tag:blogger.com,1999:blog-6570061087276796800.comments2015-11-24T10:13:40.126-05:00Out In Left FieldKatharine Bealshttp://www.blogger.com/profile/02838879769628392605noreply@blogger.comBlogger4478125tag:blogger.com,1999:blog-6570061087276796800.post-22948355885667569092015-11-24T10:13:40.126-05:002015-11-24T10:13:40.126-05:00Some level of explanation should occur in class, a...Some level of explanation should occur in class, and orally. Written explanations should be no more than some descriptors on showing one's work. And for the record, <a href="http://oilf.blogspot.com/2013/04/i-sneak-in-back-window-and-teach-how-to.html" rel="nofollow">Chapter 14 of "Letters from Huck Finn"</a> does recommend that some explanation be explicitly taught:<br /><br />"Of course, the students would not know that there are people who view those who can't "explain their reasoning" (however correctly they solve a complex problem) to be doing "rote work" and lacking "understanding." But it seems to me that if we really want students to do such explaining, then we should tell them how. Simply telling students to "explain your reasoning and attend to precision" is not likely to accomplish much. Knowing how to explain something precisely doesn't come automatically with understanding. And students are not likely to pick it up by themselves working in groups and the like."Barry Garelickhttp://www.blogger.com/profile/01281266848110087415noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-38034742560889974562015-11-23T19:33:53.295-05:002015-11-23T19:33:53.295-05:00Educrats are just so clueless. They're consta...Educrats are just so clueless. They're constantly underestimating how difficult things are.<br /><br />"Kids learning to read? No problem! We'll just put them in a text-rich environment and they'll teach themselves!"<br /><br />"Kids stressed out, with mental and emotional health issues? No problem! We'll sit them in a circle and have them talk about their feelings!"<br /><br />They seem to live in a much simpler world than the rest of us.FedUpMomhttp://www.blogger.com/profile/00951858601020687242noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-83126874029076492292015-11-23T16:52:36.501-05:002015-11-23T16:52:36.501-05:00With my experience as a substitute I can tell you ...With my experience as a substitute I can tell you point blank it is a waste of time. Not only does it take time away from actual teaching of subjects, it does not change the emotional outlook of the students. There will always be poorly behaved students disrupting the class. This has more to do with unstable home environments and nonexistent disciplinary policies. If anything this kumbaya fad makes things worse because it is quite dull and meaningless. <br /><br />Bookish BabeAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-36177055860121232672015-11-22T10:06:56.921-05:002015-11-22T10:06:56.921-05:00I had well-taught, traditional math in my small-to...I had well-taught, traditional math in my small-town 1-12 school, in the 50s-60s. Most of the town was poor, but all kids came from respectable, intact, two-parent homes (widowhood aside)and were all well-socialized at school entry (no pre-k-k available). Of HS classes of 30-36, only 3-4 went to a 4-yr college and there probably weren't 10 adults with college degrees in town. At the end of 8th grade, however, everyone was decently literate, numerate, could write correctly and had decent general knowledge. ES teachers taught science, history (including art and music hx/appreciation), geography and civics. In no way was this a high-IQ or highly-educated population. The ability to use math for everyday purposes (this was before calculators) came strictly from school (perhaps some flashcard math facts at home, but only a few kids). We were taught well enough that we could calculate tax, interest, gas mileage, amount of materials needed for household projects and the like. Traditional math worked for all.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-88205467929834934032015-11-21T20:04:41.457-05:002015-11-21T20:04:41.457-05:00First sentence should have read: "I agree wit...First sentence should have read: "I agree with that statement and add that some also seem..."Barry Garelickhttp://www.blogger.com/profile/01281266848110087415noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-28491163703923827852015-11-21T17:52:38.866-05:002015-11-21T17:52:38.866-05:00"What is the basis for this argument? I fear ...<i>"What is the basis for this argument? I fear it is as simple a logical error as causation reversal: Some seem to believe that open-ended instruction (etc) CAUSES students to be advanced."</i><br /><br />I agree with that statement and add that also seem to believe that traditional math worked only for a small group of students who happened to be advanced. I.e., the advanced nature of the student CAUSED traditional math to work. It failed for everyone else. The logic of this breaks down when one stops to define "advanced" and takes a close look at the other factors at work with traditional math. E.g., did the teachers teach it poorly or well? Of those for whom traditional math worked, what was the breakdown of IQ's and "advanced" nature of these students. For many if not most of the truly "advanced" students, the factual and procedural foundation for their success was predicated on their obtaining that foundation through the traditional teaching of math. <br /><br />In the various online discussions of traditional vs reform math no one ever bothers to look at what is happening with the students who manage to make it through to HS calculus and major in a STEM field. For many of these students, they do a lot of practice and drills and memorization, either at home, or at a learning center, if they're not getting that through school. One has only to look at the Asian countries to see that Jukus and similar organizations are providing foundational skills to make these students excel at what appear to be inquiry-based assignments