Archive for October, 2010
Strategic Competence is extremely important to build in our students. When students have strategic competence they are flexible with numbers and can think about a variety of efficient solutions. Strategic competence requires conceptual understanding. Students have to know what they are doing, in order to think about it in a variety of efficient ways.
When I say efficient, I am referring to the fact that some strategies are slow and others are fast. So for example, let’s take 12 x 14. One strategy is to draw 12 groups of 14 and count them up. This is a strategy but it isn’t very efficient. There are much more efficient ways to solve this problem. One way is to multiply 10 by 14 and then add 2 by 14. This would be much more efficient than the previous method. Another way would be to multiply 12 by 12 and then add 12 x 2. The point is that when students can think of multiple ways to approach a problem they are exhibiting strategic competence. Below I have listed some examples of ways to think about teaching double digit multiplication, so that students build conceptual understanding and strategic competence.
Double Digit Multiplication
Explanation of Lattice Multiplication
Cross Hatch Multiplication
Other Resources:Read Full Post | Make a Comment ( 1 so far )
Here are a couple of research based ideas on assessing mathematical dispositions. Try one of these out and let me know how it goes!
I. McIntosh(2009) created a prompt similar to this (I’ve adapted it a bit):
As a math student in this class, I rate myself on the following scale (put an X on the scale where you rate yourself).
What kind of math student are you:
1. Probably the worst in the class _______
2. Pretty bad______
3. Not bad/ not good ________
4. Pretty good ______
5. Totally awesome! Maybe the best in the school __________
The reason I rated myself as a/an ____ on the scale above is because:
II. See the NCTM article and samples of assessments of mathematical dispositions here: (http://www.fayar.net/east/teacher.web/Math/Standards/Previous/CurrEvStds/evals10.htm#f2802341).
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In 1933, Dewey suggested that:
When the teacher fixes his attention exclusively on such matters as these [the acquisition of skills and knowledge], the process of forming underlying and permanent habits, attitudes, and interests is overlooked. Yet the formation of the latter is more important for the future. (1933, pp. 57-58) (cited in Merz, 2009).
Do you agree? Isn’t it at least just as important to shape habits, attitudes and interests as it is to teach them their multiplication tables or how to divide fractions? How we teach is just as important as what we teach? At the end of our lessons, do our students feel like, “Whew, I’m glad that’s over with.” Or do they walk away wanting more, desperate for the next lesson?
I love this quote because I think Dewey reminds us that we do teach and touch the future. What we do daily, will affect them for the rest of their lives. And they will either walk away from your class, thinking they are capable, that smart is learned, that they can do it if they try, or that they can’t and they hate math.
If as Dewey states, “the latter is more important for the future”- habit, attitudes and interests, how do you then begin to think about teaching more than 3 x 4 =- 12?
How do we teach this too? We set up spaces to cultivate great habits, attitudes and interests. All of our moves, many of them implicit shape our students attitudes. We have to be attentive to how each step we take shapes this for them.
Guided math groups provide us a special space to cultivate these aspects of learning math because we can give more individualized attention. We can attend to these in our groups and coach our students more one to one.
Any thoughts? Please share:)Read Full Post | Make a Comment ( 2 so far )
October is National Cookie Month. There are so many great interdisciplinary activities to do around this celebration. Here are a few that focus on math.
- Survey the Class on their favorite cookie
- Make all kinds of graphs with data: frequency table, tally chart, bar graph, pictograph, pie chart (see graphing post in this blog)
- Have a Chocolate Chip Cookie Taste Test
- Find range, mode, median and average number of chocolate chips in a bag of cookies
1. Make cookies in class
2. Make a timeline of the invention of the chocolate chip cookie
C. Number Sense
- Estimate the average number of chips in a chocolate chip cookie/ compare brands
- Estimate the number of cookies in a jar
Have a Family Cookie Festival. At one of the schools I work with in the Bronx, we usually throw a big Family Math Cookie Day. We set up several cookie stations and the families come and do math, eat well and have fun. One station is to make a cookie in a jar (the children measure out all the dry ingredients and then go home and add the wet stuff (eggs,milk, etc…); another station is to make spider or monster cookies with frosting and various candies; another station is to make s’mores; another station is to guess the amount of cookies in a jar; another station is to graph your favorite cookies. We usually get about 200 to 300 people who come out and have a great time!
