Wednesday, September 30, 2015

The Power of Explanation

A Closer Look at MP.3

First, let's look at a brief classroom scenario based on a practice HiSet math question from ETS:

A tank for mixing chemicals solutions is 1.5 meters long, 0.6 meters wide, and 1.5 meters deep. Which of the following represents the maximum number of cubic meters of solution this tank
will hold?
         A  (1.5 + 0.6) × 1.5 
         B  (1.5 + 1.5) × 0.6 
         C  1.5 × 0.6 × 1.5 

         D  1.5 × 1.5
         E  1.5+0.6+1.5 

Teacher:  Who would like to share their answer?
Student 1:  (raises hand) I got C because the tank is a rectangular solid and we multiply all of the sides.
Teacher:  Great. Does anyone else have a different answer?

If you have ever found yourself in this situation and knew you should be saying more to Student 1, this blog is for you.

By now you are at least familiar with the College and Career Readiness Standards for Adult Education, but have you stumbled across the Standards for Mathematical Practice yet?  There are eight practices; go ahead and look them up on page 48 of the College and Career Readiness Standards for Adult Education document.  These standards are the same for all of adult education's Educational Functioning Levels and describe the behaviors that the student should exhibit when working with mathematics.  The practice standards describe how content should be taught in order to ensure that our students are mathematically proficient.  Several MP standards should be evident in each lesson as they interact with each other, however, you will find that some are more common than others.  MP3 is one of my personal favorites.  MP.3 is not an audio download of your favorite song, it represents the third standard for mathematical practice and should be one of the practices to consider as you design your lessons.

What I'm learning is that the even the national experts view the Standards for Mathematical Practice (MP) as somewhat vague compared to the mathematics standards.  We can wrap our heads around concrete mathematical standards like:

Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. (7.G.1)

However, 'constructing viable arguments and critiquing the reasoning of others' requires a bit more interpretation.  See what you can make of the MP.3 standard below:

Construct viable arguments and critique the reasoning of others. (MP.3) 
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Less experienced students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later. Later, students learn to determine domains to which an argument applies. Students at all levels can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 

Now skim MP.3 again.  This time, note any key words that jump out at you.

Justify.   Plausible arguments.   Compare.   Logic.   Analyze situations.   Assumptions.   Respond.

You would probably summarize the paragraph with phrases like 'explaining what to do,' 'why it works' and 'making sense of others' mathematical thinking'.  Communicate.  Back it up.  Prove it.  So, what does this look like in your classroom?  I can tell you one thing for sure, it does not mean a quiet classroom nor does it mean solely focusing on computational procedures.  Your classroom should be a place where students are encourage to think, talk, explain, disagree, and challenge around the mathematics.  Trust the experts; facilitating student discourse will build problem solving and conceptual understanding.

I see you nodding your heads.  You're on board.  Great.  Now how do we do it?

I love teaching math, but I have a deep, dark confession.  A few years ago, I didn't know where to take the discussion after I asked students, "so, how did you solve it?" and listened to a correct response.  I thought I was doing enough by asking students to explain their procedures/answers.  I knew I should be probing, comparing, and building on what they shared, but I wasn't sure what that looked like or what to ask and I wasn't sure if my students would be ready for such a shift.  I couldn't expect my students to have profound mathematical discussions with one another if I didn't know how to do it myself.  Today, a quick Google search for MP.3 will yield many quality websites and resources to help you improve your practice.  I'll save you some time and give you a two basics ideas that you can implement in your classroom tomorrow.

1.  Find meaningful math tasks.  This seems obvious, right?  Let's look at the HiSet example above.  There's very little wiggle room for discussion, comparison, and debate because there's a simple solution with a single answer.  How can students explain/support/argue for a particular strategy when there is only one? They can't.  The only thing this question allows us to do is discuss the procedure.  We need to shift our focus from how to get the answer to understanding mathematics.  That is difficult to do when the question we are asking don't require critical thinking.  I found an example from the GED practice test that is slightly more interesting, but still not perfect:

Ideally, you want to select tasks with multiple solution strategies.  While this question includes drag-and-drop responses, there's still room for problem solving and student-student discussion.  Students cannot rely on a formula sheet, they have to analyze multiple sources of information, and they have to think about what a linear equation means.  The CCRS and new HSE tests are forcing publishers to update their textbooks, but we still see the math equivalent of the one-trick-pony (computational procedures) on most of the pages.  Unfortunately, many our current HSE textbooks lack the rigor required to prepare students for the GED/HiSet/TASC or college math placement tests.  Your HSE textbooks provide many practice questions to increase a student's math fluency and procedural knowledge, however, if you are basing your lessons on those textbooks, you will need to look elsewhere (or modify) to avoid the, "here's the formula, here's how to do it, now let's practice" routine.  If you're looking for really thought-provoking lessons, check out the MathematicsAssessment Project website for lessons arranged by grade level [equivalent] and mathematical practice standard.  You can also click on MP.3 and it will direct you to lessons designed to get students talking.

Additionally, you can increase the rigor of your existing materials with minimal effort.  I participated in Cynthia Bell's COABE webinar, "Shifting from Mathematical Worksheets to Meaningful Tasks" (her PowerPoint and resources can be found here) and learned how to give a simple math worksheet a CCRS upgrade.  Cynthia chose an existing worksheet used in her math class:

Instead of asking her students solve for the variable, she gave them the following task:

This simple modification changed a hum-drum worksheet into a meaningful task with cognitive complexity.  Students can explain and compare their solutions and strategies with peers.  They can also challenge their peers or develop joint solutions.  This task provides an opportunity for students to engage with each other's ideas and that's where the magic happens.

2. Give students opportunitites to engage with each other's ideas.  This means giving students time to work with one another.  Ask students to justify their solutions rather than recount procedures and then have them practice asking each other the same.  Teach students how to ask questions, "I disagree/agree because ________" or rephrase what another student has said and explain your thinking.  You will have to model productive 'math talk' and monitor their progress in pairs or small groups, but the benefits will be well worth the time invested.

Question stems are a good starting place, but one shouldn't solely rely on this list.  In time, you will modify your technique based on student needs and in-the-moment decisions and things will flow more naturally.

Additionally, here are a few "look fors" to help you get started:

These are far from perfect, but do provide a starting place from which to expand your MP.3 knowledge and practice.  Each classroom is different and every student within that classroom will require different supports.  As an experienced educator, you will navigate the standards for mathematical practice and find the perfect strategies to foster more lively, engaged student interaction.

Leave a comment below and let us know about the CCRS successes or challenges you're having in your own classroom.

1 comment:

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