Friday, July 25, 2014

Dividing Fractions

I attended professional development (PD) this week entitled, Developing Proportional Thinking with Connections to Linear Algebra. The PD has inspired me to further explore the division of fractions.  How, you might ask, is the division of fractions related to proportional thinking? Well the ratio is simply a secret division problem in disguise.  The deeper your understanding of fractions (and the division of fractions in particular) the deeper you will be able to go with ratios and proportions.

Take the picture above as an example.  What we are really trying to figure out here is how many one halves fit into three fourths.  When we look at the fraction bars provided, we get a better sense of how many one half servings fit into our new total of three fourths.  This visual model offers an alternative to the commonly used 'invert and multiply' trick.  There are even more ways to divide fractions while maintaining a conceptual understanding of the mathematics involved. Take a look at the Math Playground video, How to Divide Fractions, for a glimpse into the common denominator method.

Our final resource for the week is homemade.  When we were math students we learned a rather simple trick when dividing fractions.  When given two fractions to divide, we were told to invert the second fraction and multiply straight across.  I was never told WHY this trick works every time.  Were you?  The answer is likely no.  For this reason, I felt inspired to create a ShowMe video entitled, Why Does the Invert and Multiply Trick Work? Check it out to see an algebraic proof of this childhood trick.

Thursday, July 17, 2014

Formative Assessments for the Win!

There is more and more research coming out in support of formative assessments in the math classroom.  Due to the sequential nature of mathematics, it is imperative that we as educators have data to prove when our students are ready to move on to new concepts and explorations.  The following three resources range in their uses, but all relate to formative assessment.

In a recent Formative Assessment Podcast from the Bill and Melinda Gates Foundation website, ‘College Ready’, Stephanie Emmons talks about her experience as a special educator in co-taught math classrooms in Kentucky.   This podcast is a testament to the impact that formative assessments have made as far as student ownership is concerned.    

The Dylan William book above is an excellent summer read.  William references learning targets and success criteria throughout and offers more than 50 classroom tools and techniques to support teachers and coaches in formatively assessing students.  If you'd like more information, check out the Dylan William's Website.

Many of our districts have a large English language learner (ELL) population and that population continues to grow.  Formative assessments can be used in powerful ways when working with ELLs.  The Department of Elementary and Secondary Education (DESE) is currently offering a WIDA 101 course in Taunton that focuses on the formative language assessment process.  This DESE PD Offering runs from August 4th-August 6th and is free to attend.

Thursday, July 10, 2014

How DO We Problem Solve?

Mathematical Practice #1 might be the most daunting of all the Mathematical Practices!  How can we possibly teach students to make sense of problems and persevere in solving them?  Problem solving has always been one of those skills that we know good mathematicians acquire along the way.  Are students born with these skills or can we support our young people in cultivating them?  This week we will talk about the various ways to help scaffold the art of problem solving. 
Our first resource is a quick two minute video from the Teaching Channel. In this video, instructional math coach, Audra McPhillips, shows the audience how powerful hint cards can be as a scaffolding tool.  Take a look at the video, Hint Cards.
Our second resource is an article from the Association for Middle Level Education website.  The article highlights the benefits of using a graphic organizer in the math classroom and includes samples of student work as evidence.  Students Use Graphic Organizers to Improve Mathematical Problem-Solving Communications is worth the read.
Finally, the graphic above is an example that helps students scaffold their own thinking when exploring longer denser tasks.  The DEEP acronym is a rather easy acronym to remember and acts as a self-created graphic organizer that gives students many different ways to first approach a task.  Students can start by redefining the question in their own words, by sketching a picture, by highlighting key details and evidence, or by writing out the steps using prose.  The more students practice using these tools, the less they will need the tools in the future!