Essential Mathematics
In the Essential Mathematics Ask-A-Mentor panel, Changelab members are invited to ask questions and share thoughts on how to improve and re-envision mathematics education, how to align mathematics to the
Ten CES Common Principles, exchange ideas related to multi-cultural math instruction, and share ideas related to one’s own practice teaching math. This panel is open to any and all questions and is designed to not necessarily address one specific issue, but rather open up the discussion of the teaching of math to allow for collaboration and sharing amongst all CES members. To help spur thoughts and ideas, we invite you to view a clip from EssentialVisions DVD Disc One which highlights math instruction at Francis W. Parker Charter Essential School in Devens, Massachuesetts.
In addition to the Ask-A-Mentor forum, there is a space for further conversations located in the Forum area located under the Exchange banner.
If you would like to view and download curricula resources and lesson plans, please visit our resource sharing pages under various green tabs at the top of the page.
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does this work?
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The panel is working and we welcome your questions. Panelists will respond within 12 hours.
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What materials do you find effective in high school classrooms? I know that you probably adapt and modify, but do you start with a curriculum, or are you developing everything as you go along? What curricula work well in Essential school math teaching and learning settings?
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I have been teaching high school mathematics for 13 years and have always used the College Preparatory Curriculum (CPM). I use it as a foundation for all other project based curriculum that I design and implement. I use the CPM curriculum for many reasons but here are my top three: 1) the spiraling of topics keeps students in constant development of mastery 2) the opportunities for investigation before memorization are substantial, and 3) there are many instances in which algebraic themes are represented geometrically and vice versa. The curriculum is also very much in line with CA content standards and I can see exactly when and where I am developing mastery of math CAHSEE topics. And the students notice the correlation as well (they often comment…. “ that stuff we did in class was on the test!”). But ultimately, I do adapt the curriculum to include use of Geometer’s Sketchpad, Fathom, Calculator Based Learning Units (motion and temperature detectors), and graphing calculators while also incorporating project-based lessons that I design and implement in collaboration with teachers of other subjects (see Colors of Algebra in the resources section of Change Lab for an example).
-Joanne da Luz
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I have taught for the last 4 years at Eagle Rock. I spent most of that time writing my own curriculum. Exhaustion and questions about my own effectiveness have led me to the point of questioning the effectiveness of my approach.
Eric Gutstein recently published a book entitled TEACHING MATHEMATICS FOR SOCIAL JUSTICE: READING AND WRITING THE WORLD WITH MATHEMATICS. In that book, he describes an 80%/20% model of curriculum use and development. He used MiC (Mathematics in Context), a NSF funded curriculum, 80% of the time and supplemented with mathematics projects emphasizing application of mathematics in real-world, social justice areas 20% of the time.
His model resonates as both realistic and effective. It seems manageable for teachers (new and old), and it seems flexible enough for schools to design curriculum initiatives that reflect their individual values and interpretations of our 10 Common Principles.
jimmy
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I too have spent a great deal of time writing curriculum, which I think is an important aspect of my teaching. At times though it can feel more than overwhelming, and that it may take away too much time from focusing on how to actually work with students. I am just wrapping up my first time teaching an IMP unit, called "Do Bees Build it Best?" While I have re-written or revised many of the activities, I think the progression of concepts and types of problems posed are great. I have also tried to incorporate some problems/activities from more traditional texts both to vary my instruction. I have also recently found a few software programs with curriculum that have been great for doing class investigations. Overall I would echo others about the 80/20 split - as a new teacher I can imagine getting burnt out by trying to create too much. At the same time, it feels important to me to constantly bring new ideas, learning, and energy into the curriculum that I use. I imagine it will be an ongoing process to balance the two, but I am glad to hear of some great resources people have found!
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Parker originally thought teachers would develop all math curriculum, then moved to adopting a text series (COMAP: Mathematics Modeling Our World), then moved back to a more teacher-designed series of units. We do, however take inspiration from a number of sources, particularly when thinking about how to build coherence into a unit or how to assess. EDC (www.edc.org) in Newton, MA has a great set of resources on how to choose a standards-based curriculum. In our middle school program, we really like Connected Math, and in high school, we like IMP, COMAP, Discovering Geometry, and some other resources from Key Curriculum Press. Our Calc class uses Contemporary Calculus Through Applications which was developed at the North Carolina School of Science and Mathematics, and we also like Foerster's Alegbra and Trigonometry as a source for amazing problems. I agree with the 80/20 split others have mentioned--every hour spent deciding WHAT to teach takes away from critical time spent thinking about HOW to teach!
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To be perfectly honest I spend a great deal of my time developing my own curricula that I believe will most engage my students. A lot of these materials come from existing texts like "Rethinking Mathematics," "Freakonomics," "Cognitive Tutor," "IMP," etc. I also pull materials from online, magazines and experiences in my own life. The key for me is to create a curriculum that is engaging and strikes at the heart of the essential learning outcomes. As a side conversation, I believe it was Jimmy Frickey who cited a great math teacher whose belief was to use 80 % of an established curriculum (which millions of dollars and countless hours went into developing) and 20 % of a curriculum that you develop to ensure relevance and social value. It's easy to get exhausted when you spend all of your time developing materials.
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In your experience, is it possible to develop and maintain a constructivist approach to the teaching of mathematics, given what some might argue is the discipline's inclination toward procedures and prescriptions? If so, how is this done and how can teachers balance the teaching of necessary concepts while giving students the latitute to pursue their interests?
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If the constructivist theory of learning is accurate (or at least more accurate than other models, namely behaviorism), then there better be a way to teach math informed by contructivist thought!
Two other thoughts:
1) Math's "inclination toward procedures and prescriptions" is more representative of how it has been taught than how it has actually been developed and how it is done. Also, the "inclination toward procedures and prescriptions" in all disciplines has been linked with different curricula provided to different students: critical thinking for students from higher socio-economic background and rote procedures for students from lower socio-economic backgrounds. That is, pedagogy is an equity issue in my opinion.
