Posted in Algebra, Curriculum Reform

What is algebra? Why study it?

I’m doing some  literature review for my research and I came across this article by L.A Steen in Middle Matters. He was arguing about the Algebra for All standard in the US and part of the article includes description of what is algebra. I thought I should share them in this blog because it is something very important teachers should be aware of when they teach algebra or what they conceive what algebra is and for. Oftentimes, when students ask what algebra is and what they are going to need it for, teachers lazy answer is “Algebra is just like your math in the grades only that this time you work with letters instead of numbers!”

  1. Algebra is the language of mathematics, which itself is the language of the information age. The language of algebra is the Rosetta Stone of nature and the passport to advanced mathematics (Usiskin, 1995).
  2. It is the logical structure of algebra, not the solutions of its equations, that made algebra a central component of classical education.
  3. As a language, algebra is better learned earlier and harder, when learned later.
  4. In the Middle Ages, algebra meant calculating by rules (algorithms). During the Renaissance, it came to mean calculation with signs and symbols–using x‘s and y‘s instead of numbers. (Even today, lay persons tend to judge algebra books by the symbols they contain: they believe that more symbols mean more algebra, more words, less.) I think that many algebra classes still promote this view.
  5. In subsequent centuries, algebra came to be primarily about solving equations and determining unknowns. School algebra still focuses on these three aspects: employing letters, following procedures, and solving equations. This is still very true. You can tell by the test items and exercises used in classes.
  6. In the twentieth century algebra moved rapidly and powerfully beyond its historical roots. First it became what we might call the science of arithmetic–the abstract study of the operations of arithmetic (addition, subtraction, multiplication, etc.). As the power of this “abstract algebra” became evident in such diverse fields as economics and quantum mechanics, algebra evolved into the study of all operations, not just the four found in arithmetic.
  7. Algebra is said to be the great gatekeeper because knowledge and understanding of which can let people into rewarding careers.
  8. Algebra is the new civil right (Robert Moses). It means access. It means success. It unlocks doors to productive careers and gives everyone access to big ideas.

And I like the education battle cry Algebra for All. Of course not everyone is very happy about this. Steen for example wrote in 1999:

No doubt about it: algebra for all is a wise educational goal. The challenge for educators is to find means of achieving this goal that are equally wise. Algebra for all in eighth grade is clearly not one of them–at least not at this time, in this nation, under these circumstances. The impediments are virtually insurmountable:

  1. Relatively few students finish seventh grade prepared to study algebra. At this age students’ readiness for algebra–their maturity, motivation, and preparation–is as varied as their height, weight, and sexual maturity. Premature immersion in the abstraction of algebra is a leading source of math anxiety among adults.
  2. Even fewer eighth grade teachers are prepared to teach algebra. Most eighth grade teachers, having migrated upwards from an elementary license, are barely qualified to teach the mix of advanced arithmetic and pre-algebra topics found in traditional eighth grade mathematics. Practically nothing is worse for students’ mathematical growth than instruction by a teacher who is uncomfortable with algebra and insecure about mathematics.
  3. Few algebra courses or textbooks offer sufficient immersion in the kind of concrete, authentic problems that many students require as a bridge from numbers to variables and from arithmetic to algebra. Indeed, despite revolutionary changes in technology and in the practice of mathematics, most algebra courses are still filled with mindless exercises in symbol manipulation that require extraordinary motivation to master.
  4. Most teachers don’t believe that all students can learn algebra in eighth grade. Many studies show that teachers’ beliefs about children and about mathematics significantly influence student learning. Algebra in eighth grade cannot succeed unless teachers believe that all their students can learn it. (all italics, mine)

I shared these here because in my part of the globe  the state of algebra education is very much like what Steen described. You may also want to read about the expressions and equations that makes algebra a little more complicated to students.

L.A Steen is the editor of the book On the Shoulder of Giants, New Approach to Numeracy, a must read for teachers and curriculum developers. The book is published by Mathematical Sciences Education Board and National Research Council.

Posted in Curriculum Reform, Mathematics education

Understanding by Design and Pedagogical Content Knowledge

If you know where you are going, it doesn’t always follow that you also know how to get there. Understanding by Design (UbD) supports the first part of the statement: knowing where you are going. Its three-stage curriculum planning framework is useful as a general guide for identifying where to lead the students in terms of understanding what they are supposed to be learning.

In Stage 1: Desired Results, teachers think about what goals they would like their students to learn. They also jot down the “big ideas”, “essential questions”, and what students should know after the lesson.

Continue reading “Understanding by Design and Pedagogical Content Knowledge”

Posted in Assessment, Curriculum Reform

Teachers teach to the test, students study to the test

The DepEd is finally bidding adieu to multiple choice test. Better late than never, I must say. So my fellow math teachers, the next time you are required to make purely multiple choice items for periodical test or are given by the division, or by the regional office an achievement test in multiple choice, you can quote the following: Annex A – The Monitoring and Evaluation of the Implementation of the 2002 Basic Education Curriculum: Findings and Recommendations of the UbD-based 2010 Secondary Education Curriculum Guide for Mathematics 1 document released by DepEd. On page 9-10 of the said document you will find this report:

 

9. Teachers teach to the test, students study to the test.

The use of traditional assessment tools like the multiple-response, simple recall, recognition and application tests is predominant.  Rubrics, portfolios, and other forms of authentic assessment are not widely used.  Teachers are aware of the limitations of traditional tests and the need for alternative forms to measure higher order thinking skills.  However, they tend to resort to the traditional forms for several compelling reasons:

  • These are the types used in periodic and achievement examinations.
  • They are easier to score.  (Teachers teach as many as 300 to 400 students a day and scoring non-traditional measures like rubrics could be an ordeal.)
  • They are easier to prepare than the non-traditional forms like portfolios, rubrics, and other authentic measures.
  • These are what everybody else is using.
  • Teachers have inadequate knowledge of authentic learning and authentic assessment.

