Topic: "MSPnet Academy: Learning to Teach the Common Core"
MSPnet Academy Discussion
November 28 - December 14, 2011
The Common Core Standards are an important first step to unprecedented agreement about what students should know and be able to do in mathematics. Join Deborah Ball in a discussion about supporting teachers with the skills and knowledge needed to teach the common core curriculum effectively.
Learning to Teach to the Common Core
posted by: Joni Falk on 11/28/2011 1:20 pm
If you missed the webinar, you can see the archived recording of it, which will be available on MSPnet beginning November 29th. The recording captures both Deborah's slides and her presentation.
To view the video that was shown during this webinar go to:
post moderated on 11/28/2011
Starting the discussion
posted by: Deborah Loewenberg Ball on 11/28/2011 1:27 pm
post moderated on 11/28/2011
posted by: Richard Askey on 11/28/2011 11:35 pm
seriously used. In the presentation today, you said that students
were working toward a definition of odd numbers. In a paper
you wrote with Hy Bass, in Constructivism in Education, you
wrote: "His argument has challenged the group's previous
complacency about definitions of even and odd". So my question
is what was their definition of odd? Even, based on what a student
was quoted as saying in this article was: a set could be broken
into groups of 2, 22 is 11 groups of 2. One student said this was
a conjecture, but another said it was not, "That's a definition".
You may not have a lesson where serious conclusions are
obtained from a definition, but Practice Standard 3 starts with
"Mathematically proficient students understand and use stated
assumptions, definitions, and previously established results in
constructing arguments." We have high school algebra books
which define lines to be perpendicular when the product of their
slopes is -1. That is a theorem, not a definition. How about an
example where a definition is essential to the work being done
and used to help bring closure to the lesson. There is a very
nice algebra example, a function defined on a symmetric interval
around 0 can be written as the sum of an even and an odd
function. All you can use is the definition of even and odd functions.
posted by: George C. Viebranz, Sr. on 11/29/2011 9:22 am
Learning Trajectorisis of Mathematical Practice
posted by: Steven Kramer on 11/29/2011 3:59 pm
But I don't think the eight "practices" are laid out in the Core Standards in a way that normal teachers (folks like me, who don't have the natural talent of a Deborah Ball or Maggie Lampert) can implement. Worse, because there is not a coherent progression for students to learn mathematical practices across K-12, even students with gifted teachers like Deborah are likely to view mathematical practices as something to be done "in Ms. Ball's class" and not applied elsewhere or in later grades.
Each year, grades K-8, the Common Core "standards for mathematical content" lay out the critical areas of focus, which are then broken into sub-clusters of medium-sized ideas. Each cluster is broken into specific standards for the grade level. The explicit intent of the Common Core authors was as best they could to have the standards organize "learning trajectories" across grade levels.
For example (according the CCSI report "learning trajectories in mathematics"), the third grade content standards introduce two concepts of fractions:
"1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b."
"2. Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal
parts. Recognize that each part has size 1/b and that the endpoint
of the part based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off
a lengths 1/b from 0. Recognize that the resulting interval has size
a/b and that its endpoint locates the number a/b on the number
The Common Core writers viewed the second (number line) definition as important in preparing students for work in later grades relating fractions to rational numbers including negatives, as well as for later work in algebra, etc.
Yet, the writers also received research evidence and expert advice indicating that number lines are very abstract and hard for third graders to understand. So they tried to build up to the third grade number-line standard with measurement standards in grades 1 and 2. Here are the Grade 1 and Grade 2 standards building up to the number line concept of fractions to be taught in Grade 3:
"Measure lengths indirectly and by iterating length units.
1. Order three objects by length; compare the lengths of two objects indirectly by using a third object.
2. Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the numberof same-size length units that span it with no gaps or overlaps. ( Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.)"
"Measure and estimate lengths in standard units.
1. Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.
2. Measure the length of an object twice, using length units of
different lengths for the two measurements; describe how the two
measurements relate to the size of the unit chosen.
3. Estimate lengths using units of inches, feet, centimeters, and meters.
4. Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.
Relate addition and subtraction to length.
5. Use addition and subtraction within 100 to solve word problems
involving lengths that are given in the same units, e.g., by using
drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.
6. Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram."
The content standards in later grade levels build on the foundation laid in grades 1-3.
Now, it could well be that the "learning trajectory" for fractions and number lines hypothesized by the Common Core is not ideal. It may be too fast, there may be alternate trajectories that work better for significant subgroups of students or even for most students, there may be critical steps left off, etc. But at least the Common Core does not list the same set of vaguely-defined Content Standards each year, grades K-8, and again for high school. It would be the epitome of mile-wide inch-deep for the Content Standards to summarize the above standards as: "Measure and estimate Lengths" and "Develop an understanding of Fractions as numbers, using concepts of equipartioning and the number line"-and then expect students each year to study these plus a list of similar key concepts.
But that is pretty much what the Common Core does for the Practice Standards: all eight Practice Standards are to be taught each year, grades K-8, and in each high school conceptual category. There is little articulation across the grade levels, beyond some general guidelines about what might be expected in elementary school vs. what might be expected in high school.
One key area where the elementary school students we saw in Deborah's video (second graders?) were working was "attending to precision," with a particular focus on the concept of definitions. The Common Core defines this as:
"6 Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students
give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions."
