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BSIE 2010: Session 25

Paul Grobstein's picture

Brain, Science, and Inquiry-Based Education
K-12 Summer Institute 2010



  Session 25

Case Study: Modeling, When Did the Cougar Die?

Rebecca Vandiver

Rebecca Vandiver is Howard Hughes Medical Institute Post-Doctoral Fellow in Mathematics and Biology.  Rebecca's research involves computer modeling of biological systems and she has been working with colleagues in biology to develop curricular materials that help enhance the mathematical sophistication of students in biology and in the sciences generally. 

Picking up  from Tuesday

I never had chemistry and I've always been lousy at math so I couldn't become as engaged as I would have liked too ... cdivo39

The lesson was hard for me to understand . However,  I enjoyed watching the teacher using all kind of techniques in order for us to understand the lesson.  He used videos, stories, discussions, dialog, and the survey...all to keep us on track ... Shoshana

I also love chemistry but I also find that students know matter what age have a reluctance when you mention this word ... we must allow the student to become aware that chemistry is a natural part of our life.  Terminology can come second and if we allow the students to explore, first, I think their stories about chemistry will change ... Judith

Often we look for familiarity in  relationships; something we can relate to; something the learner already knows something about, and which may be useful in gaining a better grasp of the concept.  When one explanation is not clearly comprehended, we "explain it another way." ... Mattie

The issue of whether "story" (or "explanation" or "construction") is or is not different from "facts," "concepts," etc is a point of conflict which (like all points of conflict) can be the grist for the development of new ways of thinking about things ... Paul G

I appreciate Paul G.'s comments about storytelling vs. story telling. However I would argue that students do both in this exercise. As they write the storytelling stories that I read (and enjoy), they also are story telling to themselves (and maybe students around them). To me the essence of this was the conversations that pairs and trios around the room were having as they tried to incorporate concepts into their stories. The conversations I heard were ones where as Paul puts it "people were telling the story that they have heard differently." So both story telling and storytelling are occurring ... Paul B


Cougar cooling lab materials



Judith Lucas-Odom's picture

Cougar Problem

Interesting, this was a difficult one to get a handle on.  Math and science should have a smooth transition and getting students to see this transition can be layered but do able.  I need to see this idea again.

Paul Grobstein's picture

From math to reductionism as an aid to co-constructivity?

Rich/generative session facilitated by Rebecca, as evidenced both during the session and in conversation, after and below.  What I want to think more about is the issue of how to get students (at all ages) more engaged/comfortable with .... chemistry, biology, math, etc.  Intrigued by the notion that there is, in Keith's words, a tendency in some people to experience a frustrating disconnection with all of these because of ... a common tendency to try and make sense of things reductionistically?  Given a problem, one can try and make sense of it by thinking of it in relation to other somewhat similar things ... what I would call "lateral" thinking/inquiring/processing.   Alternatively, one can try and make sense of it by breaking it down into smaller things and those in turn into smaller things etc etc and then building things back up to the problem.  My sense is that many people aren't familiar with that strategy, lose track in the successive reductions of what the problem was that was is exploring the reductions to achieve.  If so, the problem that needs to be addressed isn't "math" or "science," but instead the inclination/ability to suspend disbelief through more focused inquiry while maintaining a sense of what is to be made sense of at a broader scale.  And the antidote is experiences that help one acquire confidence in such a procedure.  Like very much the suggestion that it would help to have "stories" at each step of reduction that help people keep in mind the relation between levels of reduction and the broader problem one hopes the reductions will help with.

All this connects, in interesting ways, to an evolving systems group discussion of "formal systems."  The unconscious always has understandings without one being able to say where they come from.  The formal systems process provides a set of conscious underpinnings for those understandings which can in turn both be more effectively tested and manipulated to yield alternative possible understandings.  Maybe what we need for the institute, in addition to a session on inquiry/conversation skills, is a session on formal systems?  As an additional way to enhance the generatory of co-constructive inquiry/conversation?

alesnick's picture

confidence in successive reductions

As one of the people who struggles with breaking problem down into smaller things then building them back up, I think way of naming the problem (not as engagement with "math" or "science" but rather imaginative suspension of resolution without losing track of that which we seek to resolve).  For me, it's important to get farther to a sense purpose (not necessarily, or even primary, practical relevance, which I often find a rather boring, scolding appeal -- like, I don't really care about the design of elevators or the lift airplanes) -- of import -- what is at stake in thought -- than is often clear to me.  And the smaller and smaller bits also come across to me as dry as a bone.  Too sheer, no handholds.  I want handholds that are meaning-oriented, not thing- or process- or achievement-oriented. 