2) Much of the NSF funded curriculum developed in the last decade address this issue. However, many people mistake a superficial exposure to these curricular efforts, or the existence of a bad implementation of the reform efforts, as a condemnation of the pedagogy and material. In a current course in the nature of mathematics and mathematics education, I had the privilege of reading an amazing article about this issue. "Facts and Algorithms as Products of Students' Own Mathematical Activity," by Koeno Gravenmeijer of the Freudenthal Institute and Vanderbilt Universtiy. Email me and I can send you a copy of it.
"The core idea is that the students develop mathematical concepts, notations, and procedures as organizing tools when solving problems. In such a process, informal algorithms may come to the fore as forms of well-organized routines for solving certain types of problems. with guidance form the teacher, these informal algorithms can be developed into conventional algorithms. The teacher, however, may also opt for fostering the informal algorithms as valuable end goals in and of themselves" (p. 117).
A main issue for me and, I think, CES math educators is, "Under what circumstances can we realisticly develop the professional knowledge necessary to implement a shift in the nature of mathematics education that is at once more reflective of the nature of the subject and equitable?"
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It is definitely possible to maintain a constructivist approach to teaching mathematics. I have been using a constructivist approach for the past 7 years at Vanguard. Our math department has really developed to the point where we now have offer a Calculus course where students will be able to take the AP exam. Keep in mind that we are also a NYC Title 1 school with 75% of our freshmen entering below grade level in mathematics. We have double blocks in math to really catch kids up and get them to at or above grade level by their senior year.
We use a lot of Complex Instruction, a specific type of groupwork, to facilitate good mathematical conversations between students. We also use a curriculum (College Preparatory Math) that has a good balance of discovery as well as explicit emphasis on important concepts. What I mean by this is that the curriculum allows students to discover a formula or theorem but later on emphasizes that that formula or theorem is important. If you have ever seen the TIMSS (Trends in International Mathematics and Science Study) videos of countries with high math scores you'll notice the approaches of these teachers, like the Japanese teachers, are extremely construvist. They will generally emphasize a main concept at the end of the students' discovery to clarify the most important concept of the lesson.
In summary I definitely believe that a constructivist approach is a great way to teach mathematics to students, coupled with curriculum, pedagogy, and assessments that support this type of learning. Please refer to my response to the question about "demonstration of mastery" for ideas about assessment.
Another good resource of video is Cathy Humphreys and Jo Boaler's "Connecting Mathematical Ideas" which is a book with DVD's of Cathy's teaching. Also, check out http://www.stanford.edu/~joboaler/equitable.pdf for another example of a school that uses a constructivist approach. (I did my student teaching here.)
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Part of a student developing mastery in mathematics is in her developing her own relationship with numbers and mathematical concepts through trial and error. That takes time. When you've discovered that a student has an erroneous way of thinking about a concept or a mathematical relationship , what are some of the effective approaches you've used to re-direct them conceptually or procedurally ?
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A thought that may, or may not, address what you have in mind:
If the student's conception is erroneous, in theory it should be possible to create a problem situation that will expose the erroneous nature of the student's understanding. Posing the new problem becomes a way for the student to experience the nature of the misunderstanding/misconception and becomes the impetus for developing a new, more generally applicable understanding.
Apply?
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I'm with Jimmy...it's all about guiding them to seeing their error in thinking without telling them, "hey, you're thinking is flawed and here is why." As you commented in your question this can take time so what can we do to maintain our philosophy while being time sensitive. That's what responsive teaching is all about. I see that this misconception exists in my students thinking. What lesson or activity can I design that will move our thinking forward while addressing the misconception? The biggest argument in support of this comes from our Japanese counterparts who allow students to linger in confusion for much longer than we do. The struggle for really understanding conceptual or procedural ideas is where the greatest learning comes from. We have to be willing to sacrifice some time to ensure "true" learning that sticks.
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The best way for me to answer questions like these is to provide a specific example....
Today I had my Geometry students do an investigation from the CPM Geometry Connections textbook where an opportunity for developing their own relationship with numbers and mathematical concepts. A mirror was placed on the floor and the challenge was to “catch each other’s eye” in the mirror while standing a certain distance away. The topic being explored here is similar triangles (this took place after three two hour periods of investigation with Geometer’s Sketchpad and sketches of similar shapes and triangles, testing out “zoom factors.”). In any case, students were able replicate themselves in sketches of two right triangles and have “aha!” moments when they realized they could compare corresponding measurements to check for proportionality and calculate the hypotenuse and verify proportionality. But, when it came to the actual calculations necessary to find the “zoom factor,” most kids were stumped. Instead of saying “just divide!” I guide students through the following….
So, would you agree that Andrew’s triangle is probably an enlargement of your triangle since you’re shorter? Okay, so about how many times bigger is 175 cm than 150cm? [At this point, some students respond, “25 more” and I usually repeat to them “Not how much more, but rather how many times more? One time bigger? Two times bigger?
When students finally get to the point that they realize 175 is about “one point something times bigger” than 150, I tell them to get on a calculator and guess and check until they get the zoom factor.
When the students finally find the zoom factor, I guide them through check their result by saying, “okay, if 175 is about 1.17 times bigger than 150, then how many times does 150 fit in to 175?” And that question usually gets them to the division that is necessary to calculate the answer while allowing them the opportunity to develop a sense of numbers. I feel that while students are following these steps, they are getting a better sense of why 150 divided by 175 should give you a result different than 175 divided by 150. And, more importantly, students have a better sense of what size number the result should be for 175 divided by 150.
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So often in math education, students are asked to "demonstrate their mastery" by taking written tests - there's a problem with a specific answer and the steps are completed to get to get to that answer; it's a sopecific process. What ways have you found for students to "demonstrate their mastery" of math without a pencil and paper?
