Jean Lave and situated learning

In chapter 6 Boaler states that one of her aims in this study was to investigate ‘situated learning’ and she cites Lave in her research throughout chapters 6 and 7. Jean Lave is a social anthropologist with a strong interest in social theory. She teaches geography and education at the University of California, Berkeley. A lot of Lave’s work in situated cognition and communities of practice has been done with Etienne Wenger. Lave received her Ph. D. in Social Anthropology from Harvard University in 1968.

In 1991 Lave worked with Wenger to propose the situated learning model of learning. This theory suggests that learning involves social interaction, and that knowledge should be presented in an authentic context. In one of Lave’s projects she compared the way ‘just plain folk’ learned to the way students learned in the classroom and she found that apprentices experienced great learning success through authentic activity without actually being taught to. This is similar to what the teachers at Phoenix Park School were trying to do. They gave students projects that were open ended and similar to real life and allowed the students to work on these projects without actually teaching them the skills and concepts.

When designing learning experiences from the situated learning perspective, one believes that knowledge is acquired through situations and this knowledge is transferred only to similar situations. That the knowledge gained out of context is more difficult to generalize to an unfamiliar situation. Students at Amber Hill seemed to have run into this difficulty. When they were asked to answer a question or solve a problem that was unfamiliar to them, they had difficulty applying the math they had learned in isolation.

Looking at learning from a situative perspective means looking at the group of learners as more important than individual ideas. The individuals in a group consider, question, and add to each other’s thinking so that important mathematical ideas and connections can be produced as a group (Brodie, 2005). According to Brodie (2005) situative perspectives argue that what a learner says and does in the classroom will make sense from the learner’s perspective of knowing and being, from the learner’s identity in relation to mathematics and to the learner’s past experiences of learning math, both in and out of school. She states that if learners have a particular expectation of ways of working in math classrooms and of what counts as appropriate contribution in the classroom, that they will continue with this outside of the classroom.

Situated learning theory seems to apply more to Phoenix Park School where the students are given freedom to work with others at their own pace, using their own constructed ideas and skills to solve relevant, and authentic problems. When they learn this way they are more able to transfer this learning to other situations. Amber Hill School on the other hand seems to be following a more traditional learning method where students are given closed end questions to work on individually using learned formulas and procedures which they find difficult to transfer outside of classroom use.

“Situative perspectives argue that a focus on conceptual structures is not sufficient to account for learning. Rather, interaction with others and resources are both the process and the product of learning and so learning cannot be analysed without analysing interactional systems.” (Brodie, 2005)

References

Boaler, J.(2002). Experiencing School Mathematics: Traditional and Reform Approaches to Teaching and their Impact on Student Learning. Lawrence Erlbaum Associates:Mahwah, New Jersey.

Brodie, K. (2005). Using cognitive and situative perspectives to understand teacher interactions with learner errors. Retrieved from the internet October, 2011.

http://www.lifecircles-inc.com/Learningtheories/constructivism/Lave.html

http://www.ischool.berkeley.edu/people/faculty/jeanlave

http://elpea.tripod.com/jlavebio01.html

GAMES

Games – Just a fun fill in OR an effective instruction and/or assessment activity?

Just last week I had a substitute teach my class for a day while I was away.  The sub worked on equivalent decimals and renaming decimals as fractions. When I returned the next day I realized that after this day my students would have a long weekend and because I wasn’t in class yesterday I didn’t know how well they had grasped the decimal concept. I decided to have them play decimal snap so that I could circulate and get an idea of their understanding of what an equivalent decimal was. Each pair of students was given a stack of cards which had either a decimal number written in the tenths, hundredths or thousandths or a picture of a grid divided into tenths , hundredths or thousandths with part of the grid shaded. The only instructions I gave the class was to share the cards equally with your partner, each person turn over a card and if you get a pair of equivalent decimals say ‘snap’.

Circulating the class it was so much easier to get a grasp on who understood the concept than it would have been if I had given them written text book questions and had taken these in to correct one by one. An added bonus was that the students loved math class and had a ball reviewing and learning a math concept.

A couple of things I noticed that I had not planned to assess or ‘look for’ happened that I found interesting.

