tag:blogger.com,1999:blog-3784276984960421233.comments2018-11-20T12:54:05.511-08:00Henri's Math Education BlogHenri Picciottohttp://www.blogger.com/profile/06875198126877279937noreply@blogger.comBlogger123125tag:blogger.com,1999:blog-3784276984960421233.post-12293251781140414242018-11-20T12:54:05.511-08:002018-11-20T12:54:05.511-08:00I can see that you're in a frustrating situati...I can see that you're in a frustrating situation. I am guessing that what you're seeing is the result of tracking. Your students were placed in low-expectations classes in high school, where the goal was not understanding, but the memorization of specific "steps". So, to take the example of linear equations: it is in fact impossible to memorize steps for every possible linear equation, so the students are over and over made to work on what can be memorized. I taught high school math for 32 years, and NEVER taught "one-step, two-step" equations. Instead I delayed equation-solving until students had a solid grounding in basic ideas such as like terms and the distributive law. Once you have that foundation, you can solve any linear equation, and don't need to memorize steps.<br /><br />One possible way forward for our profession is for high school math departments to discuss the ideas in NCTM's Catalyzing Change, which aims to help rethink high school math, and end the tracking of students into dead-end classes where what you describe is all too common.<br /><br />Of course, I may be wrong, and your students maybe were not in low-tracked classes. But they're still the victims of a low-expectations, memorization-based approach to learning math.Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-61469066595183753852018-11-20T11:04:21.801-08:002018-11-20T11:04:21.801-08:00Example:
The 1st time "solving 1st-degree equ...Example:<br />The 1st time "solving 1st-degree equations" comes around, students work on one- and two-step equations.<br />The 2nd time "solving 1st-degree equations" spirals around, they work on two-step equations.<br />The 3rd time "solving 1st-degree equations" spirals around, they work on two-step equations.<br />The 4th time "solving 1st-degree equations" spirals around, they work on two-step equations.<br /><br />Rather than:<br />The 1st time "solving 1st-degree equations" comes around, students work on one- and two-step equations.<br />The 2nd time "solving 1st-degree equations" spirals around, students work on equations with variables on both sides.<br />The 3rd time "solving 1st-degree equations" spirals around, students work on equations for which students need to distribute and/or add like terms first.<br />The 4th time "solving 1st-degree equations" spirals around, students work on equations involving all of the above with rational coefficients.<br /><br />I see this within a grade level, and, more importantly between grade levels.<br />The 8th/9th grader is beginning to be proficient when solving two-step equations, maybe with variables on both sides.<br />The 12th grader is proficient when solving two-step equations, maybe with variables on both sides, but has had no opportunity to develop proficiency with needing to distribute and/or add like terms first, nor with rational coefficients.<br /><br />Then when those students arrive at the two-year college I teach at, they end up being placed into our lowest level algebra class because by the end of that course we expect them to be able to solve all of those equations by the end of that semester. Also, the first half of the unit on solving first-degree equations, they don't pay attention because they believe they already know all of the material. Then toward the end of that unit they are surprised to find out the equations they are expected to solve are much more difficult than the ones they are familiar with.<br /><br />I see this over, and over, and over, for almost every topic.<br /><br />Another example:<br />In 3-5 grade students learn to add fractions. The State test specifications say that students whould be able to add two fractions with denominators amoung 1,2,3,4,5,6,8,9, 10, and 12. Students in grades 3-5 never see a fraction with a different denominator, and for the next seven years students don't get any practice adding fractions whose denominators may factor into 2-4 prime factors, including say 7, or 11, or 13. I see just-graduated students who have no idea what is necessary to add two fractions whose denominators are say 35 and 105. (And others who do know, but who also choose to use 35*105 as a common denominator.)<br /><br />etc.