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Another method for moving beyond "pen and pencil" assessment is to co-grade rubrics with the students through a conversation. In other words, you've completed some written work that may be a typical thing to do in math class. i.e. here's a set of linear data...your job is to construct a scatter plot, draw a line of best fit, create a linear model, and make predictions with that model. Rather than sit in isolation with the students work and assess it why not sit down with the student, put the rubric in front of both of you that illustrates what "good work" looks like and have an honest conversation about where this piece of work falls on the spectrum. I have found these conversations to be
enlightening regarding where the students greatest strengths lie and where their misconceptions still exist. Beyond my own take aways the students are asked to be metacognitive about their learning and really think about what it is they know and how they know they know it.
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At Vanguard students graduate using Performance Based Assessment. Students must complete 5 Graduation Portfolio Presentations where they prepare a presentation and defend their understanding in front of a committee of teachers, students, parents, and guests (professors, teachers from different schools, members of our community, doctors who work nearby, etc.) They complete a Math, Science, Literature, History, and Autobiography Portfolio Presentation. In my experience, nothing assesses a student's understanding better than an individual Portfolio Presentation where students are asked to defend their understanding to a committee. Committee members are able to really push the student's thinking by asking questions like, "Why does that equation work? What justification can you give? What does that +5 in the equation tell us about the situation? What if the radius of the circle was changed to 10 feet, how would that affect the equation and graph?" Interacting with the student's thinking while they are presenting gives an authentic assessment of the student's understanding.
Another means of assessment that we use at Vanguard are Round Table Group Presentations. A group of 4 students are asked to think through one long, complex problem. The problem is generally a more open ended problem and we assess their mathematical thinking based on what questions and comments they make with their teammates. We can really look for their mathematical thinking in their ability to make and test conjectures, generalize, make connections between different representations, working backward, justifying, etc.
We also use individual and group projects that may consist of a written report and/or presentation. The more we vary the modes of assessment (paper and pencil test, presentation, group round table, written report) the more we will give kids with various strengths opportunities to demonstrate their understanding. Thanks for reading my response to your question!
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I love using an observational assessment tool that I call "Discourse Time." It starts by having 5 or 6 students sit in a circle and engage in the solving of an open ended problem that strikes at the heart of the essential learning outcome. Listening to them debate about possible strategies for solving the problem and think out loud about counter arguments to their ideas gives me a great sense of where their level of learning is. The grades on this assessment don't come from the outcome of their work, but the engagement in the conversation. What did the student say that lets me know they "get the big ideas."
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Trends seem to be towards requiring 2 years of algebra and 1 year of Geometry for ALL high school graduates. Technical math, applied math, consumer math and business math are no longer considered required math courses; they only count as electives. Students maybe able to “factor a quadratic” (for a month or two) but don’t do well with %, proportions and conversions – things they will need routinely in life. Is this good educational policy?
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In my opinion, the traditional sequence is becoming more and more antiquated. Geometry needs more visualization, 3-D concepts, and work with software if it is interested in preparing students for economic participation. Also, statistics, particularly inferential statistics, is of ever increasing importance to effective citizenship. Imagine a community of people with the ability to skillfully apply inferential statistics to locally relevant issues. Inferential statistics is not particularly difficult and is universally applicable, especially by anyone making a case for change.
The ability to work with "%, proportions and conversions" is absolutely necessary, but those skills are no longer sufficient, in my opinion. I believe students and society will benefit from everyone being expected to do more in math than merely the "applied basics." For that reason, I'm glad to see some of those courses only count as electives. However, there is much in the traditional conception of a math curriculum that can be overhauled to be more useful, relevant, AND rigorous.
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Requiring ALL high school graduates to complete algebra and geometry seems to me like a trend towards "high expectations for all students." While the notion of equal expectations of all graduates seems good to me, I would question the content or nature of those expectations. I think we need to begin by considering what we hope all graduates will be able to do, what they'll be prepared for, and what they'll know upon completing high school. This would hopefully include some of the things Jimmy mentioned such as statistical literacy and 3D modeling as well as the basic concepts they will need that you mentioned - %'s, proportions, and conversions. Once those goals have been articulated, then it seems like we could go back and figure out what types of learning experiences best support students in attaining those skills and knowledge. It may very well be that we decide algebra I and II and geometry do seem to be the best way to reach those goals - or we may not. I think it is hard to judge whether such policy is "good" or not without first deciding what the aim is. On a slightly different note, I share your concern about whether students truly understand some basic concepts in math. Two thoughts. First, I share Jimmy's goal that we need to have expectations beyond those basic skills for all students in today's world. The other is that I think many of those basic concepts are fundamentally related to the topics traditionally covered in algebra and geometry (among others) and I am constantly wondering how to best support the development of those skills within the context of other topics in mathematics.
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What tensions exist in teaching mathematics between the "equity" and "high expectations for all students" principles in mind? What specific things do your schools do to address these principles?
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Eagle Rock does not group students according to ability or prior experience. Each course is designed to "stand alone," and we have students in nearly every course with a wide variety of prior math class background. In the last 4 years, for reasons motivated mostly by my belief that tracking is inherently inequitable, I stuck with this model.
At the Fall Forum in Chicago, a principal from another CES school (I don't remember which one?) presented work his school did to detrack their middle school. He presented the most compelling definition of tracking I have heard:
"Tracking is the practice of assigning students by measured or perceived ability/performance to coursework of differing curricular and instructional rigor, having differing completion expectations/requirements but carrying equivalent credit/value towards graduation."
They, the teachers at the school, articulated clear expectations for everyone at every grade level. They provided an extra course to support success in the main stream course that some students were dual enrolled in. They allowed students to take the next level course when appropriate. Finally, they created a "bridge" course for students who were determined not to be ready for the grade level material, even with the extra hour of instruction. This "bridge" course carried no credit toward graduation/promotion, and it was designed for students to be "ready" in one year.
