Category: Algebra

Categories
Algebra

This Week’s Required Reading for Algebra Teachers!

Mid-April, that time of year where teachers and students start to see the finish-line of the school year.  Everyone feels the burdens…state testing, class distractions, covering all the “material”….teachers have a lot on their plate.  But it’s also a great time to reflect upon the past year, work in teams to consider best-practice, and plan changes for next year.  Two intriguing blog posts by Grant Wiggins this week should be required reading for all secondary math teachers.

First, Grant Wiggins rants against courses we call algebra 1.  What could be wrong with Algebra 1?  We all took it, we all agree kids “need” it, and isn’t a proven gate-keeper to college success?

Algebra, as we teach it, is a death march through endless disconnected technical tools and tips, out of context. It would be like signing up for carpentry and spending an entire year being taught all the tools that have ever existed in a toolbox, and being quizzed on their names – but without ever experiencing what you can craft with such tools or how to decide which tools to use when in the face of a design problem.

Amen, brother.  In algebra, we move from the unit of linear functions, to the unit on systems of equations, to the unit on exponents, then the unit on polynomials. At the end of each unit, we duitfully give the unit test, get some number score back, then move on to the next unit.  We have trained students to think this way:  that algebra means mastering one skill, then the next.  How often do we provide rich tasks which allow students to reflect upon their cumulative skills set?  I appreciate the work of many math folk out there to change the nature of Algebra 1 from a rigid sequence of skills to a course which encourages application and reflection, driven by interesting, authentic problems.  Some examples of outstanding math educators working to promote inquiry in math class are listed at the end of this post.

worksheets

For many special education students, chunking is a device used to “help” students in algebra.  By continued pounding of square pegs into round holes, using worksheets of similar problems (i.e. solving a one-variable equation, with variables on both sides), students can achieve temporary, recordable “success”.  The students most in need of seeing auhentic problems are often those least likely to move past the chunking, and into authenticity.  Fortunately, to help sort out the madness, Grant Wiggins provided a second great article of required reading for math teachers this week:

Grant Wiggins on turning math classes into bits of disconnected microstandards.

What’s so harmful about taking a broad subject like Algebra and breaking it into pieces?  What is the consequence?

Take a complex whole, divide into the simplest and most reductionist bits, string them together and call it a curriculum. Though well-intentioned, it leads to fractured, boring, and useless learning of superficial bits.

Hallelujah!  Make sure you check out Grant’s driver-ed analogy for the full effect.  More ammunition for us to develop math courses rich with interesting, relevant tasks, where algebra is the tool, not the star of the show.

Fortunately, there are many educators out there working to develop tasks which develop algebraic thinking, and encourage the use of algebra as the tool, rather than the exercise.  Keep them in your toolbox for future planning.

Dan Meyer: the king of perplexity.  If you aren’t visiting Dan’s blog at least semi-regularly, then start now.  And check out his spreadsheet of tasks for the math classroom.  In the same theme, visit Timon Piccini, and his many on-point 3-act tasks.

Sam Shah:  Sam leans more towards the pre-calc, calc end of the math spectrum, but I apprecaite Sam’s constant self-reflection and great ideas for engaging kids in math discussions.

Kate Nowack:  sometimes task-oriented, sometimes ranting on policy, but always interesting.

NCTM’s reasoning and sense-making task library has a number of problems around which algebraic ideas can be wrapped.

Categories
Algebra Middle School

Follow-Up on Math Term Expungement

In a comment from my recent post “3 Phrases From Math Class we Need to Expunge“, Tina from the blog Productive Struggle shared a Google Doc she has been assembling of terms and “tricks” we all could evaluate in our math courses.

Tina is requesting 3 categories of entries:

What tricks do you hate when students shout out?
What words do your students use without understanding?
What notation do you wish students started using earlier?

Here is the link to the document: Tina’s Google Doc

My favorite so far is Tina’s idea to introduce subscript notation for sequences and series earlier in math courses.  It always surprised me how much trouble that was for my 9th graders….silly almost.

Categories
Algebra Middle School

3 Phrases from Math Class we Need to Expunge.

A brief twitter exchange last night between myself and the great NY math educator Mike Pershan caused me to get off my rear to assemble a post which I had kicking around my head for some time now, a list of terms and shortcuts we use in math class which, while well-intentioned and used everyday by many math teachers, aren’t necessaily helpful in causing kids to understand their underlying math concepts.

Twitter

In a recent in-service with middle-school math teachers, I used a video by Phil Daro (one of the authors of the Common Core math standards) to have colleagues reflect upon the practice of “answer getting”, short-term strategies employed by teachers to get students through their immediate math assessment, but with little long-term value in math understanding.  Click on the “Against Answer-Getting” tab for the video.

So, here is my first list of nominees for elimination, and some strategies for helping students develop underlying algebraic ideas.  It probably won’t be my only list, and I welcome your candidates and thoughts.

SAME-CHANGE-CHANGE (aka KEEP-CHANGE-CHANGE):

This is a device I often see in pre-algebra classrooms, often times as a poster for easy reference, other times as a mantra for the students to help complete worksheet problems.  From the site Algebra-Class.com:

TIP: For subtracting integers only, remember the phrase

“Keep – change – change“
So, we have a short and snappy device which helps us with just one specific type of integer problem.  It’s not wrong, just too specific, and do students understand why it works?
What to do instead:
Let students develop their own summaries of integer problems, and create their own posters which describe their findings.  Use integer zero-pair chips or online applets, like from the National Library of Virtual Manipulatives (search for “chips”).  Number line applets can also help students visualize addition and sibtraction problems.  Have students write stories about given integer and subtraction problems, and have students peer-assess work for proper use of math terms.  Eventually, have students debate the possible equivalence of integer pairs:
  • 5 – (-2) and 5 + 2
  • a + (-b) and a – b
  • a – b and b – a

FOIL

The ad-laden math site Coolmath gives its own snazzy description of foil:

We’ve got a cool little trick called “FOIL” for multiplying binomials….it’s really just an easy way to do the distributive property twice, which would be really messy and confusing to do.

YEY!  You mean I can multiply stuff without that nasty and scary distributive property, without actually talking about the distributive property!  Yey shortcuts!  I’m in! {insert sad face}

Folks, ditch FOIL, and use the opportunity to talk about the double-distributive property.  Re-write the binomials as an equivalent expression and multiply.  Set the stage for factoring and note how much more understanding factoring by parts takes on.  And, now we can tackle those “messy” trinomials too.

CANCEL (LIKE) TERMS

Try this exercise tomorrow: take a class tht has been through Algebra 1, and as an opener tomorrow ask them to explain what the phrase “Cancel Like Terms” means when dealing with a rational expression.  Or, if that is a bit too scary, simply ask your students what it means to recude a fraction.  This is a nice activity to do as a Google form, and have students assess the explanations.  Many students will give an example as a definition, which is not what we are looking for here.  How many students discuss factors, GCF’s, numerators or denominators?

Reducing a rational expression means to divide both the numerator and the denominator by the greatest common factor of both numerator and denominator.  (Incidentally, also try having your students provide steps for finding a GCF.  This one also reveals what your students understand.)  The great part about this procedure for reducing is that it works equally well for each of the following expressions:

To many of our students, cancel is digested as “cross-out stuff”.  We have better vocabulary for it, so let’s encourage its use.