Week 9 Discussion

CMESG40 Years – Edward Doolittle – Mathematics as medicine

This is an interesting read from Doolittle. I had a good laugh reading some of his jokes and stories (UFO abduction, the long fart by the old man, the Blackfoot creation story) and at the same time got me thinking how we can pull mathematics into indigenous culture (p. 127). It was intriguing to see how he explained mathematics into indigenous thought (p. 128)., that although mathematics is all about simplifying, analyzing and breaking down, Indigenous thought is all about developing and building up sophisticated complex responses to complex phenomena such as the weather, healing, and human behavior. The author in all his life had been searching for something he felt was missing in his life as an Indian. He realized how incompatible being an Indian and a mathematician was (p. 122). He finally found enlightenment after listening to some Indian elders at the Indian Health Careers Program. He finally became an indigenous person connecting deeply to his Indian root and mathematics.

The author made mention of ethnomathematics which according to him is an approach between indigenous thought and mathematics as well as reflective and respectful to indigenous tradition. He questioned the role of mathematics in non – indigenous culture. Throughout his writing, he portrays how Indigenous thoughts and mathematics are spiritually connected to one another (the Black Elk’s reference, Ramanujan and Blackfoot creation story, p. 129).

STOP 1

With reference to Black Elk and the Ramanujan story (p. 129), perhaps, mathematics is a spirit connected to one’s indigenous culture. It makes me wonder whether a person is naturally gifted and understanding math comes off easily.

STOP 2

The author on page 122, made mention of his identity as an Indian and being a mathematician in conflict with each other. I wonder why’s that? Can one’s identity affect his ability to do mathematics?

QUESTIONS

1.  Can a student be considered mathematically gifted?

2.  Relating to your own experience, can you connect a person’s mathematical ability to his indigenous culture?

Week 8 Discussion 

Competitive comparison and PISA bragging rights: sub-national uses of the OECD's PISA in Canada and the USA - Laura C. Engel & Matthew O. Frizzell

Generally, competitions are good in education, more specifically in subject areas like mathematics. Mathematics is a subject less adored by most students. Therefore, such international competitions like the PISA bring together unique ideas from all over the world and ensure that students have foundational understandings and skills. It also serves as a benchmark for on-going reform. As one provincial leader from Ontario quoted in the article:

“…PISA has indirect impacts along with other sources in informing education policies and programs…and provides a sound research base for reference”.

Despite the advantages PISA brings as the ‘bragging rights’ among states and ‘healthy state-to-state competition’ (p. 676), participants in Canada however, also reflected on some of the risks of increased international exposure if perhaps the results were not good (p. 673).

Stop 1

·         Is it important how a country or a school fares in a competition if it is considered as healthy and necessary for educational reforms?

Stop 2

·         My District Education Service organized math and science quizzes among schools within the district and it was highly competitive. I also held several interclass quizzes with my students in Science, Math, and English. Students enjoyed the competition and learned a lot from it in terms of academics and socially (how to handle defeat and try to work harder next time).

I was alarmed to know from the findings of the research that, ‘little has been done with the state-level results outside of the ‘a big splashy announcement when the results came out… little to no data analysis has been done other than the use of PISA for benchmarking purposes or rhetorical support for the validity of existing reforms’ (p. 678). I am wondering how PISA is organized the subsequent years without any analysis.

Question

·         How does your province or school prepare for PISA?

WEEK 7 DISCUSSION 

The “What – If – Not” Strategy in action. By Brown and Walter

This is an interesting strategy to extend questions to a problem or going further to engage the students. Brown and Walter carefully explained the ‘What – If -Not’ Technique using four levels: Breaking up attributes, What – If – Not for the attribute, Asking Questions, and Analyzing questions.

Example: Ivan and Maria are playing cards using a special deck containing only 1,2,3,4,5,6,7, 8 and 9 each appearing once. Maria dealt Ivan a hand of three cards. He said that the three cards had a sum of fifteen. What are all the possible hands Ivan could have?

