Homegenous Grouping, Population: One

This week’s discussion forum on ability grouping seemed to garner more responses than any other topic to date in the course. Boaler (2002) certainly wasn’t kidding when she referred to this topic and one of the most contentious in education. Both mixed-ability and same-ability grouping structures come with their own issues and undesirable side effects. I think we are all in agreement that how we structure our classes is less important than what actually goes on in the classroom. However, decisions have to be made and one format has to be chosen moving forward. Unlike many debates in education, there really is no mixed or hybrid approach in this regard – it’s either one or the other. Some of the more outstanding issues surrounding each grouping structure are as follows:

Mixed-Ability

- High-attaining students have less opportunity to interact with other high-attaining students, thus reducing their achievement

- Teachers often focus on the “middle of the pack”, leaving little attention for both low- and high-achieving students

Same-Ability

- High-achieving groups are often taught in pressurized, fast-paced environments resulting in student stress

-            Low-achieving groups are unmotivated and develop negative self-concepts as part of being in a lower level

-            Students at the cut-off points often feel cheated or disheartened from just missing out on a higher set

-            Students can be placed in sets based on factors other than achievement, such as behavior and social class

This list is far from complete, but serves as a sample of the trends in research findings, the key word here being “trends”. All of the above points have been challenged or refuted by other research studies in the area. What this tells us is that the structure of our classrooms is but one single factor that affects achievement, and that many other dynamics are in play, the most glaring being the classroom practices of the teacher.

It is my belief that the largest possible group of students that can truly be of homogeneous ability is one. No two students are of the exact same ability level. When we track students into same-ability groupings, we are merely narrowing the range of ability. These classes still have their high, low and average achievers. Mixed-ability groupings are often associated with differentiated instruction, while same-ability groups tend to bring to mind a more traditional transmission of knowledge approach. As Boaler (2002) indicates, it does not have to be this way. Differentiated instruction can just as readily be used in same-ability groupings. As is always the issue, restrictions of time and stringent outcomes in curricula often keep teachers from implementing the classroom practices they feel are most valuable to learning and understanding. 

References

Boaler, J. (2002). Experiencing school mathematics: traditional and reform approaches to teaching and their impact on student learning. Malwah (NJ): L. Erlbaum. 

 

Men are From Mars, Women Need to Understand...

This week in the discussion forum there was a lot of talk about the apparent gender gap in mathematics. What factors contribute to the underachievement of females in mathematics classrooms? Where and for whom is the gender gap most prominent? Does the gender gap even still exist? With these questions in mind I think one thing we can all agree on is that there is nothing about our biological make-up that makes males better at mathematics than females. There is no inherent ability gap when it comes to mathematics, and whatever has caused the gap in the past was external and societal as opposed to internal and chemical.

Jonathan Mauger cited Nicholson (2010) whose findings suggest that the gender gap that was once evident in the 1980’s has all but vanished. I agree with Jonathan’s notion that this was in all likelihood a culture gap, and that women have never been any less capable of achievement in mathematics. Traditional gender roles certainly had an impact on attitudes, and consequently achievement of females in the past. I wonder what the assessment results from home economics classes looked like during that same time period? I would be willing to bet on finding a gender gap of a different sort.

Ashley Kinsella cited Fennema et al. (1990) to support the idea of the gender gap being a self-fulfilling prophecy, a point I believe is worth considering. Sometimes the more attention we draw to an issue, the more of an issue it can become. As Margaret Senior posted in response, a professor at Northwesten University “investigated whether a global gender gap exists and whether it was the result of social engineering rather than intrinsic aptitude for the subject” (Lipsett, 2008). That same study found that in countries where gender roles are less defined, the gender gap in mathematics does not exist. As a result, teachers must be conscious of the fact that their own beliefs and values transcend the classroom environment and have a significant impact on student learning. I would perhaps challenge Ashley’s comment that differentiated instruction may create more of a separation between genders. I feel that differentiated instruction allows students to choose from a myriad of learning resources that cater to their individual learning style, thus removing mode of instruction as a possible contributor to inequity. As Sherida Ryan stated in her post, “According to Small (2010) differentiating instruction in mathematics has multiple benefits. More students experience success with meaningful tasks, more students are engaged, more students see themselves as competent in math, and more students enjoy learning math.”