I left that presentation thinking hard about the effectiveness of my heterogenously grouped classes. I still believe in heterogeneous groupings, but I think my old notions of differentiation allowed me to give the same credit in those classes with vastly different academic expectations. Based on that school's work, I now distinguish between two kinds of differentiation: 1) students in same class working on the same topic/problem but with different academic goals, and 2) students in the same class with identical academic goals but with variable methods of attaining those goals.
I believe that there are times when both notions of differentiation are appropriate; however, I believe my practice of differentiation, mostly the first type, was potentially creating the same negative benefits of the programs I was trying not to participate in. All of this is to say that I think I (we?) have a lot to learn about effectively supporting all students in meeting high expectations. I echo Kari's recommendation to read the work of Jo Boaler. Complex instruction is a compelling model. I didn't know she had published a book and classroom videos. I'll be checking that stuff out myself!! Thanks for the recommendation Kari!!
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I have been facilitating a problems based curriculum for the last 5 and 1/2 years. I value discovery learning and allowing students to construct their own meaning and understanding but at times I become weary. The idea behind developing mathematical tools is indeed valuable; the master knows how the tools are crafted. My concern is every time a tool is needed it is ineffective to have to craft the tool. Sometimes you just use what is in your tool box. In your perspective, how can a teacher effectively allow students to develop mathematical tools on their own but at the same time give them guidance on how to select the appropriate tool from a large tool box? I have seen a deficiency in tool selection and implementation even though my students rock at creating tools.
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Good question! I am excited to read other folks' answers to this one. I have a couple thoughts, but I offer them as thoughts more than "answers."
Are you talking about students selecting tools that are different than the tools that you or I would select because there are more efficient tools available that we want them to use? Or are you talking about tools that are inappropriate for the task because they do not contribute to a solution?
If it is the former, students using inefficient tools longer than seems necessary, then I'll offer an idea I've heard about more than I've used. If the goal is to move students toward a particular algorithm/skill/technique, then it is on me to design problems that "frustrate" the less formal, or general, tool they are currently using as a means of creating the need for a "better" tool. Who wants to use "repeated addition" in a table to find the 1000th input?
I also think that I readily underestimate the time necessary for a student to be ready to move on to a next level of generalization, or formalization. Freudenthal Institute folks call this "rushing to formalization."
If it is the latter, then ... I'm thinking planning lessons about "selecting appropriate tools" could be effective. In literacy trainings they advocate think alouds as a means of modeling "what effective readers do." I've recently been thinking that a similar idea could work while modeling thought processes useful while solving problems. Hope that is helpful. Can't wait to hear from others.
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The first thought that comes to my mind is: selecting the appropriate tool is itself a tool. I try to articulate this type of more general skill (more general than, say, applying the Pythagorean theorem tool) alongside the more specific learning outcomes for my courses. I share these learning outcomes with my students and encourage dialogue about them as well as the more specific content/skills we encounter along the way. Perhaps intentionally teaching toward a "tool selection" tool (part of problem solving?) could give students the opportunity to utilize one of their strengths - creating tools. Although I have not formally included that as an outcome, I have a student who struggles greatly at retaining new tools. After some informal work together, one strategy she has devised for herself is keeping very careful track of her previous work so that she can reference it when she gets stuck and figure out on her own what previously learned tool might be useful. Now I am wondering what formally including such a tool could look like...I think having this as an articulated learning outcome could provide students with a framework to put related learning experiences into and also provide instructors with a natural space to provide guidance about such skills. I am very excited to hear what ideas others may have about this question since it is something that I hope to get better at as well.
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Any recommendations/advice/suggestions of a math intervention that is customized for individual needs and is good for students who need to make immediate progress? We are not interested in software programs, but a service that could come to our school in the Bay Area and provide instruction in a small group setting. Thoughts?
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I have two totally different thoughts. First, I can put you in touch with someone I know who has started a business that does exactly what you are asking for. I have not seen them do what they do, so I have no opinion about the efficacy of their practice. Email (jfrickey@eaglerockschool.org) me and I can put you in touch with them. I'm hesitant to post their info and have it seem like a plug, but I am happy to share the contact with you personally.
Second, I've always been concerned with interventions that one more time do the same thing and have been curious about effective ways to intervene that are effective, engaging, and empowering. At ERS, we have some experience forging relationships with an elementary school to put our students in tutoring relationships with elementary kids. We teach the material the kids are learning but with a focus on empowering them to help the other kids. One example of the benefit of this is that many of my students who struggle with math are more effective small group facilitators of young children than I am! A different take on the same tutoring idea is to have students shoot mini-tutoring lessons on video that can be used by middle and elementart school teachers as a resource in their classrooms. Again, I hope an approach like this would help utilize students' strengths and creativity in addressing their challenges.
The second idea doesn't really address the staffing and resource issues I attribute to your question. Hope this was helpful.
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Educators who don't focus on math in CES schools still often need to work with students on mathematics concepts as advisors or within cross-disciplinary studies. Have you ever worked with non-math teachers on your schools' faculties to help them grasp math-related habits of mind and/or specific skills and concepts? How do you do it - on the fly, or in specific professional development time? And are there any books or other resources out there that help adults who don't think about math a lot get into the right frame of mind to work with kids on math?
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Eagle Rock's World Languages instructor and I collaborated to incorporate proportional reasoning into his world language curriculum. We did this with a couple of things in mind. (Outlines of the lessons we developed are under resources attached to my name in the list of panelists.) First of all, incorporating math into his curriculum needed to add value to his instructional objectives. We used ratios and proportional reasoning to discuss and compare exchange rates and the numeric work with different exchange rates really helped deepen his class's conversation about why exchange rates might be different between countries and over time, as well as discussing who benefits from these differences. Second, we choose proportional reasoning because of its central importance as a necessary concept and skill for continued success in "higher" mathematics. Also, I thought choosing a specific concept, like proportional reasoing, would simplify the outreach and professional development efforts necessary in engaging other teachers in mathematics education. Of particular importance in this case is that proportional reasoning is NOT developed by use of the standard algorithm. Rather, proportional reasoning requires the flexible use of multiple informal strategies over a period of time. While I believe that flexible reasoning skills should be in our classrooms as well, other teachers' classrooms are a perfect place to forget about the algorithm for a while and discuss the mental strategies that support comprehension of issues related to proportional reasoning, which are numerous!! A major resource for me in supporting this endeavor has been John Van de Walle's book, ELEMENTARY AND MIDDLE SCHOOL MATHEMATICS: TEACHING DEVELOPMENTALY. In high schools, using proportional reasoning across the curriculum supports, but doesn't replace, remediation efforts. Middle schools could choose a significant topic from number sense in the elementary curriculum, like place value or the meaning of operations. The final argument for this practice is to actually demonstrate how useful math is to students by incorporating it authentically into the curriculum verses merely claiming its significance. I'll let other folks talk about the ways in which to effectively engage other instructors in mathematics education...
