” would prove more useful than lightly throwing around a loaded term like “elitism.”
“There is no end to learning.” [Robert Schumann]
😎
Personally, I don’t read elitism in the post.
But I do wonder about my reading of the post. It appears to me that the author’s problem is not with the CCSS-m themselves, but perhaps with the assessment and accountability system. While it’s great to help the kids think deeply at these topics and try to make sense out of them, perhaps it is still too soon to expect mastery in an inflexible assessment system?
Yes, he does question the developmental appropriateness for 14-year-olds, but he offers no research to say for sure if that is the problem or if formalized and symbolically abstract assessment is the problem.
Again, I am not certain in my reading of that. But most things I read that are critical of CCSS-m tend to focus on assessment and accountability policy but not much on the merit of the standards themselves; with one exception: I have read some good critiques of K-2 mathematics that raise some valid points about developmental appropriateness there.
Jonathan works at the middle and high school levels. It would be presumptuous of him to critique the K-2 content standards based on his professional work. And he wisely chose not to do so.
A little more intellectual honesty and modesty would go a long way in the current debates about K-12 mathematics education and the good, bad, and ugly aspects of the CCSS themselves as well as the more disturbing (to me, at least) ‘big package’ that I refer to as the Common Core Initiative. We can debate each and every specific content standard (in math, literacy, etc.), as well as each standard for mathematical practice (the ones I pretty well like as written and wonder why some “progressive” opponents of the Common Core seem to ignore, overlook, or actually oppose. I GET why reactionary critics would hate the Practice Standards – they’re so similar to the Process Standards NCTM has promoted since 1989 and which are the theoretical basis for what I term “progressive math teaching,” and critics call, dismissively, “fuzzy math.”)
But what isn’t debatable in my mind is the overall Common Core package. The way it was put together. The way it’s been pushed and packaged. The use of Race to the Top as the carrot and stick combined. The high-stakes testing and enormous abuse of psychometric principles. Pushing charter schools and, no doubt, vouchers down the road. Phony teacher and school evaluation schemes. VAM. The sinister influence of big testing and big publishing companies, private foundations, billionaires, ad nauseam in the whole sick initiative. And of course the data mining. The actual content standards could be marvelous and I would still have to oppose this initiative. And I think some of the reactionary attacks on various math problems based on a complete ignorance of basic mathematics and its teaching just muddies the waters. It pains me to see some otherwise insightful critics of the Common Core Initiative to go down the dark alley of anti-progressive math education thought. To be specific, I mean Mercedes Schneider, who just gets it wrong every time she writes about math education issues and the standards, even as she continues to nail the deformers and their lies. It’s more than a bit strange to see such internally contradictory writing from the same person.
But to get back to Jonathan, there is no question that his primary focus is on the insanity of the assessments, their misuse, and how they are destroying the lives of kids and educators. If that’s not enough for some readers, so be it. We can’t all mouth one particular orthodox line in which ANYTHING and EVERYTHING connected to the Math Standards “must” be trashed. Because when the hysteria ends, people will realize that we’re just back to the Math Wars arguments that were NEVER settled, despite various claims to the contrary. That debate is fundamental to the direction math teaching and learning will take in this country for the rest of the century.
Intriguingly, there is a young, vibrant subculture of math teachers who are ignoring both Math Wars thinking and Common Core debates. Instead, they are communicating with each other and anyone else who bothers to come play with them, via Twitter, weekly on-line presentations (see the Global Math Department page), blogging, etc. These folks are so creative and passionate about math and its teaching, so open about their ideas and classroom practices, that they are effectively immune from the negative bullshit that characterizes most Math Wars debates and the recent spate of Facebook attacks on individual math lessons and problems by people who, frankly, don’t have the first clue as to what it means to think about math at the K-3 level or so. Many of these teachers are quite young, they are very attuned to various resources that the average math teacher or parent or politician is utterly ignorant of (often proudly so), and they aren’t about to let the dumb 18th century thinking that informs most people’s take on math education to get in their way. Bravo! They’ll be doing fine work despite the Common Core, and long after this particular pile of manure is gone.
Michael Paul Goldenberg, I agree with your bottom line. Even if the content of the standards were perfect, the process by which they were developed and imposed is fundamentally anti-democratic; the fact that they cannot be revised is absurd. The fact that they must be tested online is a bonanza for the testing and tech industries at a time when schools are suffering budget cuts. The fact that Pearson and Knewton boast openly that the tests permit a new era of data mining is very problematic. The fact that the testing is tied to evaluation of teachers and schools locks into place practices that are highly dubious. The standards are not a standalone “reform.” They are the linchpin of a much larger package which brings together teacher evaluation, school grades, data mining, and the whole Bush-Obama era practices that have proved so counterproductive.
Diane,
I think that’s the crucial “big picture” view that I wish more people would focus upon. Reading so much of the reactionary analysis, such as it is, of specific math problems – which in and of themselves may be terrific, horrid, or somewhere in between – gets us no where for the simple reason that none were written by the people who created the math content standards (most of whom are very reasonable people, even if I may disagree with their understanding of the role that their work is playing in the larger scheme of education deform), and none of which can fairly be called “Common Core Mathematics because the ideas that inform them don’t come from the Common Core, but rather have been developed by mathematics teachers, educators, and researchers long before the very idea of a Common Core” existed, at least in any meaningful political sense.
I’m less sanguine about the literacy standards because of David Coleman’s direct involvement with them and the fact that many early critics are English teachers I deeply respect (one of whom, Phyllis Tashlik, is a colleague of Jonathan Katz’s with whom I had the pleasure of working and from whom I learned a great deal about effective literature teaching with underserved students in NYC). But it is unlikely that there are NO good ideas in the entirety of those standards. And it is absurd to look at the math content standards and try to paint them as either just perfect or a tool of the devil. They are neither.
