Monday, April 7, 2014

Principles to Actions: Teaching

An excellent mathematics program requires impactful teaching that engages students both individually and collectively in challenging mathematical tasks that promote their ability to make sense of mathematics and reason mathematically.

  • How do you see productive struggle?  What is it?
  • How are lessons best developed - task leading to the skill or the skill leading to the task?
What are your thoughts on the unproductive vs. productive beliefs on page 23?

This is a vital chapter for us.  How are we doing?  What can we do better?  What are your thoughts?

21 comments:

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  2. productive struggle is watching an 8th grade math student in my class look at a math idea and hopefully figure it out with minimal support from the teacher...it might involve a struggle of sorts between the student and the teacher...but...in the end the student finds the means to solve the problem...i keep trying to find ways to avoid 'giving out the answer' directly and i am constantly searching for ways to assist the student to think through the problem strategically...it is really interesting to try to 'snoop' in on small groups/partner work times in class and to listen to how students come up with the answers and how they explain how they did it too! i learn alot through this process too!! i think earlier in my career i would've been really nervous to watch a student struggle and ultimately would supply them with the answer...now leaving things open for the student to think through and 'walking away' when the time is right are ok and really more helpful for student thinking skills. the only idea that makes me nervous these days is that there does not seem to be enough time allowed in the day for students to engage in productive struggle...this might be a trap that may lead to more teacher directed instruction :(

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  4. i think that different math units may lend themselves to different lessons developed...
    some units may work great with a task leading to the skills...some units otherwise. i think that a variety approach is important too especially with junior high aged learners...it seems that after every unit is completed i think about other interesting ways that the unit could have been studied!!! at this point however we are rushing into another unit (topics to be covered!!!) and these important unit follow up ideas are quickly lost and only parts of them may be remembered next time...having a PLC to share these discussions with is a huge positive!!!

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  5. my computer is not publishing the correct posting times!!!

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  6. Todd mentioned how productive struggle is watching a student figure out a challenging answer. The student may figure it out on his/her own or work collaboratively with other classmates to arrive at a mathematically sound method. Productive struggle becomes quite a challenge at this time of year. The students are worn out and do not want to work for the answer. It becomes difficult to use probes or questioning to spark an idea for the students to explore.

    Then the question you mentioned of how are lessons best developed - task to skill or skill to task. I think that depends on the material. Some standards require a great deal of prior knowledge or skill building. Other standards can be accomplished by using productive struggle to arrive at a method where different skills can then be emphasized.

    I know I have a long way to go on encouraging productive struggle. I continue to struggle to balance all of the material and allow students the time to struggle. I always seem to be pressed for time. I then cut corners on time for the students to struggle. However, I do agree with the author that the struggle is vital. The students take ownership in the learning and acquire the deeper understanding of the material. It is just difficult to move out of your comfort zone.

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  7. Luke CoenenApril 10, 2014 at 4:51 AM
    Welcome back! Do we have the right direction? Heck yes, However on page 22 line 9 it says"Teachers must employ instructional strategies that support this vision of teaching and learning." However I agree with Steve that it seems we have more collaborative time then ever before, however we have more hoops to jump through and we really don't get the time to discuss instructional strategies that we use. We hardly ever get instruction from administration on the instructional strategies they want us to use. I feel it is all about assessment and how students do on their test, especially now when the comment in our department was made that in three years our evaluation will be partly based on how our students do on the ACT test. I want to make it quite clear : WE DO NOT CONTROL WHICH STUDENTS WE GET IN OUR DISTRICT! Please show us what you want! How do we motivate students to Persevere? How do we " actively engage students with their peers"Are there good tasks out there that we should be using? Where are they? What are they? Do we need to create them? I feel I do not have the skill or time to create great motivating task that excite student to come up with great products? Please teach me how to do this. I feel the CPM curriculum is rich, however the discovery learning takes time. The recommendation the the authors state we need 60 minutes of class time per lesson ( and that is without going over any homework!) We have 45 minutes! What do we cut??? We get shorten time but I feel no instructions on what and how we change our instruction when we have less time.What are teachers doing? " Allocating substantial instructional time to making connection among representations? I do not feel we are achieving this. I do feel we have a math staff that is willing to work! We need instruction and examples as to how this works! The reading gives examples at to low a level. WE need examples and instruction for the classes we teach.
    "Selecting tasks that encourage students to use multiple representations in solving problems?" The students and teachers are really struggling with this. Both of us need help in this area. However, if I amgoing to be evaluated on an ACT exam. Why wouldn't I spend more time on ACT type questions? What are the ACT questions going to look like? My goal is for every student to get the concepts and score a 100%! However I do not think that is their goal all the time! For a lot of good reasons and a lot of not so good reasons. This is a very complex problem! this is all I have time for now! More questions than answers!

