Problem Solving as a Basis for Reform in Curriculum and Instruction.

Hiebert et al made many references to Dewey's views on problem-based learning (PBL).  For this, I was very grateful.  Like Dewey, I found the reading easy and insightful.  Overall, Hiebert et al were discussing the importance of making the subject problematic.  This is opposed to learning a skill so that you can find the answer as quickly as possible.  The difficulty lies in attempting to foster a desire to solve a problem in the students.  Numerous times the idea of making the problem a "real-world problem" came up.  This idea in itself can be difficult to wrap one's head around.  Is a real-world problem one that only occurs outside of school typically?  Does it have to be related to an occupation, that isn't a mathematician or mathematics teacher?  It was decided that perhaps we don't need to focus on the real-world problem, and instead focus on the process.  By establishing a culture in a classroom where reflective inquiry, and PBL are fostered and have become second nature, any type of problem can produce effective "residue".  What is residue you ask?  Residue is a term that has been defined in this paper as the "understandings that remain after an activity is over" (p.17).  If there is no residue, then what was the point?

I find that it can be very difficult to produce PBL without the necessary basic skills.  I believe that it is a fine balance between being a teacher who stands at the front of the classroom and provides knowledge to their students, and being the teacher who provides a classroom with an embedded culture that promotes reflective inquiry and PBL.  I feel that this is perhaps one of those things that appears scary to try out, knowing that it could very easily crash and burn, but at the same time produce interesting results, whether it goes the way you think it will or not.

The question I have is:

Do you think that this style of teaching is more appropriate at certain age levels than others?  On a spectrum does PBL lead towards understanding of relationships, or does teaching of basic facts/relationships lead to PBL which leads to deeper understanding of said relationships?

When Learning No Longer Matters

In this article, Boaler examines a school in California that has been labeled as "underperforming" despite her research saying otherwise.  The school is found in an area that is typically represented by a low socio-economic-status population.  The school itself receives minimal funding, however the mathematics teachers have developed strategies that do not necessarily rely on monetary resources.  "The mathematics department plans lessons collaboratively, and the teachers meet every week to discuss and improve  their lessons.  They visit one another's classes frequently, and every new teacher is given the opportunity to watch every lesson that he or she will teach being taught by an experienced colleague first" (p.502).  From Boaler's research, she has found that these strategies have made vast improvements in the students' mathematical learning.  These students have surpassed their fellow peers in other California state schools.  The reason for their low grade on performance from the state is due to the standardized testing (SAT-9) that takes place.  Many of the students have poor language reading fluency and comprehension, and many of the questions on the test are very wordy.

We have been encouraged, as teachers, to incorporate more society-contexed questions into our teachings.  Boaler makes an interesting point that these types of questions may be useful when first learning a new concept, however in the end we are assessing their mathematical ability and not their English language comprehension.

The labels placed on schools can be demoralizing to both student and teacher.  Despite fantastic improvements in student learning, more and more teachers are feeling the pressure to 'teach to the test' so that their school receives a higher score/ranking.  It seems that more students are only satisfied with an assessment of the test-taking skills, rather than their ability to learn.

How can the system change to acknowledge and reward schools such as this one?

Unpacking Pedagogical Content Knowledge

Hill, Ball, and Schilling created a large-scale study involving thousands of teachers.  The purpose of the study was to analyze two major factors in effective teachers: knowledge of curriculum, and knowledge of student learning.  Teachers analyzed student work and attempted to make sense of where they went wrong and for what reason.  A large portion of the research results was based on a multiple choice platform.  It was found that the wrong answers in the multiple choice were almost too wrong, making it easier for the teachers to choose the correct answer.  Upon reflection, the researchers decided that further research should be accomplished through more open-ended questions.  It was very difficult to differentiate what specifically allowed a teacher to answer a question correctly.  It was decided that for many of the questions, teachers drew from both their content knowledge, and their knowledge of student learning.  Even teachers who supposedly had minimal content knowledge of a particular strain of math, were able to effectively determine where the student went wrong, because their knowledge of how students think compensated.  In the end, Hill, Ball, and Schilling were able to say that knowledge of student learning was important in being an effective teacher, however they were not able to specifically analyze solely.

One takeaway that I had from this article is that I appreciate even more so the benefit of really analyzing student work, and not just marking whether it was right or wrong.  For this reason, it is very important to continue to require students to show their work.  By viewing the problem-solving process, teachers are able to make adjustments to their teaching.

One problem that I had with the article, is that it was very data-dense.  I understand a need for data analyses in the beginning of an article, however I did not gain a full understanding of what was achieved until the conclusion.  There were many references throughout the article of the questions that the teachers had to analyze, but the researchers did present one of these said questions their paper.  This made it very difficult to understand what the teachers in the study were analyzing.

My question that I have is:

If a multiple choice style of research produced inaccurate results, and a large-scale study is required to produce a desired degree of validity, how can a large-scale study be achieved through open-ended research methods?

Apologies for the lengthiness of the response.

Assessing Understanding in Mathematics

Romero and Mari analyzed examples of children's work of mathematical problem-solving.  Through analyzing the work itself and interviewing the children, an assessment of mathematical understanding took place that was largely formative.  In order to analyze one's understanding of a specific mathematical concept, the posed question must be carefully constructed.  An appropriate problem would require the subject to use their understanding of the mathematical concept being analyzed, and it should minimize the subject having to use many other concepts.  This way a specific concept can be analyzed without external factors influencing the outcome.  For example, if one is attempting to assess whether a student can simply multiply, the question probably should not incorporate exponents, decimals, fractions, etc.

One of the concepts that Romero and Mari looked at was multiplication of whole numbers.  The question was initially written in vertical fashion so as to imply that long multiplication is how it should be solved.  STOP! This article was written in 2006 which was only nine years ago, however since then several interesting ways to multiply are being taught and encouraged that are not long multiplication (window method, fibonacci method, Chinese line method, mental math).  Some of these methods have been used in other countries for many years, however I have noticed that they seem to becoming more popular, at least in my school, because some children find them easier to understand.  Because of this, I believe that when a question is posed in a vertical fashion, it creates an instant advantage to students who use long multiplication.  Perhaps the questions should be written out in words or even spoken orally instead.  Then again, this may benefit those with a better understanding of the English language.  Clearly creating a problem that specifically analyzes a single mathematical concept is more difficult than it would seem.

There were two different ways of posing a problem: exclusive or non-exclusive.  An exclusive problem is one that must be solved with a specific algorithm, while a non-exclusive problem can be solved in different ways.  STOP!  I cannot think of a single math problem that can be identified as 'exclusive' under Romero and Mari's definition.  Even the simplest of problems can be solved numerous ways.  One plus one may be solved with use of a number line, pictures, manipulatives, etc.

Question.  Should we be focusing on what students 'understand', being can they reproduce what was taught to them, or should we be focusing on what can a student do when presented with a problem that is foreign to them?  Are they able to use their available resources?  Can they collaborate?  Can they critically engage in solving a problem and be persistent?

Red leader standing by