Your point about the rigid way in which they seem to be pushed by the powers that be is a vital one, I fully agree. That is one of the more disturbing aspects of their implementation, one that gives support to some of the paranoid fantasies of the right, but which nonetheless tell me that the folks at the top are truly afraid of losing full control over THIS set of reforms (regardless of whether they have a secret Communist/ Socialist/Muslim/Zionist/Illuminati agenda develop by the Elders of Zion, the Gnomes of Zurich, the Bavarian Illuminati and the Mafia, all in the pay of the Pope, as opposed to the more obvious fact of their commitment to making beaucoup megabucks, which I think is the driving force behind most of this nonsense.
I still have a hard time holding my peace when the tin-foil hat theories appear, but I think by now most people of good will recognize them when they see them. Less clear is whether average Americans have sufficient knowledge of math or math education to wade through the smokescreen being thrown up against the perfectly reasonable Math Practice Standards because of old Math Wars gripes coupled with new Tea Party hatred of anything affiliated with Barack Obama. I should probably learn to count to ten, a lot, but I’m still a growing boy with a lot to learn about self-control. 😦
Thanks so much for airing Jonathan’s essay. He’s been giving a great deal of thought to what’s going on nationally and especially in New York City and State with the enforced curriculum and assessments (ENGAGE-NY, he told me last year, is a horror show of the worst kind). He’s a person of great reflection and deep feeling about education, kids, teachers, and schools who deserves to be taken seriously.
Regardless of one’s position on CCSS, I don’t think I’ve read a more arrogant piece than the MPG comment. Makes me feel really good to hear about the big picture. I’m more concerned about my own children’s education.,maybe that’s the picture he should look at. And realize there are are some people out there who really do know about mathematics and really do know about actual children.
Well, Peter, then I really suggest you “get out more” in your reading.
I agree that there is heavy conflation of NCLB infrastructure and the standards themselves.
Like you, I am not a blind defender of the CCSSM. I spent a lot of time in urban schools and saw horrifyingly low expectations for kids. Flags are raised for me any time folks make sweeping generalizations about what is possible and appropriate for kids in these settings.
One major problem with CCSS is the fairly consistent poor quality of the “Common Core aligned” materials that our kids are bringing home. The rhetoric around raising standards sounds great, but where the rubber hits the road- the actual math work that real kids are bringing home- this is real junk. The problems are often flawed- things like un-labeled right angles, misleading diagrams, attempts to make a problem authentic but so poorly done that they serve only to mislead- these things are causing many actual students to fail to learn math and learn to hate it in the process.
thank you. I was hoping, instead of the name calling and defending of peoples who do not even post here, that people with knowledge, as opposed to opinions, would reply to my questions. The materials provided (or mandated for use by) to the teachers for teaching the standards seem to be poorly designed or miss the mark. So we could have good common core standards at the appropriate grade/developmental level, and we could have, could have mind you, as the tests are only now being field tested and the only other field test, that of NYC, seems to have shown a poor alignment somewhere in the system, tests aligned with the standards, but the materials and teaching practices are not proper. I am not blaming teachers. This is a system, where are the flaws, the breakdowns, the glitches? No one responded concerning my question regarding developmentally appropriate standards at grade level–other than someone who said all children learn and develop differently and so we can never devise standards at grade level (such an approach is a non starter unless we adopt a system of education where each student is assigned (or chooses) a teacher and the two set off on their learning experience all alone). And someone did tell me that NYC developed a test and it seemed from the response that the field test was the first time it was given to all students, so we have no idea if the exams are aligned, are grade level, are well written etc. Again, we need to be specific in the response, this is why gatesrheemindtrustersetal are winning–they have a set of talking points and stick to them. They are lies, but repeated often enough they become accepted and are used to frame the issue and discussion.
I did get a kick out of the algebra problem. In my day this was asked (what is the shortest route Johnny could take between home and the grocery (or baseball field). The better question, given their premise of broadcast range–thrown in as something to confuse, something not of importance–something the student needs to be able to recognize as not part of the problem–would have been to include that info and ask if any of the given points (lets call them lat and longitude) were outside the range of the broadcast. And no gps devices or calculators are allowed. (on a canoe trip to Nunavut, Canada and the Northwest territories, my buddies brought along their gps devices with pre-marked coordinates of various important sight seeing things (the cabin where the body of poor young christian was found with his desperate diary and his two frozen buddies, or the rock cairn where travellers leave messages, or ancient graves/campsites or hunting blinds). I had done the readings as well, and made crude marks on my paper maps. As we paddled along, I would see them trying to get a reading on their devices (instead of looking at the wonder of nature) and then listen as they all tried to figure out why their devices did not agree with each other or their pre-marked coordinates. Each morning I would say that in a certain time that day we should look along the left or right side of the bank for a certain landmark ( a river/stream entering, or a slight waterfall, or a stand of trees, or the bear that looks like a rock) and that once we saw that we should park the boats and hike inland to find our site–I would then put the map away and listen to them argue about coordinates. I used my algebra stuff to learn to read maps and figure direct lines and such, no gps, calculator, or even for that matter, much algebra was required–the same way us carpenters use algebra and geometry every day to measure, cut, and stack wood into long lasting structures (rule of thumb, if we have not lost them yet to the blades). Hint–those devices rely upon satellites and the magnetic north–which are not always available and it changes/moves around. Our 1958 Beaver pilot had a small cb radio, a gps device on his dashboard and his planes system. But what he used was a map and he kept looking at the river and certain landmarks to guide us back to Yellowknife, NWT, Canada.