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    1. Luke gets to one of the notes I made while reading. Allowing students to struggle through things can take a lot of time! Sometimes it is time that we don't have. I had a group of 7th grade MM students the other day who thought 7x - 5 equaled 2x (even after weeks of solving equations etc...). I had about 10 minutes left in class. I tried everything I could to help them see the error in their thinking. Finally as the bell rang I had to keep them in class. I didn't let them leave until I felt like they had an better understanding. Still I know half (or more) of the group probably left not knowing their error in thinking. If I had more time this may not have been as much of a problem.

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    2. I agree Luke. With all the items that are rated most urgent, where do we begin...continue. Do I think we are doing some good things. Definitely! But, the plate labeled HIGHLY IMPORTANT is getting fuller by the moment. I feel that if we put great emphasis on instruction, the test scores, etc. will follow. But, if test scores are the resounding measure of success it can make trying new methods, etc. more nerve-racking. Our time continues to wither in the classroom as new waves of testing are expected of us. Not to mention if a student requires remediation or "progress monitoring". I feel that I have sound instructional practices, but, many days I wonder if I'm headed in the right direction with the wide range of demands.

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    3. Point made Mark. If ACT or SBAC or ABCDEF...? is the standard of success perhaps time spent experiencing and demonstrating this level of questioning and understanding is the best option. Much of the information being discussed refers to many of the topics we at DCE have begun focusing on and implementing to some degree: higher DOK levels, scaffolding questioning, tasks that require deeper thought, etc. Again, restating the obvious here. Just because all of this is OUR agenda, the students' agenda needs to parallel this for great success to happen. For some kids, no matter what we try, they are going to need T & M (time and maturity). That being said, looking at any standardized test as a gauge of my effectiveness is a worrisome. Also, the students with other agendas in return demand great amounts of our most precious resource, time. We spend exorbitant amounts of time on a small percentage of our student body. I fear this is or could be creating a greater "what could have been" for our students who fall into the middle and/or upper levels despite our efforts to meet all needs.

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  8. My thoughts on the unproductive vs. productive strategies goes back to one of the additional points they made. On page 30 they state that tasks with high-cognitive demands are the most difficult to implement well, and are often transformed into less-demanding tasks during instruction. What I read here is even when we think we are implementing rich problems that will foster higher level thinking our instincts will tend toward lessening the activity.

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    1. You are correct Luke, rich is a very subjective thing. To truely be rich it would need to have a specific context to each of your 100+ students. That will never happen.

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    2. Luke
      I hear you on the time thing!!!
      Yesterday our class politely waited out a student responding to a math problem that took 12 minutes to generate. We did not let the student 'off the hook' either...perservere right? We did compliment the student for sticking with the problem and not giving up in the end. TIME IS OUR ALLY???

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  9. Okay, so I agree with a lot of what has been published so far, especially many of Coenen's concerns on time, and now this effective educator system, and what that will mean to us as individuals.
    As to the productive vs unproductive beliefs, we do well in some areas and at some moments during a "unit" . I do always wonder sometimes when they are much more concerned about the "reasoning process" then if a correct answer is produced. The details of getting a correct response are important. It would be like saying it okay for a writer to have good ideas, but who cares if they can write a complete and well constructed sentence!
    If a task should lead to a skill or vice-verse, is really dependent on the material and what the pre-understanding of that concept is. When you hit ALg. II and Pre-Cal. most material is so foreign that some intro skill work is a most before an application or task.

