Reflection on "Three 20th century reform movements in mathematics education"

Here is a brief list that contains all the big issues mentioned in the article. All items are categorized into three different eras:

Progressive Reform (1910-1940)

• Early days of public school in NA 
• Criticisms of school math as a process of meaningless memorized procedures
• Targeted on memorizing rules than understanding the meanings 
• Cares more on answers than reasons
• Alternate ways are ignored
• Push for quick, correct answers and a short circuiting of uncertainty and doubt

• Reform addressed to unify algebra and geometry and to bring both pure and applied math to the secondary school
• Addressed the nature of mathematics teaching and learning and questioned the concept of covering topics
• John Dewey proposed that students must engage in doing math as part of reflective inquiry if they were to increase their intelligence and gain knowledge
• Process of experimentation and inquiry was difficult to control than simply lecturing to large groups
• Students need challenge of doing and experimentation in math
• Sense-making activities of reflective practice
• High quality mental processes and scientific attitude
• Problematic and a degree of uncertainty as the incentive to form and test hypotheses and perceive patterns and relationships

The New Math (1960s)

• National anxieties in US 
• School math was not keeping pace with changes in research-level math at universities 
•  US face a shortage of qualified college students 
•  New math 
•  Inclusion of modern math areas in school math 
•  Set theory, abstract algebra, linear algebra, calculus, etc 
•  Regardless of local conditions, cultures, or educational traditions in other countries 
•  Imposed western, collegiate, highly abstract math in school

-          Early 1970s, new math was being denounced

• Misguided experiment 
•  Similarity to the progressive:

o   Inquiry and sense-making over absorbing and applying facts

• Differences:

o   Conservative as university teaching

o   Presentation, precision and correctness

o   Deductive method than others

o   Math is infallible and authoritative

o   Individual tests

o   Knowledge was assumed to be in the head of learners;

o   Aim to produce teacher-proof materials that could be delivered by teachers anywhere in the world

o   Aim to educate future elite scientists and mathematicians but refused the rest students

Math Wars over the NCTM Standards (1990s - present)

• Back-to-basics curriculum, standardized tests, rein-in of teachers’ autonomy in the name of accountability 
•  NCTM standards 
•  Flexible problem-solving skills 
•  To present math relationships in multiple forms 
•  Emergent technologies like graphing calculators and personal computers 
•  To communicate mathematically 
•  Appreciation for the power and beauty of math 
•  Depth of understanding 
•  Make math connections above calculation skills 
•  Although fluency in calculation was still considered an important goal 
•  Different opinions 
• Anxieties

o   3^rd TIMSS: bad result (28^th) of US math

o   Deeper conceptual understanding of math was key

o   Continuing

Comments:

    The old style of teaching in that era was “appropriate” in the era before. With limited educational resources and “low” need for most kids to understand mathematics, mathematics education tended to put more efforts on calculation and procedures so that graduates can perform basic works in factories. During WWI and WWII, more and more mathematics knowledge was applied in arms and wars (and of course in factories and almost all the industries), requirements for workers were raised correspondingly. The old-style products could not perform well in the changing environments. Thus educational reform became a necessity.

    But, why was Dewey? A simple answer could be the mainstream social values required a sound like Dewey that emphasized individual values and developments, especially at the time the world war was over and everyone returned to a relatively peaceful environment. The problem was that the cost of implementing Dewey’s educational methods was (and still is) so high that it was almost unrealistic for most families (in the US, not to mention the world).

    Similar situation happened in the NEW MATH era and the concurrent MATH WARS and NCTM STANDARDS era. Teachers and schools are not the elements in society which stay in the first line of use new knowledge. Therefore, they cannot be the first one who calls for changes. It is not surprised to me that all these reforms were not first advocated by teachers. 

    Then can teachers become the early birds in the next reform? 

    The current situation of use of mathematics in industries is extremely polarized. On one side, only a proportion (vary based on industries) of high-school mathematics knowledge is largely used; on the other side, high-school-and-university-level mathematics can only be a small component of a qualified worker’s knowledge base: other related knowledge like statistics, economics, accounting, physics, chemistry, biology, and so on, are the necessities that support the worker.

    Then, what is the matter here? My answer is the ability to learn (and communicate, but this is of topic so I would not elaborate it here). Learning from problem solving is important, but self-learning through reading and analyzing should be placed at a higher rank. In my case, I would encourage students to read textbooks as a daily pre-reading assignment, so that I would have more class time for solving problems and exploring ex-curricula relevant topics.