<br /><br /><br /><br />.Ms. Hentgeshttps://www.blogger.com/profile/00816175522118444479noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-28065084359724514832018-11-19T16:12:52.503-08:002018-11-19T16:12:52.503-08:00What are you referring to?What are you referring to?Henrihttp://www.mathed.pagenoreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-7906561139779749092018-11-19T14:25:29.536-08:002018-11-19T14:25:29.536-08:00I find that the spirals have become circles with t...I find that the spirals have become circles with the "next" level of the spiral not any deeper than the current one.Ms. Hentgeshttps://www.blogger.com/profile/00816175522118444479noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-52501066104846831372018-11-15T12:08:04.031-08:002018-11-15T12:08:04.031-08:00You're making want to write more about my stru...You're making want to write more about my structure, Henri! So glad I landed on your posts. I think I'm finding it easier to talk about it now! https://lazyocho.com/2018/11/06/i-have-trouble-talking-about-my-teaching/Brianhttp://www.lazyocho.comnoreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-39345832795662672312018-11-11T10:52:12.531-08:002018-11-11T10:52:12.531-08:00More and more university math profs complain that ...More and more university math profs complain that for many if not most students, earlier calculus is counterproductive, as it results in students with a superficial knowledge of calculus, and with weak understanding of the algebra prerequisites. See, for one, this guest post by a professor from Penn:<br />https://blog.mathed.page/2016/02/more-calculus-less-understanding.html<br />Unknownhttps://www.blogger.com/profile/00358216623813307745noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-72041622010170672612018-11-10T19:42:09.355-08:002018-11-10T19:42:09.355-08:00A responder asked, “But what about when parents se...A responder asked, “But what about when parents see taking Calculus early as a method to improve their child’s college admissions chances?”<br /><br />First of all, on some level, improving college admissions chances is behind most parents’ concerns about acceleration. Parents may say that they want their children to “not be bored” or “to be more challenged,” but really, parents want to be able to help their child demonstrate in the college admissions process, that he is better than other students, and parents (possibly subconsciously) think that advocating for accelerated math is the way to do this. <br /><br />Second, if this is truly the parent’s main concern, it is also short-sighted. I’m actually not against putting students in advanced courses in high school. I have done this, many times. But I have done this for students who are exceptionally cognitively mature, and I have purposely waited until a child is 14 or 15 instead of 11 or 12. If a parent needs to overly advocate with me, it is because their child is likely not ready for the acceleration, and should I back down from my stance, I am offering no gift to the student in terms of “college admissions support.” Because our math sequence will just be too hard for this student, the student is likely going to really struggle and get much poorer grades than he might have taking math with his peers. In the end, the parents’ over-advocation ends up hurting the child’s overall mathematical growth, power, skillsets, and confidence, and also ends up hurting his college prospects (that is if we believe that college prospects are directly correlated to GPA). <br />Rachel Chounoreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-56158174649910303122018-10-09T16:05:26.370-07:002018-10-09T16:05:26.370-07:00Wow! That is an advanced use of the participation ...Wow! That is an advanced use of the participation quiz! I'm impressed. Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-65838032319545659322018-10-09T11:39:25.794-07:002018-10-09T11:39:25.794-07:00Just to chime in on this most excellent conversati...Just to chime in on this most excellent conversation, I've done the participation quiz many times and my students and I love it. It sends a clear message that the focus is on doing math, not the "nit-picking accuracy". I've even gone as far as to use the 8 Math Practice Standards as a "rubric" for what I look for in these participation quizzes: persevering in problem solving, construct and communicate viable arguments, reason abstractly and quantitatively, make models, use tools, examine structure, find patterns, be clear and "say what you mean".<br /><br />Of course, we don't focus on all of these every time. We'll usually pick one or two to watch for. But it is neat to see the MATH that goes on.<br /><br />Thanks, Henri, for a great post. And thanks, apm, for the comment that lead to the posting of that link to the assessment post. :DThaslamhttps://www.blogger.com/profile/16890604336141114110noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-78200891674377074792018-10-08T14:45:17.287-07:002018-10-08T14:45:17.287-07:00I appreciate the reminders about lagging homework ...I appreciate the reminders about lagging homework and the importance of extending student exposure. I'm looking forward to your next article about over-spiraling as well as the compilation! Thank you for sharing your work and wisdom.Tammy Lallyhttps://www.blogger.com/profile/17679777447094859338noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-81151116141794773342018-10-07T21:02:01.163-07:002018-10-07T21:02:01.163-07:00Thanks for the compliments!
