In her article, Noddings addresses the issue that not everyone uses mathematics in their particular vocation when they grow up. She questions why we force every student to take mathematics throughout the K-12 system (with the exception of grade 12). Her article is in response to Everybody Counts which states that, "Over 75 percent of all jobs require proficiency in simple algebra and geometry, either as a prerequisite to a training program or as part of a licensure examination" (p.90). Quite astutely, she points out that while this may be true, no where does it claim that these same jobs actually use this math. These mathematical achievements are only being used at gatekeepers to attain jobs. Once the job has been acquired, it is noticed that algebra and geometry are no longer required.
The article was written in 1994. Noddings points out that present jobs do not require the same arithmetic skills as they once did due to the advance of technology. For example, where clerks used to have to perform basic arithmetic on a regular basis, they now simply scan an item into a computer. I would argue that now, over 20 years later, required math skills are being replaced with required technology skills.
Another interesting point that was brought up was that due to the perception of fewer and fewer math skills being necessary in the work force, mathematicians and mathematics teachers are simply teaching students math to provide work for themselves (in a manner of speaking). If math really isn't all that necessary for the majority of society, why are we forcing it on all students?
One solution that Noddings brings up is to offer a greater variety of courses that can be alternatives to math. This way there will be options for those who wish to pursue their enjoyment of math, and there will be options for those who don't. This was one of the first points of the article that I was able to appreciate without feeling that teaching math was a doomed pastime. Despite this hope, I cannot imagine a scenario where we are able to offer this plethora of courses. It would require larger schools, more teachers, more classrooms, more professional development, etc. In British Columbia at least, this does not seem very feasible.
Noddings wrapped things up with a glimmer of hope, in that we must attempt to make more meaningful connections with our students by relating the math in class with the students' lives outside of class. This statements suggests that she realizes how difficult it would be to achieve her previous suggestions, and that perhaps this is next the best thing.
Sunday, 21 February 2016
Sunday, 14 February 2016
A Linguistic and Narrative View of Word Problems in Mathematics Education
In this article Gerofsky discusses the reasoning behind including word problems in mathematics education. Most word problems, she mentions, are fictional and have a "false truth" aspect to them. It interesting that most of the time when we discuss why there should be word problems in math, it is because we want to make some sort of connection between the math and the world outside the classroom. More often then not, this "real world" is not real at all.
Gerofsky shows how word problems follow a basic formula involving 3 components. The first component creates the story, the second component provides information, and the third component identifies the goal. I take note of the fact that my students will almost always skip the first component, in search of the other two components. I find it especially astonishing when students can not figure which operation to use in the problem. By the time they have reached grade 5 many students, I would anticipate, understand that textbooks are broken up into chapters of content. If the class is learning about division, then you can safely guess that the word problem will involve division. Is it because they truly don't understand the question, or are they not interested in the problem at all?
It was interesting to read about that use of tense can also play a large role in word problems. It appears that in the English language many tense words are thrown around and used in all tenses. As someone whose first language is English I am able to get by despite these anomalies. However, I can imagine how hard it would be for an ELL student to understand a word problem. Even if they could read it fluently, it might not make an sense.
My questions are:
1) Do you think word problems should continue to be used in math curriculum? Why or why not?
2) Susan stated, "All this leads me to a question for which I have no answer as yet, the question of the purposes of word problems as a genre" (Gerofsky, 1996). My question for Susan is, after 20 years have you found an answer yet? Have you found purposes for word problems?
Gerofsky shows how word problems follow a basic formula involving 3 components. The first component creates the story, the second component provides information, and the third component identifies the goal. I take note of the fact that my students will almost always skip the first component, in search of the other two components. I find it especially astonishing when students can not figure which operation to use in the problem. By the time they have reached grade 5 many students, I would anticipate, understand that textbooks are broken up into chapters of content. If the class is learning about division, then you can safely guess that the word problem will involve division. Is it because they truly don't understand the question, or are they not interested in the problem at all?
It was interesting to read about that use of tense can also play a large role in word problems. It appears that in the English language many tense words are thrown around and used in all tenses. As someone whose first language is English I am able to get by despite these anomalies. However, I can imagine how hard it would be for an ELL student to understand a word problem. Even if they could read it fluently, it might not make an sense.