By the end of Deborah's lesson, the students had developed a definition of "even number" and experienced its usefulness. Should second graders focus on developing an understanding of why clear definitions are important in mathematics? If so, what other parts of the second grade curriculum might be a fruitful area in which to explore this? Is such an objective sufficient for dealing with definitions in second grade, or should more be expected of second graders-or less? How should third graders follow up on this? What foundations should be set in first grade?
The students were also working on giving "carefully formulated explanations" to each other--but what specific communication skills (listening and expressing) should be expected of second graders' explanations, what groundwork should be laid in first grade, how should teachers in third, fourth,and fifth grade build on this?
I fear that until we do this kind of work, breaking down the Practice standards into reasonable progressions assigned to grade levels, the Practice Standards will be largely ignored.
updated: on November 29, 2011 at 3:56PM
posted by: Deborah Loewenberg Ball on 12/4/2011 1:27 pm
Capitalize on new opportunities to make the practices core
posted by: Christine Tuckerman on 12/1/2011 11:07 am
After watching the webinar, I intend to make the mathematical practices central to those collaboration and articulation efforts.
I will be looking for resources that describe the learning progressions for the practices at primary grades, and also for preschool if such resources exist.
Perhaps others beginning new projects can also make the practices central to the work. The work we develop and document should be helpful for others as they begin common core implementation.
posted by: Ruth Parker on 12/2/2011 10:08 am
The Standards for Mathematical Practice has become the lens through which we examine most of our PD experiences with K-20 teachers. I was excited to hear Deborah identify them as basic skills.
mathematical practices as basic skills
posted by: Deborah Loewenberg Ball on 12/4/2011 1:38 pm
preschool learning progressions
posted by: Steven Kramer on 12/2/2011 12:44 pm
You said, "I will be looking for resources that describe the learning progressions for the practices at primary grades, and also for preschool if such resources exist."
I can't help you with learning progressions for PRACTICES at the pre-K level, but there is a pre-K curriculum specifically developed to implement a learning progression of CONTENT. It is called "Building Blocks" and the authors are Doug Clements and Julie Sarama. I haven't used it, but the research I've seen indicates that it may be the best thing currently available for preschool math.
Of course, this still doesn't address the need for a progression of PRACTICES.
New grant project prek- 3
posted by: Jill Schenck Waffensmith on 12/2/2011 1:50 pm
posted by: Christine Tuckerman on 12/6/2011 2:44 pm
sharing ideas about trajectories and ways to connect practices to the rest of the curriculum
posted by: Deborah Loewenberg Ball on 12/4/2011 1:29 pm
New Grant Project
posted by: Connie Doorlag on 12/5/2011 9:51 am
We are looking at the possibility of a mathematics professional development project for Grades K - 3 teachers with some of our districts in Michigan and have found little information on the web for these grades. We would be very interested in finding out more about your project. Do you have a website that gives more information?
posted by: Linda Hardin on 12/6/2011 2:07 pm
Jacqueline Labate <firstname.lastname@example.org>
She consulted with us two years ago and is OUTSTANDING. She is in Vermont, so a little closer to you than she was to us. She also did a program called Family Math
New grant project
posted by: Christine Tuckerman on 12/6/2011 2:41 pm
The grant will be starting up in January, so there is no website yet, and much to figure out before any PD happens.
If you contact me this spring, I may have something.
Learning to Teach the Common Core
posted by: Mary Govan on 12/2/2011 9:07 am
two ideas at the HS level
posted by: Al Cuoco on 12/4/2011 9:13 am
(*) Teachers: Working through examples of how the standards for mathematical practice can bring coherence to the disparate topics in high school is a very compelling way to show that these things really *are* useful basic skills that cut across all content areas. In study groups, seminars, and workshops, we've found it most effective to take ordinary tasks that show up in almost every high school program---things that are common in the daily work of teaching---and look at the value that's added by employing various general-purpose mathematical habits (like the practice stds) in their investigation. Something as simple as finding the rectangle of fixed perimeter that maximizes area has been very generative in this regard (for this example, it's been the precision, structure, and regularity standards). A detailed example of this approach is in Hy Bass's wonderful paper ``A Vignette of Doing Mathematics'' in the online Montana Mathematics Enthusiast, 2011.
(*) Teachers, Administrators, and others: The recently released ``Model Content Frameworks''
puts more detail into the specifics of how the standards for mathematical practice connect with the content standards. I confess that I'm only familiar with the HS sections and the intro, but my daughter (who teaches 6th grade) says that the elementary sections are also quite good.
helping teachers incorporate good mathematical practices in lessons
posted by: Amy Cohen on 12/5/2011 2:18 pm
Incorporating mathematical practices
posted by: George C. Viebranz, Sr. on 12/6/2011 7:21 am
As advocates for improved mathematics education, we must be consistent with the message that these are mathematical practices for students. Having teachers demonstrate the practices in a didactic manner is not likely to give us the results. With your thoughts in mind, I'd suggest that the use of rich and relevant problem scenarios, coupled with ongoing formative assessment/evidence of student understanding, would be a practice that teachers need to experience, personally, then practice in the classroom. Once the assessment samples from PARCC or SBAC are available, we will get a picture of the complexity of the assessment. Students will need to be given regular opportunities to learn and apply both mathematics content and mathematical practices. It is going to be a big shift for many experienced teachers for whom didactic demonstration of textbook exercises is the primary mode of instruction.