Monday, September 14, 2015

Thoughts on the Week of iMath...

In lieu of asking everyone to take a survey from youcubed (Jo Boaler), feel free to make any comments about the Week of iMath right on this blog.  Some prompting thoughts...

  • How was engagement?
  • Which activities seemed to work well?
  • What about students with needs?
  • Did the tasks help form group norms?  Group cohesiveness?
  • How well were you able to integrate the Math Practices?
  • Others?
Youcubed at Stanford University

Wednesday, March 18, 2015

Math Practice #3 - Construct Viable Arguments and Critique the Reasoning of Others

Math Practice #3 is all about listening, analyzing, and responding to others viewpoints.  In a nutshell - this is authentic classroom discourse.  

If a student is proficient in Math Practice #3 they can:
  • Analyze problems and use stated mathematical assumptions, definitions, and established results in constructing arguments.
  • Justify conclusions with mathematical ideas.
  • Listen to the arguments of others and ask useful questions to determine if an argument makes sense.
  • Ask clarifying questions or suggest ideas to improve/revise the argument.
  • Compare two arguments and determine correct or flawed logic.
In the classroom, it is important for teachers to be able to prompt this thinking without giving it away.  The article Convince Me by Wendy Petti describes the implementation of this practice in the classroom where students need to take ownership of the problem.  


Below are some actions that should be seen in the classroom by students and teachers along with some examples of open ended questions that can be used for almost any scenario.  

Wednesday, January 21, 2015

Using Math Practice #1 - Make Sense of Problems and Persevere in Solving Them

Math Practice #1 is, as Rena Sabey puts it..."Math Sweat."  The title of the practice really says everything about it.  Students are able to approach rigorous problems in a logical systematic way to help them sustain and reach a conclusion.

If a student is proficient in Math Practice #1 they can:

  • Explain the meaning of the problem.
  • Find entry points to the problem.
  • Plan a solution.
  • Compare this situation to other similar problems they may have solved in the past.
  • Keep track of their progress towards the solution.
  • Determine if their method was the most effective.
This poster was made a few years back by teachers in the district.  It sums it up pretty well.


Houghton Mifflin Harcourt put out this guide to prompting students towards MP #1 in 2012.  It helps to facilitate questioning without giving the answer away.
  • What is the problem asking?

    • How will you use that information?
    • What other information do you need?
    • Why did you choose that operation?
    • What is another way to solve that problem?
    • What did you do first? Why?
    • What can you do if you don’t know how to solve a problem?
    • Have you solved a problem similar to this one?
    • When did you realize your first method would not work for this problem?
    • How do you know your answer makes sense?

    In the blog replies below, feel free to share how you are using MP #3 so we can all grow and learn together.

    Thursday, November 20, 2014

    Why not PEMDAS?

    I know this is ahead of the game however it is coming up and I was hoping to get a jump start on rethinking the process.

    A teacher asked me if it was OK to teach order of operations using PEMDAS.  My answer...be very cautious and then ask yourself why...

    PEMDAS is inherently wrong.  It actually teaches students the wrong order of operations.  The correct order is:

    1.  Parentheses
    2.  Exponents & Radicals
    3.  Multiplication & Division
    4.  Addition & Subtraction

    This may seem similar but acknowledging that multiplication and division are inverse operations as are addition and subtraction are vital to true understanding later on.  We want them to understand what they are doing - not just follow a "rule."  

    I could go on but I would rather you hear it from others.  

    Here are some great Teaching Videos - they will help improve instruction!




    Tuesday, November 18, 2014

    How are you enhancing instruction with Math Practice #1?

    Comment below on how you are encouraging students to:
    • Explain the problem
    • Organize information
    • Persevere while working through rigorous situations
    • Monitor their work - are they on the right track
    • Change their plan if things are not working out
    • Ask themselves, "does this make sense?"
    • Check their work for correctness
    • Evaluate what worked - what could we have done better?