My questions are:
1) Do you think word problems should continue to be used in math curriculum? Why or why not?
2) Susan stated, "All this leads me to a question for which I have no answer as yet, the question of the purposes of word problems as a genre" (Gerofsky, 1996). My question for Susan is, after 20 years have you found an answer yet? Have you found purposes for word problems?
Sunday, 7 February 2016
The "What-If-Not" Strategy in Action
Brown & Walter display the "What-If-Not" strategy in action through numerous examples in their chapter. This strategy involves reading a question and then listing the attributes of the question. At first this seems as though it would be rather simple, but they point out that there are several attributes to even the simplest questions that perhaps we take for granted. For example, take the question 1+1. We may simply state that that attributes of this question are that it is an addition question and it involves small numbers. But upon further breaking down of the question we may also state that the numbers are the same, they are whole numbers (specifically natural numbers), there are only two numbers, there is only one operation, and that operation is addition, etc. The reason for analyzing a problem in such depth is so that we can then choose one of those attributes and slightly modify it. Upon changing the question, we then analyze how it changed our results. Are there any patterns to notice? Were these patterns noticed in the original question? If not, can we relate the patterns to each others? By doing this, we are asking the students to break down the question and scrutinize the framework before building it back up again. This type work can add a lot of depth to one's mathematical understanding.
After reading this article, I realized that the analysis of attributes isn't anything new. We've been doing this all throughout our schooling. Whoever came up with the idea breaking a question down into parts "a, b, and c" must have read this article. Many of the questions that I assign to my class will usually start off by assessing their basic knowledge of a particular topic, and then the question will by modified by changing single attributes. I suppose the difference is that in a text book, modifications usually make a question more difficult whereas, Brown & Walter may not agree that an attribute change needs to result in an increase in difficulty.
The three problems that I have chosen have been taken out of the Math Makes Sense 8 textbook.
1)
a) List the attributes of the problem: Copy the shapes, copy them on to grid paper, the shapes are squares, the squares are different colours, find the area of each square, rotate the squares so that they line up with the grid paper, count the grid paper squares to find the area, state which square has the greater area.
b) Possible ways to alter the attributes: What if the squares were triangles? What if one of the squares was a triangle? What if you didn't have grid paper? What if they were measured in different units? What if you were allowed to trace the shapes? What if you had to find the perimeter?
2)
a) List the attributes of the problem: Write two equations, the equations should represent what is happening on the number line, the equations should have an operation, the operation should be multiplication, the number line is in whole numbers, the whole numbers are partitioned into 5 equal segments, the amount being skipped is 4/5. 4/5 is shown visually with a red line, the number line is horizontal, the number line starts at 0.
b) Possible ways to alter the attributes: What if the partitions (1/5's) were not visible? What if the jumps were 2/10? What if you had write an equation using addition? What if you had to write an equation using division? Can you get to the same end point counting in 1/3's? If so, what equation would you use? What if the jumping starting at 1/5? How would that alter the equation?
3)
a) List the attributes of the problem: A jacket is for sale, the regular price is $89.99, after 4 weeks the price reduces by 25%, after 6 weeks the price is reduced an additional 15%, the price is only reduced if the item has not sold, tax is not included in the price, there are 2 taxes added on to the price, PST is a tax, PST is 5%, GST is a tax, GST is 6%, the manager is the one who reduces the price.
b) Possible ways to alter the attributes: What if another jacket when on sale for 40% off after 4 weeks? How would this be different then a 25% reduction followed by a 15% reduction? What if you were buying the jacket for a child and there was no PST? How much more money would you save if you purchased the jacket 7 weeks after it went on sale rather than 5 weeks?
After reading this article, I realized that the analysis of attributes isn't anything new. We've been doing this all throughout our schooling. Whoever came up with the idea breaking a question down into parts "a, b, and c" must have read this article. Many of the questions that I assign to my class will usually start off by assessing their basic knowledge of a particular topic, and then the question will by modified by changing single attributes. I suppose the difference is that in a text book, modifications usually make a question more difficult whereas, Brown & Walter may not agree that an attribute change needs to result in an increase in difficulty.
