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.