Inventing Understanding

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Happy Holidays

Melanie

Chapter 11 – Looking to the Future

When I first began this chapter I was certain that I would find that teaching methods and grouping practices at Amber Hill had changed to resemble those at Phoenix Park.....imagine my surprise to learn that the opposite had occurred....and all because of pressure from some “big wig” inspector who believed in and encouraged transmission models of teaching and the pressure put on the school by new middle-class parents.

Boaler uses her study to show “what is possible” when teachers try to deliver instruction in a way that stretches thinking and supports all students, but in this final chapter she shows what is possible when people are not willing to accept that there is a better way to teach mathematics other than a procedural approach, an approach which according to Boaler “has served few students well in the past - offering limited opportunities for understanding, identification, and affiliation with mathematics.”

Is this the same “back to basics” approach we hear of so often? An approach which Cheek and Castle question whether it was actually ever abandoned? I had an interesting conversation with a friend a while back, he works as a math itinerant with a school district and frequently hears teachers talk of adopting a “back to basics” approach in their mathematics classrooms, to correct and remediate deficiencies students bring with them from other grades.....it should stand to reason then if such an approach is successful that we will stop hearing of students who “don’t know their basic facts”, however, if everyone is using such an approach, then why then are the problems still there? Is it because math is not about knowing, but about understanding....I think so. Disconnected pieces of knowledge and rules will not serve students well in the long run, it might get the test, but it will not get them through life.....it will not allow them to transfer their knowledge to new situations, nor will it allow them to feel confident in their problem solving abilities.......why then is such an approach favoured by many? Is it because, like ability grouping, it is an easier approach for the teacher?

As I finish up this course I am wishing more and more that I had a classroom of my own, a classroom in which I could allow students to explore, inquire, create and understand...... I am certain I would need guidance along the way, but I can clearly see than such an open approach would allow students to experience greater success, much like those at Amber Hill......and it would allow them to really get back to basics.....to get back to creativity.....thank you Jo Boaler and Sir Ken Robinson for opening my eyes......

Chapter 10 – Ability Grouping, Equity, and Survival of the Quickest

If I were to provide advice to Boaler as to how she could improve this book for further editions I would suggest that she rename Chapter 10 – “DIS-ability” Grouping.

The disconnect between student and teacher needs, student and teacher beliefs and student and teacher ideas regarding ability grouping became very clear as I read this chapter and prepared to lead the discussion.

I was shocked first and foremost by the fact that the set decisions (and set implications) are often hidden from students, and that they may actually spend a great deal of time working in a class unaware of the set that they are in. The rationale for this is that students often become demoralized and unmotivated when set decisions are made known.....and with just cause I believe.....wouldn’t you be upset to know that no matter how hard you work, that even if you know 100% of the material you can still only receive a low grade on a standardized test, I know I would. I think I would be even more upset if this knowledge was withheld from me for a long period of time, if I thought I was doing well, only to find out that I was doing so well because I was actually working at a much lower level than I had thought. So who does this decision to withhold set decisions from students really benefit? The teachers of course, they don’t have to deal with students acting out or not completing work because “what’s the point if you can only get a low grade?”

This disconnect between students desire to do well and their opportunities to do well remained on my mind as I read the rest of the chapter. Students felt cheated by a system that teachers put in place because they feel it is not realistic to assume that all students can achieve A-levels......but why should the decision to decide who can try for A-levels rest with teachers? Why should the decision be made so early? Shouldn’t all students (like those at Phoenix Park) be given the same opportunities to learn and succeed, and then if the set-up of the exam requires that students be placed into sets can’t the decision be made much closer to the exam (much like at Phoenix Park).......oh, wait.....to teach in such a way requires much more work on the part of the teacher.....which brings me to my next point....

The nature of the lessons in ability grouped classes allows teachers the ability to work at a fixed pace, to deliver lessons using a one size fits all approach.....with little regard for understanding or students ability to keep up to the pace of the work; some students will become bored, some will become frustrated, but that’s the way the games is played.....

Speaking of games, the set up of the ability grouped classes encourages competition among students, which is alright for some, for the students who are competitive by nature, who do well in a competitive environment. But what about the students who experience anxiety and constant pressure because they feel as they are constantly being judged against their peers....what about those top set girls? The competition may have caused them more harm than good, and again, the competition was for the benefit of the teachers, not the students.

The grouping of students also appears to have been done for the benefit of the teachers, with some students reporting that they had been placed in a lower set than would have been expected due to their behaviour. This is discomforting to think that we would restrict a students potential based on their behaviour as opposed to examining factors, such as boredom or frustration that might be influencing their behaviour.

It’s interesting to note that ability grouping was designed to maximize student potential and achievement, but the data presented by Boaler is in sharp contrast to this. The students at Phoenix Park performed much better than Amber Hill student on the GCSE’s even though initial assessments suggested that achievement levels ought to have been similar? And if Amber Hill students were taught using a method that was designed to increase potential shouldn’t they have done much better than Phoenix Park’s students.......they were expected to....

