We started the morning by reviewing #6 from the area sheet we worked on yesterday afternoon (p. 4.9). Beth was very brave to share with us where she was stuck and then very gracious as we practiced asking her purposeful questions to help her move ahead in her thinking.

Then Fran led a pedagogy section about “asking purposeful questions” and “eliciting and using student thinking.” We watched a video of an 8th-grade math class (from Pittsburgh Public Schools!!) and noticed a number of things about the teacher’s questioning patterns. We then examined the transcript of the video, looking at specific parts to unpack her questioning patterns further. [All of these materials are available from the NCTM Principles to Action Professional Learning Toolkit. Lin found out that you don’t need to a member to access some materials!!] Participants got a copy of the PBS Teacherline Question Starters.

Fran then shared information about the National Council of Teachers of Mathematics (NCTM.org). Lots and lots of good resources; NCTM Regional Conference is in Philadelphia this fall; K-8 schools are offered a special deal on institutional membership. Don’t forget that we have a Pennsylvania Council of Teachers of Mathematics (PCTM). Then we looked at the Illuminations website, which also contains lots of quality lesson plans and interactive applets. Another good website for interactive applets is the National Library of Virtual Manipulatives.

Then, we dove back into the math. We learned how to motivate the rules for multiplying signed numbers. We then spent a lot of time on the distributive property. We saw how this can be used to help develop students’ number sense (for instance, recognizing that 201 x 5 is 200 x 5 + 1 x 5) and also to build toward algebra (FOIL… but that only works when you’re multiplying something with two terms times something else with two terms).

We stopped for lunch a bit early in order to get our picture taken with the Lion Shrine.

After lunch, we continued with the distributive property, extending its application to situations where there are more than two terms multiplied together. We talked about extending the area model to a volume model. We showed how the distributive property, together with the adjective-noun theme, can motivate the standard algorithm for multiplication.

Just before the break, we started to consider division. We discussed how there are two types of questions that can be answered using the calculation 12 divided by 3. In one of them, we ask, “how many groups of three can 12 be divided into?” (Or: how many groups of three fit into twelve?” Beth calls this “dividing to fit”.) In the other, we’re asking, “if we split 12 into 3 groups, how large is each group?” (Beth calls this “dividing to split”.) We discussed how activities done in Kindergarden and first grade (for instance, making measurements in non-standard units), are actually building “floating capacity” for the ideas of division.

After the break, we examined how division can be considered as “unmultiplication”. We recognized that division by zero is undefined because there is no answer to the corresponding multiplication problem.

Homework:

Complete the workshop evaluation. Don’t worry about the topics that we didn’t cover (since we didn’t do week 2 of the workshop).

We began the day by reviewing the Math homework from last night. Many people had realized that the “100 days of Christmas” problem was giving the same result as the triangular number problem from yesterday. However, some people expressed the concern that if they didn’t have the formula from yesterday, they wouldn’t have been able to derive it for themselves. Fran showed how using a different representation can help you to get from the numbers to the formula. Lauren mentioned that for her, this raises the point that she should have the physical manipulatives more readily available to her students, so that they have easier access to the tools of multiple representations.

For the handshake problem, many had noticed that it’s very much like the 100 days of Christmas, but with a key difference. On the nth day of christmas, we’re adding the numbers 1 through n, but in the handshake problem, for n people we’re adding the numbers 0 through (n-1). Fran also showed a group of 3rd graders working through this problem. The video is loaded into our file on google drive.

We then moved on to talking about the P to A reading from last night about building procedural fluency from conceptual understanding. One big take-away from this discussion is that procedural fluency is much more than just being procedurally competent.

Fran then led a discussion of signed numbers. We discussed how to model positive and negative numbers in terms of the number line, and also using chips of different colors to represent positive/negative numbers. We used these models to think about adding and subtracting signed numbers. To make sense of subtracting signed numbers, we re-wrote the subtraction problems as addition problems.

At the end of this section, we wrote word problems for 6 + (-9) and 6 – (-9). Here are pictures of the word problems that participants developed:

After lunch we began to think about multiplication. We discussed that thinking of multiplication in terms of repeated addition or skip counting is limiting when we want to approach multiplication of fractions, but that the area model is a bit more powerful in this regard. We examined various properties of multiplication: the commutative property, the multiplicative identity, the zero property, and the associative property, and justified them using the models of multiplication. We saw that using the Adjective-Noun theme, to multiply, we must multiply the adjectives and multiply the nouns. Finally, we explored how to calculate the area of shapes such as rectangles, triangles, parallelograms and trapezoids.