Margaret Senior also indicated by citing Tomlinson et al. (2003) that gender is just one of a variety of factors including race, culture, socio-economic status and motivation that can impact the way a student learns. I am in agreement with her that being aware of the diverse and unique needs of each individual student is more important than finding the reason as to why each student learns differently. Call it chemistry, biology or hormonal balance, males and females are different and there is no contesting that. Boaler (2002) suggests that female students have a stronger desire for understanding whereas males are more likely to abandon understanding in the shallow pursuit of correct answers. Is it so hard to believe that our gender may affect our preferred learning style? I don’t think so. Let’s focus on making it as easy and comfortable as possible for all of our students to learn instead of how or why they ended up with the learning style they prefer.

References:

Boaler, J. (2002). Experiencing school mathematics: traditional and reform approaches to teaching and their impact on student learning. Malwah (NJ): L. Erlbaum. 

Fennema, E., Peterson, P. L., Carpenter, T. P., Lubinski, C. A. (1990). Teachers’ Attribute and Beliefs about Girls, Boys and Mathematics: EDUCATIONAL STUDIES IN MATHEMATICS Volume 21, Number 1, 55-69.

Lipsett, A. (2008) Boys not better than girls at maths, study finds. Education Guardian.  Retrieved Nov 17, 2011 from http://www.guardian.co.uk/education/2008/may/30/schools.uk1

Nicholson, C. (2010). No gender gap in math. Psychological Bulletin.

Small, M. (2010). Beyond one right answer. Educational leadership.

Thomlinson, C., Brighton, C., Hertberg, H., Callahan, C., Moon, T., Brimijoin, K., Conover, L. & Reynolds, T.  (2003).  Differentiating Instruction in Response to Student Readiness, Interest, and Learning Profile in Academically Diverse Classrooms: A Review of Literature.  Journal for the Education of the Gifted: 27(2/3), 119-45.

Now What?

This is the fourth course I have completed online towards my degree thus far. Each previous course leading up this one, I have noticed a particular phenomenon that occurs in the online postings once, maybe twice per term. As we are all teachers attempting to better ourselves as educators through higher education, we regularly tend to share opinions on many of the major issues we currently face in education. Often the general consensus on any debate involves open-mindedness, balance, and whatever is best for the student. The discussion forum bursts with agreement and support. It seems that point has now arrived in this course as well.

Let me be clear when I say that in no way am I implying a negative “here we go again” attitude toward this situation. In fact, I think the fact that the majority of us are on the same wavelength only exemplifies our passion to become better teachers for our students, and that what impedes us are often barriers of the external nature. In other words, we know what needs to be done but are in on position to do it.

It’s probably safe to say that as of now, most everyone in the course would agree on certain aspects of Boaler’s study. Phoenix Park students were taught in a guided discovery-based environment, viewed math as exploratory and adaptable, saw no difference between school math and real world math, and were able to adapt their mathematical knowledge to new situations. They were active in their learning and active users of mathematics. Conversely, Amber Hill students were taught in a traditional, instruction-based environment, viewed math as inflexible and inert, saw school math and real world math as two separate entities, and could make little use of their mathematical knowledge outside of the classroom opting instead to rely on inventing their own methods. They were rule followers and passive receivers of mathematics. With that, we all acknowledge that there were successes and failure in both schools, and that one instructional approach will never suit the needs of every student in the classroom.

When I get to this point in an online course where it seems a harmony has been reached, I remind myself to take a step back and think about why I am a part of this discussion in the first place and ask myself some key questions. What does all this mean for my ‘daily grind’? How do I get better? How does this make me a better teacher? What can I do right away, in the classroom tomorrow?

Researchers are becoming increasingly aware that knowledge cannot be separated from the environment in which it was acquired (Greeno & MMAP, 1998, as cited in Boaler, 2002). If we want students to solve real world mathematics problems with discourse, we must provide them with opportunities for classroom discussion. If we want them to be able to work collaboratively on an inquiry, we must give them opportunities for group work. These are things that can be done relatively easily in the classroom to enhance the learning experiences of all students.

