Category: Curriculum Reform

Posted in Curriculum Reform

The nature of math vs the nature of school math

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The mathematics experienced by students is so much different from the real nature
of math. What a tragedy!

What is the nature of mathematics?
  1. Mathematics is human. It is part of and fits into human culture. It is NOT an abstract, timeless, tensely, objective reality…
  2. Mathematical knowledge is fallible. As in science, mathematics can advance by making mistakes and then correcting them…
  3. There are different versions of proof or rigor. Standards of rigor can vary depending on time, place, and other things. Think of the computer-assisted proof of four color theorem in 1977…
  4. Empirical evidence, numerical experimentation and probabilistic proof all can help us decide what to believe in mathematics…
  5. Mathematical objects are a special variety of a social-cultural-historical object …They are shared ideas like Moby Dick in literature and the Immaculate Conception in religion.

The above description of the nature of mathematics is by Reuben Hersh,  from his article “Fresh Breezes in the Philosophy of Mathematics published in American Mathematical Monthly Aug-Sept, 1995 issue. He is also the author of the now classic What Is Mathematics, Really?.

What is the ‘nature’ of school mathematics?

The following is a 2002 critic of the US k-12 mathematics by Paul Lockhart in A Mathematician’s Lament.  It’s also true in my part of the globe.

The Standard of K-12 mathematics according to Lockhart:

LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not something you do, but something that is done to you. Emphasis is placed on sitting still, filling out worksheets, and following directions. Children are expected to master a complex set of algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part, and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables are stressed, as are parents, teachers, and the kids themselves.

MIDDLE SCHOOL MATH. Students are taught to view mathematics as a set of procedures, akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are handed out, and the students learn to address the church elders as “they” (as in “What do they want here? Do they want me to divide?”) Contrived and artificial “word problems” will be introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison.

ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this course instead focuses on symbols and rules for their manipulation…. The insistence that all numbers and expressions be put into various standard forms will provide additional confusion as to the meaning of identity and equality. Students must also memorize the quadratic formula for some reason.

GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy and distracting notation will be introduced, and no pains will be spared to make the simple seem complicated. This goal of this course is to eradicate any last remaining vestiges of natural mathematical intuition, in preparation for Algebra II.

ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety of standard formats for no reason whatsoever. Exponential and logarithmic functions are also introduced in Algebra II, despite not being algebraic objects, simply because they have to be stuck in somewhere, apparently.

TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and obsolete notational conventions, in order to prevent students from forming any clear idea as to what the subject is about. Students will learn such mnemonic devices as “SohCahToa”…

Posted in Curriculum Reform, Teaching mathematics

What is reasoning? How can we teach it?

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The  world does not give us complete information that’s why call on our power of reasoning to complete this information the best we can and produce new knowledge.  Mathematics is one of its most famous product.

What is reasoning? When do we learn it?

Reasoning is defined as  the capacity human beings have to make sense of things, to establish and verify facts, and to change or justify practices, institutions and beliefs. We can make this definition more specific using Ol’eron’s:

“Reasoning is an ordered set of statements, which are purposefully linked, combined or opposed to each other respecting certain constraints that can be made explicit.” – Ol’eron (1977; 9)

Teachers’ knowledge of learning trajectory for reasoning is as important as their knowledge of students’ typical learning trajectory for specific content topics. In this post I will share a framework that I think will be useful for teachers in developing the reasoning skills of learners. I cannot anymore trace where I got this idea but I know it’s from a Japanese lesson study document I was reading last year. Reasoning is a skill highly emphasized in Japanese mathematics lessons. They have developed a framework for analyzing their students ‘reasoning trajectory’. This is applicable even in non-mathematics context. The framework even specifies the grade level to which a particular way of reasoning and arguing it is expected.