1)      A student of ‘lower ability’ became bored with the game and decided not to play after a few minutes. She spent the rest of the class just ‘wandering’ while others played. This was similar to the students at Phoenix Park who were off task.  I wonder, would I allow this to happen in all my classes if this student decided not to participate? I wonder how my administrator would react to this?  I don’t think this would fly as a continual practice in elementary especially and not likely in high school at my school.

2)      I was pleased with the number of different ways my students chose to play the game. As I circulated the 12 groups I noticed that about half of the groups had devised another method of play. Some had combined rules from other games they had played such as ‘go fish’ and others had decided to only deal out half the cards and keep a center base pile. In the end it didn’t matter how the played the game, only that they were reviewing and learning about equivalent decimals while having fun.

Boaler, J.  (2002). Experiencing school mathematics:  Revised and expanded edition.  New York:  Lawrence Erlbaum Associates Inc.

Thoughts on Schoenfeld's article - Good Teaching Bad Results

It is important to be able to accurately complete computations in an adequate time frame but also important to understand. In Schoenfeld’s article Good Teaching, Bad Results, he gives a couple of examples of how students were able to answer mathematical problems using procedures and operations but without a deep understanding of the ‘underlying substance’. (p.5)

In one example students were given problems such as

274 + 274 + 274          or        812 + 812 + 812 + 812 + 812           (p.5)

3                                                          5

A lot of the students who could use all four of the basic arithmetic operations solved these problems by working out the addition and then dividing rather than looking at the problem and seeing that the number they had to divide by was the same as the number of addends and the addends were the same numbers. If they had used their number sense and understood the problem being asked they would have completed the problem without having to work out all the operations. “By virtue of obtaining the correct answer, the students indicated that they had mastered the procedures of the discipline. However, they had clearly not mastered the underlying substance.” (p. 5) Schoenfeld explains that this shows that being able to perform the appropriate operations does not necessarily indicate understanding.

The second example Schoenfeld gives is Wertheimer’s example of ‘the parallelogram problem’. Students had been taught how to find the area of a parallelogram by cutting off and moving a triangle to change the parallelogram to a rectangle which they could easily calculate. The students could easily do this problem but when they were asked to find the area of a parallelogram that was not in a standard position they couldn’t do it. The students had memorized a formula but did not understand the reasoning.

Understanding why we do the steps we do in math and why they work is a very important aspect of understanding mathematics. From my own experience in school I remember doing ‘long division’ where I knew how to put a number ‘into’ another and how to bring down the digits until I got the correct answer but didn’t really understand what division was. This made it more difficult when given a ‘story problem’ in which I needed to figure out what operation to use to solve the problem.  

In our provincial curriculum in Newfoundland and Labrador I am proud to say we do now teach our students to understand division and not rely on step by step formulas to do computations only.

Pondering ability grouping- a new thought....

I’m very interested in learning from the findings for the study of the two schools and the different approaches to teaching mathematics in the text by Boaler (2002). I’ve been questioning the effects of ability grouping with students and through my readings have come across some interesting ideas and findings. Marsh (989) interpreted such findings as “big fish little pond” (BFLP) effect: An individual’s self-esteem is strongly influenced by the group that individual uses as a reference.” (Kemp & Watkins, 1996).

I’d been aware of how low-ability students may compare themselves to high-ability students in a heterogeneous classroom and how this may affect their self-esteem. I’ve also been aware of how low-ability students develop high self-esteem when they get to work with students who have a similar ability to theirs and them working at a level which brings them success. However, in reading this particular article this week I have for the first time began to think about how students in a high-ability class have only high-ability students to compare themselves to and therefore, “views their own academic competence less highly than they would if they were in a class with students of varying ability levels.”  (Kemp & Watkins, 1996). I'd never really thought that grouping high-ability students together would affect 'their' self-esteem. Something new to ponder.

Reflection on nature of mathematics

What is mathematics and why do we teach it?

The three required readings for this week, raises the questions of what mathematics is and why do, should, or would we teach it?

 The first reading The Theory of Embodied Mathematics from the book Where Mathematics Comes From by Lakeoff and Nunez,  discusses how some have a world view that is a part of their identity in believing what the authors have called Romance of Mathematics. Some points of the Romance are that mathematical truth is universal, absolute, and certain; that mathematics would be the same even if there were no human beings; and that mathematicians are the ultimate scientists. The authors state that beliefs like these intimidates people and leads many students to give up on mathematics as simply beyond them. The author’s goal is to make mathematics more accessible and give a more realistic picture of the nature of mathematics.