But the measures put in place to bring about those high expectations did not meet the needs (or expectations) of students. The methods would put in place largely to assist teachers in maintaining structure in lessons, in planning one lesson to deliver to all students; they were not designed to stretch and support thought and understanding as did the lessons at Phoenix Park, they were not designed to maximize student enjoyment and success. The design of lessons at Amber Hill and the decision to ability group results in DISabled thought, DISabled achievement levels and DISaffected learners.......

Chapter 9 - Girls, Boys and Learning Styles

Girls just wanna understand........no wait.......girls just wanna have fun........but can they enjoy mathematics and "have fun" if they don't understand?


As I read this chapter (and many others) my own mathematical experiences started to make a little more sense and I gained a new level of confidence that had been lacking for many years. Boaler asserts that “women tend to value connected knowing, characterized by intuition, creativity and experience, whereas men tend to value separate knowing, characterized by logic, rigor and rationality.” This is especially interesting because when I took advanced mathematics in high school the class was almost entirely females, yet the teaching style seems to have catered more towards the learning style of males, lessons were presented much like those at Amber Hill, in a series of disconnected ideas, sequential steps to practice and “know”, things that you didn’t “need” to understand, you just had to “remember”.......but I couldn’t remember, I couldn’t “know” something without understanding it, so rather than stay in a program which I knew I was unlikely to experience success in I switched to the academic math program. This class consisted of far more males than females, yet the teaching style was directed more towards that of females. There was more time allowed to make connections and build understanding, math was no longer simply a game with rules that had to be followed. The pressure and competition of the advanced class had been removed, and there was time to ask questions that related to understanding, time to try our own way of doing things or to figure out just why “the rules worked”.

I sympathize with the girls at Amber Hill, aware of the fact that instruction was not meeting their needs, but unable to do anything about it. Students at Amber Hill were assigned to sets and were not able to make the choice to move between them, I had that choice, but in doing so I had to make some sacrifices, for example, I knew that by switching to the academic class that the math classes I could take in university would be restricted would be restricted; however this was just fine with me at that point, as I had no desire to take math classes beyond high school, looking back now I wish I had taken more math classes in University and that I had remained in the advanced program, but alas hindsight is 20/20.....

A question that remains with me after reading is why should the girls at Amber Hill have been expected to compromise their desire for understanding and replace it with a desire for speed akin to that of the boys? Why should they change who they as learners to meet the teacher’s needs? Is there an approach that would allow both genders to be successful........enter Phoenix Park, the open ended inquiry projects allowed boys and girls to experience success with mathematics, why then is such an approach not emphasized and utilized everywhere? Is it because it is an approach that requires more of the student, but far more of the teacher that the direct instruction method?.......this leads to the next chapter and ability grouping.....



I wish I had been present for the discussion on this chapter, it would have been interesting to hear what others had to say on this topic, to hear their experiences and ideas related to gender and learning styles. Hopefully the blog posts will allow me to gain a sense of the discussion.

Chapter 8 - Developing identities

Mathematical empowerment. What a loaded phrase from chapter 8. What does it mean to be “mathematically empowered”? Boaler suggests that the students at Phoenix Park were mathematically empowered because they were flexible in their approach to mathematics, because they were prepared to take what they had learned and adapt it to new situations. The flexibility in their thinking and approach to mathematics was reliant on two key features, one being that students believed the math they were learning was adaptable, that it had many uses and could be applied in numerous situations and secondly it relied on students ability to adapt and change the math methods, to think mathematically. These students do not see any separation between the mathematics they are learning at school and the mathematics they are using in real life, they feel that they have knowledge and skills that are of use to them, as opposed to being in a possession of fragmented and disconnected ideas about mathematics, much like the Amber Hill students.

Who we are as teachers will determine who our students become as learners and users of mathematics. If we provide them with challenges and aim to have them construct meaning for themselves then our students are likely to develop this sense of “empowerment” that Boaler describes in the Phoenix Park students. On the contrary, if we provide our students with set procedures and methods, rules that they must follow without having a true understanding of the mathematics behind “the rules” then we are likely to produce students of a similar calibre of those at Amber Hill. Students who are led to believe that they will do well in mathematics if they “follow the rules” but who find that in the real world they are not prepared to think about and make sense of problems of a mathematical nature, students who have a false sense of security that has been provided by cuing systems, rules and practice.

I know that I would like to prepare my students to face “the real world” of mathematics; I would like them to approach problems with the certainty that they are capable of arriving at a solution through their own thinking (without having to rely on long forgotten rules). I would like my students to believe in themselves as learners and users of mathematics but I would also like them to believe in the mathematics. In constructing their own knowledge, taking time to investigate and make sense, taking time to understand, time to see why “the rules” work without just practicing rules. I believe students can become more skilled in their use of school mathematics, whether it is for school use of real world use. How can I achieve this? How can I ensure that my students are not just becoming passive users of mathematics, that they are not simply “playing the game” or “following the rules” much like the students at Amber Hill? I think that even at a young age students can assume some of the responsibility for the construction of knowledge, knowledge that is built through hands on investigations rather than worksheets or text book questions that follow a chalk and talk session directed by the teacher. Can we really expect young students to grasp some of mathematical concepts at a symbolic level when they have not had enough time to experiment with it at a concrete level? We see from Boaler’s work that it didn’t work at Amber Hill, so it is likely to fail our own students as well.