Documentary analysis showed that schools in general lack an institutionalized system of utilizing test results for diagnostic and remedial purposes.

Teachers tend to teach to the test; students tend to study to the test.  This culture is reinforced by supervisors who specify units to be taught and tested for each grading period and use test results more for judging rather than improving teacher and student performance.

Recommendations:

Schools should review their present assessment practices.  The teacher appraisal system and the kinds of tests used in the classroom as well as those, in the division and national examinations, should be evaluated against the goals and objectives of the Basic Education Curriculum, among which is the development of critical thinkers and problem solvers.

Schools should also consider the use of alternative assessment tools and techniques that would provide opportunities for students to experience learning as an enjoyable, delighting process of inquiry, discovery, construction and creation of new knowledge, rather than as a tedious process of cramming to pass examinations.

While schools should double their efforts for students mastery of the basic competencies they should also never lose sight of the fact that their ultimate goal should be the development of functionally literate citizens of a democratic community.

I think the DepEd forgot to include another reason why teachers use multiple choice test. The sixth bullet should be: The National Achievement tests  in all subject areas are 100%  in multiple-choice type form and the test results are used more for comparing schools rather than as basis for developing programs for improving teaching competence and performance.

The day the National Achievement Test (NAT) of the DepEd will include constructed-response type questions should be declared a national holiday because it will really mark a turning point in the history of education in the Philippines.

Posted in Curriculum Reform, Mathematics education

Curriculum change and Understanding by Design, what are they solving?

Not many teachers make an issue about curriculum framework or standards in this part of the globe. The only time I remember teachers raised an issue about it was in 1989, when the mathematics curriculum moved from compartmentalized (elem. algebra, intermediate algebra, geometry, adv. algebra & statistics) to spiral-integrated approach. The reason behind the change was the poor performance of the students. Many teachers didn’t like the change in the beginning not only because it’s the first time that the mathematics curriculum is organized that way, hence new, but also because it demands re-learning other areas of mathematics which they have not taught for years.  Also, teachers were not taught mathematics in high school nor in college that way. But the curriculum was pushed through just the same and eventually teachers complaints about it died down. Why? No one knows. They just continue teaching what they know in the way they think best.

Sometime in late 2001 or was it 2002, the then secretary of DepEd made a phone call to one of the country’s math education consultants. The country’s students seem not getting any better. Something’s got to be done about it. So one day, in 2002, the country’s basic math community woke up with a new curriculum, back to the compartmentalized system. The identified culprit according to the sponsor of the compartmentalized curriculum was that teachers are not that capable yet to implement the spiral-integrated curriculum that is why the still low students’ achievement. Clearly teachers need upgrading in their content knowledge and pedagogical knowledge and they need a lot of support resources for teaching.  The solution made to this problem? Change the curriculum. In fact not only to change it back to where it was but DepEd reduced the content further to minimum competencies consisting of learning of facts and procedures, a sprinkling of problem solving and an inch thick of content for mathematics. Did the teachers like it? Did it work? No one knows. They just continue teaching what they know in the way they think best.

It’s 2010. The minimum learning competencies lived up to its name. It provided minimum knowledge and skills. The students’ achievements did not get any better.

By June this year, the Math 1 (Year 7) teachers will be making their lesson plans based on UbD. UbD or Understanding by Design is the title of a book which proposes a new way of doing curriculum planning. In the school level, its in the way the teachers will be preparing their lesson plans. UbD is based on backward design. The main difference between backward design and the usual way of writing the lesson plan is that you spend time first formulating how you will assess the students based on your identified goals (aka enduring understanding and essential questions using UbD lingo) before thinking about the activity you will provide the class and how you will facilitate the learning.  I’ve yet to see and read a report from the proponents and users of UbD for evidence that it really works. And working in what aspect? in which subject area? and, whether it is better than the usual way teachers prepare their lesson plan?  Some schools who have tried it reported that at first, teachers had a lot of difficulty in making a UbD-based plan but they eventually got the hang of it. Are they teaching any better? Are the students doing well? Silence. I’m asking the wrong questions. For indeed, a great distance exist between way of preparing lesson plans and students’ achievement. So why are schools all over the country mandated to adopt UbD? I don’t know.

But this is what I know.  I know that teachers need support in upgrading and updating their knowledge of content and pedagogy.  I know that teachers teach what they know in the way they know.  These are things that cannot be addressed by simply changing the curriculum or changing the way of preparing the lesson plan, much more its format. The book The Teaching Gap which reports the TIMSS 1999 video study tells us what we should focus our attention and resources more on:

“Standards [curriculum] set the course, and assessments provide the benchmarks, but it is teaching that must be improved to push us along the path to success” (Stigler & Hiebert, The Teaching Gap, p.92).

I couldn’t agree more to this statement. I’m not very good at memorizing so to commit it to memory I paraphrased Stigler & Hiebert’s statement to: It’s the teaching, stupid.

Click here for my other post about UbD.