Happy Mathing!Read Full Post | Make a Comment ( None so far )
Teaching staying power, sticktoitness and the right attitude: Mathematical Disposition and Guided Math Lessons (Part 5)
Polya (1969) [the grandfather of problem solving] states: This is the general aim of mathematics teaching – to develop in each student as much as possible the good mental habits of tackling any kind of problem. You should develop the whole personality of the student and mathematics teaching should especially develop thinking. Mathematics teaching could also develop clarity and staying power. It could also develop character to some extent but most important is the development of thinking. My point of view is that the most important part of thinking that is developed in mathematics is the right attitude in tackling problems, in treating problems. (Part II, pp. 5-7) (cited in Merz, 2009).
Polya provokes us to think about what we do everyday. He says that we are charged with developing in students ” the good mental habits of tackling any kind of problem.” We have to come up with rich math tasks so students can engage in this type of thinking. And, dare I say most of those types of problems are not on page 47 in problems 3-10:) Real problems, with real contexts helps students to see that math is a real subject.
What do you think of his statement that we should “develop the whole personality of the child and math should develop thinking?” This idea makes math class look very different from many teach, test and move on scenarios. If we are teaching to develop personality and thinking, then what on earth does that look like?
Polya goes on to state that teaching math is about teaching character, staying power and the right attitude.
Let’s all think about how we write that into our lesson plans!
Merz, A. (2009). Teaching for Mathematical Dispositions
as Well as for Understanding: The Difference Between Reacting to and
Advocating for Dispositional Learning. Journal of Educational Thought
Vol. 43, No. 1, 65-78.
Polya, G. (1969). The goals of mathematics education. Retrieved March 3, 2005, from http://www.mathematicallysane.com/analysis/polya.asp.
Unpublished videotaped lecture presented to T.C. O’Brien’s
mathematics education students
Mathematical Disposition: More than an attitude (using guided math groups to foster the many aspects) Part 4
Much has been written about mathematical dispositions or ways of thinking and being NCTM (National Council of Teachers of Mathematics, 1989, 2000) and others (e.g., Maher, 2005; De Corte, Verschaffel, & Op’T Eynde, 2000: Polya, 1969). The research tells us that mathematical disposition is much more than an attitude. It is about ways of thinking, doing, being and seeing math. It includes confidence, flexibility, perseverance, interest, inventiveness, appreciation, reflection and monitoring (Merz, 2009).
Given that, how do you cultivate each one of these in your class?
- What do you do to boost students’ confidence? Name 2 things. What could you do? Name 1 more. How might you do this in a guided math group? Since you only have a few students in a group, you can attend more individually to each one. One way to do this is to give problems that they can do. Success breeds success and confidence.
2. What do you do to help foster flexibility? This idea of thinking in many different ways? Do you engage in ongoing strategy talk? Do you have a culture of sharing in your class that goes beyond the answer but talks about how people got the answer or didn’t get it?
3. What do you do to build perseverance? How do you teach that? How do you do that? So that students’ perseverance levels increase over time?
4. What do you do to spark interest? How do you connect math to their lives? Where is the math in Pokeman or Dragonball Z?
5. What do you do to encourage inventiveness? Do we publicly celebrate inventive thinking? How do we get our students to think hard about the math their doing and take risks?
6. What do you do to cultivate appreciation of math? Do you make connections to real life situations that are important to them so students see that math really does matter?
7. How often do we get them to reflect about the math they are learning? Do we consistently use entrance and/or exit slips so they can think about their learning? Do we use individual pupil responses like thumbs up, thumbs down or sideways to check in with them? Do we use red, green, and yellow slips so they can give us immediate feedback about speeding up the lesson, slowing down the lesson or stopping to explain further? Do we ask them to do oral and or written reflections on their quizzes and tests and make plans to learn what they are still struggling with? You can do this in small guided math groups! This is an excellent space for these types of discussions.
8. How do we get student’s to monitor their learning? Do they have action plans that they reflect on? Who is responsible for knowing where they are? Just us? Think about how powerful it would be if they knew too! And if their parents knew, more than just a few times a year. The more people who know, the more likely the student is to get there! Think of the power of everybody being on board. What does a consistent inclusive monitoring system look like?
Journal of Educational Thought
Vol. 43, No. 1, 2009, 65-78.