Mattie Davis's picture

~~~~ M E M O R I E S ~~~

How much of what we learn is actually remembered and "ready to be used (consciously) and recalled" at a moment's notice?  I have come to understand that a significant amount of observations and stories are located within the unconscious recesses of the brain.  The unconscious contains a wealth of "observations and stories".   Certain stories and observations are easier to list (on paper or verbally) because of the conscious brain.  There are more complex memories which we are not able to list as easily.  We are not always able to connect-the-dots.  As a middle school student, I recall studying fractions.  I could compute all the basic functions by plugging numbers into a formulas and following patterns to solve the problems.  I recall doing this, but did not fully comprehend exaclty what I was doing.  During my first year of college,  I was able to add, subtract, multiply, and divide fractions, but the difference was that I actually understood what I was doing and was able to see patterns.  Sometimes we may think that our students are grasping stories and observations at a particular level, when in in fact they may be just going through the motions.  The students in any given classroom are very diverse human beings.   It is of the utmost importance that we relate  to each one individually as a unique being.  A learner's ability to perform or 'shut down' may be directly related to expectations or lack of expectations of the teacher, and the relationship between teacher and student.  But there are those times when stories in the unconscious (given certain conditions) become conscious.

Regina Toscani's picture

The cougar problem

Rebecca's approach to this problem was interesting.  I don't think I ever had a class where I had to derive an equation to solve the problem.  However, once we started the data collecting process, I felt very comfortable.  I enjoy manipulating numbers and plotting them.  Like many others, I had a diificult time (at first) with the rate of change.  Intuitively, I knew that the rate changed as the temperture changed.  Yet the first attempt to plot this was wrong.  Rebecca had to point out the the "x" axis should be the temperture data.  It took me awhile to understand but I eventually did.

I thought that I might incorporate the concept of rate of change (of temperture) into my Life Skills curriculum.  The first lesson would be having each student taking their body temperture and recording this.  Of course some of the students will have a difficult time with this so first I'll model the action, and pair the students to help each other.  The lesson will have a discussion about what this number means (in terms of health).  The next group of lessons will be having the students outside ,taking and recording the air temperture.  Then a general discussion of how that temperture makes us feel (Hot, cool, good, etc.).  After several days of this (and hopefully there be a range of tempertures) we'll review the data and comments.  Some students might be able to relate that the smaller the number of the temperture, the more people become colder.  Again we'll have a discussion, (including how a person's clothing will change as the temperture number gets smaller).  The next lesson I'll reintroduce the body's temperture and ask the children to compare that number to the air temperture.  The next part is to subtract the 2 numbers, (and still writing comments about how the air's temperture make us feel) and after several days, hopefully a pattern will form.  Ther bigger the difference between the numbers the more the temperture affects us. 

Geneva Tolliferreo's picture

3rd Wednesday PM

So many questions and comments...

Paul gets the train running and then pulls the brake in the name of 'tabling conversations' until or for the forum.  Argh-----------frustrating!

However...if we teach the lesson without using the correct terms, at what point to do introduce and teach the terms so students are able to assimilate in the real world?

joycetheriot's picture

Facilitating Problem Solving

In today’s math-aligned workshop, Paul G. commented that many of us were frustrated because we were forced to think along these very narrow avenues of process and then expected to connect some of the small parts to a different set of values and make meaning. Many in the workshop experienced anxiety or irritation because the usefulness of that next course of action was not apparent.

As I notice an increasing math disconnect in my science classes, I am very interested in this line of inquiry. I can recognize that assembling the finished small parts into a new process with yet new featured problems that need a more detailed representation is quite complex and I want to help my students develop the skills that they could use to find meaning in each succeeding part.

A model to represent this narrow to large problem-solving approach might be similar to solving a large jigsaw puzzle that has no “finished” picture on the box cover. We work on clumps searching for related parts and then finding connections to other clumps that finally lead to a connection of every piece. However I suspect that this is too linear to represent the complexity of this issue.

I’m trying to relate this to Sudoku and thinking that instead of running helter-skelter around the entire puzzles looking for connections, that we instead have better results if we look within single boxes and solve the connections to related rows.