Up until now our main interest of comparison between the two schools has been the instructional methods and students’ achievement in, and perceptions of mathematics. With the focus narrowing on the issues of gender and ability grouping in the coming weeks, it almost feels as if I should try and wrap things up into some kind of a “pre-conclusion” of my thoughts on Boaler’s study. The only thing that comes to mind is to offer a variety of instructional approaches through a variety of social contexts to maximize student learning. Pay attention to each individual student’s strengths and needs, and cater my instruction to the dynamics of the classroom. It may sound complicated, but in the end it’s just good teaching.

Boaler, J. (2002). Experiencing school mathematics: traditional and reform approaches to teaching and their impact on student learning. Malwah (NJ): L. Erlbaum. 

Greeno, J.G., & MMAP. (1998). The situativity of knowing, learning and research. American Psychologist, 53(1), 5-26.

Attitude

On the first observation day of my B.Ed. program at MUN, I observed a French immersion class that was the picture of a perfect classroom. Students were engaged, behavior was a non-issue, the atmosphere was comfortable and relaxed, and the teacher seemed to be enjoying himself. When I spoke to the teacher afterward, I made a comment about how it must be nice to teach French immersion because the students in your class are the “cream of the crop” I believe were my exact words. The teacher pointed out that the students were not in French immersion because they were highly capable, but because their parents valued education. He believed that if parents valued education, the students in turn valued education, which was what made for the idyllic state of his classroom. This short conversation has stuck with me over the years and has been validated time and again in my own teaching experience.

I’m sure this teacher was not trying to say that parents who did not put their children in French immersion did not value education. Rather, he was simply pointing out that in order for a student to end up in French immersion, the parents must believe in the importance of education. While there is much research showing a positive correlation between achievement and socio-economic status, there are surely parents and students from all walks of life that view education as valuable. The point of the story is not to highlight these particular trends in research, but rather to illustrate how a positive attitude towards learning can heavily impact the value of that learning.

A study by Mura (1995) showed that university professors viewed mathematics as “either a formal abstract system ruled by logic, or a model of the real world.” This dichotomy also seems evident in Boaler’s (2002) study of Phoenix Park and Amber Hill. While Amber Hill delivered their curriculum in a more abstract, traditional behaviorist manner, Phoenix Park chose a constructivist project-based approach. These two approaches had significant implications for the attitudes students developed towards mathematics. Phoenix Park students saw no difference between their school-learned mathematics and the mathematics they encountered outside of the classroom, whereas Amber Hill students struggled to connect the two. At Phoenix Park it was their attitude towards mathematics that allowed them to perform comparably to Amber Hill students on standardized assessments, as well as be able to adapt their knowledge to a variety of other practical situations.   

Last week’s blog focused on the idea that a mix of both traditional and reform methods would be the most effective approach in mathematics. After some more thought and trying to imagine what a Phoenix Park math class would actually look like, I wonder if the open-ended, project-based approach carried out in this school is not already a hybrid of sorts, containing snippets of traditional instruction, as students require them. Even without considering the three months spent at the end of their final year, the Phoenix Park students would have at some point been directly instructed in some form of traditional manner throughout their educational careers. It would not make sense to lean so far in the other direction that students were left without some reasonable level of guidance. Then I consider perhaps it is not so much the mode of instruction but rather the practice time that is holding us back from becoming fully and completely constructivist. While the social context of their learning allowed Phoenix Park students to easily adapt their knowledge to a variety of situations, these students will still require the basic procedural skills necessary to manifest conceptual knowledge into the solution to a problem. Project-based learning helps students in planning towards a solution, but procedural knowledge is a requirement for carrying out that plan. As I said last week, our scales currently feel unevenly tipped in favor of traditional approaches necessitated by a mainly procedural curriculum. Again the discussion leads back to the issue of standardized testing, and its omnipresent erasing of teacher and student creativity. To echo the concerns of one of my fellow classmates Margaret Senior, we have to look at whether these standardized assessments are actually testing what they should be testing. On the GCSE taken by the students at Amber Hill and Phoenix Park, only 30% of the questions were deemed to be conceptual by Boaler. What does this say about the type of knowledge that is valued by the education system?

In terms of the best approach, it comes down to knowing each student individually and constantly transforming the balance of the approach to best fit their needs. Focusing on student engagement and developing positive attitudes towards mathematics should be our primary focus. The challenge is doing so within the confines of time, money and curriculum.

Mura, R. (1995). Images of mathematics held by university teachers of mathematics education. Educational Studies in Mathematics, 28, 385-399.