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I have co-taught two math-related courses with non-math instructors in the past year, once with a science teacher and once with a world languages teacher. Both teachers had completed traditional high school/college mathematics programs and that made up almost all of their math education experience. There were not any formal PD sessions, although the science instructor participated in one of the lesson tuning protocols I mentioned above mid-way through our course. Based on my experience, and similar to what Jimmy mentions below, I think the most effective way to bring teachers from other disciplines in is by having them participate in class. Both instructors I taught with would regularly complete activities and solve problems right alongside the students. Because many aspects of my approach were new to them - such as having students derive their own methods for solving a problem before introducing them to a more efficient formula - I think they furthered their understanding of both math and math education. This hands-on experience, combined with many conversations outside of class about my intentions and ideas seemed to work well. I would also add that I have learned a great deal from these experiences, since adults who do not often think about math pose very insightful questions and contribute a fresh perspective to my thinking. Their thinking can at times be similar to the thinking of students who are learning math or unfamiliar with a non-traditional approach. I think that perspective is extremely valuable when working with students and would encourage educators to make the most of those prior experiences in addition to what they gain from participating in the math classes already happening at the school.
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I regularly work with my non-math teacher colleagues at LLA. Sometimes it’s on the fly…. For example, when the culinary teacher mentioned how difficult it was for his students to comprehend the size of fractions when measuring and working with recipes, I described to him the “Aha!” moment I had when talking to Li Ping Ma at The Carnegie Foundation. She described to me how, while working part-time doing data entry in graduate school, she noticed that many U.S. teachers responded “2” to the following question: How much is 2 divided by one half? She went on to describe how teachers in China would teach the concept of 2/ (1/2) by posing a scenario similar to the following: If you have a 2lb. bag of rice, how many times will you dip a half pound measuring cup into the bag to empty it completely. When I described this story to the culinary teacher, he had an “Aha!” moment as well. And, a further conversation ensued about how to design mini-lessons in the kitchen that would simulate the experience described by Li Ping Ma.
There are other instances when I have collaborated with the Chemistry/Physics teacher to ensure that we are using the same language during our lessons that involve proportions. We both noticed how challenging it was for students to understand when proportionality exists and how to calculate values when the proportions were not set up with nice, neat, whole numbers. And, I have collaborated with teachers of other disciplines to create project based curriculum and as a result, the non-math teachers were exposed to a new way of learning math that they had never encountered.
Regardless of how it was done, I have to say that two important aspects of mathematics and teaching seemed to stand out: 1) number sense was always the instigating topic and 2) using the same language to communicate math concepts across disciplines has made a significant impact on our students’ overall development, both academically and in terms of confidence.
I also have to mention that the Number Sense strand in the Math CAHSEE has always been the weakest performance for our students and paying attention to it through collaboration and project-based learning has made the most impact.
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So much of math is procedural knowledge (manipulations – rearranging expressions and equations). Tests (state graduation and college entrance) tend to be based on these skills. How can procedural mastery be developed in a PBL setting?
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The great thing about teaching skills in context (through problem-based learning, inquiry, or student projects) is that kids see the need to refine a particular skill. Ideally, kids will then have a reason to want to practice and build the skills beyond "it will be on the test." As kids at Parker get older, they do start to say things like, "I still have a hard time figuring out the first step with an algebra equation," or "Oh--it's a system of equations, and I never remember how to do those." That's when, as a teacher, we point out how nice it would be for that skill to be more automatic, and we start to practice. If you came to my classroom, you might see a bunch of kids doing practice exercises, just like you would see in another program. They don't like it when the procedures are not automatic!
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I support a math department in a large, very urban, public high school. Our achievement results are low and all of our dropouts are failing at least one math class. We are working to be personalized, differentiated and honor the 10 Common Principles - our challenge? 1) How do you support students in math in addition to offering them problem-based and inquiry approaches (students who are well below grade level in their numeracy and literacy)? Question #2) What do you recommend as far as professional development for math teachers who have great math knowledge, traditional pedagogy, but are committed to changing their practice to support students?
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1) We have had the same issue with math failures at our school. As a result we have done a few things to address the need for support. We give all students a pre and post math assessment at the beginning and end of every year to look at few things. What experiences are students coming in with, how are they progressing through a school year and with what skill set are they leaving when they graduate. We look at individual students performance as well as classes as a whole. This has informed a few curriculum decisions as to specific learning outcomes that need reinforcement. In addition to this we started offering a school wide math "coaching" program that is open to all students on Mondays. We invite engineers and scientists from a local lab to tutor students (food is a big incentive). This is to establish a mathematics learning community that spans grade levels.
We also have a lot of students coming below grade level and one thing that has seemed to help those students is using an outcomes based assessment structure. We were able to pinpoint specific mathematical targets for each quarter at each grade level so that we could assess some of the core skills. Using this system seems to allow for more flexible differentiation because a specific learning outcome can be made more challenging or a teacher could make it more accessible to students who have specific learning needs. Either way, we know what the "it" is that we are differentiating because we have named it. Implementing this can be done with a problems based curriculum; we use IMP. This is only our second year implementing this assessment structure so our only data so far is decreased failure rates and observation of students williness to dig in more. Another idea is perhaps instituting an 8th grade to high school transition institute in the summer. Our school has talked about wanting to do this.