at school.Barry Garelicknoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-49316366557500006332015-11-21T16:30:35.406-05:002015-11-21T16:30:35.406-05:00I'm not sure why that commenter wants to cherr...I'm not sure why that commenter wants to cherry-pick PISA scores rather than all of the various other measures employed in the Quebec study, but the raw scores are not relevant anyway since you have to control for various student characteristics that may change between provinces over the course of the intervention. (Especially relevant here because the intervention took place over so many years.) And indeed the authors do just that in their DID/CIC methods. I will grant that the econometrics involved are complicated but these particular "compare to the rest of Canada!" and "PISA!" objections seem to be based in misunderstandings of the authors' methods and the right way to use test scores. <br /><br />If only fans of reform math were this concerned with rigorous controls and falsification exercises when considering their preferred education research.Paul Brunohttp://twitter.com/mrpabrunonoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-67893558045767404702015-11-21T16:02:57.707-05:002015-11-21T16:02:57.707-05:00The edworld has been assigning causation wrongly f...The edworld has been assigning causation wrongly for decades; self-esteem, Latin, 8th-grade algebra (when it was only honors), foreign languages, honors and APs, debate, music etc. When it was found that kids who had high self-esteem were very successful, the edworld immediately jumped to the conclusion that the former caused the latter. When data showed that kids who took Latin, 8th-grade algebra etc. did better on a variety of measures (graduation, college etc), they jumped on the Latin-for-all (a local MS did this) etc. as a causative factor for success. In fact, those courses merely served as a proxy variable for the identification of the most able, prepared and motivated students. Inability to recognize that has lead to placement of kids into courses where they lack the background knowledge and are unable to do the work. No, the same approach does not work for all students. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-12909036654239095332015-11-21T15:02:58.449-05:002015-11-21T15:02:58.449-05:00Hi Katherine. I liked your response to #7 especia...Hi Katherine. I liked your response to #7 especially. You make a very good point. Cognitive scientists speak of a well-established pattern called the Expertise Reversal Effect. Essentially what it says is that discovery-based teaching/learning is ineffective with novices but effective with experts. That's why, for example, we tell PhD students "Here's your subject matter and the question you must solve. Good luck because nobody's ever solved it before. Go dig up all the relevant references and become and expert. Then consider how you plan to attack the problem and come back to me and we'll discuss whether you are likely to make a successful thesis out of this." That's discovery learning of a sort that blows Boaler, Meyers, Mitra etc out of the whole discussion. Even when they imagine themselves leading such a thing they have no realistic idea of how to bring it to fruition. But we do it all the time with PhD candidates, and they are SUCCESSFUL. Why? Because you don't even get INTO the PhD program without establishing your expertise. Then the first thing we do is put you through a barrage of Comprehensive exams to test your mastery of that knowledge. Then, and ONLY THEN, do we say, "okay, now lets get started on your thesis work." (Well, I'm obviously generalizing and glossing. But this is the essential principal of the matter as pertains to our success rate in producing PhDs)<br /><br />When my children were in an elementary school in Fresno California they were put in the pull-out GATE (Gifted and Talented Education) program. There they did more open-ended exploration. To me that's a no-brainer. You do that with kids who have mastered the "canon" and are ready to build upon it. Nevertheless every one of those kids still sat in regular classes and did the timed drills etc with the rest. Had they not done so, and their skills fell behind, their GATE experience would have been a millstone around their educational necks.<br /><br />Now I'm no rah-rah "my kids are better than yours" elitist on these matters and that's not why I bring this up. It is that I am enraged at the tendency for some to argue that because some program dealing with very talented students with a strong background is able to accomplish something with their demographic that somehow this means that it is an ideal way to teach average students. What is the basis for this argument? I fear it is as simple a logical error as causation reversal: Some seem to believe that open-ended instruction (etc) CAUSES students to be advanced. Uh, no. The observation of students having advanced abilities or backgrounds, in contrast, does open the door ("cause" is a bit of a strong word) to these possibilities with them. Causation reversal on this point is cargo cult deception.R. Craigenhttp://www.blogger.com/profile/13025432606771258184noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-27696240685123471612015-11-21T14:02:56.556-05:002015-11-21T14:02:56.556-05:00Brian,
The Weeks and Adkins book can be ordered at...Brian,<br />The Weeks and Adkins book can be ordered at http://batespub.com/geometry.html<br />It has some supplementary materials that has worked answers. My copies are at work, so I do not remember which of the supplements are best. I do not believe there are answers in the back of the book.<br /><br />I will also say that the First Course in Algebra and Second Course in Algebra by Weeks and Adkins are excellent, for anyone looking for Algebra textbooks.Thomas Treloarhttp://www.blogger.com/profile/07442325260358729264noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-56756908885008204272015-11-21T13:34:23.770-05:002015-11-21T13:34:23.770-05:00Hi Paul.