Boaler, J. (2002). Experiencing school mathematics: traditional and reform approaches to teaching and their impact on student learning. Malwah (NJ): L. Erlbaum. 

Balance

When peering deeply into the issues surrounding mathematics education, it is important to remember that the nature of research is without endgame. Every question answered uncovers several more that need asking. Each controversy has two or more schools of thought all with their own merit. The dynamic essence of teaching and learning does not allow for certainties and absolute truths. The search is for balance.

Anderson (1996) critiques the four claims of situated learning, which are:

1. Action is grounded in the concrete situation in which it occurs
2. Knowledge does not transfer between tasks
3. Training by abstraction is of little use, and
4. Instruction needs to be done in complex, social environments.

Anderson analyzes each claim individually, in each case finding evidence to both support and reject the theory. He cites cases both where the claims are upheld and where they are challenged. This type of struggle is common in the research in mathematics education that I have encountered.

Boaler's (2002) study of the Amber Hill and Phoenix Park schools also contains findings that both reinforce and refute the claims of situated learning theory. Receiving a traditional form of procedural and abstract instruction, the Amber Hill students in general performed poorly on the conceptual questions on their GCSE examinations, but did well on the procedural questions. These findings support the third claim of situated learning, that training by abstraction is of little use. In turn it also supports the second claim, that knowledge does not transfer between tasks.

Phoenix Park students were taught mathematics in an open-ended, project-based approach until the final few months leading up to the examination, when they switched to an approach comparable to that of Amber Hill to make sure all examination content was covered for all students. As a result, these students did significantly better than Amber Hill students on conceptual examination questions, and their performance on procedural questions was comparable. The knowledge they had acquired during their project explorations was transferred to the situations they encountered on their examination, thus challenging the second claim and supporting the fourth claim of situated learning theory.

It is easy to get caught up in theories of practice, to uncover something that sticks with you and jump on the bandwagon. Situated learning theory is an acute example of this ever-complicating search for the be all end all of teaching and learning that will bring profound achievement to each and every student. However we are all aware that such an ideal cannot exist, and that theories are meant to be changed and altered and built upon over time, never to be quite finished. After reading Chapter 6 in Boaler, I was close to fall swoop to the apparent power of project-based instruction. After all, the Phoenix Park students developed a more adaptable and therefore useful knowledge of mathematics than those of Amber Hill. Yet I thankfully realized that it would be foolish of me to choose one type of instruction over another, accepting one and rejecting the other. With every individual student possessing his or her very own cryptic learning style, it makes no sense to teach in a single particular manner.

Right now I feel that in our province, the scales of instructional methods are tipped unevenly, due in part to a mathematics curriculum that is too rich in content and lacking in opportunities to make meaningful connections. If the only way to transmit the curriculum content in the time allotted is to stand at the front of the room and introduce objective after objective in an abstract, disconnected manner, it may be time to take a look at the situation.

The search is for balance.

  

Anderson, J. R., & And, O. (1996). Situated Learning and Education. Educational Researcher, 25(4), 5-11.

Boaler, J. (2002). Experiencing school mathematics: traditional and reform approaches to teaching and their impact on student learning. Malwah (NJ): L. Erlbaum. 

Good Teaching, Bad Results

The article by Alan J. Schoenfeld entitled “When Good Teaching Leads to Bad Results: The Disasters of Well-Taught Mathematics Courses” was very intriguing and a truly revelatory read that I would recommend to all teachers of mathematics. It has forced me to question what I have long considered effective mathematics teaching and I have a feeling I will remember the reading of this article as a turning point in my educational career.

In what other subject would students be asked to complete a task like the construction problem without having to provide any explanation as to why the construction makes sense? The emphasis on the physical skill of using a straight edge and compass, particularly the speed and precision with which the skill is performed, devalues the importance of the theoretical framework underlying such operations. In the case of the class where the six students take turns using the one compass to display their solutions instead of simultaneously free handing a sketch and explaining their reasoning is an explicit example of sacrificing the “why” for the “how”. In reflection, this is something I have surely been guilty of in the past, namely in the graphing section of Physics 2204. I consider myself lucky to have read this article and to be consciously aware of this issue.