Like the teachers at Amber Hill and Phoenix Park I too have my student’s best interest at heart when I plan my mathematics lessons. The challenge now is to combine best interest with best practice, to use instructional methods that allow students to construct knowledge and apply knowledge, in the classroom and in the real world.

Chapter 7 - Exploring the Differences......

What stuck with me the most from this chapter was the false sense of success the Amber Hill students must have experienced in mathematics, particularly those in the higher sets. Students worked through the classroom assignments and textbook work with little difficulty, if they followed the steps and memorized the rules then surely they could do well in maths. However, when the time came for the GCSE’s and the questions were posed in an unfamiliar way, when cuing systems were removed and the math had to be applied as opposed to memorized the Amber Hill students found they were ill-prepared for the test. They realized that when their memory failed them they had little else to turn to; they had no real understanding of the math.

On the flip side of this coin we have the Phoenix Park students, while they may not have been prepared for the test entirely in that there were some topics that they had not encountered or been taught through their project work, these students still fared better in the GCSE’s because they could think about the math. These students could make the connections between mathematics and its real world applications. They did not rely on the cuing systems which Amber Hill students struggled without. They were more willing to try questions when the answer or method wasn’t instantly apparent.

Towards which end of this spectrum are our own students lying? Are they prepared for new and challenging tasks? Are they willing to take risks in mathematics to increase their understanding or are they reliant on us as teachers to provide them with the answers to “difficult questions”?

How often do we hear our colleagues talking about classes that just don’t “get it”, classes that don’t “know their basic facts”? Why is it that year after year we hear these same questions, even though we know that we have taught these skills and ideas? Amber Hill’s teachers were especially concerned with their students’ success, so much so that they provided all sorts of “helpful hints” to get them through the math, all sorts of rules that they “just had to remember” and we can see in the end how these students fared. Let’s set things up differently for our students, let’s encourage them to think, to make sense on their own and to truly understand the mathematics we are “teaching”.

Who's really in charge here......

Discussions in class lead me to question who is really in charge of the learning in our classrooms? If we are supposed to meet our students at their level and take them through the curriculum over the course of the school year, why then are school districts imposing rigid schedules and  outlines for us to follow in our classrooms? What if our students are not ready, what if they need more time? How can we truly do justice to a math program and our students math education if we are so caught up with "staying on track" of a schedule imposed from above, the schedule should come from the grassroots level, from the students themselves. It only stands to reason that if we move on to newer and more complicated areas of mathematics before our students are ready, before they have a good understanding of "the basics" that they will continue to struggle, that they will never "catch up", that they will never enjoy mathematics or come to see it's usefulness.

I have to say I never experienced this strict regime when teaching in Alberta, nobody imposed a schedule on me, and I was actually encouraged to spend additional time on areas where students were struggling, knowing that in the end it would allow my students to grow in mathematics as a good foundation had been put in place. Nobody ever told me what HAD to be assessed at each reporting period, I also exercised choice over the resources and assessment methods used within my classroom, if the other grade level teacher chose to give pencil and paper tests and I chose to assess in a different manner that was ok, we did what worked for us, and for our students. Here we have teachers who not only must assess in the same manner, but on the same day.......whatever happened to differentiation? Since when does one size fits all work for students? for teachers?

If we are to provide our students with a truly rich mathematical experience then we must tailor that experience to meet their needs and to meet our needs as teachers. We need to make the choices that are best for our students, choices about when and how to assess, how quickly to move through the curriculum, choices about the types of activities and resources we use, choices that allow our students to make sense of mathematics and build their own understandings of concepts; as opposed to rushing through a series of text book lessons, expecting students to make sense of abstract symbols and concepts without the hands on, concrete experiences that although they may take more time and effort to complete can provide our students with a much deeper understanding than a teacher provided demonstration and pages of pencil and paper practice.

It's time to give the control back to those who need it, the students. Let them dictate the speed of our lessons, the activities we use and the assessment tasks. Let's give them the opportunities they deserve, the opportunities that many of us were not afforded during our own math education.

Melanie

Show What You Know - Ch 6

The results presented in this chapter brought to light the reality that the traditional teaching methods employed by the mathematics teachers at Amber Hill proved to be of disservice to the students when the time came to show what they know.....which in fact was very little. Students demonstrated that they actually understood very little of what they had covered in math lessons and heavily relied on memory and on obscure cueing systems to help them to figure out the answers to both procedural and conceptual problems. When memory and cueing systems failed them students had no idea how to go about solving the problems, no idea how to make sense of it on their own, and in many cases couldn’t even make sense of what the question was asking.



