I Highly Recommend Math Misconceptions!

With the roll-out of Common Core in math, teaching and modeling how to recognize the way mistakes made in math can actually strengthen students' (and teachers') learning is very key.  But many of us only know the types of mistakes we see year after year...not necessarily why students make them.  A lot of times, those mistakes are engrained in them before they reach us in the intermediate grades.  So what can we do to tap into those students' mistakes and begin to figure out why they continuously run into walls in their learning?  How can we anticipate those misconceptions as we lesson plan?  Let me tell you!!!...because I am seriously carrying this book around back and forth from school like my own little teaching Bible.

For the first time in my 9 years of teaching, I started to realize why my students have trouble understanding why the denominators do not change when adding or subtracting fractions!  I attest this amazingness to the book I discovered and read this past summer...

This past summer, I was a part of a Math Lab School in which a small group of teachers from all over my district were able to learn about theory and put it directly into practice the following day for 2 weeks.  While reading a text we were assigned to use...I found a recommendation for Math Misconceptions and became extremely curious...as I was also studying for my National Boards Assessment Center portion. It is available on Amazon for $40, although I could have sworn I got mine for under $20 this summer. 

That's besides the point.  With a highlighter and plenty of post-it tabs on hand, I dove head first in to Math Misconceptions!  And as you can see, I found quite a bit that I LOVED throughout the entire book.

Math Misconceptions is for PreK-5th grade and it was insanely interesting to read up on things like number sense and beginning addition in the primary grades.  I instantly understood why so many of my students do not have a firm grasp on number sense and place value.  Why can't college math courses in the teaching major make this a required text and class?!

So, not only does this book help educators understand common misconceptions (as identified by NCTM)...but it does it in a way that makes it enlightening and simple to understand.  Why oh why wasn't I told about this book when I first started teaching?!!!

Here's how it is put together:

First...

Each chapter has multiple sections.  For example:  Number and Operations has sections on counting with number words, renaming and regrouping when adding and subtracting, and understanding fractions, etc. The different chapters also cover algebra, geometry, measurement, data analysis and probability, and assessing children's mathematical progress.

Second...

At the beginning of each subsection, you can read a classroom vignette in which a teacher is bombarded with student misconception(s) (mostly through the use of story problems).  Aside from the instructional ideas, this is probably my 2nd favorite part since it allows me to peek in on how to use these ideas in my classroom.  AND an "Identifying the Error Pattern" follows each of these vignettes.

Third...

One would think the research portion would be super boring!  But because you get reeled in so much by the all too familiar misconception you see year after year, it actually helps the entire "mess" make sense and helps you realize that you weren't imagining things!  Not to mention, it is explained in a user-friendly fashion....none of that scientific mumbo jumbo.  

Fourth...

The best part of this book?...the Ideas for Instruction portion chalked full of ideas for activities.  All of the activities are very well connected to the philosophy of Common Core.  They allow students to represent, construct, visualize, draw, connect, explain, and describe.  We are able to identify the misconceptions and address them while implementing these activities because it allows for so much exploration and explanation opportunities.  Love this!!

Fifth...

Last but certainly not least...questions that force you to think about what you have possibly seen in your own classroom and how you can work on changing how misconceptions help not hinder your students (and you!).  

Not only will this book create awe and lightbulb moments, but you can also buy an accompanying text that gives black and white reproducibles (also on an included CD) for each section...most of which they mention in the ideas sections.  There are 2 versions for primary and intermediate grades.  While I only own the grades 3-5 book, I'd like to grab the PreK-2 since some of my intervention students could use the primary activities

Here is one of the first activities and documents I implemented in my classroom when I was getting ready to teach adding and subtracting fractions and everything that comes along with it.  I wanted to make sure my students had a solid base understanding of part to whole relationships and equivalent fractions.  

I already have been recommending this to everyone I know, especially my 5th grade team.  But I am now singing from the hilltops like Julie Andrews in Sound of Music because I just finished grading my common assessments.  We were running extremely behind in our fractions unit due to still working with students on their behaviors and expectations.  So, I was not able to get to everything on our assessment.  But for the first time in my 9 years of teaching...I have students who demonstrated mastery of the skills I did not even get to!  This is not because they are my higher math students.

The models and work they provided immediately showed me they have a solid understanding of the basics of fractions!!!!  Can I even begin telling you how ecstatic I was as I sat on my couch correcting test after test?  I had to pick my chin up off the floor.  

These students (not all of them yet) got those unknown skills correct because they applied their knowledge from the meaning of part to whole in improper fractions and unlike denominators or equivalent fractions to adding and subtracting mixed numbers with unlike denominators AND they simplified their answers!!  

From Math Misconceptions:

Fraction... "lessons are too often focused on procedures and memorizing rules rather than on developing conceptual foundations prior to skill building." (pg. 41) "Before students study how to add and subtract fractions, they need to understand the meaning of fractions through various models, as well as how to use the language of fractions."  (pg. 42)

Using the Ideas for Instruction section, I planned accordingly:

1. We tackled fractions by starting with pattern blocks, something so many students are already familiar with!  When we used those to show part to whole relationships, my students had lightbulbs going off like crazy.  
2. Then, we used our fraction kits and the whole bar to create number lines.  My students immediately saw how equivalent fractions match up with each other.  But as we added 2 more twelfths pieces to one twelfth, they were already adding fractions without even needing the algorithm.  The buzz in my classroom between partners was insane and their enthusiasm as they continued discovering different equivalent fractions was inspiring.
3. We then used the triangle activity you see in the picture above in order to figure out why our denominators stay the same when we add or subtract...those of you who have taught intermediate grades know this misconception that haunts us every single year.  Kids keep adding and subtracting those denominators because they were always taught to do so when they are given two numbers and they see + or -.  Not any more!!

I really wish I had some photos to show you of all these activities in my classroom, but I was so deep into watching my students make all these connections I had never seen my kids make before, so I didn't have a chance.

But I do have some photos of the day we were working with greatest common factor to simplify fractions.  This of course went a lot smoother this year due to a stronger base in number sense and factors from the beginning of the year (we spiral it over and over in our calendar math and standard-based homework).

I can honestly say, the results I'm seeing in my classroom are completely connected to the use and my implementation of Math Misconceptions!  The writing is on the wall after receiving my district benchmark testing results and, of course, my own common assessment results.

Run, don't walk over to Amazon to buy your copies!

Have you read this book and/or tried it in your classroom?  What has worked?  Or, how would you like to implement this in your classroom?