Association of Teachers of Mathematics in Maine

Week 5 (February 8-15) Chapter 4-Building Sensible, Sense-Making Mathematics: What to Encourage and Implement

  • 16 Feb 2013 3:57 PM
    Reply # 1210263 on 1203886
    Evelyn Krahn
    I agree with Steve’s lists of "non-negotiables." His list is quite thorough and lines up with the 8 Mathematical Principals of the CCSSM.

    I also agree with Steve’s thinking that our goal as educators is to ensure that our students posse these basic skills. Which,  in turn will give them the tools needed to become stronger problem solvers and efficient mathematicians.
  • 17 Feb 2013 1:24 PM
    Reply # 1210764 on 1203886
    Katrina
    Reading through the list of non-negotiables has created two dimensions of thought for me.  

    The first being that this a prime example of how we are seen as having lowered the bar for our math students. I do not teach at the high school level but there appears to be a lacking of focus on higher level mathematics is this list of basic skills. 

    The second thought is on the other extreme.  If teachers focused on such non-negotiables then our students could progress at the higher level of mathematics and those who choose not to would at least have the basics.

    One topic that struck me as not necessarily a prime focus is that of transformations.  I would rather see the focus on geometric measures such as area, perimeter, volume and surface area.

    I would also be interested to learn about the approaches which would be non-negotiable.  Would alternative methods of solving equations be accepted rather than the traditional approaches?  
  • 18 Feb 2013 9:50 PM
    Reply # 1211956 on 1203886
    Bobbie P

    In the resource room for upper elementary, most of my students have recently become readers and still struggle with reading below grade level which has and continues to impede their access to the math curriculum. Many also have difficulty receiving instruction in the general ed setting and struggle to learn new concepts, because of learning differences and processing disabilities, even in the small group setting of the resource room. Given these learning differences, most of my students are working below grade level in math.


    Each of these 'basics' is a daily struggle and even if they achieve one today, by tomorrow we often need to reteach to bring them to what they could do yesterday. Now we are being expected to write grade level goals for these students based on the Common Core Standards. The gaps from where they are to these non-negotiables are amazing, the gaps to grade level are daunting.


    I am truly struggling to visualize how this all works for students with special needs. Many districts around the country are developing resources to assist general educators to align to the Common Core, but the resources are not yet showing up for students with special needs, yet the requirements to meet grade level Common Core Standards are already there.

  • 20 Feb 2013 5:03 PM
    Reply # 1213834 on 1203886
    Tom Light
    When I first read the prompt and the list, I thought, this is easy.  His list is great.  Then I started reading other people's responses and started to wonder.  Many felt that important things were left out (areas, volumes, algebra, create/derive formulas, fractions, etc.).  I thought, of course these are also important.  And, as Lisa points out, why are transformations on the list instead of what seems to be more basic geometry skills.Then I read again that this is a list of "non-negotiable basic skills for mathematics" and I started to wonder what would make an item a basic skill.  As Lisa and others' point out, this is what we want for all students.  But, isn't the list of what we want for all students the Common Core? 

    As Ruth pointed out, Leinwand says these skills are to achieve the goals of solving everyday problems, communicating understanding, and representing and using mathematical ideas.  If that is the case, leaving out skills for finding volume/area would be needed.  (Plenty of everyday problems there!) 

    I have decided that basic skills might best be defined as  the skills that are necessary to achieve the conceptual understandings presented in the common core and to be fluent (efficient) in solving mathematical problems.  When I look at the list through that lens, it seems to make more sense.  For example, if a student understands the "attributes of 2 and 3 dimensional shapes", they understand volume and area and so could find methods for finding the volume/area.  I'm still not totally satisfied that his list is the "best" or most complete or least redundant list of basic skills, but it is a good place to start and get me thinking.  What is obvious to me is that my childhood exercises in long division are not necessary in the world I live in  now.


    As an aside, I decided to do an estimating check on one of C. Wolfram's statistics (from the TED talk pointed out by  Angela).  Please check my math and reasonableness of my assumptions.

    He said that 106 lifetimes were spent teaching hand calculating today.
    100 yrs/lifetime x 100 lifetimes = 10,000 yrs x 365 days/year=
    3,650,000 days (4,000,000 days) x 20 hrs/day=
    80,000,000 hours teaching hand calculating in a day.
    There are about 7 billion people on earth, perhaps 1 billion in school
    If each child spent .1 hours (6 minutes) practicing/learning hand calculating we would reach 100,000,000 hours.  Seems to me that Conrad's statement is reasonable.
  • 24 Feb 2013 3:10 PM
    Reply # 1225282 on 1203886
    Karen Morton

    What are the basic non-negotiable skills that all students should have?  I nodded while reading the list that Steve outlined. I would agree as others have mentioned that some basic measurement skills should be included as well as an understanding of area, perimeter, and volume of basic shapes. Formulas for area, perimeter and volume of basic shapes wouldn’t need to learned if these topics were understood. In addition to learning the basic facts I agree that the ability to mentally multiply by .1, 10, 100, 1000 is essential. If this were true, students would be able to mentally figure out the savings when an item was 10% off when shopping.


