Thursday, June 27, 2013

Chapter 4: Increasing Comprehnsion by Asking Questions: Book Study (BMC)

"Questions lead children through the discovery of their world..."  (Keene and Zimmerman, 113) children progress through elementary school, something squashes this curiosity.
"Children enter school as question marks and come out as periods." (Postman, 116)

These opening thoughts in Chapter 4 definitely left me with something to think about. How can we continue to pique the curiosity of our students and foster that endless stream of questioning that was second nature to them in earlier years? Easier said then done, yet questions really are at the heart teaching and learning (117).

I wonder why... What if... How can... are just a few of the questions we want our mathematicians to ask in order to develop deeper understandings. Another interesting point made was "it may be more important to find the right question than to find the right answer (118)." How can we flip our students' thinking to this mindset? A key take-away was that students NEED to be the ones creating the questions, not the teachers. Students need to be explicitly taught to ask viable questions that will enhance mathematical comprehension.

Laney Sammons goes onto explain the types of questions and how they are related to math: Right There, Think and Search, On My Own. Sound familiar in the reading world? Teaching students the difference between thin questions and thick questions also can improve the quality of student questioning (128).

Skill: multiplying a whole number by a fraction
Thin question: Is the product greater than or less than the original whole number?
Thick question: Why is the product less than the original number?

When teaching about thin and thick questions, Sammons recommends using different size sticky notes for students to record their questions. 3" x 3" sticky note for thick questions; 1/2" x 2" sticky page markers for thin questions. Another recommendation was to have students write in a thick marker when recording thick question or a thin marker when recording a thin question. The goal is to help students become more automatic and independent at creating questions that develop deeper thinking and understanding.

Questions that linger also were mentioned. How often do we ask questions that motivate students to continue to ponder and revisit those questions? It these types of questions that leave a trail of "math residue" that leads to invaluable learning (129).

Laney talks about strategy sessions and how they differ from a typical mathematics lesson (131). During these strategy sessions the focus needs to be on helping students to develop rich mathematical questions that enhance mathematical comprehension. To use as part of a bulletin board or as a bookmark for students, I created a few visuals called "Get to the point..." to help in the scaffolding and building of students' independence. If you think your students can benefit from these tools, click on the image and grab a copy.

Check out an earlier post on questioning in the math classroom for these color coded question cards.

The next chapter is on visualizing mathematical ideas. Come back after July 3 and see what Sammons has to say about visualizing. In the meantime, feel free to comment or link up and share your ideas about student questioning in the math classroom. Don't forget to click on the schedule below and visit some other blogs hosting this chapter.

Sunday, June 23, 2013

Chapter 3: Making Mathematical Connections (BMC Book Study)

As I was reading this chapter, I thought is important not to lose sight that teaching students in the mathematics classroom to make connections is not the learning goal, rather the means to an end. The end being activating schema and opening entry points of learning to solidify mathematical understanding (105).

I like how Laney Sammons described making mathematical connections as building bridges from the new to the known (100). To many students math is viewed as a textbook with isolated units of study and not necessarily a part of their everyday lives. We need to take where students are at and build from there. Sammons spends some time talking about schema. Schema being the students' prior knowledge. Throughout the chapter, Sammons makes it clear how important it is to link new knowledge to existing schema in order to provide for a richer learning experience. Prior knowledge, or schema, consists of:
  • attitudes: How do students perceive math? 
  • experiences: What role does math play in the daily lives of our students and the world around them?
  • knowledge: What mathematical concepts do students understand and know? (88-89)
There are three types of connections in math much like reading: math to self, math to math, and math to world (92-94). Math to self connections focus on one's own life experiences involving math (allowance, time, measuring). Math to math connections focus on connecting past mathematical concepts and procedures to current units of study. Students need to see how math builds on prior math. Math to world connections dispels the misconception that math is only taught in school with a textbook. Math to world connections help students to understand the bigger picture of math.

Once again explicit instruction is needed to help students learn this process of making connections so it becomes automatic for students. As in reading, Sammons goes on to explain how important it is to focus on meaningful connections and not the distracting ones. This, I know from reading, is sometimes hard for students. And when introducing making connections in math, I feel caution needs to be taken so that the making connections does not steer learning away from the math concept at hand.I do feel making math to math connections is a place I want to focus on next year to help students see the bigger picture.