Beliefs are Big. Students beliefs about math and themselves as mathematicians “exert[s] a powerful influence” on what students think they can do, they are capable of doing, on how much they will try, how much they will stay engaged with the math and ultimately on their mathematical disposition. (NCTM, 1989, p. 233).
Because we know that there is a definite relationship between what students believe and teaching and learning, we have to think about how we as teachers take this into account when teaching math. By third grade, students have these really ingrained beliefs about themselves as learners in all curriculum areas because they have had ample experiences. So, they either believe that they are really good at something or that they suck!
Given these mathematical autobiographies, what do we do? How do we shape or in many cases un/shape their mathematical dispositions? How do we help shift students’ negative belief systems about math and themselves as learners?
Spangler (2000) did a study where she asked a series of interesting questions, in order to unearth some beliefs and engage in a deep discussion about math springboarding from the answers. Ask your students these questions, and compare their responses with the ones from the study. Think about how to use these to spark discussions about math. Here are some of the questions.
If you and a friend got different answers to the same problem, what would you do?
Spangler found that most of the students said they’d check to see who was right and often times they would default to the student who was believed to be more capable. She points out that this highlights a common belief that math is about getting the one right answer. This conversation can lead the students into thinking about how there can be more than one right way and one right answer in math.
If you were playing “Password” and you wanted a friend to guess the word “mathematics,” what clues would you
give? (“Password” clues must be one word and may not contain any part of the word “mathematics.”)
Spangler found that the most common responses were add, subtract, multiply and divide. She points out that this highlights a common belief that math is about computation. This conversation can lead into a discussion about how math is so much more than arithmetic.
How do you know when you have correctly solved a mathematical problem?
Spangler found that students said they would recheck their work to make sure there were no errors. She points out that few students thought about if their answer made sense. Students failed to think about the context as a way to reason about their answer. This answer leads directly into a discussion about thinking about the problem solving situation and the answer in a given context.
Close your eyes and try to picture a mathematician at work. Where is the mathematician? What is the mathematician
doing? What objects or instruments is the mathematician using? Open your eyes and draw a picture of what you imagined.
Spangler suggests, “Ask students a variety of questions about the mathematician they imagined. Was their mathematician male or female? How old was their mathematician? What was their mathematician wearing? What did their mathematician look like? In what types of activities was the mathematician engaged? Were there other people around?
Spangler found the results very similar to the ones when this same question is asked about science. She noted that
“In the case of the mathematician, the students generally picture an older male with grey Einstein-like hair, wearing glasses, and sitting at a desk. He is usually using pencil and paper, books, a calculator or computer, and sometimes a ruler. The mathematician is often in a nondescript room, and there are no other people around.”
One way to get them to see math as a real world endeavor is to talk about the following questions.
Do you suppose McDonald’s has a mathematician on its corporate staff? What might that person do for McDonald’s?
Spangler found that the students said things like count how many people came or how much money was made. She writes that “To stimulate additional thought, the teacher can pose some real-world questions: How does McDonald’s decide where to build a new restaurant? How does McDonald’s decide on new food products to offer? How are the promotional games created? These questions open the doors for discussions about data collection, statistics, and probability.”
Questions I might add:
Do you think Disneyland hires mathematicians? What do you think they do for Disneyland?
Do you think ABC, NBC and CBS hire mathematicians? What do they do?
Do you think that the secretary, the custodian and the cafeteria director use math daily? How?
How do you use math daily?
These types of questions are a great way to spark conversations and unpack beliefs about math. Try them and let me know what you find out.
http://cimm.ucr.ac.cr/ciaem/articulos/universitario/concepciones/Assessing%20Students%E2%80%99%20Beliefs%20About%20Mathematics.*Spangler,%20Denise.%20*Spangler,%20D.%20Assessing%20Students%E2%80%99%20Beliefs%20About…%201992.pdfRead Full Post | Make a Comment ( None so far )
- It is important that we as educators have the language to describe the mathematical dispositions of our students. The Center for Language in Learning (2002) has created a scale to describe the mathematical dispositions of our students www.fairtest.org/files/LR%20-%20Math_Disposition.pdf . There are 5 levels. Look at these levels (see summary below and click on above link for original chart). Think first about where you fall and then where your students fall. How will you address this in your teaching this year?