Level 1 Breaking up attributes. Cards contains numbers I,2,3,4,5,6,7,8 and 9 each appearing once

Level 2 What- if-Not for the attribute. What if the deck has numbers 1,2,3,4,5,6,7,8 and 9 each appearing twice? (Answer: 9,3,3; 7,4,4; 6,6,3; 7,7,1)

 Level 3 Asking questions. What hands are possible, each appearing twice, and Ivan has four cards that add to fifteen? (Answer: 9,4,1,1; 9,3,2,1; 8,3,3,1; 8,4,2,1; 8,5,1,1; 8,3,2,2; 7,6,1,1; 7,5,2,1; 7,4,3,1; 7,4,2,2; 7,3,3,2; 6,4,4,1; 6,5,3,1; 6,6,2,1; 6,5,2,2; 6,4,3,2; 5,5,3,2; 5,4,4,2; 5,4,3,3; 5,5,4,1; 5,4,3,3)

Level 4 Analyzing questions. The exercise deals with the decomposition of numbers for grades 1-2. What if the cards contained numbers up to only 7, could we still have summed up to fifteen? What highest possible number could have been formed? What if each number had appeared thrice, could we still have formed up to fifteen?

Discussion 6 Dancing Mathematics and the Mathematics of Dance – Sarah-Marie Belcastro and Karl Schaffer

Summary

I love dancing and enjoy watching others dance but haven’t realized yet that math and dance could be intertwined as vividly explained by Belcastro and Schaffer. It is fascinating to know that mathematics could inspire dance and vice versa.

The authors through practicality showed several examples of how the two are intertwined. For example, music could be divided into counts and use counting to mark the times at which movements are done. They argued that complex patterns arise when dancers play with rhythm.

The authors also gave examples of traditional dances such as Classical Western ballet and Bharatya Natyam having a strong sense of line. I would say the same of my Ghanaian dances Kete and Adowa. I am trying to figure the mathematical rhythm in them. Can you guys help me out? Please, find the videos on youtube below:

https://youtu.be/sBPomwXuK7c      and  

 https://youtu.be/ff7zvV2C54I

I liked John Conway’s “hop-step-jump” terminology to describe the seven linear patterns for ambulation and tried to do it. The dizzy sidle is difficult to do because of the constant going back and forth almost rotating and hop is the easiest, I think.

At a young age, my friends and I always did the finger geometry for fun (Karl, Scott Kim, & Barbara, 1995), but not as complicated shown by Karl, Scott Kim, & Barbara (1995). I don’t think I realized the mathematics in it.

Stop 1

How do we guide our students to identify math in dance or vice versa?

Short Paper Proposal

Rote Learning in the math class

For teachers and parents and the rest of the world, it is such a joy to see our children recall facts from memory with little or no mistakes. For example, imagine a four-year-old being able to recite 4x table or 7x table. Who would not be awed at such ‘intelligence’ at that age? Repetition is fun especially when it involves very weird gestures or actions which depends on the teacher to make it interesting and playful. Eventually, children can understand the concept at that moment and may easily recall overtime, but how well can they relate what they have memorized to real life experiences or logic?

The traditionalists often support rote learning which is a memorization technique based on repetition to help students get to understand a concept. Most students tend to grasp the concept but are unable to generalize, cannot tell when an answer is wrong or cannot see alternate ways to get work done. 

  However, contemporary research has shown that there are other several ways to help children retain what they have learned forever other than memorization.

For my paper, I would like to discuss the implications of rote learning and how effectively teachers can use it to benefit their students in the math class. My discussion will center around these three articles:

Main Article

SKEMP, R. R. (2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 12(2), 88-95.

Two other articles

Mayer, R. E. (2002). Rote versus meaningful learning. Theory into Practice, 41(4), 226-232. doi:10.1207/s15430421tip4104_4

Akin, E. (2014). In defense of "mindless rote". Nonpartisan Education Review, 10(2), 1-13.

A Linguistic and Narrative View of Word Problems in Mathematics Education – Susan Gerofsky

Summary

The author critically described word problems as linguistic genre considering its pragmatic structure and observed the problems associated with the use of word problems in mathematics education even though other writers such as Burton (1991) and Nester & Katriel (1997) supposedly found the value of word problems used in mathematics teaching and testing and encouraged students to become more proficient at solving them. The author talked about the three components structure of word problems though there may be variations, the question of truth value, linguistic and metalinguistic verb tense, Tradition: “I did them and my children should do it too” and word problems as parables.

