The November issue of Teaching Children Mathematics features 3 very good articles. The first, “What Knowledge does teaching require?” by Thames and Ball mention that a clear description of such knowledge in research over the past 40 years has been rather elusive. Just because you know a lot of math or can do math very well, doesn’t mean you can teach it well. Of course, the authors do not doubt that you have to have a good grasp of mathematics. Whether you’re a math major, however, does not make a difference according to their findings. What they argue is that teachers need to “uncover the mathematical issues that arise in practice. By better understanding the mathematical questions and situations with with teachers must deal, we would gain better understanding of the mathematics it takes to teach.”
They broke up what they called, “Mathematical knowledge for teaching” into several domains in relation to teaching. There’s “subject matter knowledge (SMK)” and “pedagogical content knowledge (PCK).” Each of these two can be divided further into three sub categories each. SMK can be grouped into these three: common content knowldedge, knowledge at the mathematical horizon, and specialized content knowledge. PMK can be grouped into these three: knowledge of content and students, knowledge of content and teaching, knowledge of curriculum.
The authors also mention that these are some math skills good teachers or mathematics need:
- posing mathematical questions
- giving and appraising explanations
- choosing or designing tasks
- using and choosing representations
- recording mathematical work work on the board
- selecting and sequencing examples
- analyzing student errors
- appraising students’ unconventional ideas
- mediating a discussion
- attending to and using math language
- defining terms mathematically and accessibly
- choosing or using math notation
Just today, we introduced the children in my class to base 4. The purpose is to solidify the concept of place value and to understand the construction and deconstruction of numbers so that the children have a better understanding when taught the traditional algorithm. Just going through the list above, we did several things differently from last year. This year, I chose to create something that we could represent on the board. It made a huge difference – not only in understanding, but in engagement. Of course having kids interact with the board was great. So what if the architects decided that the board had to be a minimum of so many feet off the ground, I stuck a chair nearby and those who needed it, could step on and still reach. Letting the children give their explanations as to why they were rounding up groups of 4 and putting them in the next column seemed to help a lot too. Posing questions was also great. About halfway through the activity, one of my students raised her hand and said, “I think that with the materials we have here, the highest number will be 133.” She was absolutely right.
This article says that mathematical knowledge in the way that is required for reasearch in mathematics is not what is needed. What is needed is someone who has excellent “mathematical knowledge for teaching which itself is a kind of complex mathematical understanding, skill, and fluency used int he work of helping others learn mathematics.”
The second article, titled “Balancing Act” focused on high-quality professional staff-development. First of all, that development needs to be ongoing. According to the author, it needs to be focused on three essential things:
- Content focused on instructional design and implementation
- Process characterized by inquiry, observation, research, and collegial interactions
- Contextual support for job embedded professional training
This development has to be long-term and sustained in order to make a lasting change. If you think about those three criteria above combined with the need to customize learning to the needs of each individual student, there is no way that two classes will ever be the same, and while a textbook may be a useful tool at times, one must resort to using other tools as well. The author of this article says the challenge is to find the time to collaborate and reflect on what worked and what didn’t. The author goes on to say that textbook providers tend to provide training to teachers once they’ve purchased their materials, but that training goes away after a year or two and teachers, especially new teachers, need continual training. Another challenge that lies in changing how math is taught is the resistance many teachers have to changing what they’ve been doing for years. I have always been open to change, but I have to admit that I am resistant to thinking that a new textbook alone can be a panacea to improve the teaching of mathematics. You need to also know HOW to teach it.
Finally, the third article from this issue I liked was titled, “A reflection framework for teaching math” by Merrit, Rimm-Kaufman, Berry, Walkowiak, and McCracken. While these authors offer a guideline with 8 dimensions in their “framework for reflection, not every lesson will be ‘high’ on all dimensions.” Below are their 8 dimensions. As one reflects on a lesson, here is what to consider:
- Lesson structure
- Multiple representations
- Mathematical tools
- Cognitive depth
- Mathematical discourse community
- Explanation and justification
- Problem solving
- Connections and applications
After we finish working in different bases in my class, we will move on to addition and the use of traditional algorithms. There are some children, solid in their understanding of place value, who already know how to use the traditional addition algorithm where you carry (regroup) into the next column. If these children have already mastered this, how do I extend their learning while teaching the rest of the children the traditional base-10 algorithm for addition? At least I’ll have until after Thanksgiving break to ponder this.
My opinion on why so many districts have failed to implement a textbook series successfully is not because of the content within those books (as I mentioned they make a nice tool – and there are many online tools available for free, if one is willing to take the time to find them), it is because there hasn’t been sustained development in making better teachers of mathematics regardless of the text or tool. Whether it be Everyday Mathematics, Singapore Math, or other programs (each has it’s pros and cons), it boils down to how you use it with your students. The concepts behind the math may not have changed over the years, but the way we teach it, engage our kids, and make it stick certainly have.