Note to math teachers: sure you emphasize how important all those equations and strategies are, but as someone with an amazing amount of high end graduate and post graduate stastistics and other maths learning, I find I seldom have a need for it, what I do need is included in software (SPSS or SAS or even crude attempts included in Excell etc). What I often need is the ability to read technical instructions and manuals, to know what matters and what does not matter to resolve the issue or complete the project. Most of that book learning never gets us to that point. And I need the ability to explain the models and analysis that use these complicated mind numbing equations and assumptions to the populace and their not so bright elected leaders. To find people such as myself who can actually understand how those equations work (or the software–the built in assumptions, constraints and starting points) is a very difficult task as we are in limited supply–(see VAM–only two or three posters on Ravitch’s blogs have ever gotten to the core reason it is a poor model, most start by saying it was developed by someone from agriculture, as though not being an educationalist somehow negates the math–hint: the equations sanders developed and now sells through SAS institute and used by several states to rate teachers uses a circular iteration that defines and estimates and labels a teacher as high or low, good or bad, and then uses that label to determine the impact of that teacher on the data that were used to create the data (yeah, a bit more complex than that, but that is how it works (I think Mercedes S finally provided a good critique wherein she spelled that out). To find those who can put together a cogent, clear and jargon free response is even more difficult.
Again, I do not like the people behind the common core/vouchers/charters/choice, I do not respect them for their obvious lies and greed, I do not respect them nor trust a thing they say given their ability to fund researchers who provide them with the evidence they want (jp greene, peterson from Harvard, the group at stanford (credo?) or any of the other think tankers (friedman foundation, heartland, fordham people, the stuff ravitch used to espouse) who produce pieces that purport to find how great charters/vouchers/choice is at raising student achievement. No wait, that was the old promise. They are good at educating students for far less (friedman said half) of what we waste on public school students–no wait, all the ed privatizers have come crawling back asking for money for transportation, buildings, technology, more more more. They are good at givng parents a choice!! Yeah, that’s the ticket!!
My twenty dollars worth
mathman is rightonman
rheetoric and rheeality are two different things
the math rubber belongs in a tire fire
As a friend and colleague of Jonathan Katz, I suggest you tread a little more lightly before jumping to conclusions. Also, speak for yourself. You are not Alan or Jo, and shouldn’t presume to speak for them. You know them? Super. So do I. And I don’t know how either would react to Jonathan’s article. But accusing someone of “grossly underestimat[ing] kids’ ability to grapple with complex mathematical idea [sic]” is a kind of dog whistle tactic I don’t find appropriate, particularly when speaking about the career and work of someone you don’t know well enough to make such leaps about.
But not satisfied to just hint, you then resort to the second-best buzzword in the lexicon: “elitism.” To accuse Jonathan of that smacks of incredible chutzpah on your part.
Do I sound a bit pissed? You bet I am. I know Jonathan’s dedication to working with poor, underserved students throughout New York City and the greater metropolitan area. I know how deeply he feels about the testing madness that has been destroying the souls of teachers, students, and parents in New York under the Cuomo/Bloomberg regimes.
As David Mamet has written, don’t crack wise when you don’t know the shot. And it is clear that when it comes to Jonathan Katz as a teacher, coach, and human being, you haven’t the vaguest idea what the shot is. It’s one thing to question what someone’s intent is, and quite another to start slinging about words like “elitism.” Better to ask questions than to shoot your mouth off in a very personal way as you’ve done.
Well said, Michael. I do not know Jonathan, but given his post and concerns, I would love to. The accusations that he a) underestimates kids and b) is elitist are both entirely uncalled for.
Obviously, I agree, Bob.
No conclusions jumped to about Jonathan Katz. I was just posing a question and voicing what was for me cognitive dissonance about people he was leveraging to make his arguments and to raise concerns about possibly lowering expectations for kids.
I will ignore your personal attacks, because pissed or not, they are not productive to this discussion.
You wrote in part: “I will ignore your personal attacks, because pissed or not, they are not productive to this discussion.”
Maybe you should have thought of that before you accused Jonathan Katz of elitism and underestimating kids’ abilities in a way that certainly skirted the edges of simply calling him a racist (“soft bigotry of low expectations, anyone?) You could have raised reasonable questions without getting personal, and any honest reading of what you wrote would recognize that you were irresponsible in what you did say. How you can take umbrage at what I wrote while justifying your own comments as anything but dirty pool eludes me, Illona.
The classy move would be to retract the unfounded accusations and apologize to Jonathan. Absent that, my comments about what you wrote and intended stand. If that means we won’t be on one another’s holiday card list, I can live with that. Jonathan’s one of the people whose integrity as a professional and as a person I hold in the highest regard. I don’t know you well enough to give you a pass, not that you care. And I still find it terribly presumptuous of you to try to speak through Jo Boaler and Alan Schoenfeld in attacking Jonathan Katz. Don’t you think you carry enough weight to fight your own battles?
We obviously work from different metaphors.
I thought I was having a discussion, not fighting a battle. Too bad because you obviously have insights to share that you feel strongly about, but none of that sticks amid all the combativeness.
I wish you all the best.
If I were to write something akin to what you did about a friend and colleague, I couldn’t exactly act surprised if you replied with ire. If this were simply a disagreement in the abstract, that would be one thing. But you’re speaking about someone I know well and respect deeply. What would YOU do were the shoe on the other foot, Illona?
So let me get this straight. I say that my colleagues are being misused in response to this blog post, and that entitles you to assail me with insults. But when I question some of the premises of the author’s argument, this is cause for said insults? Again, I say that you and I are playing by different rules of civil discourse.
Oh, I see. You were defending the honor of Jo Boaler and Alan Schoenfeld. My mistake. By all means, that gives you the right to accuse someone – utterly absent evidence as far as I can see – of elitism, with, I suspect, a thinly-veiled suggestion of racism and/or classism to boot. Perfectly good rules, and how DARE I retort with anything but the greatest respect. I’m going to send this to Alan and ask for his reaction. Should he agree with you about Jonathan’s essay, I’ll stand corrected and bow to your insight. I expect, however, that should he not agree with you, no adjustment in attitude will be forthcoming from you. Bet?