Gerofsky claimed that the stories in word problems are hypothetical and does not necessarily mean they are real-life problems or could be such. The author got me interested in the ‘throwaway’ stories which seem to be quite useless in our time but somehow still found in our textbooks and used as exercises for practicing algorithms. In my opinion, I do not consider any mathematical question to be a ‘throwaway’ unless there’s Missing, Surplus or Contradictory Data Problems (MSCD) (Puchalska & Semadeni, 1987, p.10). Even with that, it generates a sort of debate among students for discussion, for children with little mathematical experience to analyze the story more carefully and older students to try and find a solution to the problem by trial and error (Puchalska & Semadeni, 1987, p.10).

Stop 1

Reading this article reminds me of how uncomfortable I always was when my teacher gives word problems, either his own set questions or from textbooks. Sometimes, the problem statement may be too long and incomprehensible. And before I finished reading, I had already forgotten the first sentences. My mathematics teachers taught us Mildred Johnson’s (1992) procedure and it helped most of us. How long should a word problem be in order not to distract students from the translation task at hand?

Stop 2

How has your experience been during your school days and now as a teacher in terms of word problems?

Stop 3

Do you consider any mathematical question to be a “throwaway” and made your students skip?

Stop 4

Would you consider a parent’s request to teach or give examples of supposedly ‘throwaway’ questions or what the parent learned in his time?

Discussion 5

Children’s Reactions to Verbal Arithmetical Problems with Missing, Surplus or Contradictory Data – Ewa Puchalska, Zbigniew Semadeni

Summary

I really enjoyed reading this article partly because it brought back some childhood classroom memories and partly because the article was quite comprehensible. The authors discussed how important it is to sometimes intentionally leave some information out from word problems to unlock the powerful reasoning of our students. They termed this as “Missing, Surplus or Contradictory Data Problems (MSCD). Consider the following questions:

·         There are 26 sheep and 10 goats on a boat. How old is the captain?

·         A 7-year-old was given the following problem: “You have 10 red pencils in your left pocket and 10 blue pencils in your right pocket. How old are you?

·         Pete had some apples. He gave 4 apples to Ann. How many apples does Pete have now?

These questions in my view, are ridiculous because some information is missing, needs more information and quite contradictory. According to the authors, such questions raise debate among students, they get to talk about the illogical reasoning behind the questions. These questions, the authors believe, help students to realize that mathematics is not always soluble.

Ewa and Zbigniew also disclosed that most students notice something odd about a word problem, but prefer to hide their doubts largely due to the teacher’s authority which confirms the problem to be unquestionable.

Stop 1

Have you felt at any point of your teaching that, your students needed more information to a word problem or question the problem but are unable to? How do you make yourself available or approachable for your students to ask doubtful questions?

Swan (1983 as cited by Ewa & Zbigniew) described two interesting approach to teaching decimal place value (not only that I think, but all topics in Math):

1.       The Conflict teaching approach which involves pupils in discussion and reflexion of their own misconceptions and errors. Conflict approach may lead to deeper conceptual understanding.

2.       The Positive only approach made no attempt to examine errors and avoided them wherever possible.

I believe when teachers intentionally use the conflict teaching approach, they could tap into the powerful reasoning of their students to solve math without fear. Deducing from this article, it is important to include MSCD in our lessons to unlock our students’ reasoning skills and create a lively and enthusiastic environment.

Stop 2

I am wondering how often to include MSCD in lessons? Will it not lead to students not seeing the authenticity in math questions or make math questions look ridiculous? (anyway, who says we can’t have fun in math? (lol).

                                       Rote Learning in the math class

I met an elderly man, probably seventy-five years old quite recently at Safeway, MacDonalds and had a long chat. He pursued Chemistry at the University of British Columbia (UBC) several years ago. I told him my program at UBC and excitedly disclosed what I was learning especially the ideas from Jo Boaler and how we could make mathematics more attractive and easy like making a cake with our students. He signed deeply and was quick to add, "Ah! Canada is always changing the way of teaching mathematics, yet children are not able to do simple calculations without resorting to the calculator". I asked how he was taught, and he mentioned drills with a firm teacher. He believes children should be able to memorize facts.

For teachers and parents and the rest of the world, it is such a joy to see our children recall facts from memory with little or no mistakes. For example, imagine a four-year-old being able to recite 4x table or 7x table. Who would not be awed at such ‘intelligence’ at that age? Repetition is fun especially when it involves very weird gestures or actions which depends on the teacher to make it interesting and playful. Eventually, children can understand the concept at that moment and may easily recall overtime, but how well can they relate what they have memorized to real life experiences or logic?