My sense is that not ...Thanks for the compliments!<br /><br />My sense is that not much depth can be expected from homework, and that most significant learning happens at school. Thus for me homework was short, and its main purpose was to trigger good conversations in class the next day, as groups "went over" it. I wrote about that here:<br />https://blog.mathedpage.org/2013/07/more-on-homework.html<br />Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-85399682817128809362018-10-07T12:21:04.237-07:002018-10-07T12:21:04.237-07:00I appreciate your identification of extended expos...I appreciate your identification of extended exposure as a crucial aspect of instruction. It helps me to think about other aspects of instruction and how they relate. For example, one effect of making connections between two mathematical ideas is that it creates exposure to both, thus extending. <br /><br />I believe that one of the reasons that homework typically is not as effective as we'd like it to be is that while it is a way of extending exposure to the mathematical ideas and skills, it rarely serves to deepen the ideas. It sometimes deepens the skills, but usually all that happens is that the problems get messier as the homework assignment goes on. It is difficult to provide homework problems that a student can do in isolation, and that also deepen their understanding. Textbooks traditionally make little or no attempt to do this. Technology provides some ways of supporting homework that deepens the exposure to mathematical ideas, but I am unaware of much progress on this front, and I hope that this will get more attention. <br /><br />I'm really enjoying your blog posts Henri! Your thoughts resonate with me, and they prompt me to think. Thanks.<br />--Scott FarrandAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-42868061826124950732018-10-03T19:19:31.239-07:002018-10-03T19:19:31.239-07:00I just read your post, and it leads me to add this...I just read your post, and it leads me to add this to the list above:<br />- *Recognize its connections with related concepts, and integrate it in a coherent mental framework*<br />However what I was trying with the list was to suggest teaching, formative assessment, and curriculum development strategies. This additional item (at first sight) seems harder to put into practice. Still it adds some depth to the list, and it may work as a way to tie the other items together. <br /><br />Thanks for directing me to your post!<br /><br />-- HenriHenri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-7017219617046322702018-10-03T15:33:52.668-07:002018-10-03T15:33:52.668-07:00Henri,
Have you by any chance read my recent blog...Henri,<br /><br />Have you by any chance read my recent blog post on conceptual understanding? https://davidwees.com/content/what-is-conceptual-understanding/<br /><br />I'm wondering what overlap there exists between our ideas here and to what extent we are focusing on different things?<br /><br />DavidDavidhttps://www.blogger.com/profile/08098221991466148258noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-17325577182551188242018-09-27T05:31:21.771-07:002018-09-27T05:31:21.771-07:00I didn't make up the participation quiz! I lea...I didn't make up the participation quiz! I learned about it from Carlos Cabana, a legendary math teacher in the Bay Area. It is indeed a powerful technique!Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-41573408415224833332018-09-26T20:35:09.239-07:002018-09-26T20:35:09.239-07:00Thank you for the link to your deep dive on assess...Thank you for the link to your deep dive on assessments. I devoured it and wanted to paste here some of your points that were especially powerful to me:<br /><br />- “Do not over-penalize students for small computational errors that could be eliminated by the use of technology such as calculators and computer algebra systems. Prioritize evidence of understanding, not nit-picking accuracy.”<br />- “Use participation quizzes, during which you watch the class work and make notes on students' desirable behaviors. This is an amazingly effective technique to clarify what you consider the most productive ways to function in a math class. Students are being assessed on work habits, not math understanding, but one leads to the other.”<br /><br />I am a big believer that struggling learners often need to be explicitly taught and provided models of effective learning skills, including how to take effective notes in a maths classroom. Your participation quiz is spot on for providing that needed modeling. <br /><br />apm<br />Twitter: @autismplusmathapmhttps://www.blogger.com/profile/14117129826488304886noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-8991024211921234252018-09-26T06:38:59.740-07:002018-09-26T06:38:59.740-07:00We don't disagree. (In the post, I suggested t...We don't disagree. (In the post, I suggested that not all assessments need to be graded.) <br />Grades often undermine learning. I wrote about this here:<br />https://www.mathedpage.org/teaching/assessment/index.html<br />(also follow the link to what the research says.)<br />The challenge is to be clear on our priorities and to balance societal pressures with our commitment to authentic student growth.<br /><br />-- HenriHenri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-294048436550948762018-09-26T06:32:10.534-07:002018-09-26T06:32:10.534-07:00This comment has been removed by the author.Unknownhttps://www.blogger.com/profile/00358216623813307745noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-14323011888804226572018-09-25T22:24:37.369-07:002018-09-25T22:24:37.369-07:00Greetings, Henry.