Levels of reasoning
  1. At the end of 2nd grade, students begin using expressions such as “because…” to describe their reasons and support their ideas.
  2. In 3rd grade, they begin comparing their own ideas with others and use expressions such as “my idea is similar to so-and-so’s idea…”
  3. In 4th grade, students use expression such as “for example…” and “because…,” more frequently Moreover, they begin to use hypothetical statements such as “if it is… then…”
  4. In 5th grade, they can become more sophisticated and make statements such as, “If it is … then it will be *, but if it is # then I think we can say @” by looking at different conditions.
  5. Finally, in 6th grade, students can start describing in ways such as, “It can be said when it is … but in this situation # is much better,” and starting to make decisions about how to choose a better idea.
In teaching mathematics, reasoning need not always be restricted to that of formal, logical or mathematical forms of reasoning. Words and phrases such as those listed above should be part of the students communication. It is therefore important to listen to the way students make their arguments or reason out in whole class and small-group discussion. If these are not yet part of the everyday communication of mathematics in our classes then its time for us to design the lesson that creates the environment where these kind of thinking and communicating is encouraged. Problem solving and mathematical investigation activities are great context where this can happen.

 

Posted in Curriculum Reform

Explore, Firm Up, Deepen, Transfer

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When we were just being trained to be teachers of mathematics it was emphasized to us that in planning our lesson we should think of manipulative activities whose results will eventually lead to the concepts to be learned. The teacher will make use of the students results to introduce the new concept through another whole class activity to tie together the results or through question and answer discussion. This leads to the definition of the concept by the teacher or to a certain procedure or calculation with the help of the students, depending on the topic. The teacher then gives exercises so students can hone their skill or deepen their understanding of the concept. A homework, usually a more difficult version of the one just done in the class, is given at the end of the lesson. I don’t remember my supervising teacher requiring me to always give a test at the end of my lesson. I think I was on my third year of teaching in public school when this ‘bright idea’ of giving a test at the end every lesson was imposed. Failure to do so means you did not have a good lesson because you do not have an evaluation part! Anyway, let me stop here as this is not what I want to talk about in this post. I want to talk about the latest ruling about “Ubidized lesson pans”.


image from art.com

When I first heard about the DepEd’s “Explore-FirmUp-Deepen-Transfer” version of UbD  I remember the framework I followed when I was doing practice teaching at Bicol University Laboratory High School. The lesson starts with activities, process results of activities to give birth to the new concept, firm-up and deepen the learning with additional exercise and activities and then use the homework to assess if students can transfer their learning to a little bit more complex situation. So I thought EFDT must not be a bad idea. I have observed as a teacher-trainer that over the years teachers have succumbed to the temptation of talk-and-talk method of teaching. Reason: there are too many students, activities are impossible; too many classes to handle, too many topics to cover. With this scenario I thought EFDT may turn out to be a much better guide in planning the lesson that the one currently being used: “Motivation-LessonProper-Practice-Evaluation” because EFDT actually describes what the teachers need to do at each part of the lesson. But it turned out that EFDT was very different what I think it is and is being implemented per chapter and not per topic or lesson in the chapter!

I don’t know if the teachers simply misinterpreted it or this is really how the DepEd wants it implemented. If this is how UbD is being done in the entire archipelago then we have a BIG problem.

  • The chapter is divided into four parts: First part- Explore; Second part- Firm Up; Third Part – Deepen; Fourth Part – Transfer. There are many unit topics in a chapter so it means for example that what is being ‘deepened’ is a different topic to what has been ‘firmed-up” or “explored’! I think this is a mortal sin in teaching.
  • EFDT is used in all subject areas.  The nature of each subject, each discipline, is different. I don’t know why some people think they can be taught in the same way or to even think that within a discipline, its topics can be taught in the same way. Or that the same style of teaching is applicable to all year levels in all kinds of ability. UbD, the real one, not our version, does not even promote a particular way of teaching but a particular way of planning. Stges 1 and 2 dictates the teaching that you needed to do.
  • Activities for Explore part always have to be done in groups and with some physical movement. A math teacher was complaining to me that her students no longer have the energy for their mathematics class especially during the “explore’ part because all subject areas have activities and group work so by the time it’s math period which happens to be the fourth in the morning, students no longer want to move. The explore part alone can run for several days. All the while I thought the “explore part” of EFDT can be done with a mathematical investigation or an open-ended problem.
  • The prepared lesson plans given during the training consists of activities from explore part to transfer part and teachers implement them one after another without much processing and connection. Most activities aren’t connected anyway.
  • The teachers can modify the activity but they said they don’t have resources where to get activities.
  • The teachers cannot modify the first two parts of the UbD plan. The teachers said they were told not to modify them. I asked “how does it help you in the implementation of the lesson?” They said “we just read the third part, where the lessons are. We don’t really understand this UbD. Our trainers cannot explain it to us. They said it was not also explained well during the training.
  • The teacher have this cute little notebook which contains their lesson. So I asked “so what is your lesson at this time?” She said it’s 3.5. Indeed that’s the little number listed there. So what’s it about. I think we are now on Firm-up. I have to check the xerox copy of the lesson plan distributed to us. Well, I thought UbD is a framework for designing the lesson. It was proposed by its author with the assumption that if teachers will design their lesson that way, then perhaps they can facilitate their lesson well. How come that teachers are not encourage to design their own lesson? How come we give them prepared lesson plans which have not even been tried out?