The three problems that I have chosen have been taken out of the Math Makes Sense 8 textbook.
1)
a) List the attributes of the problem: Copy the shapes, copy them on to grid paper, the shapes are squares, the squares are different colours, find the area of each square, rotate the squares so that they line up with the grid paper, count the grid paper squares to find the area, state which square has the greater area.
b) Possible ways to alter the attributes: What if the squares were triangles? What if one of the squares was a triangle? What if you didn't have grid paper? What if they were measured in different units? What if you were allowed to trace the shapes? What if you had to find the perimeter?
2)
a) List the attributes of the problem: Write two equations, the equations should represent what is happening on the number line, the equations should have an operation, the operation should be multiplication, the number line is in whole numbers, the whole numbers are partitioned into 5 equal segments, the amount being skipped is 4/5. 4/5 is shown visually with a red line, the number line is horizontal, the number line starts at 0.
b) Possible ways to alter the attributes: What if the partitions (1/5's) were not visible? What if the jumps were 2/10? What if you had write an equation using addition? What if you had to write an equation using division? Can you get to the same end point counting in 1/3's? If so, what equation would you use? What if the jumping starting at 1/5? How would that alter the equation?
3)
a) List the attributes of the problem: A jacket is for sale, the regular price is $89.99, after 4 weeks the price reduces by 25%, after 6 weeks the price is reduced an additional 15%, the price is only reduced if the item has not sold, tax is not included in the price, there are 2 taxes added on to the price, PST is a tax, PST is 5%, GST is a tax, GST is 6%, the manager is the one who reduces the price.
b) Possible ways to alter the attributes: What if another jacket when on sale for 40% off after 4 weeks? How would this be different then a 25% reduction followed by a 15% reduction? What if you were buying the jacket for a child and there was no PST? How much more money would you save if you purchased the jacket 7 weeks after it went on sale rather than 5 weeks?
Sunday, 31 January 2016
Problem Solving as a Basis for Reform in Curriculum and Instruction.
Hiebert et al made many references to Dewey's views on problem-based learning (PBL). For this, I was very grateful. Like Dewey, I found the reading easy and insightful. Overall, Hiebert et al were discussing the importance of making the subject problematic. This is opposed to learning a skill so that you can find the answer as quickly as possible. The difficulty lies in attempting to foster a desire to solve a problem in the students. Numerous times the idea of making the problem a "real-world problem" came up. This idea in itself can be difficult to wrap one's head around. Is a real-world problem one that only occurs outside of school typically? Does it have to be related to an occupation, that isn't a mathematician or mathematics teacher? It was decided that perhaps we don't need to focus on the real-world problem, and instead focus on the process. By establishing a culture in a classroom where reflective inquiry, and PBL are fostered and have become second nature, any type of problem can produce effective "residue". What is residue you ask? Residue is a term that has been defined in this paper as the "understandings that remain after an activity is over" (p.17). If there is no residue, then what was the point?
I find that it can be very difficult to produce PBL without the necessary basic skills. I believe that it is a fine balance between being a teacher who stands at the front of the classroom and provides knowledge to their students, and being the teacher who provides a classroom with an embedded culture that promotes reflective inquiry and PBL. I feel that this is perhaps one of those things that appears scary to try out, knowing that it could very easily crash and burn, but at the same time produce interesting results, whether it goes the way you think it will or not.
The question I have is:
Do you think that this style of teaching is more appropriate at certain age levels than others? On a spectrum does PBL lead towards understanding of relationships, or does teaching of basic facts/relationships lead to PBL which leads to deeper understanding of said relationships?
I find that it can be very difficult to produce PBL without the necessary basic skills. I believe that it is a fine balance between being a teacher who stands at the front of the classroom and provides knowledge to their students, and being the teacher who provides a classroom with an embedded culture that promotes reflective inquiry and PBL. I feel that this is perhaps one of those things that appears scary to try out, knowing that it could very easily crash and burn, but at the same time produce interesting results, whether it goes the way you think it will or not.