Then we made iceberg models by grade level:

Homework:

Math:

Complete number 6 on page 4.9.

Pedagogy:

Read Principles to Action: “Pose Purposeful Questions” (p. 35-41) and “Elicit and Use Evidence of Student Thinking” (p. 53-57)

In your notebook, write a response to these three prompts:

1. In questioning small groups of students working on a problem, a teacher noticed that when she asked a “focusing” question, the students continued to look at their work and continued to engage in their own dialogue. When she asked a “funneling” question, the students looked up at the teacher. Comment on these observations.
2. If Fran came to your classroom during math time, would she see you using a funnelling pattern of questioning or a focusing pattern of questioning?
3. Suppose a student presented you with the following piece of work. What questions could you ask that would elicit the student’s thinking?

Welcome back! We hope you all had a good evening last night and were able to dig into the homework. Please don’t spend more than half an hour on the Math homework each night. If you’re not able to finish in half an hour, write down what you know, and some questions that would help you to continue before setting it aside.

Throughout the day today, we discussed various resources that people might find helpful. These are listed (and linked, where appropriate) further down in this post.

We began this morning by introducing Fran Arbaugh, a professor of Mathematics Education from the Penn State main campus in State College (or University Park) who is joining us to lead some pedagogy discussions. We looked at the “Burning Questions” from yesterday’s evaluations. Many of you had questions about how to implement the ideas we’re discussing in your own classrooms – Fran will be addressing many of these questions during pedagogy discussions today, tomorrow, and Thursday.

Several people had burning questions about the formula from the end of yesterday –  how do we calculate the sum of the first n numbers? Kelly showed how to pair numbers – 1 and n, 2 and (n-1), etc., to get the formula n*(n+1)/2. Mark used counters to show the same result geometrically.

We then discussed the Math homework problem, “Woodworker Ken”. Groups shared their solutions with one another, and we discussed a few solutions and how different methods of solution were related to one another.

We next dove into an examination of subtraction. We started by thinking about subtraction conceptually – what does subtraction mean? We defined it by converting a subtraction problem to an analogous addition problem. We also discussed how thinking of subtraction only as “take away” can be limiting, and examined various models of subtraction – in addition to “take away”, we can think of subtraction as a “distance between” on the number line, by “counting up”, etc.

Then Fran led a discussion of pedagogy focused on the readings from last night. Participants discussed the content of the “Smarter Than We Think” message from Cathy Seeley (see book information later in this blog) and the introductory material from Principles to Action. Fran also pointed out the diagram of “multiple representations” on page 25 of P to A. She talked about how important it is for student learning that they make connections among the representations – that the understanding occurs in the arrows of the model.

Fran then shared the model of Mathematical Proficiency from Adding it Up: Helping Children Learn Mathematics (see book info below). Here’s the figure of the five strands of mathematical proficiency and the definitions of those strands from the book:

• Conceptual Understanding – comprehension of mathematical concepts, operations, and relations.
• Procedural Fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
• Adaptive Reasoning – capacity for logical thought, reflection, explanation, and justification.
• Strategic Competence –  ability to formulate, represent, and solve mathematical problems.
• Productive Disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

After lunch, we returned to the consideration of subtraction.  We moved on from thinking about subtraction conceptually to examining the standard algorithm for subtraction, and then looked at a few less familiar (at least to most of us) algorithms for subtraction. Although we might not expect our students to use all of these algorithms, being able to interpret what is happening in each of these algorithms

We started the afternoon pedagogy session by reading the chart about supporting productive struggle on page 49 of P to A and then generating a list of classroom conditions for supporting productive struggle (which you wrote into your notebooks). We moved from there to examine the levels of cognitive demand charts on pages 18 and 19 of P to A. Fran made the point that kids need to have opportunities to think mathematically across all of the levels. Participants then examined their curriculum materials through the lens of cognitive demand.

We ended the day by watching My Favorite No video from the Teaching Channel.

Resources that were mentioned today:

1. http://www.abebooks.com/ for cheap books

2. Putting the Practices Into Action, Implementing the Common Core Standards for Mathematical Practice, K-8,  Susan O’Connell and John SanGiovanni http://www.heinemann.com/products/E04655.aspx 

3. Smarter Than We Think : More Messages about Math, Teaching, and Learning in the 21st Century, Cathy Seeley http://store.mathsolutions.com/product-info.php?Smarter-Than-We-Think-pid775.html

4. Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching, Jo Boaler http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470894520.html 

5. Creating Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools, Ron Ritchhart  http://www.wiley.com/WileyCDA/WileyTitle/productCd-1118974603.html 

6. Adding it Up: Helping Children Learn Mathematics, Jeremy Kilpatrick, Jane Swafford, and Bradford Findell, Editors http://www.nap.edu/catalog/9822/adding-it-up-helping-children-learn-mathematics [you can download the whole book for free at this site]

Homework:
Math Homework:
Complete problems 6 and 7 from the “Long Sums” worksheet. (The 100 days of Christmas, and The Handshake Problem). These problems are reproduced at the bottom of this post.