Kim Fuller's picture

I still do not know when the cougar died?

I found this interesting and challenging at the same time. I would have been so lost without the girls helping me to do what needed to be done. But I did manage to get something out of it as the day moved on. As an educator we must try to understand what a child feels when he/she is not getting it. Today I did not get a lot of it. I was with it up to a point, although it was wonderfully presented and the facilitator was wonderful in her presentation I just did not know how to get to the result that I was hoping to find. The other thing is that I was hoping to find out when the cougar died? 


Educators must be very mindful of their students,  to try to catch when they go on shutdown. It will not be hard to spot just watch their behavior and you will know who is not engaged and who is. If students can be caught at the point of (I do not get it) before they go in to the shutdown mode then we would not lose so many student to miss behavior or just sheer boredom.  I applauded all educators that have to teach and be the class room watch person as well.  


I am that person that has to break it down so that I can understand it then I will learn the technical terms.  I so agree, break it down to the students and then teach them the technical term.


RecycleJack Marine's picture

Breaking up the story/Endangered Cat Species

I caught the "tail" of the cougar lesson. I sat down next-to and understood how Joyce plotted the different temperature redings through a specific time. But, as Keith pointed out, I too began to fade as we plotted negative number relationships. I'm glad Paul began the discussion about not enjoying math, and that all students could better understand mathematical concepts if the story was broken down into pieces. I think Susan's various metaphors for "slopes" is an interesting and probably effective. But I am afraid that if I break down a large concept into many various stories I may never get to the end of one subject's unit.

I plan to direct, or present a constructive science lesson to the class on Thursday. Based on what Paul suggested about breaking up the big story into parts, I will use this in my lesson. I hope that all of the participants who complete my activity will then be able to answer this question: How do all of these small parts lead to a whole?

Keith Sgrillo's picture

Response of Emotion

This was the first time during the institute where a particular subject matter presentation really provoked an emotional response to the activity and content.  I was very grateful to have had this experience. It really helped me to make connections to content that I don't think were intended (I guess this is what you could call the hidden curriculum).  I found it very interesting that when I hit my threshold I just could not process the information. It began to create feelings of frustration, irritation, shutting down, and even a question of self-worth as it pertained to my ability to contribute to the activity.  I then found myself feeling empathy for my partner who was breezing through the activity and thinking of many alternative ways to figure the problem out. 


As I reflected further, I analyzed one particular emotion that was brought out that I had not felt in a very long time.  This was the feeling of self-worth in terms of the value I had in contributing.  I began to associate this to what I will call an "Americanism" to be the best, to out-achieve others.  When I was unable to attain this, I felt left out.  In every aspect of my life, from sports to school, to social settings that I should stand out and be on top (as I think we continue to perpetuate this message in schools through standardized testing).  The "Americanism" to out-do everyone else created this impulse to become intensely competitive, even if it was just with myself, that burnt me out because my conscious and subconscious just were not clicking.  I was saying to myself "it's ok and no one really expects you to get this" but inside I felt this almost burning feeling of agitation in my brain that seemed to be a self defeating notion saying "you should be able to get this.  Why can't you get this?"  That conflict ultimately lead to shutting down and a total disconnect.  I feel as thought this has really brought me closer to my students and will encourage much more reflection on my lessons in the future.  So my question then is what message are we really trying to convey to our students and the educational community.  "Are we trying to educate our children to be the best or are we trying to teach our children to be better?"  jI feel these two messages are really in direct conflict with one another. 

alesnick's picture

"out-achieve the others" or fail?

Hi Keith and others,

This post really reminds me of something you said, Keith, the day I was there about how our ways of schooling today are essentially creating failures as people are promised jobs and futures, if only they follow the directions, that actually don't exist.  The idea that the only way to be okay is to out-achieve others seems tied to this, and I agree it has a distinctly American feel to it!  My daughter (age 10) is conflicted because she likes to be on the swim team but didn't win any prizes.  She's not sure it's okay or possible to be part of the team without being the best on the team.  This is such a difficult place to be! And yet, I think many of us dwell there much of the time!  I agree with you that opening space for reflection and dialogue about this offers a way to manage and possibly change it.