I don't quite get the objection eith...Hi Paul.<br /><br />I don't quite get the objection either. But here is the rest of what the commenter wrote on Dan Meyer's blog:<br /><br />"I find their discussion of the PISA to be particularly problematic. To hear them tell it, Quebec either held steady (which would not support their findings) or declined. However, when you look at the data (http://cmec.ca/Publications/Lists/Publications/Attachments/318/PISA2012_CanadianReport_EN_Web.pdf), specifically Table 1.6 in the link, you notice that PISA scores grew pretty steadily (though by small amounts) over that time, while other provinces declined. That does not build confidence in their case.<br /><br />"In fact, the fact that they fail to consider the performance of other Canadian provinces over the same time is a warning signal on its own. The lack of a control comparison when one is available suggests an unwillingness (or at least a lack of interest) to account for possible historical threats to validity."Katharine Bealshttp://www.blogger.com/profile/02838879769628392605noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-73515321559800600972015-11-21T12:59:53.998-05:002015-11-21T12:59:53.998-05:00In addition to international comparisons being unf...<i>In addition to international comparisons being unfair, a comparison within a province of one country of student performance before and after a student-centered discovery-oriented curriculum was introduced is also unfair. Why? Because it ignores what was going on concurrently in the rest of the country at large. <br /><br />Then what kind of comparison is fair?</i><br /><br />I don't think I understand this objection. In the study of Quebec in question (this one, I think: http://www.sciencedirect.com/science/article/pii/S027277571400034X ), the authors used differences-in-differences and changes-in-changes methods precisely to compare Quebec to the rest of Canada. So they don't "fail to consider the performance of other Canadian provinces over the same time."Paul Brunohttp://twitter.com/mrpabrunonoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-24454500078356004392015-11-20T09:48:59.788-05:002015-11-20T09:48:59.788-05:00Oops, the first part of my comment got truncated. ...Oops, the first part of my comment got truncated. I said that the commenter on dy/Dan's blog referred to in the post, laments that students in algebra classes are given the formula for sum of consecutive numbers without proof, and sniffs at that. But whether one knows the proof or not, solving problem 13 of the FInnish test still would not be any easier knowing the proof. It is a challenging problem that makes use of a student's knowledge of the formula.<br /><br />A similar thing happens in other courses. In a textbook of real analysis for engineers, the definition of "compact set" is given as "a set that is closed and bounded". THis is actually a theorem that derives from the formal definition of compact set which is a set K ... (the rest follows from the above comment).Barry Garelicknoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-12411534808381656522015-11-20T09:45:55.797-05:002015-11-20T09:45:55.797-05:00set K is said to be compact if an only if eery ope... set K is said to be compact if an only if eery open covering of K contains a finite open subcoering of K." <br /><br />Using the definition of compact set as "closed and bounded" will suffice for the less rigorous course of real analysis. Similar things are done in introductory calculus courses; for example, students learn an intuitive definition of limit and continuity, and then later learn the more formal delta-epsilon definition and do proofs of limits and continuity using the more formal definition.<br /><br />Thus, providing algebra students with the formula for sum of consecutive numbers is not egregious; particularly if students are given challenging problems to solve that depend on knowing that definition.<br /><br />I have seen the famous so-called "young Gauss" proof of sum of consecutive integers. It is sometimes pointed to as "evidence" that we are not teaching young students to think mathematically if they cannot come up with it on their own. Barry Garelicknoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-21443169796197369362015-11-19T20:19:20.896-05:002015-11-19T20:19:20.896-05:00Sorry: syntax error! Speaking as a linguist...Sorry: syntax error! Speaking as a <i>linguist</i>...Katharine Bealshttp://www.blogger.com/profile/02838879769628392605noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-23858465962033367962015-11-19T20:00:13.002-05:002015-11-19T20:00:13.002-05:00Speaking as a linguistic with a background in foun...Speaking as a linguistic with a background in foundational logic/model theory, I'd say that symbolic expressions and symbolic manipulations are the syntax of math, and numbers, sets, and graphs the semantics. Both are key. Symbolic expressions and manipulations vs. semantics do, indeed, apply to other fields, including natural languages, where syntax is, again, pretty darn important. One could answer this particular problem symbolically or graphically; graphics aren't necessary. In the math I see mathematicians doing (I'm not one myself, but am closely connected with many), the work is mostly symbolic rather than graphical.