The question I have is:
Do you think that this style of teaching is more appropriate at certain age levels than others? On a spectrum does PBL lead towards understanding of relationships, or does teaching of basic facts/relationships lead to PBL which leads to deeper understanding of said relationships?
Sunday, 24 January 2016
When Learning No Longer Matters
In this article, Boaler examines a school in California that has been labeled as "underperforming" despite her research saying otherwise. The school is found in an area that is typically represented by a low socio-economic-status population. The school itself receives minimal funding, however the mathematics teachers have developed strategies that do not necessarily rely on monetary resources. "The mathematics department plans lessons collaboratively, and the teachers meet every week to discuss and improve their lessons. They visit one another's classes frequently, and every new teacher is given the opportunity to watch every lesson that he or she will teach being taught by an experienced colleague first" (p.502). From Boaler's research, she has found that these strategies have made vast improvements in the students' mathematical learning. These students have surpassed their fellow peers in other California state schools. The reason for their low grade on performance from the state is due to the standardized testing (SAT-9) that takes place. Many of the students have poor language reading fluency and comprehension, and many of the questions on the test are very wordy.
We have been encouraged, as teachers, to incorporate more society-contexed questions into our teachings. Boaler makes an interesting point that these types of questions may be useful when first learning a new concept, however in the end we are assessing their mathematical ability and not their English language comprehension.
The labels placed on schools can be demoralizing to both student and teacher. Despite fantastic improvements in student learning, more and more teachers are feeling the pressure to 'teach to the test' so that their school receives a higher score/ranking. It seems that more students are only satisfied with an assessment of the test-taking skills, rather than their ability to learn.
How can the system change to acknowledge and reward schools such as this one?
We have been encouraged, as teachers, to incorporate more society-contexed questions into our teachings. Boaler makes an interesting point that these types of questions may be useful when first learning a new concept, however in the end we are assessing their mathematical ability and not their English language comprehension.
The labels placed on schools can be demoralizing to both student and teacher. Despite fantastic improvements in student learning, more and more teachers are feeling the pressure to 'teach to the test' so that their school receives a higher score/ranking. It seems that more students are only satisfied with an assessment of the test-taking skills, rather than their ability to learn.
How can the system change to acknowledge and reward schools such as this one?
Sunday, 17 January 2016
Unpacking Pedagogical Content Knowledge
Hill, Ball, and Schilling created a large-scale study involving thousands of teachers. The purpose of the study was to analyze two major factors in effective teachers: knowledge of curriculum, and knowledge of student learning. Teachers analyzed student work and attempted to make sense of where they went wrong and for what reason. A large portion of the research results was based on a multiple choice platform. It was found that the wrong answers in the multiple choice were almost too wrong, making it easier for the teachers to choose the correct answer. Upon reflection, the researchers decided that further research should be accomplished through more open-ended questions. It was very difficult to differentiate what specifically allowed a teacher to answer a question correctly. It was decided that for many of the questions, teachers drew from both their content knowledge, and their knowledge of student learning. Even teachers who supposedly had minimal content knowledge of a particular strain of math, were able to effectively determine where the student went wrong, because their knowledge of how students think compensated. In the end, Hill, Ball, and Schilling were able to say that knowledge of student learning was important in being an effective teacher, however they were not able to specifically analyze solely.
One takeaway that I had from this article is that I appreciate even more so the benefit of really analyzing student work, and not just marking whether it was right or wrong. For this reason, it is very important to continue to require students to show their work. By viewing the problem-solving process, teachers are able to make adjustments to their teaching.
One problem that I had with the article, is that it was very data-dense. I understand a need for data analyses in the beginning of an article, however I did not gain a full understanding of what was achieved until the conclusion. There were many references throughout the article of the questions that the teachers had to analyze, but the researchers did present one of these said questions their paper. This made it very difficult to understand what the teachers in the study were analyzing.
My question that I have is:
If a multiple choice style of research produced inaccurate results, and a large-scale study is required to produce a desired degree of validity, how can a large-scale study be achieved through open-ended research methods?
Apologies for the lengthiness of the response.
One takeaway that I had from this article is that I appreciate even more so the benefit of really analyzing student work, and not just marking whether it was right or wrong. For this reason, it is very important to continue to require students to show their work. By viewing the problem-solving process, teachers are able to make adjustments to their teaching.