Susan Dorfman's picture

The Cougar Connection

Rebecca Vandiver's model lesson was complex and multilayered, allowing for various adaptations to subject specific courses and grade levels. I could see the cooling experiment with the math component in a high school physics class. I plan to use the cougar connection and the cooling experiment with the graphing exercise but without the math derivation in my grade 7 biology class. As a result of the debriefing discussion, I might try the change in rate with manipulatives. To develop this part of the activty, I will consult a math teacher teaching middle school.

As Rebecca stated, the cougar story motivates the students by presenting a problem to solve. Is is the hook to gain their interest and attention. Our colleagues suggested using a human forensic problem, but my students would be interested in either the animal or human example. They are already interested in conservation issues. I will have them read a conservation article from the grade appropriate magazine, Science World, prior to the experiment. A discussion of the article will connect the ideas we discuss to the problem of limiting cougar hunting and how to enforce the law. Rebecca's model included stopping points in her presentation to allow students to solidify their understanding before going on to the next step in the lesson by solving a specific problem. This also allowed time for students with questions to gain the help they needed to go on. These opportunities are important to encourage students to continue even if they begin to feel out of their comfort zone. For this reason, Rebecca's model needs to slow down at the point of the introduction of rate of change. Wil and Paul both suggested that a new story needs to be introduced at this point. I agree. In grade 7, rate of change is understood well with the example of sledding down a hill. The students make the connection, and without prompt, will use the word slope. We can then extend the idea to lines having slopes.

Today's experience reminded me that students come to a class with a variety of likes and dislikes based on abiltities and past experiences, both positive and negative. Our approach to problem solving including the words we use can create an encouraging or inhibiting experience.

Jessica Watkins's picture

Thinking "Up"

As Jenna, Kate and I discussed in a meeting with Wil and Paul yesterday, Rebecca's presentation proved that starting out with an interesting, far-reaching question that everyone can understand makes more complex, mathematical material easily accessible.  Relating something so concrete, such as graphs and slopes, to something emotionally-connecting like wildlife conservation was a good decision, particularly within our group of mostly nonmath teachers.  Her continuous rounds between groups during the activity was encouraging because she took the time to answer all questions and stressed that fact that there was no "right" answer.  However, even after all of that, there was still much confusion floating about the room at the end of the activity when we began our follow-up discussion.

It seems as if some people just can't learn certain subjects, no matter how hard they try (example: Keith's lifelong struggle with math).  Why is this?  If it is true that we are all born with the capability to learn, a notion that we have pushed throughout the entire Institute, why can't some people learn those subjects at which others excel?  Does personal background/culture play a role?  The idea of breaking down a larger story into smaller ideas and questions sounds like it would work if, after each step, the entire group paused to make sure everyone was understanding the material.  Then, students could work their way back up to the larger picture and the original question by first going through the smaller ones ("thinking up").

Keith Sgrillo's picture

Culture as a factor

I think that culture clearly plays a huge role, as I suspect it did in this experience for me.  But I also think that there is a degree of "wiring" that takes place during development that may predispose some limits.  Take the professional athlete for example.  Thier bodies just seem to be connected and wired in such a way that they are able to produce incredible physical feats that the "average" person cannot.  In most cases, no matter how hard you try, you will never be able to run a 40 yard dash in 4.2 seconds (I know, I've tried).  You can definitly improve your time, but it is doutful we would reach the level of a world class sprinter.  So if all parts of the body are connected through the same neurological pathways, then why should it be any different for the brain to have certain limitations for some and not for others.   If you read my response to the presentation I tried to outline one cultural aspect that I feel got in the way.  Great post.  Very insightful.  

Geneva Tolliferreo's picture

3rd Wednesday AM

Interesting, doable, fun...and this would be the sentiment of my students too.

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