I took some of Sammons ideas and used them to create these reminders for thinking stems/questions related to making connections. I can make a copy, cut them out, and put them on a ring as a visual reminder to use them during modeling and think alouds. I also can make them for students as bookmarks to help them practice making meaningful connections in the context of what we are currently studying in math. Click on the image above and see if you think this might be useful for your students.

As students begin to develop an understanding that math impacts and exists in their daily lives, incorporating real world problems create authentic learning experiences. It is through these experiences that math becomes more meaningful and relevant (104). This reminded me of an activity I did during a geometry unit, Architectural I Spy. Once we started talking about architecture and the different shapes we could find in the different architectural structures, I am telling you students no longer looked at buildings in the same way. Talk about connecting to geometric shapes in the real world. Check out the activity below by clicking on the image if you think you could use something like this with your students. Just something to think about...

What are your thoughts on making connections in mathematics? Leave a comment or link up and share your thoughts. The next chapter focuses on asking questions. Asking questions is indeed an art in itself. Hope you stop by after June 27 to see what new ideas I find. In the meantime, don't forget to visit the other blogs who are hosting this chapter to get some of their ideas too: Teaching with a Touch of Twang and Smiles and Sunshine. You can click on the schedule below to see other blog posts from previous chapters as well!

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Saturday, June 15, 2013

Chapter 2: Recognizing and Understanding Mathematical Vocabulary (BMC Book Study)

The language of math, you gotta love it! This chapter begins by Laney Sammons emphasizing that explicit instruction is needed to teach the specialized vocabulary of math. Sammons notes a key difference of vocabulary acquisition in reading versus math. For reading, students have incidental exposure to words through daily conversations and reading, whereas in math specific vocabulary is rarely used during everyday conversations (81). If you take a look at the chart below, you can see the different categories of math words and how they can hinder math comprehension if not explicitly taught in the context of math.

Clipart by Phillip Martin
Vocabulary instruction DOES NOT mean copying definitions. Sammons refers to Beck and Marzano who explain that the most effective type of instruction that increases vocabulary instruction needs to be robust. It needs to be thought-provoking, playful, interactive, and REVISITED. Who ever knew that learning vocabulary could be so much fun?!

Sammons goes on to include Marzano's eight research-based characteristics of effective direct vocabulary instruction:
  • Vocabulary instruction does not rely on defintions.
  • Representation of knowledge should be shown through linguistic and nonlinguistic ways
  • Gradual shaping of word meanings occurs through multiple exposures
  • Teach word parts (milli-, centi-)
  • Different instruction for different words - in context and through concrete experiences
  • Verbal practice of word usage in authentic contexts
  • Play with words - KEY!
  • Focus on words that have a high probability of enhancing student success (50-53)
How do you know which words to teach? This is never an easy question to answer. One starting point Sammons recommends is teaching those words that have a high probability of enhancing student success. Choose words that are mentioned in the standards. These are the must knows that students will need to understand the concepts, skills, and practices for that academic year (57).

Immersion in math vocabulary is key. Whether that means involving parents to "talk math" at home (60), offering time in math huddles to talk math in authentic contexts, or writing math (63-66), multiple exposures in multiple settings will help help students understand and grasp the language of math.

 Other recommendations by Sammons...

Math Word Wall (67-68) Make it a "living word wall." Keep it current. Revisit it. Have students use the words to talk math.


Graphic Organizers (69-76). Graphic organizers help students to organize their thinking and show conceptual understanding of math vocabulary. As was Sammons recommendation in the previous chapter, it is important to model and do teacher think-alouds when introducing a new graphic organizer. Keep in mind the gradual release of responsibility back to the students. You might find reading this previous post interesting if you want to read more about graphic organizers. Do you have a graphic organizer that works well in math?

Games and Word Play (77-80). Games and word play are motivational ways to help your students become more word conscious. If you click on the image below you can find a few activity cards with word play recommendations from Sammons along with a few others I have used to bring in some kinesthetic practice when reinforcing vocabulary usage. I cut these activity cards out and ring them so if I have a minute, we can look to the math word wall and engage in some vocab play.

Well, that is vocabulary in a nutshell...sort of! How do you teach vocabulary in the math classroom? What activities have you found to be helpful. Please comment or link up and share your thoughts. Don't forget to visit some of the other blogs to learn more about Building Math Comprehension: Carol from Still Teaching after All These Years (Carol has posted some very helpful downloads.),  Beth from Thinking of Teaching, and other bloggers who have linked up below. The next chapter focuses on mathematical connections. Don't forget to stop back after June 21st to see more ideas.