Pedagogy Homework:

Read Principles to Actions: Section titled “Build Procedural Fluency from Conceptual Understanding” (p. 42-48).

Also read: Webb, Bozwinkel, & Decker. Beneath the Tip of the Iceberg: Using Representations to Support Student Understanding.  MTMS, 2008.  (Handout from class today)

In your notebook:

• Identify a procedure or skill that you consider essential for students at your grade level to learn. Describe the conceptual understandings that support students’ learning of the procedure or skill.
• Write questions you have about the readings.

*Does everybody have access to an audio-recording device? Do you know how to use this function on your phone? Practice tonight!

Welcome to the first day of the Pennsylvania Math Initiative at Penn State Greater Allegheny! We started the morning with a welcome from Dr. Kurt Torrell, the Associate Chief Academic Officer for the campus. We also had introductions from Beth Lindsey (a physics professor at the campus), Kuei-Nuan Lin (a math professor at the campus), and Andrew Baxter (the mastermind of the PMI). Tomorrow we will be joined by Fran Arbaugh, an elementary education professor from Penn State University Park, who will be leading some discussions on pedagogical issues. Participants also took turns introducing their partner to the group.

After introductions, we dove into the Kayak problem. Everyone took 5 minutes to work the problem through on their own, and then compared their solution to their groups solution. Everyone was encouraged to look for alternate methods of solution once they had solved the problem one way. We discussed number 3 in detail, with several people presenting alternate methods of solution:

1. Rhonda showed how she had created a table of values to identify the time at which the two deals cost the same amount.
2. Christa and Lauren shared how they had translated each deal into an algebraic expression, and used algebra to solve for the time.
3. Mark showed a graph which illustrated the answer.

We discussed the idea that part of the goal of thinking of Math as a Second Language is building fluency – being able to say the same thing using different words, or think about problems in multiple ways. Andrew pointed out that some methods of solution are more efficient than others – if the answer to the problem had been “100 hours”, making a table would allow you to solve the problem but would be tedious. Doing the algebraic solution takes the same time regardless of how large the answer is. Several teachers mentioned that they would expect one of these solutions from their students, but not the others, so we discussed how useful it is for teachers of the lower grade levels to see what ideas they are building toward, while teachers of the upper grade levels have a good understanding of the foundation that they are building upon.

After the break, we dove into the Adjective-Noun theme. We saw how in basic arithmetic (1 + 1 = 2), the numbers represent adjectives, and there are hidden nouns associated with each of these numbers. In addition, we add the adjectives and keep the nouns. We saw how this relates to the standard algorithm for addition by writing out the nouns (ones, tens, hundreds, etc.), and re-grouping (11 tens can be 10 tens + 1 ten, or 1 hundred + 1 ten). We also considered how the adjective-noun theme could help students to understand the addition of fractions.

This afternoon, we began by considering different models of numbers – thinking about the affordances and limitations of thinking of numbers as amounts, or measures, or locations. We then used counters to explore the geometry of numbers – thinking about how to represent odd numbers and even numbers using counters to understand why the rules for addition of odds and evens work out the way that they do.

After the break, we explored the geometry of numbers some more. We investigated square numbers and triangular numbers, and saw how the counters can help us to make sense of some of the patterns that emerge when doing long sums.

Homework for tomorrow:
Math homework:

“Kayak variations” (p. 1.5) #4 (Woodworker Ken).

Pedagogy homework:

1. Read Cathy Seeley’s chapter. Message #1: Smarter Than We Think: Helping Students Grow Their Minds.
2. Read Principles to Actions: Sections titled Progress and Change; Effective Teaching & Learning (Pages 1-12)

In your notebooks, write a response to these prompts:

• What struck you about the content of the “Smarter Than We Think” message?
• Consider the chart about beliefs on page 11 of Principles to Action. What is your reaction to this description of unproductive and productive beliefs?
• What are a few connections that you see across these two readings?

This is the new blog for the PMI workshops at Greater Allegheny. This blog will contain daily recaps of the workshops, as well as updates on any future workshops.