Level 1 :
- Lacks confidence
- Lacks strategies
- Doesn’t persevere
- Gives up easily
- Needs lots of guidance
- Has trouble explaining their thinking
- Avoids math
- Limited confidence
Needs lots of encouragement to do the math
Needs scaffolding to participate in discussions
Not good with explaining the math
Usually avoids math
- Confidence and persistence vary with task
- Still hesitant about asking for help
- With scaffolding can think about more than one strategy
- Responds to prompts
- Can describe the math
- Math notation correct
- Uses Math Vocabulary
- Can engage in mathematical conversations with others
- Usually confident
- Persistent in solving problems
- Thinks about the math
- Has some strategic competence
- Responds to prompts
- Willing to admit to “being fuzzy”
- Asks for help
- Can give mathematical explanations
- Engages in mathematical discussions
- Flexible thinker
- Has high levels of strategic competence
- Can explain
- Engages in deep discussion
- Understands self as a learner
- Seeks assistance when confused
- Can write clearly about the math
- Sees the big picture and can make connections and generalizations
- Seeks to do some math on their own
About how many students do you have in each level? What do you do about moving students to higher levels? What does that look like when it is well done? How does it influence our instructional strategies and daily engagement with our students? I’m thinking about these questions for my own pedagogy. Please send me your thoughts.Read Full Post | Make a Comment ( 1 so far )
Recently I was on a plane and I heard a conversation between a father and son that reminded me a lot of teachers and students.
Father: Lower the armrest.
3 year old son: Why?
3 year old son: Why?
Father: Because I said so. I’m your father.
3 year old son: I need a reason.
Father: What? I don’t know where you got that misconception! I said lower the armrest.
3 year old son: I need a reason. That’s the rule….
Finally, the Father won and the 3 year old lowered the armrest…never getting a reason. It was comical. It made me think of what we often do to kids in school.
We say, “Turn to page 35 and do problems 1-7.” And our students often feel like “Why? I need a reason.” And we get as indignant as that father and think/say “Because I told you so. I’m the teacher. This is school. This is what you are expected to do. Period.” So, usually, they just follow the directions, but deep down…they want a reason. And it’s a fair question…that Why?
It reminded me to always give my students a reason…and a really good one about how this math makes sense for their lives. If they get reasons, then they do math with understanding. They gain confidence. It starts to make sense. This directly shapes their mathematical disposition.Read Full Post | Make a Comment ( None so far )
We have been talking about fostering adaptive reasoning among our students. Donovan and Bransford have defined it as the “Capacity for logical thought, reflection, explanation and justification” (p. 21). So this means students are thinking, reflecting, explaining and justifying themselves both publicly and privately. We build this by doing it as a whole group, as small groups, as partners and then as individuals. We must scaffold this type of thinking. The landmark report Adding It Up notes that even very young children from the ages of 4 and 5 can “demonstrate sophisticated reasoning abilities” if given the appropriate experiences (http://www.nap.edu/openbook.php?record_id=9822&page=129).
It goes on to note that “Research suggests that students are able to display reasoning ability when three conditions are met: They have a sufficient knowledge base, the task is understandable and motivating, and the context is familiar and comfortable” (http://www.nap.edu/openbook.php?record_id=9822&page=130).
Furthermore, it adds that one important “manifestation of adaptive reasoning is the ability to justify one’s work. ” But it clarifies that there is a difference between justify and prove noting that proofs are more complete whereas justifications are less formal. So, proofs would be where we would make the students write it down and explain it through a series of logical arguments whereas justifications would be more talking about it (http://www.nap.edu/openbook.php?record_id=9822&page=130). So for example, we might have students prove that a number is even or odd. We also might have them prove that 6 + 7 is a doubles +1 fact. We could have them prove that 1/4 is smaller than 2/3.
I challenge all of us to think about how we might start doing this in our elementary math classrooms. How might it look in your kindergarten classroom? How is that the same and/or different from how it would look in a 6th grade classroom? I think there are many possibilities.
Here is a great example of having students write a CONVINCE ME paper:
Remember IT’S ALL IN THE QUESTIONING….
Accountable Talk Icons
More Question Prompts for Problem Solving:
How Students Learn: History, Mathematics, and Science in the Classroom By National Research Council (U.S.). Committee on How People Learn, A Targeted Report for Teachers, Suzanne Donovan, John Bransford Edition: illustrated Published by National Academies Press, 2005 ISBN 0-309-08949-2, 978-0-309-08949-4
Adding It Up: http://www.nap.edu/catalog.php?record_id=9822#tocRead Full Post | Make a Comment ( 1 so far )