Schedule for October 23 Pro-D day

Hey, here I just inform anyone who is relevant that on October 23, I will attend the IB cohort meeting.

The question in Sun Zi Suan Jing

Image there is a large table, and there are 12 people sitting around it. Why 12 people? Because LCM (2, 3, 4)=12.Thus for each table like this, there are 12/2=6 dishes for rice, 12/3=4 dishes for broth, and 12/4=3 dishes for meat. Therefore there are 6+4+3=13 dishes on this table. Totally there are 65 dishes. Then 65/13=5 tables are needed, and thus 5*12=60 guests attend the dinner.

What is interesting is that the most common solutions online is to consider the dish-person rate, i.e., how many dish (es) does ONE guest use. Since 2 persons share a rice dish, ONE person uses 1/2 dish. Similarly, 1/3 dish/person for broth, and 1/4 dish for meat. Thus a guest uses 1/2 + 1/3 + 1/4 = 1/3 + 3/4 = 13/12 dish/person. Since there are 65 dishes total, there are 65 / (13/12) = 60 guests.

It might be helpful if students understand the cultural background that, when dining, Chinese people sit around a table (roughly 8-12 persons/table), and share cuisines that are put in the middle of the table. They use chopsticks to move the food they want to individual plates. This "shared cuisine" then creates the concept of 1/2 dish/person.

342 in class writing

The best math teacher:

UBC Math 316 (partial differential equation) teacher. 

He always starts a topic by telling a story;

Then clearly states the importance of this topic, i.e., why are we interested in this topic;

Then explores the topic so that students can have an idea what tools will be used;

Then mathematically explain it, with a solid conclusion in the end;

Further expansion of the topic;

SMILE ALL THE TIME!

He can apply the above principles (maybe) because this subject is highly reality-related. It is not that good if he can only do so in one or two classes; however, he does follow the above in almost all his class. Thus, even though he does not apply these rules in some classes, students can still positively take his class.

The worst math teacher:

This is totally imagined so that I cannot really tell a story or a background.

Directly go into the mathematical topic without explaining the interests;

Copy notes rather than explanation;

Strict exam grading process that students cannot explain things freely;

Take students' negative comments personally.

TPI result and reflection

So first, here is my result.

As explained in the "Five perspectives" page, the five perspectives are NOT mutually exclusive perspectives.

My mean score is 37.2 with a standard deviation of 2.79. It is clear that for the two "dominant" perspectives (apprenticeship and nurturing), the sub-scores of B (beliefs), I (intentions), and A (actions) are roughly at the same level. The sub-scores deviate a lot in other perspectives. I really cannot comment on the result as there is no supporting document about the inner calculation. I am really surprised about my low transmission and development scores, but having no surprise about the low social reform score. The two apprenticeship and nurturing score are roughly the same to my expectation.

Although my B and I sub-score vary among all perspectives, my A sub-score are consistent at the level of 13 (14 in social reform). One interesting intepretation is: according to the result, I dob't believe in social reform, and I do not intend to accomplish any social inform activity, but I am really doing some social reform actions. I do not know if I have intepreted the result correctly, but it will be really helpful if the website can provide some sample intepretations.

Here is another test with all selections being "SA" (strongly agree) or equivalent options.

Notice that the test contains 15*3=45 questions. The test organizers assign 3 different weights to each question under different perspectives. It is possible that some questions have a negative weight in a perspective (thus 15 may not be the highest score for the sub-scores).

How many squares are the on a 8*8 chess board?

Students normally react immediately about the 1*1 squares, and there are 64 of them.

In some situation, they may see the big board as a big square----and actually that is----1 square contribution.

Thus it is reasonable to consider those squares "in the middle", or those squares whose sizes vary from 2*2 to 7*7.

Starting from finding a 7*7 square, students can easily see that there are 4 of them. Using similar schemes, students can see that there are 9 squares for 6*6 squares, and so on.

As a result, the total number of squares is calculated by 1+4+9+16+25+36+49+64=204.

Additional:
Look at the image:

There are 8 squares of different sizes. If students thinking through the processes of:
1) What is a square:
2) How many different styles/kinds/sizes of squares are there in a chess board?
3) How many of each kind of squares?

The following table is an informative one to show:

Then compare this table with the following 2 or 3 table(s), student can find out some patterns within the tables and the relationships between the tables and the chess board graph.

Which is More Important: Instrumental Understanding or Relational Understanding?

Reflection requirement:

For Monday at 9 AM, please write a 1-2 paragraph personal reflection on your blog that follows up on our discussion/ debate today on instrumental and relational learning. Your reflection should consider ways that "fluency" and "meaning-making" might be integrated in math teaching and learning. Please give an example related to a particular math topic if possible.

Also, please take a look at the blog postings about Assignment #1, and think about what topic(s) you are interested in and who you might like to work with in your group. If you have requests and suggestions about choice of group or topic, feel free to email them to me -- but we will not settle on anything till the end of our next class, where we will discuss this. I may also add some other possible project topics before Monday!

