Thursday, April 2, 2015

Some Things to Keep In Mind Regarding Elementary Geometry...

Back in the old days, foundational geometry units were often reserved for the end of the year in many grade levels. Browsing through the preview materials of the new Everyday Math units for grades 3 through 6, I see geometry units and lessons occur at the beginning, middle and end of the year. I don't know why geometry wound up back-loaded into March, April, May and June, back in the day, but I remember it well.  For me, geometry was fun and not difficult, whereas all other forms of math gave me greater challenges.  For others, operations, algebra and statistics come easy but geometry poses problems.

Here are some things we all should keep in mind when we are teaching the basics of geometry to prevent confusion and future geometric mishaps.  Please let me know if you have others I should add to this list! These are in no particular order:


  • Many students have trouble remembering the relationship between a square and a rectangle.  They often mistakenly use the terms interchangeably.  It may take some time for your students to know that a square is a classification of rectangle, or a type of rectangle, and not the other way around.  The more you model the correct terminology for them, the sooner they will clear up their misconceptions.  A square is a closed figure with four ninety degree angles and four congruent sides.  A rectangle only has the four 90 degree angles.  When referring to rectangles, it is important that we use the word rectangle, even if it is somewhat close to being a square.
  • Students and adults get three-dimensional and two-dimensional figures confused all the time.  Like the rectangle-square issue, there is also a square-cube issue.  A cube is not a square, plain and simple.  But nevertheless, it happens, we catch ourselves saying to a child, "Can you put those wooden blocks away when you are finished? Yes, the triangle ones and the square ones..." There is no such thing as a "square block."  It is called a cube, even though cubes have squares on them.  Using three-dimensional geometric terms appropriately is super important to encourage correct use of geometric terminology.  Rectangular solid/prism, triangular prism, pyramid, cylinder, cone, cube and sphere, those are the words to use when describing, well, rectangular solids, triangular prisms, pyramids, cylinders, cones, cubes and spheres. 
  • Don't be afraid to use other geometric terms, even if they sound too advanced.  Words like congruent, vertex/vertices, figure, base, height, etc, should all be heard by students when discussing geometry.  For example, you can say "the bottom side of the triangle..," just say "the bottom side, or the base, of the triangle..," so students become familiar with the terminology.  The same is true for all categories of math, in fact.  Everyday Math likes to use kid-friendly terminology, but the rest of the world does not often recognize EM-talk.  When you say, "number model," or "number sentence," you can also say, "or an equation."  When you refer to "turn-around facts," you can say, "also known as the commutative property..."
  • Think carefully before making definitive statements about shapes, such as, "when you put two triangles together, you get a rectangle," or, "when you cut a rectangle in half, you get two triangles," as statements like these may be dependent upon certain conditions and could lead to significant misunderstandings.  Sometimes, two triangles put together make a parallelogram, or even an irregular quadrangle, and sometimes when you cut a rectangle in half, you get two smaller rectangles. 
  • Be careful when drawing/ modeling representations of shapes and angles for students.  Your "right angle" might look right from where you are sitting off to the side of your whiteboard or easel, but from the vantage point of your students it might not look like a right angle at all.  Sometimes I am horrified at what I have drawn or written for my students when I back away from the board and have a proper look!  
  • When students are drawing polygons and circles, have them use rulers, compasses and stencils whenever possible.   The more they are encouraged to represent such figures as accurately as possible, the less they will be inclined to think their distorted and/or deformed representations are acceptable.  Also, these are important mathematical tools they should be comfortable using.

How Much Should I Trust the Spiral? Everyday Math, Engagement, and Best Practices

Recently, EdReports published a Consumer-Reports-like review of the major published K-8 mathematics curriculum materials available to schools right now.  Everyday Math was not on the list of reviewed programs, because they are in the midst of updating to their latest edition (EM4), which we are currently using in grades K-2 and will be using also in grades 3-6 starting this coming fall.  I did have a chance to look over some of the programs that were reviewed, and it left me feeling very good about the program we've got, and the changes in teaching practices that are occurring.

We refer to these changes we are seeing in our math instruction as an "instructional shift" toward greater understanding, moving beyond teaching isolated procedures that are easily forgotten.  By asking students to explore more than one strategy, critique others' thinking, and write about their own mathematical thinking, we are giving them greater opportunities to understand the math that they use in their problem solving.  For a little more on "teaching for understanding," here is a recent blog post with an interesting example of looking at using multiple strategies from David Ginsburg via EdWeek:


Ginsburg's example of looking at two distinctly different ways to solve a problem is one way to "construct viable arguments and critique the reasoning of others," and it helps our students "make sense of problems and persevere in solving them."  Almost all of the CCSS Standards for Mathematical Practices (SMP's), in fact, relate to this example.