Saturday, June 8, 2013

Get to Know Your Students as Mathematicians

Getting to know our students as mathematicians is important. Many of us may give our students reading surveys so we get to know our students as readers. Why not survey our math students to get to know them betters as mathematicians? Here are two different versions of a math survey that can help you to learn more about your students as mathematicians. Have students complete these survey charts by  filling in the height of the bars that best reflect them as mathematicians. My Math-o-Meter can be used by students to rank their comfort level with each of the concepts. To show what I know in math, I do best when I... can be used to tap into your aspiring mathematicians' learning styles. Use these charts in planning different learning opportunities for your students. Have your students tap into areas of strength and venture into areas that may not always be in the comfort zone.

If you click here you can find a post that includes a multiple intelligence survey similar to these. Start building learning profiles of your students early on and revisit them often to find multiple entry points of learning during different units of study.

What are some ways you get to know your students as mathematicians?

Chapter 1: Comprehension Strategies for Mathematics (BMC Book Study)

With the CCSS standards it has become more apparent that all teachers are READING teachers. Content area teachers are reading teachers, PE teachers are reading teachers, librarians are reading teachers...and yes, math teachers are reading teachers, too.

In Chapter 1 Laney Sammons discusses the global achievement gap in mathematics - a gap between what our students are taught and what is needed to be successful in our ever changing world. It goes on to define mathematical literacy as " individual's capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgments, and to use and engage with mathematics in ways that meet the needs of that individual's life as a constructive, concerned, and reflective citizen." (Organisation for Economic Co-operation and Development 2010, 18). Bottom line, our students need to be able to functionally use mathematics.

Understanding mathematics is much more than just number crunching. Mathematicians have to construct meaning just like readers do. Characteristics of "good" mathematicians are similar to characteristics of "good" readers (22).
  • They use prior knowledge to help them to tackle "new" concepts and problems.
  • They use multiple strategies to tackle a problem.
  • They demonstrate mathematical fluency.
  • They monitor and fix up their understanding of concepts.
  • They reason and defend their thinking to others.
So how can we help our students construct meaning as they learn math concepts and solve problems? Well, we can borrow from what has worked well during reading instruction. Through explicit instruction (31-34), we teach strategies to our math students...but with a twist on mathematical content and processes. We need to explain "what" the strategy is, "why" we use the strategy, and "when" the strategy should be used. Learning opportunities need to include teacher modeling, guided practice where students work in small groups or with a partner, and then independent practice. As in reading, we want to gradually release the responsibility of learning back to the students to help them develop a deeper conceptual understanding of mathematics. One interesting recommendation made by Sammons (33) is that during the initial modeling of a strategy, it should be soley the teacher talking. Student participation during this time should be avoided so that the focus remains on the teacher's thinking. Something to think about...

A take-away from something I read from Beth at Thinking of Teaching was her idea to incorporate math text into guided reading. Not necessarily a book, but rather reading a math problem. This would give students the opportunity to peel away the layers of a math problem much like the way they peel away the layers of a more commonly used guided reading text. This reminds me of an activity I did last year with a math task. I didn't do it in a guided reading setting, rather as whole group "close" read. See the post here if you are interested in reading more.
The instructional strategies and terminology that reading teachers use so successfully in teaching reading comprehension should be utilized in the math classroom as well. The next chapter is Recognizing and Understanding Mathematical Vocabulary. Math is a language all its own. Come back June 15 and see some ideas how to help our students talk the language of math.

One of my passions is math. I would love to hear thoughts and ideas. Comment or link up below and share your take on math comprehension. Don't forget to visit some of the other blogs hosting the chapters. Click on the schedule below and happy reading.

Monday, June 3, 2013

Get Your Math On ~ Summer Book Study

Button by Brenda from Primary Inspired
Starting June 8 a book study will begin for Building Mathematical Comprehension. The coordinators for this book study, Brenda from Primary Inspired and Beth from Thinking of Teaching, have gathered together a group of bloggers who will be hosting different chapters throughout the summer. Click on the image to get a copy of the upcoming schedule.

Whether you want to follow along or lurk to get some new ideas, link up and join the blog hop. Check out the different chapters throughout the summer by visiting the different blogs. I will be talking about Chapter 8: Synthesizing Information in July.

You can take a peek at the book at Amazon by clicking on the image.
Hope you will join us!