Teaching Math to Promote Critical Thinking

Late last year, Google added a graphing calculator to its growing list of utilities raising the age old question every teacher has faced: why do students need to learn basics if they have a calculator? For a long time teachers could effectively retort, “You won’t always have a calculator with you.” However, in a world where anyone with a smart phone has unprecedented computing power in their pocket, that response is becoming less and less satisfying.

Yet even as students’ have greater access to machines, they should remember the answers those machines provide are only one small part of being a successful math student. Because computing is everywhere, mastering, presenting, interpreting, and attaching significance to those answers is as important as arriving at the right answer.

Math Solutions’ Lessons for Algebraic Thinking provides clear and effective lessons plans that teach traditional algebraic basics like variable relationships but also takes the lessons one step deeper. The real value added by these new plans is the emphasis on interpretation dialogue between a presenter and an audience and creating meaning out of the answer. In one lesson plan, students are not only taught to graph estimated-versus-actual weight but are also asked to think of conveying the data to an audience. “What should a person know by looking at your scatter plot?” is a critical question for interpretation and presentation. Similarly these plans ask students to create meaning: what new information does a correlation line convey, what does it mean for a correlation line to slope upwards, or what is the significance of a point position lying above or below that line?

Now that all students can compute answers with technology, educators should teach the skills needed to arrive at deeper understandings. Math Solutions’ Lessons offer a framework for adapting teaching to complement students’ increasing access to technology.

What techniques or technology do you use to help students present their data and interpretations to others?

NCTM Recommendations for Common Core Professional Development

As a continuation of my exploration of professional development materials, here is a report released by the NCTM in November that provides professional development recommendations specific to the Common Core State Standards in Mathematics (CCSSM-PD). The recommendations help school leaders and professional development providers identify the best practices for teachers and how best to implement them. 

Additionally, a report entitled Gearing up for the Common Core State Standards, written last year by a group of math educators and experts, lists five initial content domains for professional development focus: (1) counting, cardinality, and numbers and operations in base ten (grades K-2); (2) operations and algebraic thinking (grades K-5); (3) number and operations-fractions (grades 3-5); (4) ratios and proportions (grade 6-7); and (5) geometry (grade 8). 

Much of the focus on the NCTM report surrounds the Standards for Mathematical Practice laid out in the CCSSM. These standards provide a framework for how math can be taught most effectively. The other recommendations adapt existing professional development practices to the CCSSM and facilitate communication between teachers and education leaders.

The five content domains identified in the Gearing Up report are useful in distinguishing the areas in each grade that will require the most training. While the CCSSM are intended to streamline math curricula, the overhaul will require teachers to update their syllabus and their teaching techniques. Some of the content domains have changed more dramatically in their treatment at various grade levels than others, and these deserve the most immediate attention. Focusing on these domains is the best way to implement the NCTM’s first recommendation: Emphasize the Substance of CCSSM-PD. The Gearing Up document provides the rationale behind each of the choices of domain. It includes the standards themselves as well as critical areas of focus for each target domain at various grade levels.

How are your schools preparing for the Common Core? Have you found any useful resources that you would recommend?

Inspiring Back-to-School Teaching Videos

As you begin planning lessons for the new school year, you may want to take a look at the Teaching Channel. The website offers free videos of teachers in the classroom delivering effective lessons to their students. You can watch the teachers' interactions with their classes as well as brief interviews with the teachers and students. The Teaching Channel has videos for all the core subjects at a variety of grade levels, including great math lessons. 

One featured video shows Ms. Warburton teaching her algebra class about "mixture problems." In one example, students start with 8 liters of a 25% saline solution and some 70% saline solution.  The question is: How many liters of the 70% solution should you add to the 25% saline solution to get a 40% saline solution?  To help her students get a handle on this challenging question, she first makes a physical mixture of two colored liquids to illustrate how the weaker solution dilutes the stronger one. Then she walks the students through a "Mixture Picture," so they can organize the information more clearly.  Finally, Ms. Warburton introduces the "See Saw" method as another way for students to visualize the information and solve the problem. 

This video offers inspiring math teaching ideas, and there are many more available. The site includes PDFs of supporting materials and a discussion board for other educators to provide feedback about the videos and share their own ideas. The Teaching Channel is a great back-to-school resource. Do you have a favorite teaching video?