The Right Answer

I have been enjoying reading Suzanne Chapin's Classroom discussions: Using math talk to help students learn.  One thing that has stood out for me so far is her discussion about our over emphasis on correct answers.  She makes the point that helping students develop the ability to talk about and through their reasoning in a given problem is more important to achieve a deeper understanding of mathematical concepts.  She does not dismiss correct answers of course, only postpones them.  She also talks about when it is important to provide "the answer" right away and that is based on the idea of knowledge that is "external" to a student rather than knowledge that can be discovered through reasoning.  Such knowledge includes the social conventions of math such as the meanings of symbols etc.

A good example of focusing on reasoning is Case 1(page 22).  In this classroom example, the teacher is introducing problems which involve two equations and two unknowns.  The unknowns are indicated by a square and a triangle.  In each set of equations the values of the squares and triangles must be the same.  This type of problem helps begin preparing students for algebra.  In Case 1, the teacher circulates as students try and solve the first set.  She notices that a number of students do not understand that the values for the squares and triangles must be the same in both equations.  She notices one student that has the "right answer."  She brings the students back together and purposefully calls on a student who did not understand the problem to demonstrate their "solution."  She asks her to explain her answer.  Then she asks if everyone agrees.  The girl with the right answer raises her hand and says she disagrees.  The teacher asks her to explain why.  A class conversation ensued in which the girl who had provided the initial answer comes to understand her error and the girl who had the right answer develops a deeper understanding of her reasoning by having to explain it.  The right answer is never actually shown to the class.  Instead, after the discussion, she gives them another problem to work on with a partner.

I admit, at first I was surprised that the right answer was not presented.  As I thought about it, I realized it was a specific strategy of the teacher to de-emphasize the right answer.  I can see how this could be beneficial for students to help them realize that the process is as important (maybe more so in some cases) than the product.  It might, over time, decrease the student's desire to just be given the answer.

Math Curriculum

I am somewhat amazed at times by how quickly school districts seem to cycle through different curriculums (in all subject areas, but math especially).  It makes me think a lot about curriculum and whether curriculum is better, than say, a teacher having a general facility with a subject area and an overview of what needs to be taught at a given grade level.  The constant cycling through curriculum makes me suspicious that we are searching for some sort of silver bullet.  A curriculum that serves every student everywhere.  This seems to fly in the face of everything we are learning in this program (that people learn differently) and probably in the face of a whole lot of plain, common sense.

I am not even sure it would be bad to have a curriculum..or at least a framework.  The problem I see (and that I have heard from a number of teachers) is that this constant cycling of curriculums means that they no sooner get comfortable with teaching a particular curriculum then they are thrown an entirely different curriculum.  One example is the Seattle Public School Districts change from TERC (more of a discovery math curriculum) to Everyday Math which spirals so quickly it leaves your head spinning.  But I digress.  Anyway, I think the issue is that when teachers get comfortable with something, if they are self reflective, they can then begin to manipulate the curriculum to fit the needs of more students.  The constant shifting however means that they are left to learn an entirely new curriculum which takes time to get used to teaching and time to see its flaws and the ways it does and does not serve students.  Unfortunately, I am not sure that the Districts adopting these curriculums understand that teachers need to have the independence and flexibility to work in and out of any given curriculum to meet the needs of their students.  Hence, in some places we get scripted curriculum.  What a travesty for students.

Death of a Subject

Yesterday we spent time in our math and science methods classes and it struck me that there was a common theme: making the material come alive.  Is science a set of discrete facts to be learned -- discovered by white men in lab coats?  Or is it accessible to everyone, alive and changing?  Is math a set of discrete facts to be learned-- discovered by white men wearing tweed coats?  Or...see above, same question.

We have talked about making the material relevant to student's lives since the start of the program so this is not a major "aha" moment.  However, it struck me that all subject matter is capable of being "killed." Math, science, history, reading, writing...you name it. They can all be taught in a way that reduces them to a set of rules and facts to be memorized.   Each can be taught in a way that makes them lack relevance to anyone -much less an elementary, middle, or high school student. Take, for example, grammar.  I remember diagraming sentences in the 9th grade.  While I never have struggled that much with grammar in my own writing, I detested learning formal grammar.  Indeed, it was something at which I failed miserably!  Math, science, history -- it was much the same.  I was not "interested" in these things.  They seemed irrelevant at the time and so, I resisted learning them or learned them without much care.

I was very involved in politics when I was young (so I was living history, in fact) and yet the two -- my history class and my budding "political;" life outside were as separate as Africa is from North America.  I think what this illustrates is another point -- that we can tap into what students are interested in from their own lives to engage them in learning.  I suppose if my history teachers had made a connection between my political activities outside of school and history, I might have been more interested.

As teachers we need to know our students well enough to make the connection between their learning and their lives.  I think it is also critical to allow students time to explore their own ideas and interests and respect their discovery process....I guess this is similar to what we talk about in math.  The right answer may not always be the most important or desirable endpoint -- the process along the way is what counts.  I like what Regie Routman states in Reading Essentials:

Unless we reach into our students' hearts, we have no entry into their minds.  Through drill and memorization, we can get students to complete assignments and pass tests.  But their is a price to pay for such short term accomplishment.  We will never inspire our students to learn for their own sake and to love coming to school (pg. 12).