Saw your request on Twitter f...Greetings, Henry. <br /><br />Saw your request on Twitter for feedback on this post, so here are my thoughts on what I liked and where we disagreed in a (hopefully) agreeable manner.<br /><br />First off, I think you did a nice job of explaining your point of view. I believe Lockhart’s Lament uses the the piano scales as an an example of understanding versus skills, and I think it’s a good analogy.<br /><br />I think that students should be afforded every opportunity to learn and to wonder, be it in maths, literature or history class, to name three.<br /><br />However, I don’t think a full understanding is necessary to getting good grades, and at times is antithetical to that goal.<br /><br />I realize the above two paragraphs are almost contradictions, but that’s the rub: there’s an obligation to provide understanding but also a responsibility to prepare students for assessments that do not require demonstration of understanding.<br /><br />Overall, I thought this was an excellent read. It was well written, in an engaging style, and it caused me to pause multiple times to consider whether I agreed with a point and why. Very meta cognition rich!<br /> <br />Best,<br />apm<br />Twitter: @autismplusmath<br /> <br />apmhttps://www.blogger.com/profile/14117129826488304886noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-11098020005042085842018-09-25T21:50:51.516-07:002018-09-25T21:50:51.516-07:00Thank you! This is so well said - it speaks my min...Thank you! This is so well said - it speaks my mind and articulates ideas that I have been having trouble saying. Kimberlyhttps://www.blogger.com/profile/04675880588194796850noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-38210851584943964722018-05-30T04:53:44.111-07:002018-05-30T04:53:44.111-07:00Thanks for this post. As a gifted/math specialist ...Thanks for this post. As a gifted/math specialist at a high achieving HS, more and more we are fighting the acceleration battle. This is yet ANOTHER piece of evidence that I can share with my parents, that unfortunately, will be ignored by many. math/gifted teacherhttps://www.blogger.com/profile/01686034436094850698noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-70570723912376449312018-05-20T07:29:52.364-07:002018-05-20T07:29:52.364-07:00I’ve been “lagging homework” for over a decade now...I’ve been “lagging homework” for over a decade now... never had a term for it, but all the reasons - yes! Thank you! Yours is the first support I’ve heard for it and I encourage other math teachers to do the same!Unknownhttps://www.blogger.com/profile/00678182996140923380noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-60896127950282929122018-02-27T05:16:26.098-08:002018-02-27T05:16:26.098-08:00Wow, Kevin, this sounds amazing! I wouldn't be...Wow, Kevin, this sounds amazing! I wouldn't be a bit sad if you blogged about it in more detail so I could learn more from you...<br />TracyTracy Zagerhttps://www.blogger.com/profile/18078005798782089280noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-65054814914782658182018-02-22T21:21:14.144-08:002018-02-22T21:21:14.144-08:00I'm a big fan of the techniques Zager promotes...I'm a big fan of the techniques Zager promotes there, and in fact have written about some of those in this blog. But even more so, I'm a big fan of learning with and from colleagues. It makes the job so much more interesting!Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-50108859719175864502018-02-22T21:13:30.879-08:002018-02-22T21:13:30.879-08:00Henri,
Great posting here and one that is dear to ...Henri,<br />Great posting here and one that is dear to my heart. We try to do a mixture during our department meetings. Over the past many years I've tried to move more and more of the "nuts and bolts" to email, google forms, etc. so that we could do more and learn more about math and pedagogy during meetings. It also helps that we have moved to a model with less frequent but longer (1 hour to 2 hours depending on the schedule).<br />During these meetings we like to start with a "warmup" (like we give the kids) - this is usually from your first point - a sharing from a Math Circle, a conference, etc. where we all get to do math together. The beauty of this is that it also allows different department members to run parts of the meeting so that the meetings are a shared experience, not just the department chair running the whole thing.<br />I wanted to add what we have been doing this year as it has been powerful. We've been doing a deep study of just one chapter of a pedagogy book - "Becoming the Math Teacher You Wish You'd Had" by Tracy Zager. I highly recommend this book and the website and materials that go along with it. We chose to work on Chapter 12: Mathematicians Work Together and Alone for the whole year - looking at how we give our students chances to collaborate, dig deeper and also when to step back. It has been amazing - different teachers have been trying on different parts of the chapter (fully randomized seating every class, debate structures, vertical surfaces, etc.) and then we report back to each other and hone the methods together.<br />Cheers my friend!<br />Kevin ReesMAConferenceAmericanPossibilitieshttps://www.blogger.com/profile/15498964993439485443noreply@blogger.com