On the flip side of this coin, students at Phoenix Park had been exposed to a dramatically different teaching method, one which relied heavily on students own understanding of topics and their own creation of knowledge. It may appear that some students were disadvantaged in that they hadn’t been exposed to some mathematical formulas or “rules” and many did not have calculators to complete the exam, however, they were prepared to face the exam with a much more powerful and useful tool than a calculator or list of disconnected and misunderstood rules. These students were prepared to approach problems without fear, they were willing to take risks, and they were willing to THINK! Phoenix Park students had experienced a type of mathematics in which the first answer wasn’t always the right answer, in which the right answer wasn’t necessarily one that could be found in 2 or 3 minutes. Though they may not have had “practiced” numerous examples in preparation for the GCSE’s, the problems they had worked on all year had provided them with practice in thinking, in understanding, in being creative and in finding a way to solve problems that makes sense to them. They had become more skilled at thinking about the math than in thinking about the answer.


How wonderful it would be if we could prepare our students to face exams such as Publics, common finals and CRT’s with the same approach as the teachers at Phoenix Park, if we could in effect stop teaching to the test and start teaching to the students. Meeting the students where they are instead of dragging them to where we want them to be, using their existing knowledge as the basis for future knowledge and having them really understand the math as opposed to sort of remembering it (because we know they don’t remember it for very long.) What would it take to get to this point? Time, resources and a willingness to step back and evaluate our own instructional methods, our own ideas about what good math teaching looks like, a willingness on our part to accept change and make change.


Melanie

Intent vs. Impact

In reading the chapters on Amber Hill and Phoenix Park one idea became remarkably clear, the intent of our teaching does not always have the desired impact. The teachers at Amber Hill cared about their students’ success, and chose their teaching methods as their way to bring about maximum success in mathematics. They felt that the structure of the lessons and the division of students in sets allowed for maximum opportunities for students to be successful. Teachers at Phoenix Park also were concerned with their students’ achievement and used a remarkably different set of teaching strategies to ensure that students were successful. The connection between intent and impact was there at Phoenix Park, but Amber Hill’s intent and impact were disjointed. Students at Amber Hill were not successful, they had difficulty with mathematics not only in test situations but in real world workings as well, they saw math as rule governed and isolated, and assumed that if you couldn’t remember the rules or figure out which rule to use, then you had little chance of succeeding in mathematics. Phoenix Park students on the other hand were far more successful in both test situations and real world application of math skills, they saw the connections and even if they did not immediately have a strategy to use they were able to work through problems and projects building on existing knowledge.


How often do we teach something, assess it, and although a large number of students may still be struggling with a concept we move on, hoping perhaps that they will eventually get it. The intent of our teaching has to change from having our students “get it” to having our students understand it and use it. Skills in isolation are of little use to our students, they must be given frequent and varied opportunities to use these skills, to think about the connectedness of mathematics and forget the rules and think about it on their own.

What was it a wise man once said about good intentions......

Phoenix Park - Breaking the "Rules"

Openness, choice, freedom, independence; characteristics that many would agree are conducive to a successful classroom, a classroom which promotes learning and encourages students to meet their full potential. Why then were these characteristics so obviously noticeable at Phoenix Park and so obviously missing at Amber Hill? One thing is certain, both schools were concerned with the success of their students in mathematics, yet the schools used remarkably different approaches in an attempt to bring about this success.

Phoenix Park, remarkably different from Amber Hill, and perhaps if we were to investigate we would find that the approach to learning was in fact very different from much of what is presently happening in our own schools.

Michelle raised some interesting questions in her presentation, of particular interest is the question, is this possible here? The question is raised in reference to the freedom and openness experienced at Phoenix Park which is unlike most school settings. The obvious answer is yes, of course it is possible, Boaler has shown that it was possible to achieve this learning style in a school in a working class neighbourhood, with students who had previously been exposed to more traditional methods of instruction, schools much like our own! The evidence in Boaler’s study clearly indicates that such an approach CAN work and CAN be successful for ALL students, how then can be incorporate these ideas into our own classrooms? Surely we cannot be expected to make an overnight switch from a method that we have become accustomed to over many years, but even small changes will help our students start to develop a deeper level of mathematical understanding.

Our current classroom situation is much like Amber Hill’s, and we are heavily influenced by “higher powers” and CRT achievement. This HAS to change; the driving force of classroom instruction needs to be the students, and not the numbers that show how a group of students performed on a series of test questions. As we have discussed in class many numbers, the interpretation of these results is driving our instruction, but the results are not being interpreted in a way that makes sense. Schools are comparing results from one year to the next as opposed to comparing a group of students in Grade 3 and then following up in Grade 6 and 9 and comparing them to their own achievement and growth. However, if the idea of following up involves drilling our students with more “practice” exercises and “practice” tests in areas that were deemed to be areas of weakness in the test, then I see no need to even look at the results if we are not going to look at our teaching methods as well. We need to take a serious look at not just WHAT we are teaching, but HOW we are teaching it.