2). I whole heartedly agree with opening doors. Making class rooms public is key to changing practice. Encourage teachers to take part of their preps to go observe other teachers. CFGs are also useful tools in allowing constructive feedback on practice.
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We recently developed a screen for our incoming 7th graders so that we could assess whether or not they were coming to us with the core knowledge and skills we assumed they had from elementary school. We responded to this data in two ways: 1) we adapted our 7th-8th grade curriculum to address a few skill issues that were affecting the majority of our students, and 2) we are working with our student services dept to devise individualized interventions as needed. For instance, if a student has not memorized their multiplication tables, this impacts the next 6 years, from work with fractions and factors to quadratics to estimating the correctness of an answer to....you get the idea. We have 11th and 12th grade students in the school willing to work one-on-one with students as they refine a particular skill, and we are also starting to ask the question: How will this student continue to have access to rich mathematical experiences while ALSO building/filling in importation skills? What does the 6-year plan look like for this particular kid? All kids are fully included in the program, even those who struggle.
The second question is interesting and tougher for me to answer. When we hire great teachers with a traditional background who are yearning to try something different, we essentially provide them with a laboratory to try those experiments and a team of interested colleagues who will help plan, observe, discuss, and support those experiments. Common planning time and work in CFGs supports this, but we also actively look for curricula and programs that will add to our toolkit. Exposure to the IMP approach changed my perspective on teaching math, and the support materials for that program include suggestions for conversations to have as a math department/teaching team. NCTM (National Council of Teachers of Mathematics) and other professional organizations also have plenty of good support--there are a lot of amazing people thinking about mathematics reform right now.
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Do you think it's possible for a mid-career teacher who doesn't have anything but enthusiasm for math to become a math teacher in a CES school, given that s/he studied math and learned it well? In other words, do you think we can become math learners at any age?
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I definitely think so. I think what may the most difficult think to shake is being the "expert". Learning to facilitate understanding is much different than teaching an idea or skill. A lot sweat goes into not doing what all of our teachers have done for years. If someone has the passion and the desire at any stage in life that is an indication of a learner which is the best model our students can have.
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I truly believe that it is possible for a mid-career teacher who has enthusiasm for math to become a math teacher in a CES school and become a math learner at any age. Enthusiasm is incredibly helpful and important. In my opinion, a willingness and interest in learning pedagogy while re-learning mathematics conceptually is just as important, if not more, especially when working with youth who have a history of failure in mathematics. I truly believe that students who have learned mathematics one way and failed should not be re-mediated, especially not with the same methods and teaching strategies that they already experienced. I cannot express how much I learned about mathematics and what it is actually doing when I attended seminars held by Key Curriculum Press (Algebra with Lab Gear by Henri Picciotto, Discovering Geometry with Michael Serra, Discovering Algebra….. Advanced Topics in Sketchpad…. Fathom…. Etc) during the summers, while I was at Stanford Teacher Education Program with Vicki and Joanne (my curriculum instructors at the time), during workshops at Asilomar, and during my first three years at San Lorenzo High School under the tutelage of wonderful educators like Phil Tucher and Carlos Cabana.
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Sure--though I think that the most critical element of teaching math effectively, particularly in a CES school, is a passion for figuring out what is going on with the students as they encounter math! Sometimes it's harder for those of us who had a talent or affinity for math to teach it well because we are less able to understand or anticipate what makes it so hard. The exciting thing about focusing on what's going on with the students is that teaching from this perspective stays fresh and interesting. There are only so many times you can solve linear equations and find it new and exciting--but it's always interesting to see things click for a student who is getting it for the first time.
The interesting thing for me as a math teacher was that there were many things I was procedurally quite good at that I never really deeply understood--and teaching the material allowed me to see it in a whole new way.
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What responsibilities, if any, do experienced educators in the CES network have to support the success of other newer programs, particularly in math? What sorts of things would be useful in making quality effective mathematics programs the norm across the CES network?
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I would agree with Diane. As a new teacher, two of the most formative experiences I have had so far were visiting other classes and collaborating with more experienced educators. It's one thing to read about teaching mathematics in a CES school and a whole different level to actually visit (and maybe even participate) in a mathematics class. Similarly, I have participated in a few lesson tuning protocols facilitated by other more experienced CES math teachers. Participating in these protocols gave me the opportunity to watch more experienced educators work through some of the very same dilemmas I encounter, have my own ideas pushed, and collaboratively generate/improve useful curriculum materials for a CES math classroom. In addition, seeing more experienced teachers share ideas that have (or have not) worked well and demonstrate continuous learning by reflecting on those ideas has been very inspirational for me. These experiences in collaboration both inspire me and provide me with tools to further my own teaching.
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One of the best things we can do, I think, is to open our classrooms and schools to each other, so we can see the messy reality of teaching math in a CES school in all its highs and lows. I was extremely discouraged in my early attempts to do things differently when my experiences didn't align with my vision. (Jimmy wrote a great article about his experience with this for Horace!) I would have really liked to see instruction that fit my vision--and to see experienced faculty messing up and having bad days, too.
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The school I am teaching in is using Understanding by Design (UbD). It has been easier to do in many other content areas than in Math. Do any of you have experience in math and UbD?What are the “big ideas” in math? What are the things that we need to go deeper on? Do you have any resources or materials that you might suggest on this topic?
