Archive for September, 2010
Here are some questions to get you jump started. Remember only introduce a few questions at a time. Picture icons are great and give non-readers more access to the poster.
The Language of Engagement
- I agree because . . .
- I disagree because . . .
- I don’t know….
- I don’t understand yet….
- I’m not sure….
- I’m still fuzzy …..
- I also noticed . . .
- Help me understand . . .
- Say more about what you mean . . .
- Can you show me/us how you did that?
- Why do you think that?
- Can you prove it?
- Can anyone add to that idea?
- Why is that true?
- How do you know that?
- I wonder . . .
- I got a different answer…
- I did it this way….
- I have a question about
- This was the tricky part
- Can you tell me more?
- Would you say that again?
- Can you give me an example?
Adaptive Reasoning is one of the key components of Mathematical Proficiency. Students with adaptive reasoning can think logically about the math and they can explain and justify what they are doing. The key to getting students to engage in mathematical discussions is to create a talk friendly environment. It is important that students feel that what they have to say will be considered worthy and important. They need to know that they will not be mocked, ridiculed or belittled.
We have to set talk as the norm in the classroom. The idea of using accountable talk in math class gives us a sturdy framework. Accountable talk in math class means that students talk about the math in a variety of ways. They learn to express their ideas as well as entertain the ideas of others; they learn to question themselves publicly as well as respectfully question the ideas of their classmates and teacher.
Here are some great resources to get you started. Cunnigham notes that “not all talk sustains learning. For classroom talk to promote learning it must be accountable to the learning community, to accurate and appropriate knowledge, and to rigorous thinking.” This is a key idea here. We want talk that promotes learning, that expands thinking, that contributes to the intellectual community.
Cunnigham has created a great graphic poster displaying various types of accountable talk- click the link at the bottom of the post – http://toolsfordifferentiation.pbworks.com/Accountable-Talk
Ramirez wrote a great teacher friendly article about getting students to be effective speakers in class when discussing the curriculum. See the article here and be sure to look at the chart she created with the specific skill sets http://www.teachersnetwork.org/tnli/research/achieve/Ramirez.pdf
Holyoke Public Schools created a very interesting curriculum map where they map out the accountable talk in a unit on fractions and decimals – see page 7 for a great list of teacher/student questions and see page 10 for an example of mapping the talk directly into lessons- http://www.hps.holyoke.ma.us/pdf/curriculum_math/grade4_fractioncardsanddecimalsquares.pdf
You absolutely must Google: Accountable Talk Toolkit and then open the toolkit (it comes up as an attached word document). It is phenomenal with very specific examples and ways to get started.
You also should Google: Lucy West – Robust conversations at every level. In this powerful powerpoint Lucy notes that students should be held accountable to the learning community, the content knowledge and mathematical reasoning. They should be encouraged and required to explain their thinking, based on the math topic at hand. They should talk like mathematicians and use math language in their explanations. Lucy quotes Tom Alec (The Tao of Democracy) – “Dialogue is the central aspect of co-intelligence. We can only generate higher levels of intelligence among us if we are doing some high quality talking with one another.”
To be continued….Read Full Post | Make a Comment ( 1 so far )
Mathematical Proficiency consists of 5 components or strands. The National Research Council published a report in 2001 called Adding It Up: Helping Children Learn Mathematics. In this report they define the 5 key components to learning math successfully as
1) Conceptual Understanding – Students with conceptual understanding know what they are doing on a conceptual level. They have knowledge and comprehension of the big ideas that they are exploring. So for example, David knows how to explain multiplication as 4 groups of 7. He also understands that if he skip counts by 7 four times he will get the answer. He can discuss the many different representations of multiplication.
2) Procedural Fluency –Students with procedural fluency are able to do the math (although sometimes they can do the math and not understand it at all). So for example, Stew can regroup but doesn’t understand place value, he only knows to “carry the one” as he mistakenly explains.
