It’s clear that in the United States we have a culture of education in mathematics (maybe in all subjects) that is a mile wide and an inch deep, too many topics and not enough time on each topic. We need to SLOW DOWN! Phil Daro agrees that our curriculum has too much junk which begs the question What are we teaching? He also says the CCSS was developed to be more coherent than our previous standards, yet teachers are still struggling to “fit it all in.” Phil Daro addresses many of the barriers we as teachers face that are issues larger than our classroom.
I myself have felt the struggle of presenting a meaningful task in my class worried that I am getting further behind because it is taking longer than I had anticipated. However, the major differences between the U.S. and countries like Japan and Singapore is that in the U.S. we teach to cover, whereas the other countries teach at the speed of learning. The Skittles task I have been working on with my students is moving extremely slowly, but I remind myself to be patient, teach at the speed of learning. The U.S. is behind because we are moving too fast through the curriculum. When we move too fast, the learning is lost because their is no depth. When we slow down we will reach higher standards, allowing for Universal Access and coherent mathematics between topics and grade levels. So how do we Slow Down with all of the pressures created by the state and districts? The standards for mathematical practices are part of the solution.
The standards for mathematical practice (SMP) are the expertise that go with the content. When implemented and utilized efficiently, it forces you to slow down. Evidence of the SMP is what we need to be looking for in each other classrooms. Daro gives an example of how we can help students construct viable arguments and critique the reasoning of others. Students need to be saying second sentences. To be able to respond to a strategy and then create an argument with evidence, it takes a minimum of 2-3 sentences. As teachers, we need to be ready to increase wait time. This is hard as it is something I still struggle with, however we need to practice patience and use open follow-up questions: Why does that make sense to you? Is that always true? If we move too quickly to the next student because we know he/she has the answer, we are showing we value speed over understanding math. What we end up really doing is making the math more accessible by taking the math out of the work of the student. If we take the thinking away, we take the math away. This reinforces the idea of needing to go slow down, we can’t learn and do math if we are in a hurry.
Daro also talks about students being precise (SMP 6). We not only need to be precise with our numbers but also in the use of language. Students need to start creating their own definitions for mathematical topics within a given context and use reasoning from those definitions to develop ideas and arguments. If teachers have students memorize definitions or concepts without context, then we have again resulted to answer getting without meaning. It’s the same as being given a procedure without knowing why it works.
We also need to change the culture of our classroom. When students respond to a question who is the audience? Typically they think it’s the teacher. The goal should be that students explain strategies so that other students can understand. When the audience is the teacher, we transition from understanding to answer getting. Answer getting results in low expectations. Obviously within the student’s explanation there will be an answer, but the answer shouldn’t be the main focus.
Changing the culture of the classroom is not easy, especially if you are trying to make changes mid-year. How do we change it? Two-thirds of the time we should:
- Present a problem. Maybe even conceal the question part of the problem so students can comprehend the context.
- Students then independently work on the problem to develop their own thinking.
- Students pair up with a partner and work in pairs (or tables), each student producing his/her own product.
- Prepare an explanation/presentation.
The teacher needs to have already prepared a summary of the lesson in advance. As the teacher selects student work to display, it should be sequenced from easiest to grade level. Then the teacher facilitates conversations to build and connect student representations. In the end, the teacher should summarize the mathematics quoting student work in the summary as examples of the teacher’s points. When the structure of the classroom is set up this way, we can bring students to readiness by gathering prior knowledge, which will prepare them for the direct instruction at the end of the lesson.
What does this mean in terms of Professional Development?
We need to overcome the idea of teachers having to know everything up front. We learn every day, every year. Let’s give ourselves a break! There have been many times I am learning the math with my students as they teach me different ways to look at things and have different ideas than what I had originally anticipated because their experiences are different. Daro says that workshops are necessary but weak, as they will never solve our problems. The only way to get better is in the school, on the job learning that is more closely related to what we are doing every day. Teachers need to create a culture where we learn shoulder to shoulder with our colleagues (Daro references David Cohen). We need to be collaborating on lessons, what happens during the lessons, and how our students responded. We need to be in each others classrooms observing each others pedagogy so we can critique and ask questions about instructional choices. Until teachers accept this responsibility, everything else will be weak. How effective is a workshop if we can’t see how the strategies learned in the workshop are implemented?