Michelle also raised the question of “What is structure?” I think we can all agree that structure is necessary for a successful classroom environment, but what does structure consist of? Is it merely the physical arrangement of the classroom? Does it include the teachers role within the classroom, the students and their actions? If we are to include all of these elements then we can clearly see that the “structure” at Phoenix Park and Amber Hill were again, remarkably different. Amber Hill relied on the teacher to maintain structure, and we know from Boaler’s findings that this so called structure did not necessarily indicate that students were spending any more time on task than at Phoenix Park, though students at Amber Hill were very well trained in playing the game, in looking like they were working.

To an outsider (or even to a teacher from Amber Hill) Phoenix Park’s math classrooms may have looked like they were lacking structure, but that is not necessarily the case. The classrooms were arranged in such a way that students were able to work on their own or in groups to work on the assigned projects, and although the teacher did not direct the entire lesson, they did provide guidance to students as they needed it, although they did not “make” students work, but really, are we actually able to do this? Can we force students to think? At Phoenix Park students were very much aware of their responsibilities and aware of the consequences that would follow if they did not complete the assigned tasks, some still wasted class time, yet this is a likely scenario in any class, despite our best efforts to “structure” the learning experience. Phoenix Park did a much better job of providing for students at a range of abilities, and all within the one classroom/lesson; something Amber Hill couldn’t accomplish even when they divided their students in sets according to ability, students still complained that they were either over loaded with work or not sufficiently challenged.

I would have loved to have been a student at Phoenix Park, to have been afforded the opportunity to make sense of mathematical ideas on my own, to make connections and develop an understanding which would have stuck with me long past the unit tests (who am I kidding, I never even understood most things then, but I could remember enough of the rules for the tests.) My past experience with math encourages me as a teacher rather than discourages me. I am guilty of having an “unstructured” classroom at times, I don’t mind noise, and mess is my friend, I know I have a long way to come as a math teacher but I feel that I have made great strides in allowing my students to experience greater freedom in mathematics, in allowing them to find the answers the best way they know how as opposed to memorizing my way. Math is not just something that comes from a textbook and there are far more ways to demonstrate understanding than pencil and paper tests, we need to embrace these ideas and put them into practice to provide our students with rich mathematical experiences.

Amber Hill - Keeping with Tradition

Kudos to Sharon for her fantastic presentation last week!

You pointed out many important ideas from the text and brought about an engaging conversation. Some of the main ideas that came through for me (and that I am still trying to make sense of or understand a little more) are:


• The difference that exists between what we “believe” good math teaching is, and how we present this so-called “good teaching” to teaching. It is safe to say that educators, including those at Amber Hill, would consider student understanding to be essential to “good teaching”. However, for the most part students in math classes at Amber Hill were not understanding, they were simply “getting on with it” or “going through the motions”, students were completing numerous practice questions in class and recreating these same types of questions on tests, yet when students had to take these ideas and apply them in context or in a way other than which they were used to they couldn’t do it, they didn’t understand...... which certainly leads to the question, did they ever understand it? The answer is an alarming “no”; the students themselves are reporting that they are simply “following the rules”. If the students themselves believe that the most important part of math is remembering, how then can we expect them to use these skills in future situations when the formulas and rules are long forgotten and they don’t have the understanding necessary to derive these “rules” on their own?

• Teachers at Amber Hill recognize that students are not developing a true understanding of mathematical concepts and are unable to transfer their knowledge (“Tim: Students are generally good unless a question is slightly different to what they are used to, or if they are asked to do something after a time lapse, if a question is written in words or if they are expected to answer in words.” p. 34). If teachers are recognizing that students do not understand, what are they blaming this lack of understanding on? Teachers are quick to point out factors related to the students background, time on task, etc, but do not readily examine their own teaching practice as a possible contributor. Why is this? Are we afraid to admit that we perhaps don’t have all of the answers? We need to see a pedagogical shift emerge that supports the idea of “Have they learned it? If not, how can I help them to learn it?” as opposed to the view of “I taught it so they better have learned it.”

• Student success appears to be one of the forces driving instruction at Amber Hill, yet the success is not evident. Teachers “care deeply about their students” yet this level of care is not such that it translates into the methods of instruction that are selected for students. Teachers rely heavily on a traditional method of instruction, they are confident that the level of structure provided by the textbooks will serve their students better. Students however, are more satisfied with the learning taking place when working on the open ended course work, something which they only experience for 2-3 weeks each year beginning at the end of year 9. There is an obvious disconnect here in what students believe to be important to mathematical understanding and what teachers believe, more opportunities for open ended work and student dialogue would serve to bridge this gap and produce greater understanding.


I would suspect that much of what is happening at Amber Hill is happening right here at home. Teachers here, there, everywhere are concerned with the success of their students, but the instructional methods being used in our classrooms are not necessarily delivering that level of success that we would like to see.

A host of reasons can be suggested as to why these instructional methods are still heavily relied upon many years after studies have proven them to be not nearly as effective as the hands-on student centered approaches Boaler describes at Phoenix Park School. Curriculums are loaded with objectives and time is something there is never enough of, teachers are pressured to complete the curriculum in a “suggested” time-frame.

