Let's return to the integers unit for grade 6 (IBL Integers Unit). The actual context doesn't really matter so much as the framework presented here, and as before I want to capture a wider audience.

We start with a traditional rote explanation of subtracting a number in traditional math settings. I sometimes see subtracting an integer as (a) change the sign ($-(-1) = +1$), (b) remember to move right on the number line. I personally experienced (a), when I was a student. I was told when you see two minuses, you change it to a plus. So I learned how to get to an answer, without having to learn the concepts that make things work.

I think it's easy to say that most of us agree that merely doing the computation in the ways presented above are a limited and not ultimately beneficial to students without a broader understanding of integers. The IBL Integers Unit uses a context, includes a

*model*

*for*

*thinking*, introduces zero pairs and mathematical equivalence, and requires students to write a justification why subtracting a negative is equivalent to adding.

Let's break things down...

LZA of the Traditional Integers content

- Computing how to subtract integers, skills practice
- A connection to the number line, but perhaps without conceptual grounding

LZA of the IBL Integers Unit

- Computing how to subtract integers, skills practice
- Context for problem solving
- Modeling numbers and equivalence (zero pair)
- Problem solving
- Argumentation and justification

It's immediately obvious the difference in the list. One misconception in the general public is that the new teaching vs. old teaching is about style and that they are assumed to have the same goals and achieve the same ends. It's clear that the goals are different, and that one is more sophisticated than the other. Moreover, both instructors can say, "I covered integers." The nature of the coverage is vastly different, and while one got through it faster, I'd like to say, "So what?" What real math was learned if all we achieved is answer getting.

Once again cultivating dispositions is done more appropriately in the IBL setting than the traditional setting. One can risk saying that a big missing piece in the general discussion about education reform is the difference in what the point of education is. It makes me wonder if unacknowledged differences in "education axioms" may be a significant contributor to the friction in public discourse.

Another point worth mentioning is that there is an interaction between the method of teaching, teaching philosophy, and the content. When we think of students as explorers and doers of mathematics, then we are more inclined to present to them tasks that are a different nature than if our view of teaching is focused on skills acquisition (or passing standardized tests). So teaching isn't just a method. It's a system. What we value is important in education, our methods, how we assess, what we assess, our beliefs about what students are capable (and not capable of doing), and the goals of education all feed into what happens in the classroom.

Once again cultivating dispositions is done more appropriately in the IBL setting than the traditional setting. One can risk saying that a big missing piece in the general discussion about education reform is the difference in what the point of education is. It makes me wonder if unacknowledged differences in "education axioms" may be a significant contributor to the friction in public discourse.

Another point worth mentioning is that there is an interaction between the method of teaching, teaching philosophy, and the content. When we think of students as explorers and doers of mathematics, then we are more inclined to present to them tasks that are a different nature than if our view of teaching is focused on skills acquisition (or passing standardized tests). So teaching isn't just a method. It's a system. What we value is important in education, our methods, how we assess, what we assess, our beliefs about what students are capable (and not capable of doing), and the goals of education all feed into what happens in the classroom.

One can argue that it is the case that one can lecture on concepts and conceptual understanding. So the traditional content can be expanded to some degree. I point out that the teacher explaining a concept is not equivalent to students actually demonstrating their conceptual understanding through a presentation or written work. How content is covered and how students engage in it are important, intertwined factors.

A highly useful application of LZA is to use it when you're teaching out of a textbook. An instructor can look at a section and make a quick list of the content and dispositions that students are likely to engage in. Then using this list, an instructor will know the strengths and weaknesses of a unit, and fill the "gaps" appropriately. Knowing students are good/not good at certain dispositions can also add valuable data for the instructor to consider. When we say, "My students normally are not good at explaining/solving...," then there exists a set of tasks or problems that should be deployed.

Short story: Get your content. Use LZA. List what's there and not there. Adjust. Win!

Upward and onward!