The traditionalists often support rote learning which is a memorization technique based on repetition to help students get to understand a concept. Most students tend to grasp the concept but are unable to generalize, cannot tell when an answer is wrong or cannot see alternate ways to get work done.   However, contemporary research has shown that there are other several ways to help children retain what they have learned forever other than memorization.

For my paper, I would like to discuss the implications of rote learning and how effectively teachers can use it to benefit their students in the math class. 

Discussion 4

The Development of Mathematics Textbooks: Historical reflections from a personal perspective. – Geoffrey Howson

Summary

Howson critically examined the role, development, and use of textbooks over time.  According to him, textbooks have not only played a great part in curriculum development but more importantly have provided teachers with a coherent framework to guide their work. From 1950 through to 1960, other sources apart from textbooks which were in the form of apparatus were used to support teaching and learning. In the 1960s, the Nuffield Primary Mathematics Project in the United Kingdom who expressed fear on basing lessons on a textbook still supported teachers with textbooks. They, however, believed teachers’ guide could be more appropriate for teachers. Questions about the desired relationships between the teacher, pupils, and textbook were also raised.
Howson also identified why authors write textbooks, that apart from financial gain, an author’s main aim of writing is to change something in the curriculum. He concluded that no matter the quality publication of a textbook, the extent to which the teacher uses the book is important. 

Stop 1 It amazes me that despite being an educational psychologist in Mathematics, Richard Skemp unsuccessfully wrote his own series of textbooks. He wished he had joined the School Mathematics Project (SMP) team and offered his cognitive ideas. This means that however good one is in Math, getting together with other mathematicians helps increase the quality of ideas. Jo Boaler (2015) termed this as ‘the Collaboratory nature of mathematicians’ (pg. 26). I can use this example to encourage group work among my students. 

Stop 2 It is an undeniable fact that the way parents were taught in their time could not be the same today. Things have evolved and sometimes, most parents wish education nowadays are back to the basics. In writing a textbook, how does the author assume that parents will be able to teach their children at home when parents’ teaching and learning are quite different? Is parents’ guide necessary just as we have the teachers’ guide? 

 ReferenceBoaler, J. (2015). What’s Math Got to Do With it? Published by the Penguin Group (USA) LLC, New York. 

 Math Wars – Alan H. Schoenfeld. University of California at Berkeley

Summary

Schoenfeld raised concern about the needless math wars that went on some decades ago in the United States between the extreme reform camp who advocated for new approaches to teaching and learning math, and the extreme traditionalist camp who believed that these new approaches in teaching math will only lead to the ineligibility of students not being able to go to college. No matter how some stand to benefit from the conflict or maybe believe in these conflicts, the author believes there is a large middle ground which is sought by many teachers. He cited Daro (2003) who offers a draft “Math Wars Peace Treaty” to stay in the large middle ground.:

• ·         Teachers, especially K –8 teachers, should learn more mathematics throughout their careers;
•        No students should be denied a fair chance to learn mathematics because they have been assigned unqualified mathematics teachers. 
•        Research and evidence should be used whenever it is available to inform decisions.

Reading this article reminded me of Goldin’s (2003, p.198) ‘Dismissive Epistemologies’ (read in EDCP 550 the last term) about how detrimental these math wars among educationists (Behaviorists, Social constructivists, and Radical constructivists) can be to teachers, students and researchers. Goldin (2003, p. 198) admonishes us, “to thoughtfully reincorporate mathematical and scientific truth, objectivity, correctness, and validity, alongside other ideas, in the thinking of the mathematics education research community”.

Schoenfeld concluded that these wars have casualties – our children, who do not receive the kind of robust mathematics education they should. We should all try and get along with our diversified ideas.

Reflection

I am wondering if these math wars are still going on? Could there be such wars brewing in our own backyard (schools)?

References

Daro, P. (2003). Math wars peace treaty. Draft manuscript, available from the author.

Goldin, G. A. (2003). Developing Complex Understandings: On the Relation of Mathematics Education Research to Mathematics. Source: Educational Studies in Mathematics, Vol. 54, No. 2/3, Connecting Research, Practise, and Theory in the Development and Study of Mathematics Education (2003), pp. 171-202 

Unpackaging Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers’ Topic-Specific Knowledge of Students.  – Heather. C. Hill, Deborah Loewenborg Ball & Stephen G. Schilling

Summary

·         The purpose of Hill, Ball and Schilling’s study was to understand and measure mathematical knowledge for teaching which they termed as ‘Knowledge of Content of Students’ (KCS).