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I think those are great questions. We use UbD at Eagle Rock and I have been wrestling with many of those exact questions. While I don’t know that I have answers to all of them, here are some thoughts. I have found it difficult to reach a good balance between conceptual and procedural aspects of mathematics in the big ideas I write. Often I think my bias is to make the big ideas as big as possible, which sometimes makes them feel too abstract too soon and leave out some of the important procedural aspects of mathematics. This might be related to what Jimmy mentioned above about underestimating the time it can take students to make the leap to a more abstract procedure. I think it is ideal to set up situations in which the learning becomes required. Imagine, for example, that students understand idea A. If I ask about idea C, they have to learn idea B to get there. Often I simply use idea C as the big idea, but I think it can be equally important to articulate the intermediary ideas as well. I would be excited to hear what experiences others have had in crafting the big ideas in a UbD based mathematics class. I would agree that crafting true essential questions in math takes me a significant amount of brainpower. When brainstorming I find it helpful to have examples and I often think back to a great EQ David Singer mentioned to me for a unit on geometry, “In today’s world, which is more important, innovation or invention?” Although not necessarily “mathematical,” it engages students immediately, it seems to provide a great context for mathematics and can also be applicable to the process of mathematics (for example, when is it better to invent new tools or use old ones when solving problems?). I think the questions used in classrooms are crucial, especially in the UbD framework, and I hope to go deeper into the process of crafting and asking questions with both my students and co-teachers.
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Our school utilizes UBD and I agree that it may seem easier to develop true essential questions in other content areas. Our team has been able to develop some essential questions around mathematical units but it did require a massive amount of brain power. Some examples of essential questions are: How does a graph tell a story? , how are reasonable predictions made, how can flight be modeled?, how do you represent a situation without words?, why is proving something important?, what does "normal" mean? (bell curve), How can you quantify how things change?
All of these are interesting questions but the truth be told when I have taught these units it is easy to start with an essential question but the culmination mathematics doesn't necessarily bring the students back to answer them unless a very deliberate facilitation is used. The idea is genius but what I struggle with is the non mathematical components of those questions.
I think developing activities and mathematical experiences around the essential questions is fairly logical; it's the assessment of the enduring understandings that's the tricky part. I believe this is where portfolios are key. You can develop a portfolio rubric that addresses the key principles of students depth of understanding related to the essential question and at the same time require mathematical evidence to support their response.
Beside the ackwardness of the essential question component I feel that working backward is very effective in developing a mathematics curriculum at least in what skills, knowledge, understanding, mathematical experiences, and performances you expect students to achieve. Having a guide/goal is always a good idea even in an inquiry based program where students develop their own understanding.
The resource our team has used is "Understanding by Design" workbook by Wiggins. We also had a consultant come to our school to guide our staff. I must admit the math team had much greater difficulty with the implementation but I think even if we don't use the design exactly the way it was intended it is so much better than just "going with the flow" of a particular curriculum. Intentionality is key.
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When I try to think of the "big ideas" in a math unit, I start by considering what I would want students to be able to remember from their learning even if almost all procedural detail is forgotten. Right now, we're doing a unit on functions and quadratics, and even if kids would have to look up/review how to factor or use the quadratic formula, here are the things our team agreed we would like all students to hold on to: 1) Functions can be classified into major families by their shapes and by patterns in their algebraic representations. Familiarity with the different families helps us to determine an appropriate model for a particular situation. 2) Quadratic functions have a number of useful features that make working with them easier, and allow us to identify with precision key points on their graphs. Some algebra techniques are specific to quadratics.
If students come away from this unit with those understandings, then they will have the big understandings necessary to reactivate the details in a newer context. This also gives me some touchpoints to revisit as the unit progresses (refrains for multiple class sessions). And it helps me explain to kids why we are doing this!
Judah Schwarz, Al Cuoco, and Paul Goldenberg are some names that come to mind when I think about people who have done some interesting thinking about the "big ideas" in mathematics.
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I have recently taken a position as a math coach in a fairly traditional public middle school. My role is to meet to preplan curriculum and instruction to focus on state indicators and standards. Our goal is to get kids more engaged and get them to "do the math." My question is, do you have any resources or ideas that could help me get better teacher buy in. Most 8th grade teachers are teaching straight from the math textbook, which is not really standards based?
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I definitely echo Jimmy's suggestions. When I was a math coach for Region 9 in NYC, we ran workshops for teachers of all grade levels K-12 and did a ton of group problem solving. We chose a lot of open ended problems that were very accessible and able to solve using a variety of methods. We tended to use a lot of problems from Mark Driscoll's Fostering Algebraic Thinking, which is a great resource for these types of problems. Teachers really enjoyed working on the problems together and building on each other's methods. Another book suggestion is "Adding it Up: Helping Children Learn Mathematics" by Kilpatrick, Swafford, and Findell. Also, watching TIMSS, Trends in International Mathematics and Science Study, videos (I know I keep suggesting this sorry!) where teachers can see teachers of countries with higher math scores using problem solving approaches to teach math rather than the traditional American approach of "chalk and talk." You can also always refer them to the NCTM standards and use some of the NCTM publications as well.
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In reflecting on my own experience learning math and learning about teaching math, I notice two things that have provided a core of my belief in the possibility to teaching math for understanding to ALL students.
1) I actually learned some math that way with other people.
2) I witnessed "a different way" working in other classrooms. Based on these experiences, I think professional development in mathematics needs to involve learning and doing math together as students so we can feel what it is like to learn it in a different model. I also think professional development needs to involve connecting teachers with each other in ways that are intended to be longer run collaborations to improve student understanding. Too much of the debate around math reform, in my opinion, throws jargon around when conversations and collaborations about actual student work and understanding in an actual lesson would be much more effective. Lately, I've been thinking of professional development models for teachig that mirror what we are trying to have happen in our classrooms. If we want to share new ideas about learning and teaching math it MUST engage teachers current understanding of learning and teaching math. We can no more tell a teacher how to teach and expect it to work in "non-routine situations" than we can tell a student how to solve a system of linear equations and expect him/her to be able to apply it in "non-routine situations." That being said, I recommend reading THE TEACHING GAP: Best Ideas from the World's Teachers for Improving Education in the Classroom, by James Stigler and James Heibert. It is an amazing book that summarizes a huge international study of math education, and it does a good job of discussing teaching as a cultural phenomenon and of discussing ways to improve practice. I think it could make an excellent starting point for a professional learning community.