I should have elaborated the last paragraph more about the logic of self-learning and reading. I would improve this article in a weekly basis and see how I reflect myself every week.

Math Wars

Learning mathematics should be considered first as a need of human development rather than a competition.

Are math contests necessary? Yes, no doubt! It can of course help capable students involve in mathematical field and get exposed to different topics in mathematics.

But are math contests good for everyone? Highly doubtable. For those who get frustrated in the abstract and hard-to-understand mathematics, mathematics becomes the last item they are willing to stay with. Although they are dealing with math everyday, they believe they are incapable in mathematics and thus hate math.

I learnt Olympic mathematics when I was young. I have to say I like the knowledge and of course the rewards won from contests. But behind me there are many students who failed in the tests. It is unfair to categorize me as "capable of math" and call them "math stupid" because I clearly understand there are things in math that I am not capable of and there are mathematical things these fellows are good at in their everyday practices.

Then as a teacher, what should I do? I would like to make my students feel two things. First, interests are of the top priority; second, competition are not necessary. For the second one, I would like to eliminate as much as possible any forms of competition: speed, right/wrong, score, etc.

Reflection of micro teaching

Here is some statistics of colleagues' comments:

This is a nice trial of teaching something I am interested in to students. In the statistical table above, s1 is my own evaluation, while s2 to s5 are the results from colleagues.

The result clearly shows that I did not provide a good activity or explanation of why this topic is important to learn: "hook" item has an average score of 2.4.

Two members (including myself) felt that the contents of the topic were not well-organized. The rules of GO are too many to be explained well in only 10 minutes (in other words I could have selected a better topic that can be taught in such a time frame).

I did not spend enough time checking colleagues' prior knowledge about the game.

All the colleagues believed I did not assess my teaching and their understandings effectively. Meanwhile, they have little idea how this game can be played.

I could teach GO better through an inquiry-based teaching rather than lecturing. My colleagues could learn well through solving problems I created rather than barely trying to figure out what those rules are. A better design may be just to teach how to capture stones.

"GO" lesson plan

Teach students the game "GO"
http://senseis.xmp.net/?RulesOfGoIntroductory 

Lesson 1/infinity, 10 minutes

Goal: game setting, basic rules, and objects of the game

Objects: at the end of the microteaching, students can

        (a) understand the basic rules of "GO";

        (b) objects of the game: occupy more territories;

        (c) how to count at the end.

Content (and equipments):

        (a) baord: 19*19 grid, back (181) and white stones (180);

        (b) two players take turns (black first) to put stones on interceptions;

        (c) capture: when a connected set of stones are fully surrounded by opponent's stones (so that there is no open point exists), opponents can take these stones and put them out of the stone pool;

        (d) players can pass a turn (not to put stones on the board). When both players consecutively pass turns, the game is over;

        (e) each empty point is counted as 1 point, and each taken stone from opponent is counted as 1 point;

        (f) the more, the winner;

        (g) idea of control;

        (h) idea of balance.

Detail:

        (1) Introduction (1 min): 

             - antient Chinese game (different rules)

             - spread to Japan in Tan dynasty and well-developped in 1700-1970

             - popular in Japan, Korea, and China;

        (2) Explain content (a) and (b) (1 min);

        (3) Show different situations of (c) (1-2 min);

        (4) Idea of control and balance (2 min);

        (5) Try and have fun (5 min);
        (6) Close and no homework.

Letters from two "students" of mine

Student1:
Hi,
    Taking your math class in secondary school is one of my memoriable moments in my life. I really enjoy your lecture because it is clear and straight forward. I still remember your "Big Word" column on the backboard. I fell like you really care about us and our learning, and I always fell welcomed to talk to you or ask questions.
    You always tell us some stories that are not included in the textbook. This is actually good to me as I can have a sense of how to use the abstract math concepts in the real world. It is of great pleasure to be in your class.

Best,
xxxxxxxx

Student 2:
Hi,
     Sorry to let you know that being in your secondary school math class is one of my worst experience in my life. I was tortured by your accented English and pointless stories. You always assigned preview tasks along with homeworks, which was too much for me. Always, I had to write homeworks and exams with a specified format, or my scores would be deducted because of "unclear writing".
    Anyway, I really hope you can improve your teaching skills and be more friendly to students.