3) Strategic Competence – Students with strategic competence have flexibility with numbers. Sue can tell you that 8 +7 is a doubles plus one fact (7 +7+1) or a doubles minus one fact (8+8 -1). She has various ways of thinking about the numbers.
4) Adaptive Reasoning – Students with adaptive reasoning can think logically about the math and they can explain and justify what they are doing. So for example, students talk about math by saying things like “I know this is the answer because…” They ask questions like “How did you get that, because I got a different answer?” They can justify their answer with mathematical talk.
5) Professional Disposition – Students with a solid professional disposition know how to persevere when the going gets rough. They have a confident math outlook and know they are capable of doing the math, if they just keep trying. So for example, Tyrone might say, “This is the tricky part for me. I’m going to have to try it a different way.” He might also say, “Right now, it’s still fuzzy for me, keep explaining…”
When all 5 of these components work together, children become proficient in math. As noted in the report, “These strands are not independent; they represent different aspects of a complex whole. … the five strands are interwoven and interdependent in the development of proficiency in mathematics” (p.116) I am going to spend the next five posts writing about how we develop these in our students in small guided math groups. Guided math groups are the perfect setting to hone in on these 5 elements and help all children to become proficient mathematicians!
ReferencesRead Full Post | Make a Comment ( 1 so far )
Number sense is the major strand that is taught in the primary grades. It is said to be like common sense, in that although we don’t have an exact definition of what it is, we know when students don’t have it:) Actually, there are some specific elements of number sense though (more on that later). This website has 15 games to build number sense concepts. Some of these games are really cool! Check them out!Read Full Post | Make a Comment ( 4 so far )
This is part of a series of problem solving posts.
The third step in Polya’s model is to CARRY OUT A PLAN. I find even when students know the plan, this can still be a tricky step. Here is where I talk with them about Ginsberg’s famous Bugs and Slips (see prior post). Students often get bit by math bugs here—they make the usual errors or they slip…making silly mistakes that with a bit more care could be avoided. So we talk about executing the plan with care…going slowly, watching what they are doing which actually segues into the 4th step: LOOK BACK (REFLECT). I make sure they stop and check their answer. I tell them to whisper it to themselves and see if it sounds right. I tell them to ask themselves: Does that make sense? Could that be possible? I tell them to double check!
This website has some great logic problems. Print these off and do them in small groups. Focus on the strategies that the students use to solve the problems.
Error Pattern Analysis:Read Full Post | Make a Comment ( None so far )
Polya’s Problem Solving Method
This is part of a series of problem solving discussions.
Polya states that Step 2 is to Devise a Plan. I find this to be the most tricky part for students. What kind of plan? Where do I start? How do I know what to use? Well there is the usual list:
Guess and check
Solve a simpler problem
Make an organized list
Draw a picture or diagram
Act it out
Look for a pattern
Make a table
Use a variable
Change your point of view
We must practice out all these various types of problems with our students if we ever want them to get proficient in using these strategies. Here are some websites and articles that have great practice problems and examples:
(Look at this interactive model making site (be sure to watch the tutorial it explains everything): http://www.thinkingblocks.com/Model_It.html)
Let’s Play Math (a great blog) has a great post about problem solving articles:
Read Full Post | Make a Comment ( 1 so far )
The Polya Problem Solving Method – Step 1 Understand the Problem
George Polya was a famous mathematician who theorized about how to teach problem solving. He identified four elements of problem solving.
- Understand the problem
- Devise a plan
- Carry out the plan
- Look Back
Sounds much simpler than it often turns out to be. Let’s start with 1.
How do we know that students really understand the problem? I have recently started asking students to do two things first.
- I want them to close their eyes and visualize the problem. Make into a mini-movie. What is happening? Who are the characters? What are they doing? Where are they? What are they trying to find out? Then I have them open their eyes and discuss/explain what they saw?
- I have them translate the problem into their own words. They share their translations with a partner. Then we share out as a group.