Money.....something else often in short supply, with large class sizes and a lack of manipulatives in many classrooms, completing lessons that stray from the traditional approach described at Amber Hill can certainly pose a challenge.

The traditional approach is one that has been tried and tested in our classrooms for many years, and while it is perhaps not providing our students with maximum mathematical understanding, it is not outright failing them (when it comes to standardized testing anyway.....) so as teachers we may be inclined to stick with what we know (as this is how many of us ourselves were taught.)

CRT’s much like standardized tests taken by students at Amber Hill have a drastic effect on instruction at the grade levels in which they are administered. There is a tremendous amount of pressure on teachers to have their students meet certain expectations, and the teachers are in turn putting this pressure on students to perform. What is the purpose of these tests? What do they really show us about student understanding? Can a multiple choice test that has in many ways been rehearsed really show us what our students truly know and understand? Is this how we define success in mathematics, the ability to do well in a standardized test situation?

Whatever the reason for the inadequacies in Amber Hill’s mathematics program and in our own mathematics programs in Newfoundland and Labrador, it is obvious that student success must be first and foremost our goal. Defining what it means to be successful at mathematics will perhaps lead us to change within our instructional strategies.

Reflecting....3,2,1

After reading Shoenfeld’s Introduction and Chapters 1-3 of Boaler’s book I was left with an alarming feeling of déjà vu. I felt as though I had been a student at Amber Hill, sitting alongside the students working through those same textbook exercises after listening to the teacher explain the procedure to be followed. Having been taught in a traditional way, much like the teaching methods at Amber Hill, I wasn’t surprised to read in the introduction that students at Phoenix Park outperformed those from Amber Hill in standardized tests, tests in which students were to take what they knew about mathematics and apply it in new and unusual circumstances. Amber Hill students were programmed to take a formula/procedure, memorize it, and plug in new numbers to find an answer.....without ever really understanding where the formula or procedure came from or why they were using it.


Students at Phoenix Park on the other hand spent a great deal of time using what they knew in mathematics to help them work towards an understanding of new concepts, they worked from the known to the unknown using their own knowledge and experiences as a guide in their discovery. Students at Phoenix Park were the driving force of the instruction in their classrooms, the goal of math lessons was understanding. Students were encouraged to think about and understand the mathematics on their own; the teachers provided guidance to the students as they worked on open ended tasks, but the students did not rely on the teachers to help them find the right answer as did students at Amber Hill.

I’m interested in reading more about the math instruction in each of these schools, I am already wishing that I had been a “Phoenix Park” student rather than an “Amber Hill” student.

Melanie

Typical Teaching = Typical Results

When I finished reading Shoenfeld’s article, my very first thought was of good teaching, what is good teaching, and who defines it? In his article Schoenfeld cites researchers Romberg and Carpenter to point to the fact that much of the instruction in schools at that time (and currently if I had to guess) was based on the “absorbtion theory of learning”. This theory is based on the premise that good teaching is defined by having many ways to say the same thing so that students will eventually “get it”.


Absorbtion theory, chalk and talk, kill and drill, call it what you like, students need to make sense of mathematical concepts on their own if they are to develop a lasting understanding of an idea.

Case in point, I had a tutoring student this week, who, although I have stopped tutoring her now that she is in Gr 7, I agreed to see her once more to help her get ready for her first test of the year. The topic being tested was divisibility rules and patterns, a skill that, if understood, can certainly be useful in completing more challenging mathematical tasks. We took some time working with the other rules, which she had no trouble with, she could write numbers that were divisible by a given number, check to see if a number was divisible by a given number and list all of the divisibility rules that applied to a number........she could apply all of the rules....except for the rule for dividing by 8. When it came to the divisibility rule of 8, “A number is divisible by 8 if its last three digits is divisible by 8” this student was having real difficulty......and with good reason I thought....I mean really, who knows their multiplication tables up to 3 digit products, yet this is the “rule” that she was supposed to use (and incidentally is the rule in the text book). I asked her what she knew about the number 8, what was it divisible by.....she quickly came up with 4 and 2, I asked her if she thought this could be of any help to her in determining if a number was divisible by 8. She didn’t see a connection right away, but when she did some more work with the divisibility rule for 6 ( a number is divisible by 6 if it is divisible by both 2 and 3), she was immediately able to make the connection that a number that is divisible by 8 is divisible by  4 and then by 2, for example if 328 is divided by 4 the answer is 82, 82 is divisible by 2, therefore 328 is divisible by 8. 292 is divisible by 4, but the resulting answer is 73, which is not divisible by2, therefore 292 is not divisible by 8. It took some time, and a little trial and error, but eventually she was able to use one rule to help her come up with another.
 She used her understanding of one rule to develop her own understanding of another, she made sense of it on her own, and has a more reliable way to find a solution as opposed to applying a rule that she doesn’t understand and doesn’t have the skills to use. Had she been forced to rely on this rule she had been “taught” it is likely that she would not be able to apply it and would not demonstrate an understanding of something that she does in fact understand quite well......when she’s given the time to think about it.

