Friday, April 24, 2009

Creating a Brain-Friendly Math Classroom

Creating a Brain-Friendly Math Classroom
Presenters: Sally Kingsley and Connie Conroy, Howard County

This session went into the latest research and information from the book How the Brain Learns Math by David Sousa. (Because of copyright laws, only highlights of the session will be posted.)

The short term memory is divided into two sections, the immediate memory (about 30 seconds) and the working memory (which also has time limits). When things are put in your working memory, they are ready to be transferred to your long-term memory. Research shows that they are best transferred by rehearsal, a "less is more" approach, and through shorter periods. More information about this can be found in Sousa’s book.

Also in Sousa’s book, you will find great information on the stages of brain development, number knowledge at different age ranges, and the best times to learn math content. This book sounded like a great resource to have. I plan on ordering it when I get back home.

Has anyone read this text?

NCTM: Think! Communicate! Justify!

Think! Communicate! Justify!
Presenter: Mickey Jo Sobierajski, Fulton County Schools

This session focused on ideas and activities to empower primary students to become confident problem solvers, good questioners, and critical thinkers.

Ms. Sobierajski opened her session with the following clip:

One of the most important things we need to do is to teach our children how to think. For example, in the clip, the operator did not just answer the little girl’s question, but asked her how much she thought it was. He was attempting to get her to think about it.

The session gave several different calculator game examples for primary students to build critical thinking skills. The games begin easy and build on each other. There is a strategy to each game and each strategy is a little different. In each game, students work in pairs to get the final sum on their calculators. A final sum is given to students as well as a set of rules for each game. Students must take turns until one student reaches the final sum and wins the game.
1. Target Number: 7. Students must start with a 0. They can take turns adding only a 1 or a 2 to reach the target number. (Sample strategy: You would want to get a four so that the next time it is your turn, you can win the game by adding a 1 or a 2.)
2. Target Number 10. Students must start with a 0. They take turns adding 1, 2, or 3 to the number to reach the target number.
3. Target Number: 21. Students must start with a 0. They take turns adding 1, 2, 3, or 4 to the number to reach the target number.
4. This is the hardest because it has special conditions and students will have to really think about each number before entering it. Target Number: 25. Students must start with 0. They take turns adding 1, 2, 3, 4, or 5 to the number only if that number or the sum of its digits is not on the screen. (For example, I put in 5. My partner may not add 5 because it is on the screen, so instead she adds 4. I add 3. Now we are on 12. My partner may not add a 1 or 2 because they are on the screen. She may also not add a 3 because the sum of the digits are 3. My partner may only add a 4 or a 5.

To become good problem solvers, students need time to struggle with the problem for teachers to see what they are thinking. Here is a sample problem:
Lauren has 84 identical cubes, each with 2 cm edges. She glues them together to form a rectangular prism. The perimeter of the base is 28 cm. Find the height of the rectangular prism in cm.
What do you know by reading the problem? What do you have to find out? To understand the math, you need to be able to talk the math.
A good problem for students to solve involves interrelated concepts and is at a high level.

Question: When there are 52 green doors, how many blocks will I need all together? Allow students the use of manipulatives to model problems like this if they would like to use them. (Some students will create a number string to solve instead.) If you give students manipulatives, give K students too many and give 2nd grade students not enough to allow them to think critically about it.

Make sure not to use key words when working with a story problem. In all does not always mean to add. If you teach students to look for key words, you are teaching them not to be critical thinkers, but rather to read only the parts of the problem that you need, then solve.
For example: Brian rode 12 rides at the carnival. In all, he spent $9 for rides and $3.50 on snacks. How much did each ride cost? What is this problem really asking us?

Beware of the vocabulary you use. Are you using vocabulary that may cause a misconception? Do you tell students “I’ll be there in a second” and take longer than a second? This does not help them develop an accurate understanding of one second.

Word Walls can also be used to promote critical thinking in the math classroom. For example, use circles to help sort rules.
Which word does not belong and why? (There could be more than one answer.)
Choose a number that has something in common with the empty circle. (Allow students to figure out what the numbers have in common.)

Another way to promote critical thinking is to give students a scenario and a solution and have them determine the question to be asked.

For example: There are 2 fish tanks in my room. One has 8 guppies and 11 goldfish. The other has 24 fish with an equal number of guppies and goldfish. What question could I ask that would give me a solution of 43? Of 12? Of 23?

Thursday, April 23, 2009

Engaging Students in Significant Math

Presented by Henry Kepner, President of NCTM

In his presentation, Mr. Kepner gave an update of the Council’s promotion of mathematics teaching through its focus on curriculum, the dissemination of teachers’ and researchers’ reflective professional experiences, professional development, and the creation of community and policy support for change.  He reinforced the overall theme of this year’s conference: “Equity: All Means All”. In addition to sharing how NCTM is working to promote equity for all students, he was also careful to include not only minority and impoverished students, but also gifted learners. One compelling statistic that he shared, in which I was unaware of, is that 20% of students who “drop out” of school have been labeled as gifted at some point in their schooling. “All Means All” certainly reflects both ends of the learning spectrum as this point proves.