So for me, Step 1: Understand the problem means visualize and translate what is happening.Read Full Post | Make a Comment ( 2 so far )
Teaching problem solving in math groups makes sense because we want students to talk about their problems and think about them outloud. We want to hear what they are saying and how they are making sense of the problems. We want to listen for their strategies and see if they are efficient or not. We can do this in small groups in a better way than trying to hear our students in the whole group.
Working problem solving into the math routine: 3 Suggestions
- Do a problem of the day that the students solve in their journal and that you go over with them.
- Do group problem work at least once a week, where they get into cooperative groups and have roles and solve and present their problems. Give each group a different problem. Roles could include the timekeeper, the recorder, the reporter and the materials collector. (I write the problems out on big paper and give to the individual groups. I then go around and listen to how they are solving the problem).
- I also show the students testing problems each week and we talk explicitly about the language of the test. What is the vocabulary that they are using? What is the phraseology? What are the tricky parts? Why are they tricky? ( I find that getting students to think about the tricky parts is really powerful. In the talking about this, they articulate many of their own misunderstandings and doubts…so if I can find out where the shaky ground is, then I can better work on firming it up!)
Problem solving is an essential part of learning to do math. During the next few posts I want to talk about some key elements in teaching this process standard. NCTM defines problem solving as “engaging in a task for which the solution is not known in advance.” The math research maintains that contextualizing problems within our everyday lives is what helps students to make connections and learn in a natural way. Moreover, students must have a variety of strategies to approach problems, including diagrams, looking for patterns and various other strategies.
CAUTION: AVOID RELYING ON THE KEY WORD STRATEGY
I remember a few years ago, NY state sent out a memo to all testing grades to NOT USE THE KEY WORD STRATEGY. Although many of us often teach the key word strategy (as in this word means….) math researchers say not to use it! Actually to heavily avoid it! Yes, I’m talking about all those key word posters we have hung up in our classrooms across the U.S:) We teach our students that when they see “altogether” and “in all” to add and when they see words like “left” and “fewer” to subtract (Van de Wahl, Karp & Bay-Williams 2010). These do work sometimes, in many of the simplistic story problems in textbooks (Svlentic-Dowell,Beal & Capraro, 2006). However, they aren’t a good strategy for the real world and even for more complicated story problems. Many researchers and math educators have advised against teaching them in this way (Burns, 2000; Sowder, 1988).
3 KEY ARGUMENTS AGAINST USING KEY WORDS FOR PROBLEM SOLVING according to Van de Wahl (2010).
1. They can be misleading, especially if students aren’t taught to read, visualize and think out the entire problem.
2. Many problems have no key words. So then, what do students do? They often get stuck!
3. It teaches bad practices. It teaches students not to think and reason, just find that word and solve it. We must teach them “a sense-making strategy.”
In conclusion, I think more than teaching students key words, teach them to think.Read Full Post | Make a Comment ( 3 so far )
Glyphs are a form of graphing that are really fun yet quite rigorous at the same time. Making and interpreting glyphs involves problem solving, representing a variety of information, communicating ideas, and discussing and interpreting all the data once collected.
Key Points about Glyphs:
- Pictorial way of showing data
- Shows a great deal of information at one time
- Symbols represent the different data
- Many types of glyphs and many teacher resources with several ideas… (shoes, snowmen, houses, horses etc..)
- Require a legend to read
Making glyphs is an involved process. First the children have to decide on various criteria. They read the legend and then make their individual glyph accordingly. They are literally transfering their data into a pictorial representation. I find that it is much easier to pull students into guided math groups and have them create their glyphs there, especially since I can focus on questioning them and hear them communicate their ideas using math language. As with all other data collection, the MATH TALK is key. Have the students justify what they are doing and the discuss the collective data.
cut and paste this url: http://www.uen.org/Lessonplan/preview.cgi?LPid=14864
Teacher Resource Books About Making Glyphs:
*Be sure to see the post on this blog about graphsRead Full Post | Make a Comment ( 2 so far )
« Previous Entries