• A teacher may have strong knowledge of the content but weak knowledge of how students learn the content. 
• Pedagogical knowledge deals with the teaching process which includes ways of representing and formulating the curriculum content which makes it comprehensible to the learners. 
• What happens in the classroom between teachers and students is the most important factor in determining quality in education (UNICEF, 2005: 36).  
• Hill, Ball, and Schilling concluded that familiarity with aspects of students’ mathematics thinking such as common student errors is one element of knowledge for learning.

Questions

On page 376, Hill, Ball and Schilling mentioned ‘how classroom practices of teachers changed and students learning improved when they studied how students learn the subject matter. However, it is not known whether teachers who did not partake in this professional development possess such knowledge’.

There is a controversial issue in my country between private schools (often recruit untrained teachers) and public schools (recruit trained teachers from Teacher training colleges) about academic excellence. Most people have come to accept that, students in private schools perform better than students in public schools because the teachers teach well.

• ·       Is it possible that teachers who do not necessarily go through this professional development may naturally possess or know special ways of teaching a topic(s) that may easily be understood by students, hence better performance?
• ·        Do such teachers still need professional training regardless of how well their students perform?

Reference

UNICEF. (2005) Improving quality education for children through reform of teaching and learning materials.

David Tall - Discussion

Teachers as mentors to encourage both power and simplicity in active mathematical learning -David Tall

Summary

David Tall elaborated on several ways teachers could adapt to make mathematics quite easy and enjoyable for students. Most students are anxious about the subject and Tall believes that it is the responsibility of the teacher to make it appear simple to the students. He listed the following as simple ways of encouraging students in active mathematical learning.

1.   Compressing knowledge. Being able to compress mathematical knowledge is “one of the real joys of mathematics”. (Thurston, 1990, p. 847 as cited in Tall, 2004.)

2.   Symbols as procepts is also a simple way to guide students to be flexible with numbers (decomposing and recomposing).

3.   Guiding students to identify different ways of interpreting symbols.

4.   Linking embodiment and symbolism

5.   Making connections in the classroom.

A connectionist classroom is ideal in helping students recognize their potentials in math. This seems to resonate with Skemp’s (1976) idea of relational teaching of mathematics, in-depth teaching of mathematical concepts, meaning making and making connections to the real world for a better understanding of concepts. However, I think a bit of the traditional transmission approach in the class may be fine. What do you guys think?

I recently met a seventy-five-year-old man at Safeway, MacDonald who pursued Chemistry at UBC. After revealing my program to him, he was quick to add that “Canada is always changing the ways of teaching mathematics, yet children cannot do simple calculations without resorting to the calculator. I do not know what’s happening”, he said.  I asked how he was taught, and he mentioned drills with a firm teacher. It seems our seniors still prefer the traditional transmission approach.

Also, I think the use of either of these approaches in the class depends on the topic on board.

Tall made mentioned that, “Imposed targets in many countries press teachers to train their students to obtain higher marks on national tests.”

In EDCP 550 last term, I read about this issue by Jo Boaler (2015) in her book, ‘What’s math got to do with it?’ Here, the author raised concern about standardized testing which harms teaching and learning. She quoted, “Almost every mathematics teacher in America will tell you that the pressure to prepare students for standardized tests harms their teaching and their students’ learning” (p. 87).  For instance, my country’s school curriculum is very much exam based which put a lot of pressure on teachers to teach students to pass national exams.

However, I was also wondering that teachers are mostly measured by how well students perform (at least in my country) in tests and not in tests. How are we to save our faces or avoid being sacked because our students failed than to teach for exams? These national tests are to usher students into the next level of their academic life and therefore teachers need to help them to pass. In my opinion, I do not see these targets as imposed but a form of motivation to best teach students to understand the concepts and to pass exams. It is also a way to ensure accountability on the part of teachers.

Reflection: In your opinion, do you think these set targets are imposed on teachers?

References:

Boaler, J. (2015).  What’s Math Got To Do With It? Published by the Penguin Group (USA) LLC, New York

Skemp, R. R. (1976).  Relational Understanding and Instrumental Understanding. Source: Mathematics Teaching in the Middle School, Vol. 12, No. 2 (SEPTEMBER 2006), pp. 88-95. Published by: National Council of Teachers of Mathematics

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