Right. The battle metaphor again. I’m not simply invoking Jo and Alan’s points of view, I’m “defending their honor.” Sigh.
I probably should just let you go the way of all trolls, but something in your discourse keeps giving me the sense that there may be some substance lurking behind your incessant snark. (I taught teenagers so I am fairly inured to this engagement strategy.) And my same eternal inner teacher holds out hope that if you see that your bark does not bother me, you might decide to actually talk to me like a real person.
Listen, I am going to sign off on this thread now. If Mr. Katz needs an apology from me, please have him contact me and I will clarify to him the intent of my comment. If he is as wonderful as you say he is, I am sure he would be a better person for me to find a resolution with, and he will understand that I was simply trying to push on his thinking as well as one can from the tiny keyboard of a smartphone, not attack his character.
Gotta love your ability to rationalize. Why don’t you reach out to Jonathan yourself, if you’re suddenly concerned that you may have libeled him (which you did)? Of course, for you, it was now just poking at his thinking. Of course, were that my intent with someone, I’d be bloody sure I included him/her in my communications. So another tack of yours that doesn’t quite wash.
I wrote to Alan and asked him to weigh in on your commentary and Jonathan’s essay, publicly or privately. Of course, I provided the appropriate link so he can judge for himself rather than take my version of things as true.
Here’s what I think: you didn’t read Jonathan’s essay thoroughly or carefully. Something set off your “equity” or PC detector and combined with your belief that you can speak for Jo Boaler and/or Alan Schoenfeld, you added 2 + 2 and got about 11. Instead of checking your work, you fired off a snide and thoughtless comment about someone who doesn’t deserve to be attacked, who in fact openly stands for equity and fairness in mathematics education. And when called on it, you decided you could weasel out of responsibility by blaming the messenger. If you had an ounce of decency, you’d admit what you’ve done, apologize, and reread the essay with a lot more care than it seems you gave it the first time around. But clearly that’s not something of which you’re capable. Speaks volumes about your character and your intellectual honesty. I asked you to provide specific evidence to support your comments about Jonathan’s essay and him. You eschewed that suggestion. Is ANYONE here surprised?
I do agree with one thing you wrote, however. You should ignore me. Because you don’t have the chops to own up to your mistake or provide ANY support that suggests you didn’t err. Continuing to obfuscate and moan about my “trolling” is the usual recourse of a coward caught in shooting off his/her mouth thoughtlessly and recklessly. The last thing you can do is actually defend your words.
On the chance that I was allowing my personal relationship with Jonathan Katz to color my reaction to your post, Illona, I just reread his entire essay. I am hard-pressed to find any justification in what he wrote for your comments. I doesn’t follow from your claim about Boaler and Schoenfeld that either of them would disagree with Jonathan’s essay, which, by the way, is focused on very specifically the situation in NY State and NYC, as viewed through the lens of the new exams and the NYC-Engage site. Last I checked, Boaler and Schoenfeld are based in the Bay Area and are unlikely to be familiar or concerned with very recent developments in curriculum and assessment in one state on the other side of the country. So there’s no inconsistency at all, except perhaps in the minds of those who didn’t read the essay very carefully.
Further, Jonathan speaks based on direct experience with a specific subset of students in a specific grade in a specific location. He makes no broad generalizations about “kids’ ability” beyond that. And even a cursory reading of what he wrote makes clear that he advocates for serious, meaningful mathematics for all students. If you can provide some direct evidence to the contrary, please share it. Otherwise, regardless of what you think of me or my choice of rhetorical style, you owe Jonathan an apology. I know you won’t give one, nor can you support your claims about his views, about which Polya would also have had something to say.
I agree with the above poster. No doubt a jump in one year to a meaningful Algebra 1 is a bit much, but the problem is that we should have implemented CCSS from early grades up.
But we should think of the finished product – when we get a curriculum that supports these concepts.
The present Algebra curriculum looks like 1950, still computational (read factoring). That needs to change.
CCSS, or whatever we implement, should be above where we are now, or what’s the purpose.
I have many problems with the origins and implementation of CCSS. But in math, there is the potential for a deeper, richer, more meaningful mathematics. Better then in most programs now
Doesn’t computational algebra come in a bit handy for physics? I have been looking at the Algebra 1 units on the DOE website and I fail to see how starting the year off with piecewise functions enhances understanding. Of course, many small high schools don’t even have physics now, so maybe this is a moot consideration.
Sheila, arithmetic and algebraic computation definitely have an important place. But hat has been pretty much the whole focus of arithmetic and Algebra.
I’ve taught physics, too, and manipulating formulas is important. But math is not there just to serve sciences. It’s a lot more.
The road to failure is paved with good intentions…and “potential”. These top-down mandates are filled with potential, that NEVER gets fulfilled. CCSS will end the same way.
Chris, rhetoric and talking points are the same ruse of the supporters of CCSS. Only he choirs hear them.
We can dump CCSS on tush origins and implementation. Maybe we should. That doesn’t mean we have to discard the content.
Always been a fan of Polya and Heuristics!
Having taught Mathematics for 42 years, there has been some opportunity to use these techniques. With the implication of NCLB, I spend many years trying to prepare students for the standardized test. I taught in a highly rated suburban HS with the majority of students attending college and a few attending Ivies. Yet, I had some 9th grade students that used their fingers to add.and could not divide without a calculator.
Let us suppose that the CCSS in Math is the way to go (which I do not believe). Then the 4th grade standards are based on students mastering the 3rd grade standards and so on! How can we possibly test students on 9th grade standards that have not been “trained” in the 8th, 7th, 6th …, standards?
I am sure we will see some 3rd grade students succeeding on their standards but will their teachers spend 90% of the time on ELA and Math preparation?