    Below are some questions that teachers can prompt students with to encourage Math Practice #1.
    • How would you describe the problem in your own words?
    • What information do you have? 
    • What do you need to find out?
    • What strategies are you going to use?
    • Can you think of another method that might have worked?
    • Can you explain what you have done so far?  
    • What else is there to do?


    Thursday, November 6, 2014

    Confessions of a constructivist/pragmatic teacher.

    I did a major application today.  It is below.  

    The goal was to engage the students in the topics we are trying to get them to understand (determining profits - new topic).  Of course, a simple application doesn't cut it.  The plan was small group work for 10 minutes, then every 1-2 minutes a new group would come up and "add to the problem" slowly forming the process and understanding.  Periodically I would interject and ask questions but mostly I just talked group to group
    never saying an answer was correct. 

    Here are my results:

    1. The students HATED the large "real" numbers.  (my response was that they were the reality, not fake school numbers - they got past this)
    2. Period 1 did awesome - fully engaged except for 1 child who is making poor choices.  They really liked it.
    3. Period 2 did fine but their struggles indicated a lack of knowledge in the pre-steps (served as a great formative - with things I can address tomorrow)
    4. The discussions in the small groups cannot be understated.  They are the backbone to why we need to do these applications.  Priceless.
    5. I was bombarded with questions after each class - the kids wanted to know the results - To be continued into tomorrow.
    6.  NO TIME WAS LOST because this method replaced a lesson.  In fact, I would estimate time was SAVED giving more time for depth!
    7. My kids stink at problem solving.  If they are not handed the process they quit.  This in unacceptable to me.  MP #1
    These applications do not always work in our content.  However, they work more than I feel we say they do simply because we are at times afraid to make that leap of faith that a lesson that is not the norm will work.  Today, was a leap of faith for me.  Not because I have never done this but the task was so rigorous for this clientele.    My class is clearly not advanced (more on the remedial side for a senior - they are not "math" kids). 

    By the way:  This problem was not found - it is not in any book - I made it on my own time using Google to find numbers.  Unfortunately, what every study says in regards to quality math instruction is not what textbooks produce.  They produce what the public wants.  We need to challenge the norm and apply our math.  

    -------------------------------------------------------------------------------------------------------------------------


    Apple Inc. sells its 16GB iPhone 5S for $649.  It costs Apple $335.00 to manufacture the iPhone including packaging, labor, freight, and warranty renewals according to a report published by UBS AG.  Apples fixed cost is in excess of $5 billion dollars.  However, they sell many more products than the iPhone.  Assume for this situation that their fixed costs are $7.4 million dollars.  The iPhone is a highly sought after phone in the SMART phone market.  The demand function is q=-3000p+5628000
    Given this information, use your classes know-how to help determine if Apple Inc. has priced its iPhone correctly. What should the price of the iPhone be in order for Apple to maximize its profit?

    Wednesday, February 19, 2014

    RTI Help - Struggling student - lots of interventions given

    I am hoping you can help me brainstorm some things for a student.

    Even though we do not need to give the winter MComp and MCap, I did it for this student.  Her computation has gotten better, but is still in the below average range.  She is practicing math flash cards (the intervention ones targeting certain strategies) and has a set for home.  She is on Xtramath.  

    But her math applications and understanding is in the critical range.  I have her working with a parent helper 3 times a week one-on-one.  She is very unfocused unless she works with someone to keep  her on task, and doesn't pay attention to lessons...hence the lack of understanding.  I check in with her for each lesson, and use the reteaching pages or modify her work.

    Do you have any other intervention strategies I can try?  There is not one area I would say is a strength or weakness for her (i.e. geometry, patterns, computation, etc.).  She is just LOW!!!

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    Additional information:  

    She is/has not been identified as special education and she might get the strategies for the day, if you are sitting there, but doesn't retain well.

    This is one of your colleagues looking for help.  Anything you can provide would be beneficial.