The second reading What Kind Of Thing Is A Number? Is an interview between John Brockman and Reuben  Hersh.  Hersh believes that mathematics is “ neither physical nor mental, it’s social. It’s part of culture, it’s part of history. It’s like law, like religion, like money, like all those other things which are real, but only as part of collective human consciousness” (Hersh, 1997). Hersh separates philosophers of mathematics into two groups which he calls Mainstream and Humanists and Mavericks. He states that Mainstream sees mathematics as inhuman or superhuman but Humanists see it as human activity. Hersh believes that “math is something human. There’s no math without people” (Hersh, 1997).  He lists three possible philosophical attitudes towards mathematics. These are:

1)      Formalism- math is calculations, has no meaning, follow rules to come up with the right answer, is connected with rote- your given an algorithm; practice it for a while; now here’s another one.

2)      Platonism- math is about abstract entities which are independent of humanity, helps to make mathematics intimidating and remote.

3)      Humanism- math is a part of human culture and human history, brings mathematics down to earth, makes it accessible psychologically, and increases the likelihood that someone can learn it.

Hersh believes that interaction, communication is essential and that students should interact not sit passively.

The third reading Why Teach Mathematics to All Students by Brent Davis addresses three questions. These questions each fall into a different question category.  Question one is a teacherly question: Why do we teach mathematics to all students? Question two is a rhetorical question: Why would we teach mathematics to all students? Question three is a hermeneutic question: Why should we teach mathematics to all students? A teacherly question already has a determined answer; a rhetorical question expects no answer whereas a hermeneutic question allows for interpretation of an answer and can be expanded on.  Davis believes that the first two cannot be questions at all because they either already have an answer or no answer can be given. The answer to all three questions however seem to come down to we teach mathematics because we have to.

After reading these three articles I believe my stance on mathematics is humanistic with a hint of formalism. I believe that math is social, cultural, historical but also physical and mental. This view comes out in the way I teach my students. I try to involve them in their own learning, always encouraging and prompting them to try different ways to approach and find a solution to a problem. I consistently use peer tutoring and small groupings allowing students to explore and discuss what they think with others. However, there is a touch of Formalism in that I believe that math does involve calculations and it is possible to come out with the right answer sometimes by following certain mathematical rules. This is one area that as teachers we have been changing our methods of teaching over the last few years. I learned long division in school by following rules.  Now, I teach my students that the algorithm is just a method of showing your calculations not how to solve the problem.

Math needs to be connected to the real world for students. It has to have meaning to their lives. I had a student in my class one year who had absolutely no interest in any subject in school. He wanted to be outside, especially on the long liner with his family. Whenever I could, I used problems with him on calculating crab catches or payroll for the workers on his boat and he would have no difficulty digging in and finding a solution. This made sense to him whereas how much would it cost to buy 20 apples at the price of .35 cents each didn’t interest him. He didn’t like apples and had no intention of buying them.  Math does have a humanistic philosophy for me. By making math interesting and relevant to the students’ real life wouldn’t we make it less intimidating and therefore more attainable for them?

References:

Lakeoff & Nunez.  The theory of embodied mathematics. Where Mathematics Comes From.

Brockman, John. (1997). What kind of thing is a number? A talk with Reuben Hersh.

Davis, B. (1995). Why teach mathematics? Mathematics education and enactivist theory. For the Learning of Mathematics, 15 (2), 2-8.

All references retrieved from http://criticalissuesmathed2011.blogspot.com/  September, 2011.

Reflection on Video of Sir Ken Robinson

Reflection on video of Sir Ken Robinson

                Sir Ken Robinson asks the question “Do Schools Kill Creativity?” As I viewed the video posted by Dr. Stordy on our course blog and the video posted by a classmate (Jonathan) on his blog I could see clearly that some of the points Sir Robinson made are true in my school.

                Sir Robinson presents the ideas that children have a large capacity for innovations and that they will take a chance. Children start off not being afraid of being wrong but then as they grow older and approach adulthood this changes and by adulthood they are afraid to be wrong. Sir Robinson wonders if we are educating our children out of creativity and I wonder this as well.