Another question that came to mind while I was reading this article is whether or not we are meeting the long term needs of our students with instruction that is intended to produce short term results? In the school in which Schoenfeld conducted his 1983-84 study, “Learning was operationally defined as performance on achievement tests” (Schoenfeld p. 4), “the primary goal of instruction was to have students do well on the exam” (Schoenfeld p.9.)

In placing so much emphasis on these tests we are influencing the very activities which take place in our classrooms. The phrase “teaching to the test” is one that is echoed by teachers everywhere, often with contempt, but is one that many adhere to even though they may be in disagreement. Our school districts are adopting new curriculums and new instructional materials that emphasize learning for understanding, yet we are still using materials for testing that emphasize memorization. I think that our assessment methods have to come in line with our instructional methods before we are able to see any real change in producing students who are able to apply their mathematical understanding in different and new situations.

On page 16 Schoenfeld states that what he observed in his study was “typical, if not better than average, instruction—with typical results” , as long as standardized test scores remains one of the driving forces of math instruction in our schools I feel it is likely that we will continue to see this “typical situation.

Melanie

Mathematics Autobiography.....

In reflecting on my early math education I can see that that it was very “traditional” in nature, the perception that “practice makes perfect” is one that I am sure was evident in many (if not most) Newfoundland classrooms throughout the 1980’s and early 90’s.

From K-6 Math instruction in my classroom generally involved pencil and paper, the manipulatives were present in the classroom, but rarely made their way into lessons and when they did we were so in awe of these “blocks” that we were not permitted to “play” with at any other time, that little was ever really accomplished in terms of mathematics within these lessons. The teacher would use the chalkboard to do a few examples of whatever skill or strategy we were working on that day and then we would set to work on pages of practice. I remember getting new workbooks in Grade 2 as part of the MathQuest series, I also remember not doing any of the pages that involved cutting and pasting as these pages were "too messy and wasted too much time." In trying to find a way to describe the role of the teacher in my early math classes I came across the following Ten Roles for Math Teachers. I don't think any of these describe the math classrooms I experienced early in my schooling. I think it could best be described as teacher as demonstrator. The teacher demonstrated a skill, we were set about on our own to practice it, if we didn't understand it the teacher would demonstate it again and we would do more practice and the cycle would continue.

(As a side note, in my classroom I generally introduce students to manipulatives long before I ever use them in the classroom, and I often let students “play” with them during centre time. In doing this I find that students get their curiosity about the materials out of their systems long before we use them in math lessons and my lessons are much more productive because students already have some basic idea of what they are supposed to do with the manipulatives.)

I don’t ever really remember math being fun, though I was quite skilled in mathematics at a young age, and was even placed in an enrichment class for math and language in my elementary grades (a class which is interesting to note that was made up entirely of girls.) The enrichment class provided some opportunity for “problem solving” but in a very structured traditional sense, we were given problems, taught various strategies for solving them and then given a number of “new” problems to solve in the same fashion...I think this for many people would be what they think of when they refer to problem solving in math class, a unit taught in isolation to other mathematical skills and processes.

My primary/elementary teachers were wonderful; some could even be described as inspiring....in every way except for math class. When I look back I can think of a memorable lesson from every other subject in primary/elementary; class elections in social studies, bookmaking in language arts, interactive science experiments....I can’t however think of one math lesson that really excited me or stuck with me. Assessment in math....much the same.....nothing too exciting, pencil and paper tests and an assignment every now and then in the elementary grades.....I don't ever remember being asked to communicate in mathematics. Sure, I was asked the answer when the teacher would correct homework round-robin style every morning, but I was never asked how I got an answer (other than to work out the computation as an example) or why I thought my answer was right.

I can say with certainty however, that my worst memory from mathematics at this time would have to be in relation to homework, I am sure my teachers at this stage in my life believed firmly that if “some is good, more is better.” I remember having heaps and heaps of homework even in very low grades, often this homework was meant to reinforce or practice strategies taught at school, I often saw this as unfair to those of us who had mastered the skills in the lessons at school and I am sure it was often a struggle for my parents to get me to complete the work at home. (I wonder if my disdain for this homework caused me to detest it even more in higher grades when I certainly could have used the practice.) While I will admit to having given homework sheets to students (even in Grade 1) I have tried my best to give hands on tasks for homework to younger students, to provide them with a challenge and to allow them to make sense on their own of concepts that have been introduced in class. (I once sent students home during a measurement unit with a container of playdough, a small ball of wool that they were permitted to cut, a box of paperclips and a deck of playing cards. Students were instructed to find out who had the biggest feet/smallest feet in their house and order the feet from biggest to smallest. It was interesting to see how students completed this task, some used the paperclips to measure the length of the feet and could use numbers to describe the length of the feet, some rolled out playdough “snakes” and ordered them from longest to shortest, and others cut the string the appropriate lengths for each foot and then measured the strings with paperclips. Students thoroughly enjoyed this challenge, some even measured the feet on their pets, their chairs....and one student even measured the feet on the claw-footed tub!)