For the whole weekend, I have been thinking about the relation between relational understanding and instrumental understanding in mathematics. As defined in the previous article, instrumental understanding focuses on how to use methods/algorithms to solve problems, while relational understanding concentrates more on why the method can solve the problem.

It is widely accepted that both of them are important in mathematics teaching. Even though in class we stand in two different groups and seemingly argued against the other side, we all believed that both of the two understandings play crucial roles in mathematics.

However, we do have different opinions toward what kind of understanding should be put more effort in class. My understanding of this question, after the whole weekend thinking and discussion, is it depends.

My logic stands in the following way. Consider our students in three different levels: (a) those who are very good at and interested in mathematics, whose scores are A level or above; (b) those who have trouble understanding the internal reasons behind conclusions, whose scores are below C level; and (c) those who are in the middle, whose scores are in the between of C level and B level. 

For type (a) students, both instrumental and reasonal understands are important, as reasonal understanding provides them interests of learning mathematics, and instrumental understanding helps them solve problems efficiently. I firmly believe the two understandings form a spirical process that carries students interests and understandings of mathematics forward. Meanwhile, with or without us, these students can find some way learning and understanding mathematics. In short, they are not the students worths of discussion in a "normal" mathematics class.

For type (b) students, it is certain that forcing them to memorize some process of solving problems can be a short-run but not a long-run method of raising their math scores.There are so many "outsider" factors influencing the results of their mathematical learning (for example, personal health problems, social-economic problems, etc.). The solution should be a mixed bundle including but not limited with the two types of understandings.

If we assume type (c) students are the majority of a "normal" mathematics class, then instrumental understanding should be an efficient way of showing students how mathematics are applied in a problem-solving process. The reality is that if a teacher spends too much time on reasoning why a mathod works (relational understanding), since most of these students need time to "catch and feel" the reasoning process----and in this case they are hard to understand what the teacher says in such a short time----they will quickly feel bored and give up. My cousin (Grade 7) mentioned an interesting story in his math class:

        "My math teacher is some how boring but entertaining. He spent 80 minutes explaining 4 questions that I can solve in 3 minutes, because most of the people in my class have not yet learned about this kind of stuff. He explained the same stuff over and over again, but always add some little examples that make us crack up." (Here is the link to his blog: http://minerbill1234.blogspot.ca/2015/09/2015-09-15.html)

And thus I believe there should not be a standard answer to the question "which is more important in mathematical learning, instrumental understanding or relational understanding". It is totally depended on students' situations and teachers' teaching skills. I personally believe 20% of reasoning and 80% of practising is a possible bundle to start with.

Reflection on “Relational Understanding and Instrumental Understanding” by Richard Skemp.

What a remarkable article!

Skemp (1976, 2) shows his new perspective of teaching mathematics and teaching methods to students. The two names of understanding clearly demonstrate how mathematics is conceptualized in students’ hearts: “relational”, and “instrumental”.

            Susan’s example of how babies learn things, from my perspective, can be an example of how instrumental and relational understandings happen. Assume a baby is told that an apple (imaging a real apple) is pronounced as APPLE (imaging the sound of apple). This 1-to-1 relationship is so simple that the baby does not need to understand what apple is. By this the baby can simply apply the rule that things with some similarities (even though s/he may not be able to tell what similarities are) are apples. This is instrumental understanding. Then when s/he becomes an adult, s/he may realize, through relational understanding, that apple is a kind of fruit with a sweet (maybe) taste and a red (maybe green or yellow, depending on species) color. The adult then realizes that apple is a general name of a kind of fruit species.

            Then here comes a question: is relational understanding superior than instrumental understanding? Well, I would say it depends. Skemp (1976, 11) summarizes four reasons of why mathematics is hard to teach, which in my understanding demonstrate a balance of educational resources and students/teachers’ needs. A costly teaching strategy may not be the best option for students with “different goals”.

For these goals, Skemp (1976, 4) summarizes them as of two kinds: (a) to understand mathematics instrumentally, and (b) to understand mathematics relationally. Here the word instrumentally means to understand HOW to finish a pencil-and-paper task, or to become competitive in exams. This situation normally happens when a test is right at the corner while students do not have enough time to carefully sit down and explore the mathematical knowledge. In this case, their driven hearts push them to find some “quick, efficient, and accurate” (according to Kevin in EDCP 550 class) ways to deal with tests.

Similar situation happened in my tutoring experience. Several years ago, I was a math tutor teaching a Grade 12 student whose math score was less than 50%. With a math test right at the corner, the student was so anxious to find some way to pass the test. It is imaginable that with a lower-than-average knowledge base he almost had no time to fully master all the test content through a relational understanding approach.

Skemp, R. R. (1976/2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 12(2), 88–95. Originally published in Mathematics Teaching.

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