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.” Two issues here: One, are students developmentally/cognitantly ready/able to hadle the rigor/level of the standard and the appropriate test question(s) at the grade level tested? Two, are students just not prepared to handle such material? I have read people criticize the standards for being developmentally inappropriate–how can we prove that? If this is the case, then creating more rigorous standards and presenting them at earlier grades will prove fruitless, frustrating the students and teachers and returning low scores. If it is a question of the latter, students could handle the rigor and processes if they had been prepared during earlier grades, then we can expect the current crop of students to continue to fail as they move through the pipeline–but the new students now entering 2nd and third grades should be passing these exams if schools are educating students using materials and processes aligned to the new expectations. While I suspect it is a combination of both but more of the latter, and we are not used to requiring such thinking skills at lower grades (do we have examples of high scoring countries or US schools that set more rigorous standards at lower grades?) we need to be careful to always separate the two issues and offer solutions to both.
Expecting students to suddenly switch paradigms and learn the standards for 8th grade and all of the supporting standards that should have been presented and learned in previous grades during the 6 months from fall till test time is downright stupid. Using these new scores to rate teachers is no less dumb. Setting low expectations for students and their learning abilities is despicable and should be a crime, but we know from the NCLB processes adapted by states that politicians, education policy people, and yes, teachers, that low expectations and low cut scores (passing scores) were implemented and praised as high (and a long history of setting low expectations of minority or low income white rural kids has been well documented in our schools, so we need to guard against that coninuing to happen). Common core, an attempt to level the playing field across all states, schools and students, is a valuable idea and goal, but we have choked on the process and implementation of the core.
My ten cents. AKLA is an Inuit word for the great one (grizzly bear). Once you have messed with one of them, taking on ed policy and reformy types is a cinch 🙂 , just much more disagreeable since the latter are not honorable, depend too much upon money, and they lie.
Reformers pretend that all children ‘are above average’, just like all men are good looking? Sure!?
Only truth is: all women are strong! Fact!!
If our children are not learning and not taught at the appropriate levels, at grade level..below or above, according to their needs, then we end up with the Huge Mess we have now.
IS THERE TRULY NOTHING TO LEARN AT THE APPROPRIATE LEVELS?
Must we constantly push? We have been doing this for years and we end up re-teaching and filling in non-mastered skills year after year. High school kids could spit – many tell you they’ve had this same stuff many times before,but still do not know it. They check out and close the learning door on many skills, especially in Math.
We talk a good game about depth of skills, mastery & deep curriculum and not miles wide. Along comes CCSS and now we push everything from pleasure reading, disjointed splinter skills & CrapTestPrep for weeks.
We are in a tsunami of ignorance and all we can do is tread water and hope to reach shore. How can anyone or should we make sense of this insanity?
I know that I can’t and won’t!
“How can anyone or should we make sense of this insanity?”
To answer the last part of the question first: YES, at least to expose the insanity for what it is-INSANITY! We would save an amazing amount of time, effort, resources, etc. . . if we would only open up our eyes, minds and hearts to what Noel Wilson has already done to “make sense of this insanity”, proving how educational standards and standardized testing as educational malpractices are completely invalid due to the thirteen epistemological and ontological errors (and there are more) he has identified in his never refuted nor rebutted study “Educational Standards and the Problem of Error” found at:
http://epaa.asu.edu/ojs/article/view/577/700
Brief outline of Wilson’s “Educational Standards and the Problem of Error” and some comments of mine. (updated 6/24/13 per Wilson email)
1. A quality cannot be quantified. Quantity is a sub-category of quality. It is illogical to judge/assess a whole category by only a part (sub-category) of the whole. The assessment is, by definition, lacking in the sense that “assessments are always of multidimensional qualities. To quantify them as one dimensional quantities (numbers or grades) is to perpetuate a fundamental logical error” (per Wilson). The teaching and learning process falls in the logical realm of aesthetics/qualities of human interactions. In attempting to quantify educational standards and standardized testing we are lacking much information about said interactions.
2. A major epistemological mistake is that we attach, with great importance, the “score” of the student, not only onto the student but also, by extension, the teacher, school and district. Any description of a testing event is only a description of an interaction, that of the student and the testing device at a given time and place. The only correct logical thing that we can attempt to do is to describe that interaction (how accurately or not is a whole other story). That description cannot, by logical thought, be “assigned/attached” to the student as it cannot be a description of the student but the interaction. And this error is probably one of the most egregious “errors” that occur with standardized testing (and even the “grading” of students by a teacher).
3. Wilson identifies four “frames of reference” each with distinct assumptions (epistemological basis) about the assessment process from which the “assessor” views the interactions of the teaching and learning process: the Judge (think college professor who “knows” the students capabilities and grades them accordingly), the General Frame-think standardized testing that claims to have a “scientific” basis, the Specific Frame-think of learning by objective like computer based learning, getting a correct answer before moving on to the next screen, and the Responsive Frame-think of an apprenticeship in a trade or a medical residency program where the learner interacts with the “teacher” with constant feedback. Each category has its own sources of error and more error in the process is caused when the assessor confuses and conflates the categories.
4. Wilson elucidates the notion of “error”: “Error is predicated on a notion of perfection; to allocate error is to imply what is without error; to know error it is necessary to determine what is true. And what is true is determined by what we define as true, theoretically by the assumptions of our epistemology, practically by the events and non-events, the discourses and silences, the world of surfaces and their interactions and interpretations; in short, the practices that permeate the field. . . Error is the uncertainty dimension of the statement; error is the band within which chaos reigns, in which anything can happen. Error comprises all of those eventful circumstances which make the assessment statement less than perfectly precise, the measure less than perfectly accurate, the rank order less than perfectly stable, the standard and its measurement less than absolute, and the communication of its truth less than impeccable.”
In other word all the logical errors involved in the process render any conclusions invalid.
