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Topic: "MSPnet Academy: Learning to Measure our Impact: Focus on Mathematics"

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MSPnet Academy Discussion
June 26th - July 10th, 2013

Learning to Measure our Impact: Focus on Mathematics
Presenters: Mary Beth Piecham, Al Cuoco, Glenn Stevens, Ryota Matsuura, Sarah Sword, and Miriam Gates

Description: Focus on Mathematics is a targeted MSP funded by the National Science Foundation since 2003. As part of this work, we are developing a research program with the goal of understanding the connections between secondary teachers' mathematical habits of mind (MHoM) and students' mathematical understanding and achievement. In this feedback-oriented session, we will share sample items from a paper and pencil assessment of MHoM. We will also share our working definition of MHoM.

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Using and measuring MHoM

posted by: Sarah Sword on 7/1/2013 3:10 pm

Do you use mathematical habits of mind in your teaching practice? If so, how? Do you have advice about how one could measure the habits you use in a paper and pencil assessment setting?

We are particularly interested in your use of seeking, using, and describing structure as well as your use of mathematical language.

post moderated on 7/1/2013


posted by: Padmanabhan Seshaiyer on 7/2/2013 8:49 am

The Center for Outreach in Mathematics Professional Learning and Educational Technology (COMPLETE) center at George Mason University has been successfully directing several Mathematics Science Partnership Programs over the years.

One of the most essential tools we introduce in our professional development programs is the how, why, when, where to improve the mathematical habits of mind that not helps to enhance the pedagogical skills but also the content knowledge in the subject. MHOM maybe introduced several ways including(a) building rules to represent functions (b) doing-undoing a problem or a concept and (c) abstracting from computation ("Fostering Algebraic thinking", Mark Driscoll 1999). These MHOM structures must be effectively embedded within the framework of essential practices that includes (a) Anticipating what students will do--what strategies they will use--in solving a problem (b) Monitoring their work as they approach the problem in class (c) Selecting students whose strategies are worth discussing in class (d) Sequencing those students' presentations to maximize their potential to increase students' learning and (e) Connecting the strategies and ideas in a way that helps students understand the mathematics learned ("5 Practices for Orchestrating Productive Mathematics Discussions", Stein and Smith). Embedded within the selection of the problems and tasks should be a framework that incorporates cognitive demand of the mathematical instructional tasks at various levels including lower-level vs higher-level ("Cognitive demands of Mathematical Instructional Tasks", Stein and Smith 1998). There must also be an opportunity for the freedom to express the solution process using multiple representations including (a) symbolic expressions, (b) graphical displays, (c) concrete or pictorial representations,(d) tabular approaches and (e) verbal/written words. These must not only be shared among everyone in the classroom but also an effective rubric must be introduced to evaluate student learning using these taking into account that children have different learning styles as well. Finally, there must a classroom professional development piece that ties this all together such as Lesson Study ( that would naturally provide a way to not only assess how well a collaboratively planned lesson unfolds but also helps to vertically align the MHOM across multiple grade levels. Problem based learning should be an integral piece of developing MHOM. Exposure to STEM connections in what they learn as well as employing 21st century skills (communication, collaboration, creativity and critical thinking) to practical situations will also help to develop the next generation of inventive thinkers that are able to apply their MHOM in practically anything!

Director, STEM Accelerator Program
Director, COMPLETE Center
George Mason University

Using structure from Mathematical Standards of Practice

posted by: Lisa Olin on 7/8/2013 3:02 am

At the middle school level, many of our students don't understand the structure of mathematics because they don't know, really know, the properties like Identity, in all its combinations, and when Commutative and Distributive can really be used to simplify calculations. Our students need to know these properties, and their names because the title seems to increase importance in their minds.

At our school, we teach simplifying fractions through the Identity property because students really just need to divide by 1 in fraction form. It makes sense to them that the value isn't changing if they are dividing by 1. We write the fraction inside a big hollow 1 to illustrate the idea. We use the same hollow 1 when we find common denominators for addition or subtraction.