Here’s my wish Explore, Firm-up, Deepen, and Transfer be interpreted in mathematics teaching.

Explore – students are given an open-ended problem solving task or short mathematical investigation and they are given opportunity to show different ways of solving it.

Firm-up – the teacher helps the students make connections by asking them to explain their solutions and reasoning, comment on other’s solutions, identify those solutions that uses the same concepts, same reasoning, same representation, etc.

Deepen – the teacher consolidates ideas and facilitates students construction of new concept or meaning, linking it to previously learned concepts; helps students to find new representations of ideas, etc.

Transfer – teacher challenges students to extend the problem given by changing aspects of the original problem or, construct similar problems and then begin to explore again.

The above descriptions corresponds to a way of teaching called teaching mathematics via problem solving which this blog promotes.

Credits: image from art.com

Posted in Curriculum Reform, Mathematics education

Why math education is failing

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A backlink to my post What kind of mathematical knowledge should teachers have?   brought me to the essay by Matthew Brenner titled The Four Pillars Upon Which the Failure of Math Education Rests (and what to do about them). Here’s the quote from the essay posted in Wild about Math.

Kids are taught math as pets are taught tricks. A dog has no idea why its master wants it to perform. With careful training many dogs can be taught to perform complex sequences of actions in response to various commands and cues. When a dog is taught to perform a trick it has no need or use for any “understanding” beyond which sequence of movements its trainer desires. The dog is taught a sequence of simple physical movements in a specific order to create an overall effect. In the same way, we teach children to perform a sequence of simple computations in a specific order to achieve an overall effect. The dog uses its feet to move about a space and manipulate objects; the student uses a pencil to move about a page and manipulate numbers. In most cases, the student doesn’t know any more than the dog about the effect he creates. Neither has any intrinsic motivation to perform nor any idea why the performance is demanded. Practice, practice, practice, and eventually the dog can perform reliably on command. This is exactly how kids are trained to perform math: do a hundred meaningless practice problems, and then try to do the same trick on the test.

Mr.Brenner’s observation is as true in America as it is here in the Philippines. This is a painful truth but something that we all must take seriously. I strongly encourage our teachers, those writing our new curriculum framework (I think this is our third within the decade), textbook publishers and our DepEd officials to read the entire essay. The author outlined the reasons why math education is failing but he also offers solutions which I believe are doable even if our average class size here is 60! Let me list the 10 point solutions:

  1. Understanding Must be Central in Math Education
  2. Textbooks Must Not be Allowed to Undermine Math Education
  3. Teachers Must Stop Teaching Math as They Learned It
  4. Curricula Must be Coherent and Cumulative
  5. Worked Examples Must be Emphasized for New Material
  6. Curricula Must Include Examples of Excellent Performance
  7. Assignments Must Draw on the Old and the New
  8. Content Must be Meaningful and Contexts Must be Rich
  9. Metacognitive Activity Must Pervade Mathematical Activity
  10. Language Must be Taught, Used and Evaluated Fairly
I do not agree with #5 proposal because I believe that mathematics should be taught in the context of solving problems but I think this is a very good list. Find time to read it. Mr. Brenner also offers very good sample lessons. You may also want to read 10 signs there’s something not right in school maths and let us know your thoughts.