5. The test makers/psychometricians, through all sorts of mathematical machinations attempt to “prove” that these tests (based on standards) are valid-errorless or supposedly at least with minimal error [they aren’t]. Wilson turns the concept of validity on its head and focuses on just how invalid the machinations and the test and results are. He is an advocate for the test taker not the test maker. In doing so he identifies thirteen sources of “error”, any one of which renders the test making/giving/disseminating of results invalid. As a basic logical premise is that once something is shown to be invalid it is just that, invalid, and no amount of “fudging” by the psychometricians/test makers can alleviate that invalidity.
6. Having shown the invalidity, and therefore the unreliability, of the whole process Wilson concludes, rightly so, that any result/information gleaned from the process is “vain and illusory”. In other words start with an invalidity, end with an invalidity (except by sheer chance every once in a while, like a blind and anosmic squirrel who finds the occasional acorn, a result may be “true”) or to put in more mundane terms crap in-crap out.
7. And so what does this all mean? I’ll let Wilson have the second to last word: “So what does a test measure in our world? It measures what the person with the power to pay for the test says it measures. And the person who sets the test will name the test what the person who pays for the test wants the test to be named.”
In other words it measures “’something’ and we can specify some of the ‘errors’ in that ‘something’ but still don’t know [precisely] what the ‘something’ is.” The whole process harms many students as the social rewards for some are not available to others who “don’t make the grade (sic)” Should American public education have the function of sorting and separating students so that some may receive greater benefits than others, especially considering that the sorting and separating devices, educational standards and standardized testing, are so flawed not only in concept but in execution?
My answer is NO!!!!!
One final note with Wilson channeling Foucault and his concept of subjectivization:
“So the mark [grade/test score] becomes part of the story about yourself and with sufficient repetitions becomes true: true because those who know, those in authority, say it is true; true because the society in which you live legitimates this authority; true because your cultural habitus makes it difficult for you to perceive, conceive and integrate those aspects of your experience that contradict the story; true because in acting out your story, which now includes the mark and its meaning, the social truth that created it is confirmed; true because if your mark is high you are consistently rewarded, so that your voice becomes a voice of authority in the power-knowledge discourses that reproduce the structure that helped to produce you; true because if your mark is low your voice becomes muted and confirms your lower position in the social hierarchy; true finally because that success or failure confirms that mark that implicitly predicted the now self-evident consequences. And so the circle is complete.”
In other words students “internalize” what those “marks” (grades/test scores) mean, and since the vast majority of the students have not developed the mental skills to counteract what the “authorities” say, they accept as “natural and normal” that “story/description” of them. Although paradoxical in a sense, the “I’m an “A” student” is almost as harmful as “I’m an ‘F’ student” in hindering students becoming independent, critical and free thinkers. And having independent, critical and free thinkers is a threat to the current socio-economic structure of society.
I am aware that there is a lot more to math than computation, but I do think this will be appreciated by relatively few students. It is not fair to the majority to shut them out, something I fear will happen. My daughter is decent at math and is perfectly capable of finishing a traditional Algebra I and II sequence and then moving on in her real interests (singer/ songwriter). Should she fail an Algebra regents designed to inspire the mathematically gifted?
Computation is NOT math. It’s record keeping.
So, my son is not using math to complete his honors physics course?
This gem is from Engage NY’s Algebra 1 module, Lesson 1:
PIECEWISE-DEFINED LINEAR FUNCTION: Given non-overlapping intervals on the real number line, a (real) piecewise
linear function is a function from the union of the intervals on the real number line that is defined by (possibly different)
linear functions on each interval.
Lesson 1!
The world needs record keepers, too. Will they not be able to get high school diplomas now? My husband has a degree from Yale and a very high IQ and even he topped out in math and realized that he was not meant to be a mathematician. My daughter may not be destined to be a mathematician but I’d rather she not check out during the first lesson of Algebra 1.
Computation in physics, unlike math class, is a means to an end. That end usually means comparative analysis or as a predictor. Felix Baumgartner did not have to guess how long he would need to be in free-fall in order to break the sound barrier (340 m/s). It was all calculated with physics formulas, well in advance, and with near perfect accuracy and precision.Unfortunately in most math classes the computed answer, completely bare of a meaningful unit label, is the end of the line. Which is why most student dislike math. They see it just as an almost completely pointless exercise; tricks with numbers. And when convoluted applications (e,g, a function to count green tea mints ) are presented to students it only makes matters worse: a beautiful analytical and predictive tool is twisted and distorted into an unrecognizable form.
But of course, the answer to your question is and emphatic “No.” This one-size-fits-all, one-track-for-all crap from the Ed Deformers is stupid and mean-spirited and uncalled for. A complex, diverse, pluralistic society needs schools that recognize and develop the unique talents of kids, not ones that treat them as parts to be identically milled.
Sorry about my hasty post, Sheila. I agree with you. It’s ridiculous for your daughter to be treated like a machine part that must be identical to all the other machine parts.
The “passing” scaled score on the “old” Integrated Algebra Regents test was 30 points out of a possible 85 points (raw score of 35%). So if the “easy” test has stdents earning a 65% by answering with a 34% accuracy, what could possibly happen with the new and riduclous CCSS algebra 9 scores?
Will NYSED dare disclose the cut score for passing this demonstrably more challenging exam? With HS graduation on the line in this truly high stakes exam they will be between a rock and a hard place. And their plan to increase the minimum passing grade to a scaled score of 75% will put them in true legal danger.
This makes a proposed moratorium seem like a reasonably sensible, proactive decision.
Should be a no-brainer for the BOR as many member seem tove lost their executive function.
Children differ. The bullet list of standards is meant to be invariant.
That’s a problem.
One of the ways in which children differ is in the extent to which they are, at early ages, capable of doing certain kinds of very abstract reasoning. Longitudinal FMRI studies of temporal lobe areas of the brain involved in abstract reasoning conducted by the Johns Hopkins Medical School show that significant portions of these parts of the brain do not start to develop in most children until around the age of 16 and are not fully developed until around the age of 25.