Perhaps my teachers didn’t feel comfortable with teaching math, although I can’t be certain I suspect that it may be partly true as I remember that there were times in Elementary grades in particular when we would often go days without a math class, and at the end of each year there were always units left uncovered, things that there simply “wasn’t time for”. (I recognize that there isn’t time for everything, and I myself have struggled with completing all of the curriculum outcomes. That being said, I would certainly not leave out an entire unit as I feel this would set students up for difficulties in future.)

In Junior High I continued to excel in math, although the battle with homework was one I continued to struggle with. In High School I took advanced math in grades 10 and 11 but dropped back to the academic program in grade 12. I remember my parents telling me that this was ok because I was more “artsy” anyway, and wouldn’t be as likely to need that math in the future. My sister was a year ahead of me in High School and I always felt that her success in math sometimes proved to be my detriment, with teachers asking how we could be so different, why I wasn’t as good as her in math, etc. I think that I heard this so many times that I eventually started to believe it and at times used it as an excuse for my poorer performance in the advanced program in grade 11.

After High School I was determined not to take any more math courses, I had done well in the grade 12 program, but still had never been very excited about math. When I began the Primary / Elementary Education pre-requisites I was required to take two math courses, not wanting to put myself through any more of what I was sure would be torture, than necessary, I decided to take Math 1050 and 1051. To my surprise I really enjoyed Math 1050, I liked that it was logic based and that it was something I could make sense of on my own. (I had always enjoyed Logic puzzles and was amazed that after so many years I was finally enjoying math.) The only other math course I took in University was Educ 3940, the math methods course.

Until a few years ago math wasn’t a big part of my life, sure, it’s useful, and I used it when I had to, but that was about it. Then I was assigned a grade 1 teaching position and I felt like I wasn’t as prepared for the position as I would like to be. At that point I made it my goal to participate in as much math professional development as possible. I began to work with other teachers as part of a Professional Learning Community developing ways to integrate children’s literature into our math lessons to provide our students with interesting lessons. I attended a week long workshop, “Math Their Way”, which allowed me to see how the effective use of manipulatives can influence math lessons in so many positive ways. I attended workshops on the new WNCP Math curriculum and its framework to help me understand where these changes in math instruction were coming from and where they will hopefully lead us.

My goal for increased professional competency has also led to an increased level of personal satisfaction in mathematics. I now feel much more comfortable in a math classroom as a teacher or student, I am hoping that this class will help me to help other students avoid some of the issues that I fell affected my own journey with mathematics.

Creativity, where does it come from, where does it go.....

The idea that creativity is as important as literacy, as put forth by Sir Ken Robinson is an idea that I found most intriguing.

Creativity is defined by www.dictionary.com as follows:
1. the state or quality of being creative.
2. the ability to transcend traditional ideas, rules, patterns, relationships, or the like, and to create meaningful new ideas, forms, methods, interpretations, etc.; originality, progressiveness, or imagination: the need for creativity in modern industry; creativity in the performing arts.
3. the process by which one utilizes creative ability: Extensive reading stimulated his creativity.


School Boards and Departments throughout the province, the country, and certainly even the world hire literacy consultants and directors, but I am fairly certain that there is not one position in education for a creativity director. But if we are in agreement with Robinson that this skill is in fact equally as important, how then can we foster this creativity in children?

Perhaps children come to school as their own creativity directors, as Robinson alludes, “kids have no fear of being wrong.” Over time however, most children develop a fear of being wrong and believe that a “mistake is the worst thing they can make.” This fear or lack of creativity comes from us as teachers if we fail to encourage our students to think outside the box, to make sense of learning on their own.

For example in problem solving in mathematics, children often find steps and strategies difficult to memorize and have trouble deciding which one to use when presented with a problem. According to Larry Buschman in Children Who Enjoy Problem Solving (TCM May 2003) “Young children want to solve problems, and their enjoyment of problem solving increases when children can solve problems in ways that make sense to them.” So, while these same children may not be able to follow the steps in a prescribed algorithm they may in fact be able to find the same answer to the same problem using a way that makes sense to them, a way that allows them to demonstrate creativity as opposed to memorization. Buschman says that he helps his students acquire the “habits, behaviours and disposition of a problem solver, such as patience, perseverance and a positive attitude.” In solving problems in this way, children aren’t faced with the same pressure to be right, they are able to use their mistakes to guide them towards their answer, this approach allows students to find many paths to the same answer, students are able to take much more pride in their work as they are now able to show what THEY know, as opposed to what they remembered. In adopting a less traditional approach to problem solving it is easy to see how creativity can be applied to mathematics, as students work on original ideas and solutions, as they work progressively towards solutions to problems, as they make sense of their own learning and form their own conclusions about mathematical rules, patterns and relationships.

Creativity, the lost skill.....definitely something interesting to think about.

It was great to meet everybody in the first class, this is my first semester in the Grad program, and this is my only on-campus class this semester, so it was nice to get a chance to connect with some "real" classmates (as opposed to my "virtual" classmates in my other courses.)

I'll try to take some time later today to collect and share my thoughts on the information/issues that were presented in last night's class.

Melanie