One problem that I had with the article, is that it was very data-dense. I understand a need for data analyses in the beginning of an article, however I did not gain a full understanding of what was achieved until the conclusion. There were many references throughout the article of the questions that the teachers had to analyze, but the researchers did present one of these said questions their paper. This made it very difficult to understand what the teachers in the study were analyzing.
My question that I have is:
If a multiple choice style of research produced inaccurate results, and a large-scale study is required to produce a desired degree of validity, how can a large-scale study be achieved through open-ended research methods?
Apologies for the lengthiness of the response.
Saturday, 9 January 2016
Assessing Understanding in Mathematics
Romero and Mari analyzed examples of children's work of mathematical problem-solving. Through analyzing the work itself and interviewing the children, an assessment of mathematical understanding took place that was largely formative. In order to analyze one's understanding of a specific mathematical concept, the posed question must be carefully constructed. An appropriate problem would require the subject to use their understanding of the mathematical concept being analyzed, and it should minimize the subject having to use many other concepts. This way a specific concept can be analyzed without external factors influencing the outcome. For example, if one is attempting to assess whether a student can simply multiply, the question probably should not incorporate exponents, decimals, fractions, etc.
One of the concepts that Romero and Mari looked at was multiplication of whole numbers. The question was initially written in vertical fashion so as to imply that long multiplication is how it should be solved. STOP! This article was written in 2006 which was only nine years ago, however since then several interesting ways to multiply are being taught and encouraged that are not long multiplication (window method, fibonacci method, Chinese line method, mental math). Some of these methods have been used in other countries for many years, however I have noticed that they seem to becoming more popular, at least in my school, because some children find them easier to understand. Because of this, I believe that when a question is posed in a vertical fashion, it creates an instant advantage to students who use long multiplication. Perhaps the questions should be written out in words or even spoken orally instead. Then again, this may benefit those with a better understanding of the English language. Clearly creating a problem that specifically analyzes a single mathematical concept is more difficult than it would seem.
There were two different ways of posing a problem: exclusive or non-exclusive. An exclusive problem is one that must be solved with a specific algorithm, while a non-exclusive problem can be solved in different ways. STOP! I cannot think of a single math problem that can be identified as 'exclusive' under Romero and Mari's definition. Even the simplest of problems can be solved numerous ways. One plus one may be solved with use of a number line, pictures, manipulatives, etc.
Question. Should we be focusing on what students 'understand', being can they reproduce what was taught to them, or should we be focusing on what can a student do when presented with a problem that is foreign to them? Are they able to use their available resources? Can they collaborate? Can they critically engage in solving a problem and be persistent?
One of the concepts that Romero and Mari looked at was multiplication of whole numbers. The question was initially written in vertical fashion so as to imply that long multiplication is how it should be solved. STOP! This article was written in 2006 which was only nine years ago, however since then several interesting ways to multiply are being taught and encouraged that are not long multiplication (window method, fibonacci method, Chinese line method, mental math). Some of these methods have been used in other countries for many years, however I have noticed that they seem to becoming more popular, at least in my school, because some children find them easier to understand. Because of this, I believe that when a question is posed in a vertical fashion, it creates an instant advantage to students who use long multiplication. Perhaps the questions should be written out in words or even spoken orally instead. Then again, this may benefit those with a better understanding of the English language. Clearly creating a problem that specifically analyzes a single mathematical concept is more difficult than it would seem.
There were two different ways of posing a problem: exclusive or non-exclusive. An exclusive problem is one that must be solved with a specific algorithm, while a non-exclusive problem can be solved in different ways. STOP! I cannot think of a single math problem that can be identified as 'exclusive' under Romero and Mari's definition. Even the simplest of problems can be solved numerous ways. One plus one may be solved with use of a number line, pictures, manipulatives, etc.
Question. Should we be focusing on what students 'understand', being can they reproduce what was taught to them, or should we be focusing on what can a student do when presented with a problem that is foreign to them? Are they able to use their available resources? Can they collaborate? Can they critically engage in solving a problem and be persistent?
Subscribe to:
Posts (Atom)