                Sir Robinson’s description of the hierarchy of education that was put in place because of industrialism and because we were educating children for what we thought would be useful in their future ‘jobs’ is chillingly the same as our education of today. Even though our world as gone through many changes, and is changing daily with regards to social changes, cultural changes, technological changes etc. our education system has not changed a whole lot. We still call our math, science, language arts and social studies core subjects, while the arts (music, drama, art….) are just considered electives or secondary subjects. We are living in a country, a world, where health problems due to obesity is growing rapidly and yet our education system is cutting back on the number of physical education classes in our schedule.

                Just like this hierarchy of education has been around for decades so has the school year from September to June. When children return to school in September (especially children who struggle), we have to spend several weeks reviewing last year’s concepts because they have had such a long break in schooling. Why are we still taking a summer break? Children used to take the summer months to help their parents on the farms or with the fishing. Across Canada today there are few families who need this assistance. So why have we not begun a semester system?

                One part of the videos that struck home to me, especially since our government is implementing an inclusion model, was the part where children are grouped according to their age not ability, or interests or any other grouping, just age. As Sir Ken stipulates, “age is not the only thing that children have in common.”  A child that is born by 11: 59 pm, December 31^st, is considered ready for Kindergarten whereas a child born 12:00 am, January 1^st is not ready until the next school year. This has never made any sense to me.  We have been implementing “inclusion” at our school for the last five years; however this year we are ‘officially’ piloting the inclusion model. I’m hoping that some direction and clarity will be brought to light on what exactly inclusion means for our children because I’m afraid that what we have been doing in the name of inclusion is really ‘excluding’ our children. Just because a child is sitting inside the four walls of a ‘regular’ classroom does not mean they are feeling included. If I were sitting in one of my son’s engineering classes, where I could not understand anything that was being taught, I surely wouldn’t feel included would I?

                I really hope that we can get to the point where we are putting the interests and the needs of the child first. Where we can allow our children to be creative and have “original ideas that have value” (Sir Ken Robinson, video).

               

Gloria

Math Autobiography

Math Autobiography

                I have very vague memories of my time in K to 6. I remember that I did like math class better than any other class. I don’t recall any particular math classes but I remember that we used text books. In K to grade three they were consumables and I loved working in those. When we got to grade four we had hard cover text books and all our work had to be completed in exercise books. Assessment in elementary consisted of unit tests which I always scored well on. My teachers explained concepts from the front of the room using the board and then assigned practice questions for the class to complete. My teachers would go from desk to desk helping anyone who needed help. I don’t remember any of my teachers showing any passion for math class. I got the impression that it was just something they had to teach.

                One of my worst memories was at the beginning of grade ten. I had changed schools and had signed up for honors math because I felt I could do it. The principal came into my class, knelt down in front of my desk and said he thought I should do academic math not honors. I was so intimidated and embarrassed that he was expressing this IN my class with all the other students present that I just nodded my head and it was done. I was put in academic math. I excelled in this math, scoring high A’s but no one ever suggested that I do honors math. Even though I did so well in academic math, this situation caused me to believe that I wasn’t good enough at math to continue with any higher level. Therefore when I began university I shied away from as much math as possible.

                When I look back at this situation I wonder if it had something to do with socio-economics. My dad had become ill and our family was forced to rely on social assistance throughout my high school years. I had to bring a ‘welfare slip’ to school to get my books. Students with slips were looked at as having lower potential intelligence and lower potential for future success.

                After my children were born, I became interested in math again. All three of my sons had a natural ability for numbers and problem solving using patterns and numbers. All three are math lovers. My eldest is working as a mechanical engineer in Medicine Hat, Alberta, my middle son is in his 5^th year of mechanical engineering at MUN, and my youngest who graduates this school year as decided he too wishes to study engineering because it is using math. These three young men are my success stories of the importance of math. Working with my own children throughout their young years and seeing their passion for math sparked my passion for math again.

                I love to teach math more than any other subject. Science is my second favourite to teach. I love how we can now use a hands-on approach. There are so many manipulatives that we can use to help children learn math and to have fun while doing it today that wasn’t available when I attended elementary.

                What both surprises and frustrates me is the number of 8 and 9 year olds who already hate math when they get to my class.  I hope that I can be a part of rekindling the love of math in the students that I teach. I think that allowing children more choice in the strategies they choose to solve a problem, the use of manipulatives, and peer/group working will help spark the interest. I can’t change the world in a day but I can help one child at a time.