From a previous course taken on the teaching of writing, I am familiar with the whole “form vs. function” debate in the area of language arts, and am disappointed to see it crop up in mathematics as well. I’m positive that while reading the section in Schoenfeld about the strict adherence to proper form in proof writing, my mouth was wide open, jaw touching the floor. Is this not ludicrous? Again, how can we so blatantly sacrifice understanding in lieu of some arbitrarily pre-determined format? I understand presenting an option that can help those in need of organization, but I just can’t justify taking away marks on an assessment for straying from said option.

I like that Schoenfeld does acknowledge that students need to learn basic facts and procedures through short fragmented exercises, and I certainly agree with him to an extent. But we are indeed in need of some balance in this highly lop-sided relationship between exercises and exploratory problems. Some problems should take longer than a few minutes to complete, and should involve complex mathematical thinking. It is quite sad to think that students would sooner give up with the impression of certain failure than to tackle a problem for longer than a few minutes.

The last point in the article reminded me of a class I was teaching on three-variable substitution method just a couple weeks ago. Of course when doing these problems one can choose any variable to start with and there are a multitude of paths to arrive at the correct answer. I chose a certain variable for my example on the board, and there was a student who expressed confusion because her workings looked different than mine but she had still obtained the right answer. She appeared to be in disbelief when I explained to her that there were several options when it comes to these types of problems. I feel like this exemplifies the issue at hand; that our students are under the impression math is mainly about memorizing what someone else has already figured out. When teaching physics several years ago a student came up with his own way of solving a gravitational potential energy problem. During final exam review I used this method in some of my examples, referring to it as the “Roberian” approach”, after his last name Robere. Reading this article has made me feel positive about that decision.

I feel like the most important point here is that we do not blame the teachers in these situations for any of this. Their intentions were always to help students succeed. Teachers are bound by the specifications of standardized assessments and objective-based curricula and are often wrongfully judged against the results of such assessments. Just when I thought my disdain for standardized testing couldn’t get any worse!

Discoverd or Invented...or Both?

Having math as the minor to my physics major, I never progressed deep enough into my mathematical studies to get to the meaty philosophical issues we are faced with in this week’s readings. However, I have often pondered on my own whether mathematics was invented or discovered, so I am excited to finally gain some real insight from scholars within the field. As I progress through my Masters studies I am becoming less and less comfortable with the objectivist view that reality is “out there” as an external entity. Consequently I intuitively reject the romantic nature of universal mathematical truths discussed in the first reading. True, we do see evidence of mathematics in physical phenomena such as elliptical orbits and fractal shapes in leaves. But is this really evidence that we are slowly “discovering” a universal truth? Or have we simply “invented” a working conceptualization of such phenomena? I feel the latter is more aligned with my developing philosophy of education.

I found the Reuben Hersh interview, perhaps my favorite of the three readings, to be quite fascinating. I definitely had an “Aha!” moment when he declares that things don’t necessarily have to be either internal or external, but can exist as social beings, or as he states, “part of human consciousness.” Here I am struggling with this discovery vs. invention issue, all the while never considering the possibility of it being both somehow. It’s a strange concept to wrap your head around, but I find the easiest way is to use the parallel of money, an example used by Hersh. Money is only real because we as humans say it is real. We can look at mathematics in the same way as only being useful because we say it is useful. The whole issue is still kind of fuzzy, but one thing I think I can commit to is that mathematics is a human activity, and would not exist without us. I also found his conception of good math teaching very valuable and in alignment with my currently held ideals. A teaching style based around interaction and communication sounds appealing. Starting with examples, a question instead of an answer, is a very practical piece of advice as well.

Brent Davis offers up some rich food for thought in our final reading for the week. What if we “are not converging onto a totalized knowledge of the universe?” What if mathematics is a tool that allows us to perceive the tiny percentage of possibilities that make it to consciousness? Mathematics has allowed us to conceptualize so much more than our ancestors of two hundred years ago. But whether these conceptualizations are snippets of universal truth depends on your philosophy. I am reminded of the example given in ED 6100 to illustrate the point that a theory need not be real. No one has ever seen an electron, but it is widely accepted that they exist. The theory serves its purpose until a better theory comes along. Tomorrow we may uncover evidence that the whole idea of electrons is no longer accurate. That said, can everything we perceive as scientific and mathematical truth potentially change at the drop of a hat?