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I have a pedagogical dilemma. Our school uses IMP as a base curriculum (we supplement as we see fit). The IMP curriculum develops the concept of rate and steepnees of a graph beginning the first year. The explicit calculation and definition of slope (formally) does not happen until the end of the junior year. I feel that if it is introduced in a way that allows students to have a deep connection with the underlying concepts then it should be introdued earlier. I also beleive that it is of detriment to delay the introduction of slope since there are two main units that deal specifically with linear functions and their applications. My other concern is by waiting until the junior year then students have less time to fully explore the idea and make meaningful connections to what they initially learned about rates and growth. Some freshmen come to our school already having a very basic understanding of slope and I hate having the thought, "wait, you will get to that in your junior year". I know concepts like ratio, proportion, rates, and growth are very developmental. What are your thoughts on when the formal definition of slope should be developed? (or how?)
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I have a friend/colleague who I met when I went through a two week IMP training a few years ago. She is a trainer for them and has been with the program since the beginning. I copied your question and emailed it to her. If I hear back before the end of the day tomorrow, I'll post her reply. I can't see the information about who posted the question, but if you email me (jfrickey@eaglerockschool.org), I'll make sure to share the response with you even if I don't hear back from her before tomorrow. That being said, I think there are many postings from this Ask-A-Mentor session that suggest intentional adaptation of materials to meet individual student, class, and/or school goals is simply good practice.
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What specific techniques and/or methods do you use to encourage mathematical talk? Do you assess mathematical communication in your courses? How? What issues/challenges can other instructors expect when trying to foster mathematical discourse and communication in their classes?
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As a graduate of Parker I think I benefited greatly from the ongoing discussion amongst staff and then revision of my own work that Diane mentions. Only now as I try to structure my own class do I realize just how complicated (and important) teaching about mathematical communication can be. I believe it is a skill that has helped me not only in math but in almost everything I do. I constantly find myself drawing diagrams, tables, etc. to try and illustrate a point - a skill I am sure I developed in some of my math classes. A challenge I have been wrestling with in one of my current classes (where I am actively working with my students on their communication skills) is the difference between showing WHAT they did and WHY they did it. When I ask students to show their work and their process, they often show all the calculations (or include them somewhere) without including any justification for the work. I've recently started having students complete formal Problem of the Week (POW) write-ups - something I found helpful in my own math education and that seems to be a good way to support both types of communication. Discourse Time (that has been mentioned on this panel a few times) also sounds like it could be a great way to foster more of the justification and "why?" questions involved in communicating mathematical processes.
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I'm currently in a math education course, and we read an awesome artical about fostering mathematical discourse by Magdalene Lampert called "When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching." One more time with the email address ... but if you email me at jfrickey@eaglerockschool.org I'm more than happy to share the article with anyone. It's awesome!
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One of the things I love about teaching at Parker is that we explicitly assess mathematical communication separately from mathematical problem solving. (The basic criteria we use are on the school's website at www.parker.org, under the Academics drop-down.) This means that a kid COULD solve a problem beautifully and creatively, meeting standards for problem-solving, but lack technical proficiency with the nuances of mathematical symbolism and vocabulary. We then get to emphasize the conventions of the discipline and the importance of communicating fully, while acknowledging the mathematical insight that the student has shown. Our faculty has engaged in 10+ years of conversation about how to distinguish between problem-solving and communication, and how to assess communication in a developmentally appropriate way. In 7th and 8th grades, we emphasize explaining your process fully and using multiple representations of a solution (tables, graphs, etc.). In 9th and 10th grade, we push for the development of a more formal algebraic communication, paring down the words and using more symbols. By 11th and 12th grade, we want students to submit written work that is both deep AND concise, with formal terminology and that omniscient third-person voice that you find in every textbook. A critical component of this is revision--just like writers need to rethink how they organized an essay or used a particular tone, mathematicians need to edit and rethink their presentations to improve clarity.
In class, we use techniques like chalk-talks, small group work, and whole-group discussion of solutions to get the kids talking with each other. That's hard! I just read an interesting article in Mathematics Teacher about a strategy called "Discourse Time" that I am hoping to think more about--I'll see if I can find the article.
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I teach in an alternative program primarily at risk kids. I have a wide range of entrance backgrounds (typically many F's and long gaps in math success). What are best practices in terms of initial assessment and then in terms of remedial needs?
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In my experience, I have not found it hard to get a sense of what students know and can do by simply working with them. I'm skeptical of the accuracy of multiple choice assessments. I do think they can be used effectively in conjunction with the huge amounts of data and evidence teachers already have right in front of them in the form of the stuff we already do in class. I think the most important thing to do with students for remediation is to raise our expectations of them, give them interesting, meaningful work, and communicate through the work we present them with that we know they are smart and capable. There is much work done with so-called "open-ended" problems that have multiple entry points for students and can lead to significant mathematical learning. In addition to adapting reform curriculum from lower grade levels (i.e. Investigations for Elementary school, MiC and CMP for Middle school), I have used two other books (ELEMENTARY AND MIDDLE SCHOOL MATHEMATICS: TEACHING MATHEMATICS DEVELOPMENTALLY, by John Van de Walle and THE OPEN-ENDED APPROACH, published by the NCTM) to help me both create my own open-ended tasks and understand how to use them. Of central importance, though, is raising our expectations of students and providing appropriate support to meet those expectations without lowering the cognitive demands of the tasks. I have found statistics to be an extremely useful discipline in bringing up "remedial" math issues in contexts that allow our students' strengths to add value and meaning to learning. Also, making real things (in art and building) have provided amazing contexts that bring up important math and draw on students' strengths in learning these concepts. Finally, tutoring situations with younger kids also allow them to draw from strengths while learning in a subject around which they usually have tremendous amounts of baggage. Check out Jo Boaler's stuff that Kari mentioned in one of her replies. Good luck!
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