Best,
yyyyyyyyy

Reflection:
Math is a perspective to observe the world. It is of great importance to show how math can be used in the real world, instead of merely solving workbook problems. Also, formation is important for both writers and readers. Unclear writing will only show that the writer has no clear mind of what the problem is. So far I still strongly believe in these two. However, I would like to adjust them based on the real situation.

The soup can problem

Here is the soup can dimensions:

Data is retrieved from the following link: http://www.walkerart.org/collections/artworks/campbells-soup-can-chicken-with-rice-soup 

Here is the bicycle dimensions:

 data is retrieved from the following link: https://www.google.ca/search?q=bicycle+dimensions&biw=1920&bih=943&source=lnms&sa=X&ved=0CAYQ_AUoAGoVChMI49nG2aTOyAIV5FCmCh1R5gJS&dpr=1#q=bicycle+length

The picture of the big can and bicycle is below:

Roughly, the length of the bicycle in the picture is 1.19 in, while the length of the can is 2.96 in. Thus the true dimensions of the can is (2.96/1.19) * (4*2.625*2.625) (all in inches) = 9.9496 inch * 6.5294 inch * 6.5294 inch.

Reflection on the math art project

The project of math/arts of our team (Jimmy, Yingting, Heijin, Iqra, and Shan) is to make a polyhedra using black paper clips.

The big ball (polyhedra) consists 12 small five-point stars----actually it is almost the same as a soccer ball of the following type:

 To simulate the polyhedra we made, one can joint two vertice with the same color. Then the polyhedra can be seen as a mix of five-point stars and equaliteral triangles. Here is the picture:

The process of making such an artifect is fun! We first used 120 clips to make 12 five-point stars (as below):

Then we bended each point a little bit and connect it with another star's point. Repeating this process, we got the polyhedra. The last step was hard as we needed to connect the last star from the inside of the ball (we actually took the last star apart first).


These artifects can be good structures to raise students' interests and engagement levels in a math class. One can tell that without taking a clip's arms off, it is impossible to make a star. This step of taking the two arms off is like assumptions of mathematical theories:  it is only that ordinarity must be broken and assumptions must be built that a theory can be constructed.

Also, the building process requires a lot of teamworks, which means that no one can solo the mission. All students----higher level students and lower level students----in class can benefit from this collaboration.

Finally, stduents can easily accept homeworks made based on this artifect. For example, how many clips are used in this project? The answer is 150, with 120 for stars, and 30 for connecting the stars.

All in all, it is a nice bridge to involve students in the math classroom.

Mathematics for Social Justice reading

Please read these excerpts from David Stocker's Math that matters ("Beyond pizza party math") textbook. Here are some questions you might want to think about as you read this:


Is mathematics 'neutral', or is it connected with social/ environmental justice?
What are your ideas about the author's intentions in writing this textbook?
Can  these ideas from middle school math inspire teaching ideas for your secondary math classes?
Are there topics in mathematics that are more or less possible to connect with social justice issues?


Mathematical knowledge itself is neutral, from my perspective, but the way we teach it can be related to social/environmental justice. However, it may be a problem that too much "relating" can be harmful to some students' interests of scoring high in exams. My understanding of this is to find a balance of different interests----neither too much relating nor too little relating is good to most students and myself.

The way mathematics is taught must be student-oriented. This ideology, if we explicitly explain its meaning through the perspective of student-oriented, means that mathematics only make sense if it is taught in a way that makes sense to students. In other word, the knowledge itself must be wrapped in a cover, through which students can relate the knowledge to their own realities, that bridges students from a comfort zone (a place in which mathematical knowledge is used implicitly, or maybe is seemingly unrelated to the reality) to a tough zone (a place in which knowledge is explicitly explained using mathematical language, which is strict and boring to some) and back.

It is absolutely possible that these ideas can be transplanted from a middle school setting to a secondary school setting, although I still want to point out that it is highly related to school environments. I was told that in some IB schools, math teachers are only given 90 hours (instead of 120 hours) an academic year to finish all the materials in HL mathematics. I like to try some strategies the book recommends, but it is risky to some extent to both me and my students.

And finally, about the social justice issues, I believe these topics can be used in math class. Minimum wage, tax policies, or any other issues are all more or less relavant to some mathematical knowledge.

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Actually I am not quite sure what "social justice" David Stocker means in his book. Here I just assume it is a real-life topic that can be related in mathematical concepts, and I do realize that my understanding ignores the word "justice" but I do not know what or how to relate it to the topic.