Tag: math teachers

Posted in Mathematics education

What kind of mathematical knowledge should teachers have?

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As a result of her research, Liping Ma developed the notion of profound understanding of fundamental mathematics (PUFM) as the kind of mathematical knowledge teachers should possess. She discusses this kind of knowledge in her book Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series). This book is now considered a classic by many mathematics educators. The ‘elementary’ in the title does not mean the book will be valuable to elementary teachers only or those engage in the training of prospective elementary teachers. The book is for all mathematics teachers, trainers, and educators. This book is a must-read to all that has to do with the teaching of mathematics.

Here’s what Liping Ma says in the introduction:

Chinese students typically outperform U.S. students on international comparisons of mathematics competency. Paradoxically, Chinese teachers seem far less mathematically educated than U.S. teachers. Most Chinese teachers have had 11 to 12 years of schooling – they complete ninth grade and attend normal school for two or three years. In contrast, most U.S. teachers have received between 16 and 18 years of formal schooling-a bachelor’s degree in college and often one or two years of further study.

In this book I suggest an explanation for the paradox, at least at the elementary school level. My data suggest that Chinese teachers begin their teaching careers with a better understanding of elementary mathematics than that of most U.S. elementary teachers. Their understanding of the mathematics they teach and -equally important – of the ways elementary mathematics can be presented to students continues to grow throughout their professional lives. Indeed about 10% of those Chinese teachers, despite their lack of forma education, display a depth of understanding which is extraordinarily rare in the United States….

Why the word ‘profound’? Profound has three related meanings – deep, vast and thorough – and profound understanding reflects all three. From the paper delivered by Liping Ma and Cathy Kessel in the Proceedings of the Workshop on Knowing and Learning Mathematics for Teaching conference, Liping and Cathy offered the following explanation:

  • A deep understanding of fundamental mathematics is defined to be one that connects topics with ideas of greater conceptual power.
  • A vast or broad understanding connects topic of similar conceptual power.
  • Thoroughness is the capacity to weave all parts of the subject into a coherent whole.

A teacher should see a ‘knowledge package’ when they are teaching a piece of knowledge. They should know the role of the current concept they are teaching in that package and how that concept is supported by which ideas or procedures.

To further explain the kind of mathematics knowledge a teachers should possess, Liping and Cathy used the analogy of a taxi driver  who knows the road system well. The teachers should know many connections so that they are able to guide students from their current understandings to further learning.

I think this is how designers of curriculum, writers of curriculum materials, and teachers should interpret the standard “Making connections”.  It is not simply linking.

 

Posted in Algebra

Teaching and Learning Algebra Blog Carnival

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I am hosting a special edition of Math and Multimedia Blog Carnival. This special edition will be all about the teaching and learning of algebra at all year levels.

A blog carnival is like a magazine. Checkout the following carnival to give you some ideas:

1.) Mathematics and Multimedia Carnival #1

2.) Let’s Play Math Carnival #24

3) Carnival of Maths

Articles, lessons, tasks and activities, algebra problems,  learning and teaching algebra in multimedia format, your experiences with algebra as teacher or student, etc are welcome. Teaching and learning prealgebra topics are also most welcome.

To submit, email me the link to your post.

The carnival will be posted in December 2010. Yes, you have time to write.

Thanks.

Posted in Curriculum Reform, Mathematics education

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

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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.