The situation, there, is not a simple one. Neural machinery for the functional equivalents of SOME very sophisticated processes (such as setting syntactic parameters) exist even before birth.
Children differ. They are on differing developmental schedules throughout childhood and young adulthood. In mathematics instruction, we have for years attempted to get kids to do increasingly more explicit abstract generalization and conceptualization at increasingly younger ages. For example, I’ve seen a lot of CC$$-inspired material that attempts to get third graders to “understand” the “concept of the variable,” by which is meant to be able to articulate the concept explicitly. And it may well be there (I suspect that it is) that for most kids, the neural machinery for that sort of explicit, metacognitive formulation simply isn’t at place that young. Again, major parts of the brain that do explicit abstract reasoning are not in place in most people until they are in their late teens to mid twenties.
We know, now, that the brain is quite plastic. In his superb Intelligence and How to Get It, Richard Nisbett reports on studies that showed dramatic increases (increases on the order of one and a half standard deviations!) in fluid intelligence scores resulting from computerized practice of pattern recognition activities.
I suspect that for most kids if, instead of attempting to get them to “understand” very abstract mathematical concepts at early ages, we instead engaged them when they were young in a lot of structured play to develop prerequisite fluid intelligence and delayed formal mathematics instruction per se until much later (beginning, say, around the age of 16), we would make more progress in three years than we now do in 12. Here’s the kind of thing we might be doing instead:
http://search.yahoo.com/search?fr=mcafee&type=A110US0&p=vi+hart+doodling+in+math+class
But, again, kids differ. There are going to be some little Eulers and Ramanujans who show up in our first-grade classes, and there need to be tracks for them, too.
The one-size-fits-all approach assumed by a single set of “standards” is entirely inappropriate. Kids are not uniform materials, like sheet metal, that we can apply invariant standards to in order to produce invariant products.
And, though there is enormous debate about the CC$$ in math, these standards basically tweak the previously existing state standards, which were themselves largely based on the previously existing NCTM standards, so there is not that much difference in the approach that they instantiate, and one cannot expect to get dramatically different outcomes by taking essentially the same approach.
A few years ago, a NEA study showed that 63 percent of adult Americans could not calculate a ten percent tip, even though all they had to do was move the decimal place!!!! Given outcomes like that, doing more of the same doesn’t make a lot of sense, but the CC$$ authors did not think through radical alternatives.
And that’s exactly what they should have done.
See Lockhart’s wonderful “A Mathematician’s Lament”:
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCcQFjAA&url=http%3A%2F%2Fwww.maa.org%2Fsites%2Fdefault%2Ffiles%2Fpdf%2Fdevlin%2FLockhartsLament.pdf&ei=Z0U8U72CBNbLsQTVsYHwAg&usg=AFQjCNFGDSuprzF62frJ9aW3y61xxH-w3A&sig2=zged8-piMVsNvyWxsoE18A&bvm=bv.63934634,d.cWc
Sorry, I included the wrong link.
I suspect that for most kids if, instead of attempting to get them to “understand” very abstract mathematical concepts at early ages, we instead engaged them when they were young in a lot of structured play to develop prerequisite fluid intelligence and delayed formal mathematics instruction per se until much later (beginning, say, around the age of 16), we would make more progress in three years than we now do in 12. Here’s the kind of thing we might be doing instead:
CX: I meant prefrontal cortex, not temporal lobes, in the post above.
We are always teaching, with everything we do in school, the love or hate of a subject. Most graduates of our schools loathe and fear mathematics.
That’s what they have been taught.
Poll adults you know about this. Most editors of trade books for U.S. adults are very, very wary of including ANY mathematical formulas in a book. It’s a truism among them that even a whiff of math will kill sales.
Does anyone really think that doing a little more of the same, with a few tweaks, is going to change all that?
That’s what they have been taught not by their well-meaning teachers but by a system of the kind that Lockhart describes in his “Mathematician’s Lament.” A system so designed that that will be its inevitable outcome for most.
HERE’S A 9TH GRADE MATH PROBLEM FROM THE ASPIRE WEBSITE
>A map of Nelson county is laid out in the standard (X, Y) coordinate plane below, where the center of the county is at (0,0). A cell phone tower is at (5,4), and Esteban’s house is at (10,–2). Each coordinate unit represents 1 mile. The tower’s signal range is 10 miles in all directions.15) The strength of the tower’s signal to Esteban’s house depends on the straight-line distance between his house and the tower. What is the straight-line distance, in miles, between Esteban’s house and the tower?<
This typifies the convoluted approach to mathematics seen during the CCSS era. the Duncan revolution if you will.
They start with the math concept (coordinate geometry) and then pretend to search for a real world application to match.
What they find is an out of context example that no one in the real world would ever use.
For locations on a map we actually have an existing coordinate system: LATITUDE AND LONGITUDE.
For finding the straight line distance in miles between two locations we have MAP SCALES AND RULERS.
Now the icing on the CCSS cake of distortion and deception:
Reminder, the test item reads:
“What is the straight-line distance, in miles, between Esteban’s house and the tower?
the correct answer to is . . .
e) [the square root of] 61
no joke.
Many’s the time I needed to drive precisely sqr(61) miles.
More seriously, I don’t think the phenomenon you describe is peculiar to math in the Common Core era, but rather reflects phony real world context problems that have been injected into textbooks for at LEAST as long as publishers were trying to align with aspects of the NCTM standards. Dan Meyer has been examining this phenomenon for several years now on his blog, and there have been some fascinating conversations there among a wide variety of people. Pseudocontext is a term he uses, one I believe he borrowed from his dissertation director, Jo Boaler.