Making Ammends with Math

         I decided fairly quickly this quarter that if I am going to teach I am going to need to make my ammends with math.  There are many times when I have hated math and many occasions when I thought I was an absolute hopeless case when it came to understanding it....like for most of my grade school, middle school and early high school years...and then, again, studying physics (college) or biostatistics (grad school).  Yet I actually like math and when the stars are aligned in the universe, I can be quite successful at it.  Usually the stars aligning relates directly to the amount of patience and knowledge a teacher has to spend time explaining to me the minutiae of how a formula actually works.  I mean ACTUALLY WORKS...sort of like at a theological level.  This is why I had such an "aha!" moment when Robin pointed out the different ways it was possible to be "smart" in math.  I guess some might say it is just a way of making everyone feel good but I can actually attest to its truth.  Take, for example, my absolute hopeless ability to understand geometry and trigonometry but, ironically, the ease and grace to which I flew through calculus.  My hopelessness at statistics was matched by an intuitive understanding of epidemiology.  My GRE math scores were not superb but my scores in logic were.  I have yet to completely figure out my own way of understanding math concepts....but I have noticed that the ways in which math is being taught now...even basic math, is helping me understand things I never understood before.  Like -- place value.  So, somewhat sheepishly, I get to be excited as I learn math along with my six year old.

"Lies, Damned Lies, and Statistics"

In class this week we looked at the Gapminder website which allows people to manipulate data to look at "relationships" between different variables.  I enjoyed playing with the data and the site has a big "ooh" and "ahh" factor.  The main issue is having enough of an understanding of how statistics works to be able to critically appraise the data you are seeing.  I like Mark Twain's quote  "There are three kinds of lies: Lies, damned lies, and statistics."  In research (and, lets not forget, politics), it is pretty well known that you can find a relationship between any two sets of data if you look hard enough.  This is why, in general, researchers start out with a specific hypothesis that they are seeking to prove or disprove based on a (hopefully) carefully structured study design.  Starting out with a question should preclude a researcher from randomly searching through the data they have collected to find a relationship.  A good design should try and control for potential confounding factors or at the very least acknowledge their potential existence.  Additionally, the availability of data and the quality of that data from any one country or source needs to be considered as well.  Certainly, some of the relationships that can be shown and discussed would be perfectly valid. Some, less so.  It is also true that we are all bombarded with data and statistics every day so the ability to evaluate this type of information is critical.  With all of this in mind, I am trying to figure out ways that this type of site can be best used in a classroom.

Calculating Density: Who Cares?

My 8th grade middle school class is studying matter.  One of the properties they are learning about is density.  They have been conducting various experiments to calculate the density of various regular and irregular objects.  They are learning the formula D=M/V (density is equal to the mass of an object divided by its volume).  One of the issues that has arisen is how students learn to manipulate equations.

Math Identity

Leatham and Hill (Exploring our Math, 2010) talk about peoples' (students, teachers) beliefs and attitudes about math as something that is generally more complex than we realize.  They have devised a series of activities to help students think about their own relationship to math as well as their beliefs about math.  Following up with whole class discussions is meant to shed light on the variety of perspectives about math. The intention of the reflection and discussion is to help "expand our beliefs about math and...reinforce the idea that math is creative".  As teachers, becoming more aware of how our students perceive math can help us design our classroom and lessons to empower rather than disempower students.

Counting by 10s: Memorizing vs. Understanding

I have been thinking a great deal about the importance of gaining a true understanding of math concepts as opposed to the "old school" way of having students memorize facts and formulas.  While I believe there is a place for memorization (memorizing times tables, for example), I thought I had come to a comfortable place in terms of believing that conceptual understanding must come first.  Students should not be allowed to simply memorize something that they have no understanding of.

Power of Ten Video

This is a great video.

Bah Humbug: Literacy in Math

The article "Using Creative Writing and Literature in Mathematics Class" has me wondering about how we decide what and how to teach as well as how we measure the effectiveness of our teaching.  On the one hand, I can envision how story telling in math could end up being useful as a learning tool if done appropriately.  On the other hand, I can imagine how it might also turn into a silly, meaningless exercise.  Reading the examples did not make me confident that the students were focused on the math as much as just figuring out how to cram  a math concept into a familiar story line (I am thinking about the rendition of the Three Billy Goats Gruff in particular)....kind of like fitting a square peg into a round hole. 

Micromanaging Math

I know.  In general, "micromanaging" is considered a negative thing.  In the context of teaching math though, micromanaging may be just what a teacher needs to do to help ensure success. The reading "Orchestrating Discussions" (Smith, M.S. et al. 2009) helped me reflect more deeply on how lessons and discussions in math need to be carefully "orchestrated" by the teacher prior to the lesson.  The authors provide a good framework for such planning called the "Five Practices Model" (pg. 550). These practices include everything from considering the range of possible student responses before the lesson to "sequencing" and "connecting" those responses thoughtfully in order to optimize student understanding. 

Discovery Math and Teacher Training

As a new teacher and a parent, math has been on my mind quite a bit. In February of 2010, a Seattle judge ruled against a Seattle Public School District curriculum known as "Discovery Math".  The math program encouraged constructivist problem solving and group work as opposed to more traditional drill methods.  According to the ruling, the curriculum favored middle class students at the expense of lower income students, students of color, and ESL students (see: http://www.seattlepi.com/local/415049_math04.html).  Indeed, after the adoption of the program at Garfield High School, math scores among poor, minority and ESL students dropped dramatically. So what does this mean?