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).