A successful mathematics program requires research, drafts, revisions and trials.  There were years when I taught my own custom-designed lessons, and was it ever frustrating how many of my seemingly great lesson ideas flopped, and flopped hard!  Sometimes what I thought was fun and engaging turned out to be embarrassing or boring to my students.  I had a difficult time deciding how much homework to assign, how long to stick with a concept when my students are not demonstrating mastery, how to differentiate, and how often to incorporate group and partner work into my lessons. I was proud of the home-made lessons that had succeeded, but the many late nights of planning and organizing had taken its toll.

When my district offered to give me a published math program to teach the next year, I was humbled at first, but eventually I welcomed the lessons, scope and sequence, pre-made assessments and professional development that came with it.  It did not feel like a defeat!  It felt like a revelation.

I realized that while a curriculum designed by mathematicians at a big university and published by some large profit-seeking corporation may not be everything I, personally, want them to be, I have the expertise and experience as a teacher to deliver them effectively, and not feel as though I am sacrificing my instructional integrity and creativity.  In addition, the lessons I designed, no matter how hard I tried to diversify them, tended to reflect my own teaching styles and preferred strategies.  In other words, I didn't do a very good job of incorporating all of the SMP's into my instruction.

So this brings us back to Everyday Math, the McGraw-Hill published program we use as our K-6 math curriculum.   In its new incarnation, it is designed as a Common Core program, which is to say, all its lessons and units were designed to teach the Common Core State Standards for Mathematics, including the Standards for Mathematical Practice.  The program spirals, which is to say it teaches concepts continually throughout the school year, and the kindergarten to sixth grade span of years.

The spiral concept seems to have come slightly under attack in some of the discussions and readings I have come across.  Older editions of Everyday Math have been criticized for not diving deeply enough into concepts before jumping ship and switching to a completely different concept, sometimes leaving students without a deep enough understanding of what was taught.  By the time the concept "spirals back again," some students require total re-teaching, while others retain enough to take their understanding to a new level.

In reality, almost every math program that spans more than one or two grade levels also spirals its content.  The Common Core itself spirals its content.  One major difference between the old Everyday Math and the new Everyday Math (EM4) is that the spiral now (more or less) parallels the spiral of the Common Core, teaching specific content within specific domains every year.  In earlier professional development this year, we looked at how important place value is throughout different points of the K-8 span.  It appears in the Common Core standards many times, and it shows up in the EM4 curriculum many times as well.

Another major difference with EM4 (which we have addressed before and will continue to address again) is that it places a much greater emphasis on the teaching and learning practices outlined in the Common Core Standards for Mathematical Practice.  In two years' time, third grade teachers will notice a big difference in their students' abilities when it comes to writing about their mathematical thinking.  Students in first and second grade are already becoming far more versed at using diagrams and words to describe how they found the solutions to their problems.  And kindergarten students are sharing more of their thinking out loud and are learning how to communicate their mathematical thinking as well.

Teachers of Everyday Math are often asked to "trust the spiral," but I think the better way to look at it is to understand the spiral, to know that it is there.  What our students are experiencing now is going to help them with their learning weeks, months and years from now.  This is true both with regards to the content standards and the standards for mathematical practice.  The place value lessons they get now, whether or not they completely master them 100%, will help them with their addition later in the year, and their ability to manipulate data next year, and their work with rational numbers two years after that.  Likewise, the oral communicating they are doing in kindergarten math lessons will help them to express themselves on paper in first grade, which will help them become more confident with problem-solving tasks in third grade.  In the end, we will see far more students demonstrating greater understanding of the math they are doing, and showing greater confidence in their solutions. Fewer students in sixth grade, when faced with the task: "Explain how you know your answer is answer is true," will respond with exasperation and fear.

Finally, when it comes to teaching a math curriculum like Everyday Math, it is best to look at the curriculum as a presentation of math concepts (standards) put together by expert mathematicians that you will be collaborating with by teaching it!  In fact, we are all collaborating together, for when we see something in Everyday Math we don't understand, or don't like, we seek each other out to make sense of the situation and come up with a solution.  Your teaching expertise, along with the support of colleagues and myself, and the research, drafts, revisions and trials that went into the authorship of EM4, make up the collaboration needed to facilitate a unique, engaging and meaningful math learning experience for our K-6 students.