- Which type of learning do we excel at? Which type are our students struggling in?
- What are your thoughts in regards to "baseless rules" as mentioned on page 62, line 21.
- These are our individual tricks that often times don't carry through from one teacher to another.
- "...given the perception that most students will not be able to think mathematically, there is little point in providing them more challenging tasks..."
- What are the ramifications of this statement?
- How are we doing on the unproductive vs. productive beliefs? What are your thoughts on them?
- Other thoughts?

## Friday, April 11, 2014

### Learning (Pages 61-71)

The 3rd section is all about student learning. Early in the section is mentions 5 types of learning. These are listed at the right in the poll question. Take a moment and rate how we are doing in DCE in regards to each type of learning.

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In response to the "baseless" rules portion. I think it is easy to say that we are just using "tricks" when actually we are reminding students of learning they have experienced in prior grades. To use their example, when we say "flip the second fraction and multiply" we are actually tying back to the learning that has happened at an earlier time. Does NCTM really want us to construct why we are multiplying by the reciprocal every time we do it? I certainly hope not! What we need to be careful of in these situations is how are we tying back to prior learning. Are we using the same terminology when we talk about a topic that we know they learned in prior grades? I agree that we can't simply use "tricks" with students and hope that they are developing a stronger mathematical sense. I also question how many times we have to construct the same thing?

ReplyDeleteTrue. They even say that "memorization should be preceded by the development of number sense." So, at what point is it safe to assume that we can stop proving why we can multiply by the reciprocal because they have the number sense, and start memorizing? At some point, for students to become more successful mathematicians, they must be able to know this phenomenon exists without having to reason all the way through it. Without fluency, progress would take a very long time.

DeleteThe type that I think we most excel at is the procedural fluency. I think that traditionally this has been the focus of most mathematical programs. As a result, through many years of developing these programs around this focus, we have become quite adept at providing students with the opportunity to gain procedural fluency. Also, since, traditionally, this is what has been measured on standardized tests that have been the measure of student, teacher, and district “success,” this has also encouraged this to be the primary focus of most math programs. More recently, there has been a shift in paradigm from the traditional emphasis on procedure and skill, which are still essential, to building a more conceptual understanding of the mathematical concepts. We are still developing in these areas. We are giving students more opportunities to struggle through problems that require them to use adaptive reasoning, develop the proper strategy to solve, and then use the procedures they learned to deduce and reason with their final answer.

ReplyDeleteThe type of learning that our students most struggle with is the Productive Disposition. Without going into great lengths about how our students don’t feel an importance for many things that they do in school, I believe we can all agree that, out of all of the types of learning, this is the most difficult to instill in our students and the biggest obstacle that we face. This is especially true in the senior high at our district. We do our best to give students opportunities to see how the math relates to their lives, but when it comes down to it, they are the ones that decide what their disposition will be. They also feel that they are incapable of doing the math that we give them, when, in actuality, they are just incapable of doing the math we are giving them without practicing it.

yes...procedural fluency tops the list for what we excel at! this seems to be the 'meat and potatoes' of our instruction at each grade level...it is what it is...

Deletestudents seem to struggle most with application and questions that now appear to relate to common core...this is certainly part of un-chartered waters to some degree...at the junior high level this requires students to sustain a task for an extended length of time...it does not relate to the go-go-go atmosphere in this building...this takes time to teach and example and develop!!! i am glad the clocks don't 'click' every single minute anymore!!! :)

The “baseless rules” that the author discusses are ways that teachers implement simple procedures that replace mathematical understanding in attempt to make a larger group of individuals successful. I think that the intentions of the teacher to do so are good. However the result is often counterproductive. Sometimes the reason a teacher has provided such “baseless rules” is because the conceptual understanding is assumed to be prerequisite and time for in-class remediation is not provided due to the amount of course-level-appropriate material that needs to be covered in such a small amount of time. Also, the proof of a formula or skill can be a painstaking process (this is especially true in the higher level math course). Even with having proven once that the area of a triangle can be found by taking the product of two sides and the sine of the angle in between them and where it comes from, having many other unrelated units and facts from other disciplines following that proof, and followed, yet, by a summer break, it is likely that the student will not remember the proof of the formula after that. Is it worth, then, proving the formula all over again? Are we given enough time for that? How do you relate it to logarithmic functions, the Great Depression, limericks, and covalent bonds so that students won’t forget the proof of this formula? More impossible, yet, how do you do it when they won’t do their homework? Any rule or formula, even having been proven or deduced, will seem like a baseless rule after that.

ReplyDeletedo this mean no more FOIL?

Deletedo this mean no more FISH METHOD?

what is this world coming too???

Despite my rant on the inevitability of students having disconnects of concept, it is still important that we provide students with the opportunities to stretch their minds and skills with mathematical tasks. I have written at great lengths, previously, about the benefits of these tasks, but I should reinstate my beliefs that these tasks benefit all who will do them, not just those in at the top of the class. These tasks are some of the few opportunities that students have to struggle, reason, and communicate about mathematics. They provide students with valuable problem solving skills, in general; not just mathematically. These deeper applications and thoughts of the math skills won’t necessarily always provide insight on where the skills come from or foundations of why they work, but might provide students with a conceptual understanding of how it applies to their live or is meaningful. Students have a better chance of retaining information if they can tie it to previous knowledge, especially if that previous knowledge extends outside of the mathematics classroom.

ReplyDeleteI think that the Unproductive and Productive Beliefs sum up the idea of memorization of skills and facts is not effective learning. For a student to be proficient they need to know the why and how. They need to know why it works and they need to how it applies to them.

I always question when I see an article say "mathematics can be learned by all students" Okay, what level of math are we talking? big difference between say 6-10th grade math and things like Pre-Calc. and above. The polar conics we did in class today are some heavy duty stuff. Are there not gifted writers and artists, and atheletes, etc... We have high and low math abilities also. It is just what level is the cut off of "acceptable" math understanding to be accepted by society.

ReplyDeleteThat and I do think math is a universal language is my only knock on ht unproductive beliefs. (Is that not why Carl Sagan put math on the voyager 1 space satilliete for any other intelligence to show our universal knowledge?)

John to go along with your statement about all students learning mathematics, there is a difference in using math functionally and using really abstract math concepts. I view many of the concepts taught in the mid grades (6-8 or even some 9th grade concepts) as usable in many careers/applications in life. If we can have all or most all young people able to at least use and apply this knowledge we are bettering our work place and communities. Many of the jobs/careers locally have a math test or need for mathematical concepts required to get the position or carry out the work in it. Saying all people can grasp and use some of the mathematics addressed by you John, I totally agree. I would like authors to be more descriptive about this...

Deletecan most students think mathematically? can we provide them with challenging tasks to help?

ReplyDeleteyes...yes...BUT...this requires special tactical training!!! if it was just me vs this person it would be a no-brainer that both of these questions above could be done!!! it is me vs 30 like this so my approach needs to be varied to be effective!!! i struggle with this at times because this juggling act in the classroom takes practice and training...in a perfect world a portion of our plc would help to improve individual instruction in this area...plc's are busy places these days...sometimes classes can feel like alot of nothing gets done when eveyone is off on their own learning tangents.

John I agree that it also depends on what level of math understanding we are dealing with too...

These "baseless rules" are important procedures the students need to possess as they move along in their education. Perhaps they begin as "baseless rules" because the child does not possess the ability to understand at the time of the initial presentation of the procedure, but my hope would be that after repeated exposure to the concept these procedures would no longer be considered baseless. It takes time and exposure for the students to understand and integrate these concepts into something more permanent and usable.

ReplyDelete