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    1. "The details of getting a correct response are important. It would be like saying it okay for a writer to have good ideas, but who cares if they can write a complete and well constructed sentence!"

      Totally agree. Getting the correct answer is important. I guess it's nice that it doesn't have to be one or the other. As with most things in life, it does not have to be either extreme (all concern on reasoning or all concern on correct answer), and the best approach is most likely a combination of the two. To go along with the last part, for a lot of the students, no one will care if the mathematician can't communicate what the correct answer means or the reasoning as to how they found it. We do a good job of letting the material dictate whether or not it is appropriate to go task-skill or skill-task. In general, it is to the discretion of the teacher to decide whether their material lends itself to a exploration of the material. I'm not sure students could "explore" their way to understanding logarithms effectively. Familiarity with similar context is paramount in whether or not students will be able to reason their way to understanding and in some cases in Alg II as well as in Pre-Calc, there isn't a whole lot of it.

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  10. There are a lot of really good things stated here. Although this is not an answer and I probably shouldn't be blogging when I am as tired as I am, however...I love the consistency of "productive struggle" and agree that do it properly we need more time in a period. I don't know how to make that happen.

    There are also a lot of thoughts about what is important to teach and the need to "move on." Is there a way to tie them together? Everything in math eventually relates but is the relationship close enough to connect? What is most important? I sometimes think that by having us define our top 10 items it basically meant we listed chapters (even though in looking at them they are not our chapters). But could we list a top 5? What is the MOST important thing we teach? Can we emphasize that all year? Can we take it to such depth that most of the other topics revolve around that one main topic.

    There is also a lot of talk of tasks, which I love to hear us in this discussion. It is clear that we understand the importance of applying the math so students can retain the information (while I write this it makes me reflect on John's statement about reasoning vs. accuracy). I believe that accuracy is very important but personally, it doesn't trump conceptual understanding. If they student doesn't understand they won't show accuracy. Although, our end goal should be accuracy WITH understanding. Bringing it back to tasks, Luke mentioned several times a need for showing the right tasks, where to find them, and what the instruction should look like. We have to take a look at what we are writing in this blog and a look at what is out there for us. We get what instruction should look like, however we are confined by time to truly implement it correctly. We understand the importance of a task but struggle to get them going either due to student apprehension/apathy or just finding the right one for that particular student. I struggle because I cannot show you the right task. Creating is among the top tiers on Blooms Taxonomy. Maybe we are not there yet. We are so bound to our resources that we don't trust what we believe is a task. However, speaking bluntly...there are several people in the the department that are very good at developing tasks. It is hard to understand why we aren't giving them a try. Furthermore, when the task is implemented often its rigor is decreased with writing portions left off, steps given to help the student which eliminates the "productive struggle" and reducing the rigor to make it more manageable. This is an overall statement and not meant to apply to all PLC's but I feel pretty confident if we all thought about it we can find times we saw PLC's that choose to make it easier - not harder.

    Where can we find these tasks? Search the net. Go on schommermath.blogspot.com and click on any of the links on the right side of the page. Most importantly, think about where something could apply and sit with your colleagues and brainstorm for 15 minutes. My guess is something productive will come out. I have never found a task online that I implemented without alteration. Typically, the alteration is removing instructions, not adding more.

    Finally, it was mentioned several times that we don't get the chance to talk about instruction. Isn't that what we are doing right now? We all know there is no one right way to teach. We all have our own spin on things. In math we would like to see a lesson that starts with an engaging thinker/activity, some concrete instruction (group, direct, or activity based), some gradual release where students have time to tackle some tasks/problems collaboratively. And we know how important homework is to student success. However, each class is a little different and should be. Not too different that students are learning different concepts and the classes have a different emphasis but...lets call it "unique opportunities to learn."