<br /><br />I question whether "the move of m and b and related sign switch" would stump a brilliant math student from any country. It certainly doesn't involve anything like what's involved in constructing the kind of multi-step proof that, here in America, use to be much more commonplace.Katharine Bealshttp://www.blogger.com/profile/02838879769628392605noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-86133367007051304312015-11-19T18:11:38.759-05:002015-11-19T18:11:38.759-05:00Your (mental) symbolic manipulation for the PARCC ...Your (mental) symbolic manipulation for the PARCC problem is more than is necessary. It also only establishes a possibility of two solutions (a bit short of full justification, but allowing for verbal explanation with no diagram...). It also almost certainly requires actual symbolic manipulation for most students. The move of m and b and related sign switch isn't a small task, even for the brilliant students of Finland.<br /><br />The problem can be done with even fewer (mental) symbol manipulations than you used, and likely would be by those with an appreciation for the graphical representations of quadratic and linear functions. The quadratic given is a parabola with vertex at (0,0) that opens up. The second equation is a line with positive slope and a negative y-intercept. Distinct real solutions will be represented as intersections. It is possible to have 2 intersections (secant), 1 intersection (tangent) or 0 intersections (line has comparatively low slope), and thus for f(x)=g(x) to have 2,1, or 0 real solutions. Still a bit short of full justification, but this thing begs for the ability to draw a sketch as part of the justification.<br /><br />Item 2 of your Extra Credit is where the vast majority of our disagreement comes from, I believe. The phrase "without involving much math," along with your comments about symbolic manipulation in item 1 suggest a willingness to conflate symbol manipulation with mathematics. But they really aren't the same thing. Symbol manipulation occurs in many other fields than mathematics. Mathematics has aspects that go beyond symbol manipulations. As is typical in many classes, my calculus students are often very good at the symbol manipulations involved in calculating a derivative and such, but they often struggle with the relationship between f, f', and f''. Both of those abilities are necessary to be 'good at calculus', even though the latter can be (and often is) assessed independently of symbol manipulation.Brett Gillandhttp://www.blogger.com/profile/12815909354554401878noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-11306681166262568252015-11-18T10:29:04.570-05:002015-11-18T10:29:04.570-05:00"They can use words, symbols, pictures, table..."They can use words, symbols, pictures, tables, poetry, videos, audio clips, the ShowMe app"<br /><br />I suppose showing them Tom Lehrer's song "New Math" would be taken as snarky?<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-65382151429355025952015-11-18T07:45:55.076-05:002015-11-18T07:45:55.076-05:00This was a comment that stood out in that thread:
...This was a comment that stood out in that thread:<br /><br />I ask my students to tell me the story of how they solved the problem. This prompt invites reflection and detail. I want them to explain their thinking, show me their understanding, and tell me how they solved it. They can use words, symbols, pictures, tables, poetry, videos, audio clips, the ShowMe app. It takes effort and practice, but the result is worth it. I get to know my students in a different way. I know how they communicate and I can push them deeper into mathematical communication and understanding. I learn who they are as learners which helps me in my planning. “Tell me the story of how you solved” is a powerful assessment tool. Try it!<br /><br />Poetry, videos . . . Definitely an efficient use of time. Much better then showing your work.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-24886605778194057232015-11-17T11:03:10.865-05:002015-11-17T11:03:10.865-05:00I'm basically in agreement with you that "...I'm basically in agreement with you that "explaining" trivial math verbally and in pictures is not as valuable as giving challenging problems. But some of your points are are weak.<br /><br />"are mathematicians who learned math in pre-answer-explaining times deficient in their communication skills?" In many cases, yes. Read Halmos's famous essay "How to Write Mathematics", which tries to teach other professional mathematicians how to communicate their math.