Day 10--Creating a Balanced Math Program

Balance is Basic: A 21s Century View of a Balanced Math Program

As the title indicates, this article is about creating a balanced math program in the 21st century.  This article is less of a "how to" and more of an outline of the importance of and reasoning for creating a balanced math program.

According to the article, the fundamental things we want students to know and do has not really changed over time.  We want them to understand concepts, be able to perform mathematical skills, and solve problems.  The problem lies not in these broad categories but in the weight and importance placed on each. The first category is conceptual understanding.  This is the area that is most concerned with students understanding of given concepts rather than their ability to do computation or solve problems.  This can be likened to the discussions we have had in class around the big idea.  If the big idea or concept is division, when thinking about this category we would be concerned most about the student understanding that when you divide, the quantity you are dividing gets smaller.  This is similar to the example in class today where Ve demonstrated how we would help students come to an understanding that a positive exponent makes a number larger, while a negative exponent makes it smaller.

The next category is concerned with the actual skills, facts, procedures of math. Using the example above, this is where teach students how do divide, show them the symbols necessary for division, and allow them to work with the numbers to figure things out.  There is some debate as to how much computational knowledge students need.  Do they need "drill and kill" where they are doing the same calculations over and over again just with different numbers, or is it better for them to know how to do a few or specific calculations and be able to use some sort of tool to figure out things beyond that?  Marian Small advocates for an approach that does less of the former and more of the latter.  She argues that given the technological world we live in and the accessibility of devices, it is easy to find a tool to help with the calculations. This ideas is furthered by the author of the article, who suggests that students might have to know how to divide two digit numbers by using pencil and paper, but calculators could be used for larger tasks.

The last category is problem solving.  This is the area concerned with moving students out of the textbook and giving them the opportunity to apply what they know to authentic and real-world contexts.  According to the article, this category is not just solving the problem, but being able to use the right tools for the problem as well.  Problem-solving is often used as the measure to indicate how well students have grasped and assimilated what they have learned.

Beyond these three key categories is what the author refers to as the "connective tissue".  These are concepts like mathematical habits of mind, flexible thinking skills and strong reasoning skills that are necessary skills for students to have if they are to be successful in the 21 century.  Most of these skills can be related to the mathematical processes that we learned about recently. For example, the connecting process would allow for mathematical flexibility because students would learn how to apply their understanding in different contexts. Another example might be the reasoning and proving processes where students would need to provide thoughtful answers and be able to support or prove their reasoning.

One of the last things the article touches on is the idea of engagement being one part of an interconnected rope that defines mathematical knowledge.  Engagement is includes skills/qualities like persistence and willingness to take on a challenge.  This can be related to the growth mindset that we discussed needing to be cultivated in our students.  Dan Meyer seems to support the idea that this is something that needs to be developed in students because according to him, we have students who are "impatient with irresolution".  That is to say that they want quick fixes and are uncomfortable with things that do not come in simple packages.  We have to teach them that sometimes they have to stick with it and sometimes that have to just accept the discomfort.

Below is a puzzle that I made to demonstrate my understanding of the today's reading.  I conceptualized it as a three set Venn diagram with a balanced math program at the intersection of doing the math, conceptual understanding, and problem solving.  All three of the categories are bound by the connective tissue that holds everything in place. I used  a puzzle to illustrate the idea that everything is connected and without any individual piece, the picture would be incomplete.

Discussion questions:
* Do you think you use one knowledge area over another? If so, which is it and how do you think you would counteract that to ensure you have a more balanced program.
* Do you think you can have an effective math program without the “connective tissue?” Why or why not? Would the program still be effective without the connective tissue?
 * Even though there is evidence to support a balanced math program, there are teachers who still favor a “back to basics” approach. What arguments do you think they might have to support this thinking?
 * Do you think you might meet resistance trying to input this program in your school? Who might you meet resistance from and how would you address the resistance?