Many students struggle with mathematics because it seems random to them. Knowing the basic properties does give structure to mathematics for our students. It gives them consistent rules to depend upon.

post updated by the author 7/8/2013

Fractions should be thought of as numbers

posted by: Richard Askey on 7/8/2013 3:38 pm

Lisa Olin pointed out one problem middle school students have of not
knowing things like the distributive property well enough to be able
to use it when no suggestion is made about using it. Let me point out
an earlier problem, knowing that fractions are numbers. To illustrate this
consider the following TIMSS-2011 8th grade question.
Which shows a correct method for finding 1/3 - 1/4?
A (1-1)/(4-3)
B 1/(4-3)
C (3-4)/(3*4)
D (4-3)/(3*4)

First, look at the answers and see that 3 of the 4 answers cannot
be correct if one just looks at the fractions as numbers. 1/3 - 1/4 is
positive, less than 1, so A=0,B=1,C<0 cannot be correct. Here are
some results and then a link so you can see more results.
Korea 2.7 6.9 4.2 86.0
Japan 15.4 11.1 8.2 65.3
Russia 12.3 18.8 4.8 62.8
Inter Aver 25.4 26.0 9.4 37.1
US 32.5 26.1 10.7 29.1
Finland 42.3 29.5 8.7 16.1
For states (9 took it along with some Canadian provinces)
Mass. 21.4 20.8 9.9 44.4
Conn 31.3 21.6 17.7 31.3
Quebec 27.3 23.0 13.0 33.0
Alberta 34.7 23.7 12.3 27.8
and then click on almanac. From there it is click on grade 8, then
math, and then go to page 15 to see these and many other results.
The Common Core uses the number line extensively. They also
do not try to do two new things at once when learning to add fractions.
It is confusing to try to teach adding of fractions and using the least
common denominator at the same time. They build up the equivalence
of fractions before doing multiplication of fractions, so those books
which teach equivalence of fractions by multiplying by the "big one"
will have to change to something which can be explained to students
or else admit they are not following the Common Core's treatment
of fractions. One should not use a procedure to do a calculation
without first having explained what is the reason or reasons for why
the procedure works.

Korea was not an outlier in the data on the fraction subtraction
problem. Singapore, Taipei, Hong Kong all had scores closer to
Korea than they had to Japan. The outlier in the scores mentioned
was Finland, with only Sweden and Chile having slightly lower scores.

Richard Askey

fractions are numbers

posted by: Amy Cohen on 7/9/2013 9:04 am

To be brief - fractions ARE numbers.

There are many ways to represent numbers, the form "k/n" where k is an integer and n is a positive integer is one standard representation of rational numbers as fractions. Of course there are other fractions that students meet in middle school and high school - for example "pi/2" which is a fraction where the "top" is a real number not an integer. There are other representaions of rationals , for example 3.5 is a "decimal fraction".

What is concpetually hard is the transition from numbers with units attached "three apples" "two thirds of a yard" ( in the sense of the sum of two pieces of cord each one-third of a yard long) to "pure numbers". Mathematicians are so comforatble with this transition, that we forget that the process of abstraction is real transition for kids and their teachers.

Mathematicians and some teachers are so accustomed to using the basic properties of numbers when doing computations that we forget to mention them. The result is that students sometimes do not learn the practical usefulness of these properties in suggestion ways to compute - especially ways to make computation easier and to make sense of the problems to set up the computation in the first place.

Common Core - Implemented Correctly - Remedies This

posted by: George C. Viebranz, Sr. on 7/9/2013 10:24 am

The Common Core is very explicit about what happens on the number line between 0 and 1, and builds from unit fractions to all fractions. They make a clear distinction, for the sake of coherence, in that they define fractions (for reading and PD purposes) as the initial part-to-whole relationships consistent with the positive side of the number line. As soon as they go to fractional representations on the negative side, they begin references to the set of rational numbers. All of this is consistent with their trajectory of building from counting numbers to the real number system across the first 8 years. Irrational numbers fill out the real number set in grade 8.