    I believe we can’t stress enough the development of number sense and the ability to estimate flexibly.  I often talk to my students about the “gong” that should be going off in their heads when they get an answer that makes no sense, particularly when using a calculator or even an algorithm.  I’m thrilled when kids say “huh???” when they get an answer like that and try again.  Unfortunately, that’s not always the case.


    I also like the idea of a quick cumulative daily review to keep these important skills sharp.  

  • 24 Feb 2013 5:11 PM
    Reply # 1225328 on 1204501
    Robyn Graziano
    Mary Belisle wrote:One of the big bugaboos that we have is understanding fractions and decimals. He mentions some basic computation of them and equivalent fractions. I find that fractions cause a lot of trouble for students because they can't quite get their head around them in number sense and later in algebra. I think Leinwand alludes to this but does not seem to see it the problem that I have seen it to be.

    I agree with you on this, also. Before we do any computation with fractions, we really need for students to understand what a fraction is. So many in high school can not even draw a picture of what 3/4 means to them. A lot of time playing with manipulatives and really getting a physical understanding of fractions is needed.
  • 24 Feb 2013 5:17 PM
    Reply # 1225331 on 1209120
    Robyn Graziano
    Lisa Russell wrote:

    In a college level class, one day we were in groups comprised of K-12 math teachers and were discussing fractions. Within the discussion it was recognized that at the elementary and middle school levels fraction skills are sometimes given up on, because it's just to hard for some students. As a high school teacher I had the opportunity to share with the others why a solid foundation in fraction skills are critical to high school mathematics. Not only do they need the background knowledge of fractions for slope, proportions, and reasoning processes, they also need them for upper level coursework as well. Without a solid foundation of number sense, algebraic processes are much more difficult for students.

    I do think teachers need more professional development to allow us to bring changing practices into our classrooms. We need support from each other and administration. We need to develop reasoning, communication, and a common vocabulary throughout our K-16 curriculum.

    Nicely said. I really wish we could get all of our K-12 math teachers together to talk about where math topics are coming from and where they are going for our students. This would give everyone an understanding of where are students started in their learning of a mathematical concept and where they are going with it.
  • 24 Feb 2013 5:28 PM
    Reply # 1225337 on 1203886
    Robyn Graziano
    I like Steve's list. So strongly agree with the first bullet. I may even start including a "mad minute" on addition and subtraction. I had to read the 4th bullet a couple of times, wondering why he included it. But, in the end, I agree that the 4th is important. Even if the student uses estimation to achieve this, I think it is still important.

    Steve does not include Algebraic representation (graphically, numerically or
    through tables). Students should have a real fundamental understanding of how these three work together.
  • 24 Feb 2013 9:48 PM
    Reply # 1225521 on 1203886
    T. Hartnett
    I also like Steve's list and agree with Tom and other posters that some of the geometry topics considered missing could very well be there (i.e. perimeter and area as attributes of 2-d shapes or these and other geometry topics falling under "evaluating and using formulas".  Recently, I felt completely discouraged to observe multiple 6th graders going through the long process of finding a common denominator by listing multiples to solve 8/8 + 3/4.  The Common Core has the potential to do away with doing mathematics robotically (without thinking) and I am looking forward to is.  Steve's words leave me feeling hopeful.

    On another note, the week 5 reading got me thinking about the importance of cumulative review.  I am thinking about how I may incorporate review during my short intervention sessions...
  • 27 Feb 2013 10:59 AM
    Reply # 1228888 on 1203886
    Susan Hillman
    I am posting very late.  Was away at a conference and then got sick and time has evaporated it seems.  So sorry and I again will be gone from March 1 to the 7 on a university trip so will be literary out of the country, so my post for week 7 will be delayed.

    I thought so many of you made fantastic comments on the non-negotiable list.  It is difficult and maybe a bit lofty to generate such a list.  Personally, as I teach my pre-service teachers--for right or wrong--they must be attuned to the CCSS-M period.  So what I think is much more important than the list are the three bullet items that follow this list:
    • solve everyday problems
    • communicate their understanding
    • represent and use mathematical ideas

    In working with future teachers the most difficult thing to get through is the emphasis on those three pieces regardless of the content while also getting kids to enjoy math.


    For those of you who made the book club happen...thank you!  I have shared aspects of the book with my students so they do not think what I am saying and trying to get across is totally crazy!!


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