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  11. This is great. There isn't a whole lot of original thought left to post after all of the other great insights. I would echo what others said about not having the time to allow students to struggle. It seems that, oftentimes, the author(s) and NCTM, in general, tend to make umbrella statements about how instruction should be, with the practicality being applicable in only the lower levels. In general, until students get to high school, there are two main objectives which are to get students to use basic arithmetic to solve linear equations and identify and analyze basic geometric figures. Developmentally, this is appropriate, as it is not possible for students to comprehend much of the abstract concepts that will come later. With that in mind, conceding that I am not teaching those levels at the moment, it would seem easier to "boil down" the standards to 5 objectives at the levels up to the Junior High. In Geometry, it gets a little more involved (though, it may be possible), and for Algebra II, since much of it is new and fairly independent concepts, it becomes nearly impossible. It would seem then that, in comparison, the early adolescence mathematics classrooms have the more appropriate material (in amount and continuity) to experiment with using time for productive struggle. However, they do not, necessarily, have the clientele capable of this type of activity. Contrarily, the high school level has the appropriate clientele capable of abstract reasoning, but the breadth of material and the unfamiliarity of something like logarithmic functions, make productive struggle incorporation more difficult.
    Naturally, in my experiences teaching in the senior high, most of the instruction has been teaching the skill, first; then, using the skill in application. NCTM deems this poor teaching. However, sometimes it seems impossible to do it any other way. As I read, I thought of two approaches, then, that could possible make both parties happy. The first was presented in Mark's most recent post: to present the contextual application as an over-arching idea that the students will reason about as they develop the skill; eventually, being able to solve it. The second idea is to flip the traditional instruction by teaching the skill through technology or other resources outside of class time allowing for the in-class instruction to be used as productive struggle and practice with support. This second alternative, of course, has the student assume the responsibility of forming the foundational knowledge needed to succeed in the course. This seems impossible to achieve with a culture of students that do not do any homework as it is. As a teacher, it would be very hard to release this responsibility, when our main focus is to see our students succeed. Perhaps, if this type of expectation is started early enough (not sure when that would have to be), students would consider this as necessary routine and not as extra work. Obviously, I’m talking of ideally. Not necessarily is this practical.

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  12. The unproductive vs productive beliefs on page 23 seem reasonable. In particular, the last row, where they discuss students investigating contextual problems and generating meaning vs. matching specific types of problems with the procedure is powerful. I suppose the philosophy should be that, as a the student’s teacher for only a short amount of time and with the rapidly-changing world we live in, it will be impossible for me to give students every type of problem they could face in their life where the mathematical skill will be relevant and applicable. So, it makes sense to instead prepare students by providing them opportunities to investigate, hypothesize, and reason ways to solve open-ended, unfamiliar scenarios by using a culmination of all gathered skills (math or not). Though, in real life, many of these situations will require a fair deal of accuracy, the process of analyzing, forming a plan to solve, and making sense and communicating the answer far more prepares students for a larger array of future occurrences than solving a specific problem to get the right answer. In fact, part of the reasoning process should be, how accurate is accurate enough?. The reality is this is a process that most students don’t get with application problems. Most of the time we rob students of this kind of thinking by saying “round your answer to the nearest tenth” because we worry about the student’s laziness affecting their accuracy; perhaps, a necessary evil.

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  14. Questions:
    Is what I am teaching actually going to provide the students with something that will benefit them in the future?
    Am I just teaching x skill to mark it off on a checklist (standards)?
    Is this lesson preparing students for life or for their next math class?
    Isn’t it possible to teach life skills not just math skills in math classes?
    Obviously, for some of my students, this class is one checkpoint on a path to higher level mathematics (engineering, statistical analysis, etc.), but for most of the students in the classes I teach it is not. Am I not doing the majority a disservice by not providing them the opportunity to reason about complex mathematical ideas within context and communicate their conclusions through writing, speaking, creating, etc; as those skills are applicable to all students, not just the top 10%?

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