Tuesday, May 14, 2013

Algebra: A Mathematical Poem


When I met the unknown
I stopped,
And cried, familiar only
With the metronomic
Repetition of numbers.
What was this letter doing
In the midst of my math book?

From one moment to the next,
Life went from simple
To impossible,
When I could no longer
Plug questions in a calculator
To get an answer.

But when I started treating the unknown
As a number
(Shaking it like a gift
That I might find a clue),
The pieces fell in place,
And I found the answers
Right beneath my nose.

Thursday, May 9, 2013

2013 NCTM Conference: "Exploring the Common Core Practices in Secondary Classrroms"

2013 NCTM Conference (4/19/13 - 3:30 pm - Kristen Bieda and Samuel Otten):

My intention in this session was to learn more about the Common Core Mathematical Practices. Kristen and Samuel provided details on these practices embedded in the standards, but even more importantly, they suggested ways of implementing the standards in a high school mathematics classroom.

The Practices are defined as follows (the links were not provided in the session):
  • Standard 1: Make sense of problems and persevere in solving them 
  • Standard 2: Reason abstractly and quantitatively 
  • Standard 3: Construct viable arguments and critique the reasoning of others 
  • Standard 4: Model with mathematics
  • Standard 5: Use appropriate tools strategically 
  • Standard 6: Attend to precision
  • Standard 7: Look for and make use of structure
  • Standard 8: Look for and express regularity in repeated reasoning

  • Kristen and Samuel noted that these practices require students to "do" mathematics rather than just following mathematical procedures laid out by the teacher. They suggested that "doing" mathematics requires the following:
    • Students participate in complex and non-algorithmic thinking.
    • Students explore mathematical concepts.
    • Students self-monitor their problem-solving processes.
    • Students access relevent knowledge.
    • Students examine task restraints.
    • Often, students will experience a level of anxiety as they wrestle with mathematics in a more real way.
    Kristen and Samuel focused on the implementation of the 3rd standard (construct viable arguments and critique the reasoning of others) which may be one of the most difficult to implement. They stressed the importance of reasoning and proving for students to meet this standard. To participate in reasoning activities, students must generalize mathematical relationships, evaluate arguments, and discuss assumptions and what they are based on.

    Kristen and Samuel promoted the importance of modifying tasks to promote mathematical reasoning. They provided several ideas for doing so:
    • Ask students what happens in the nth case of a pattern.
    • Ask students to justify their conjectures.
    • Reduce scaffolding, so that students are prompted to seek out strategies for problem-solving.
    • Ask students to identify prior knowledge that they assume is true and justify the use of this knowledge.
    • Ask students to define key mathematical objects.
    • Ask students to model a problem through visual representations.
    Kristen and Samuel suggested that if students are actively "doing" mathematics, they will cover all the Common Core Practices. The Practices are interconnected and naturally work with each other.

    Wednesday, May 8, 2013

    2013 NCTM Conference: "Take Time to Question the Questions"

    2013 NCTM Conference (4/19/13 - 12:30 pm - Mark Howell):

    I've come to believe that asking the right questions is one of the most important parts of building student understanding of new mathematical concepts. One of my goals in coming to the NCTM conference was wanting to continue to sharpen my questioning skills. I was therefore eager to learn from Mark's session about "questioning the questions".

    Mark started the session with a bold statement: he suggested that as beneficial as multiple representations of mathematical concepts are, the use of multiple representations is not sufficient. He argued that linking between and among the representations is where learning takes place, when student understand how the representations are connected together.

    Mark proposed that good questions are those that come from powerful connections between representations. These questions prompt students to explore these connections and examine what the representations actually mean.

    Mark noted that it is possible to ask a lot of questions in the classroom without building a deeper understanding of mathematics. He noted that we've all seen procedural questions that ask students to provide the next step in a familiar mathematical routine. These questions tend to be very repetitive and removed from context. But students can memorize procedures without knowing how to think about math or understanding what the procedures really mean.

    Teachers often provide shortcuts for students that allow them to solve problems without thinking about how math really works. This can be detrimental for students in the long run. For example, students may have memorized the "vertical line test" without knowing the properties of a function.

    Mark noted that questions are essential for revealing deeper mathematical realities. But it is difficult to craft good questions that prod students into these deeper realms. There are four different levels of questions:
    1. Questions that elicit feedback from students.
    2. Questions that ask for factual responses.
    3. Questions that ask students to explain a procedure.
    4: Questions that ask a student to explore math, make conjectures, explain and verify their results, or talk about connections between representations.

    Most teachers' questions remain in the first three levels, but these questions are all very shallow compared to the fourth level of questions. It is important that we consider how to craft questions that take students to this deeper level.