<br /><br />In doing math, there is a balance needed between manipulating the formal symbols and explaining what the symbols represent. The usual K–8 math class these days goes way overboard on the "explaining" and the 1960s math classes I had as kid did too much formal manipulation. The Art of Problem Solving on-line classes for gifted math students that my son took did a very good job of balancing the symbols and the explanation, improving students' math writing while giving them very challenging problems—problems whose solutions were not obvious and the students benefited from providing some verbal explanation of what they were doing.gasstationwithoutpumpshttps://gasstationwithoutpumps.wordpress.com/noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-38996749671990855112015-11-16T00:35:20.720-05:002015-11-16T00:35:20.720-05:00It seems like two of those subgroups, (the virtues...It seems like two of those subgroups, (the virtues of showing your work, and the virtues of doing math proofs,) are quickly convinced that explaining your math Common-Core style is bad.<br /><br />It seems like two of those subgroups, (the specific examples, and what Common Core actually says), also agree that explaining your math in essays is bad.<br /><br />There are only three subgroups that seem to think this is a good idea: the communication skills necessary for math professions, the counter-example anecdotes, and the metacognition/faith group.<br /><br />So it sounds like many of the supporters of "explaining your work" don't actually support it.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-22509778452730010462015-11-15T18:25:24.065-05:002015-11-15T18:25:24.065-05:00Regarding the issue that CC doesn't require ex...Regarding the issue that CC doesn't require explanations, etc. We quote something that appears on the CC website, that calls for students to justify their answers. Website is not the standards, true, but still... Then, the content standards themselves contain the phrases "students shall explain.." and "student shall understand that..." . The intro to CC standards states that whenever "understand" appears in the content standards, to link that with the Standards for Mathematical Practice, the first of which is "Make Sense of Problems and Perservere in Solving Them". That SMP states in part:<br /><br />"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."<br /><br />Such language has served as gasoline thrown on the fire of math reform that has been burning for many years, and which has fetishized conceptual understanding.Barry Garelicknoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-62761618290934374362015-11-15T15:18:11.156-05:002015-11-15T15:18:11.156-05:00Interesting! How many of each? (said the right-br...Interesting! How many of each? (said the right-brained mom...)Mariehttp://www.blogger.com/profile/01918765206178559884noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-73053316630194357072015-11-13T08:47:06.253-05:002015-11-13T08:47:06.253-05:00My son took an Art of Problem solving course onlin...My son took an Art of Problem solving course online. The discussion was live, text based. He found that fantastic as he could drop out and consider ramifications, then read and catch up to the discussion and contribute. In a live class he often has too much too keep track of, and cant participate. Removing the emotions, cutting out nonessential verbage, blocking noise, and not having to break thru the barriers of dominant personalities who retard the discussion made the experience highly effective. It looked a lot more like a work conference than a school lesson, as it was productive for all.lgmnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-66482025879475366572015-11-13T08:40:06.805-05:002015-11-13T08:40:06.805-05:00Recitation is a discussion section.
There doesnt ...Recitation is a discussion section.<br /><br />There doesnt seem to be concensus on best practices in the inclusive environment. One school may ask that everything stated in lecture be on powerpoint, with advance availability, while another may have a different method. Others expect the instructor to rework his presentation based on who is enrolled and what their particular disability is. In public elementary, for ex, my child was taught multiplication using finger techniques, to benefit the included who had insufficient short term memory or no ability to move beyond the concrete...but the courts have decided colleges dont have to drop rigor. <br /><br />My pet peeve is the DE lecturer who is winging it. Cant use cornell notes, cant outline...its just a hodgepodge. One goes to class to find out the info that should have been on the syllabus- what to read and what problem sets along with due dates. An hour at the library would be more productive.lgmnoreply@blogger.com