I think a parallel set of problems (in the pejorative sense of the word) arises that might be called pseudorelevance and pseudoengagement. When publishers try to decide what will be “cool” for kids, they invariably come off looking stupid. Things change far too quickly in youth culture. I go back to Marion Brady’s “reality-based learning” ideas as laid out on his website: http://www.marionbrady.com/
7.810249676 miles to the nearest gas station?
I
How did you guess?
That’s as the pigeon flies, in NYC.
Great response, NY! LOL!
And in the NYT today:
http://parenting.blogs.nytimes.com/2014/04/03/research-on-children-and-math-underestimated-and-unchallenged/?emc=edit_tnt_20140403&nlid=31963390&tntemail0=y
Peter, at the risk of drawing further comments about my arrogance: you appear to be suggesting that a) if only we didn’t think like Jonathan Katz, we’d be challenging kindergarten kids to do more challenging math. And then 9th graders would not only be algebra-ready, they’d have already mastered the subject by 8th grade and be doing higher math by the time they graduated.
There are a lot of problems with what seems to be a logical connection between Ms. Paul’s piece and what Jonathan wrote. First, no one, least of all people like Jonathan Katz, would suggest that we’re doing a stellar job of teaching primary grade mathematics on a national basis. However, it’s important to keep in mind, particularly if you’re going to use something like PISA scores to evaluate how we’re doing here, that such tests are flawed. (see for example http://www.tes.co.uk/article.aspx?storycode=6344672), and that they often are not comparing similar populations of students because of questionable sampling practices in some of the participating countries. The US tests everyone (that is, samples from a broad spectrum of students); other countries often do not. We could get top-notch results if we restricted ourselves to testing kids from affluent suburban schools and leaving out schools in communities crippled by poverty.
But even taking PISA or TIMSS results uncritically, you have to ask what has kept primary mathematics teaching at such a low level? Pretending for a moment that we need not concern ourselves with children who come to school having suffered from the debilitating effects of poverty (malnutrition, exposure to drugs, disease, violence, ad nauseam, lack of supportive home life, and so forth), there’s the simple truth that a significant number of K-5 teachers are weak in basic math. We could exhort them to “raise the bar” and to teach “more challenging content” all we like, but if the math is over their heads, how are they expected to teach it to young children, exactly?
People who work at the 6-12 or 9-12 level don’t get to teach the kids they wish they had or fantasize that they would have if THEY had taught them in the lower grades. They have to deal with who those students are when then enter the classroom and try to address many serious gaps in their knowledge of mathematics. It’s enormously challenging and frustrating. No one has yet come up with a magic formula for bringing students up to speed when they may be more than three years below grade level AND teaching the content that is at the current grade level without leaving a significant number of students completely in the dust.
Further, it’s unclear just what to say about developmental appropriateness (DA). The current debate of Common Core math partially hinges on widespread complaints from parents and teachers about lack of consideration for DA in the primary grade standards and in the materials showing up in classrooms (not necessarily the same thing).
Many mathematics educators have argued for decades that we spend far too much time in K-5 mathematics stressing rote learning, memorized computational algorithms, and, in general, donkey arithmetic that could be done far more quickly and accurately by calculator or computer software. Stray a few angstroms from that traditional emphasis and parents, teachers, administrators, and politicians start screaming bloody blue murder. So even if you have elementary teachers who are ready, willing, and able to teach more advanced math ideas to kids, they may find that such attempts are very unwelcome to a significant number of parents.
It’s not the Jonathan Katz’s who resist teaching more challenging and meaningful math to our young kids, but a moribund curriculum and tradition, supported by parents and educators who fear change or who don’t see the point of teaching problem-solving and more authentic mathematical thinking to kids. And at the same time, you have folks like R. James Milgram of UC-Berkeley’s math dept. attacking the NCTM Standards and, now, the Common Core Standards as being too fuzzy, not challenging enough, not world class, not competitive with Asia, and blah, blah, blah.
I’m sorry, but from what I have seen in 25+ years of work in K-16 math education in this country, the nation as a whole has a schizoid view of math, doesn’t really know what the subject is about past the level of computation, and wants two very contradictory things simultaneously: that the bar for math be set higher so kids can compete with Singaporeans and other Asians on international tests, and the bar not set higher so that kids aren’t left in tears every day. Any reasonable teacher who actually has the necessary mathematical and pedagogical chops is going to be very frustrated, unless s/he’s lucky enough to teach at a place like St. Anne’s in Brooklyn, where Paul Lockhart is able to do challenging math for all his students.
The things needing to be in place for a real revolution in US math education are nowhere near to existing at this point. Stronger teachers at the K-2 level, better-educated parents (in the sense of understanding more about what is and isn’t math and what it means to learn, know, and do authentic math), and a national commitment to change a system that is more harmful than helpful to a majority of students. All of that must be embedded in a willingness to address social and economic ills that are beyond the scope of schools and teachers to cure through what can be done in a classroom. And all of this done in an honest way that doesn’t try to lay the blame for an ethically bankrupt socio-economic system at the feet of teachers, schools, or those trapped at the bottom end of the spectrum.
Of course, I guess it’s awfully arrogant of me to say any of this. I’m merely a mathematics educator, coach, teacher, teacher-educator who is also a parent and whose work for the last quarter century has been almost exclusively with students, teachers, and schools in districts of abject poverty. I don’t know what gets into me sometimes.
Pretty vague reporting.
What specific “sophisticated” math concepts? Did the mean anything beyond counting and recognizing/writing numbers?
What “enaging and creative” activities?
Is the NYT ever not vague? Or maybe hey were beyond the reporter’s grasp.
But I get the impression they were asking kids to do more and deeper thinking than sometimes we ask them to do.
Read the post……reads like astrology….
Huh?
Amen. Teach the teachers how to use the new standards and give up the idea of individual accountability on examinations that do not incorporate the Mathematical Practices. A valid test of those practices would be very expensive to deliver to every student in America. Test the program with samples. Eliminate the individual score and VAM for teachers.