Mr. Kepner emphasized that the NCTM Curriculum Principles guide teachers in the important mathematical ideas that NCTM recommends be taught at the varying grade levels. In addition, the focal points also guide teachers on when closure of big ideas are expected for particular ideas and concepts. What a vital resource in guiding our educational systems throughout the country.  These principles also emphasize the need for simultaneous preparation for algebra, and the development of conceptual understanding, fluency, and problem solving skills.

Mr. Kepner announced an upcoming resource (book series) known as Essential Understandings, currently being developed by NCTM, which will serve as helpful professional development tools for teachers. This should be a fantastic resource.

In addition, NCTM Topic Books, which will be available Fall 2009, will provide details and examples to support teachers in how to teach the NCTM Focal Points.

Another valuable resource for educators that Mr. Kepner shared during his presentation is a link called “Briefs and Clips” which can be accessed at the NCTM website: This link provides educators with quick access to useful and practical research. One topic example is “Effective Homework Assignments”.  There are also many other topics for educators to explore and this resource will continue to grow and develop over time.

NCTM:Children’s Difficulty with Elapsed Time and Giving Change

Reasoning at Two Hierarchical Levels Simultaneously:
Children’s Difficulty with Elapsed Time and Giving Change
Presenter: Constance Kamii from University of Alabama

I decided to go to this session because the title caught my eye. Two of the things that I find some of my students struggle with a lot are elapsed time and giving change, so I headed to this session to learn a few strategies to help me teach these concepts.

Elapsed Time
The presentation opened with Ms. Kamii, a very entertaining presenter, explaining the difference in operational time versus intuitive time. (Most of her data came from a study done by Piaget.) She shared a scenario of a four year old student who was asked several questions about time.

Question: Are you older than your sister?
Answer: Yes because I am taller than her.
Question: Is your grandma older than your mom?
Answer: I don’t think so.
Question: When you are 10, will you be older or your sister?
Answer: I don’t know.

This four year old has a concept of intuitive time. Intuitive is based only on the observable things. She can see that she is taller than her sister, so she must be older than her. She is not able to know if she will still be older than her sister when she is 10 because she cannot see who is taller then. These things are all things that she has deduced as logical based on what she knows. She will eventually move into operational time which is based on temporal reasoning. This is a much higher level of thinking.

Piaget states that there are three types of knowledge:
1. Physical knowledge: objects in the external world (That is a cube. It is red.)
2. Logico-Mathematical Knowledge: Consists of mental relationships with the ultimate source being in your head not knowable with your eyes. Time is not something that is observable, but a series of mental relationships. This is why it requires an operational time in order to understand.
3. Social-Conventional knowledge: knowledge of things like the English language, that Christmas occurs each year on December 25, knowing when it is appropriate to say “please” and “thank you.”

Ms. Kamii stated that there is no way to really show elapsed time to children because it is something that they have to see mentally. A majority of students do not recognize operational time developmentally until third grade.

As teachers work through problems with elapsed time, they should allow students to create and build upon their own understanding in order to build their operational knowledge. Textbooks teach students “standard” ways of determining elapsed time such as counting the hours first, then the minutes, creating a sort of algorithm and subtracting, or counting only the minutes. These strategies do not allow students to construct their own understanding of elapsed time naturally, thus creating misconceptions as they go.

Students should learn elapsed time in a natural progression. How would most of your students solve this problem? Work it in as part of your Calendar Math in the morning and see what the most common answer is.

“I started reading at 8:15 and finished at 9:10. How long did I read for?”
Many second graders in Ms. Kamii’s study said 9 hours – 8 hours is one hour and 15 minutes + 10 minutes is 25 minutes, so the answer is 1 hour and 25 minutes. They do not think logically about the time and realize that their answer cannot possibly be correct because from 8:15 to 9:10 is less than one hour. They are reasoning sequentially.

One way students begin to construct knowledge is to think about the problems in ways that make sense to them. Take this problem for example: I arrived at a birthday party at 6:40. I left the party at 9:15. How long did I stay at the party? One way of making sense of the problem is to think about the 6:40 in terms of how long it will be until we reach the next hour, 7:00. See the thought representation below. Hours are represented across the top, minutes along the bottom.

To help students with elapsed time, we should:
1. Allow students to create their own understanding of elapsed time and ways to model it.
2. Give students meaningful opportunities to work with elapsed time in class. Children are not taught to think about time throughout the school day in a context that is beneficial to them creating their own understanding of it. As teachers, we tend to say “It’s time for lunch, get your things ready” or “at 9:15, you need to be ready to do Calendar Math.” Instead, we should say to our students “How much time do we have until lunch? Are we able to complete a 1 hour and 15 minute math workshop before lunch or would it be best to do a 45 minute science lesson right now?”

Making Change
The second part of the session opened with a quick video clip of a second grader playing store with her teacher. The teacher had a box of candies for 2 cents and a box of lollipops for 3 cents. The student had one dime and one penny in her money box. The teacher began with simple problems. She bought one candy for 2 cents and gave the little girl 2 pennies. The little was satisfied with it and waited for the next problem.
The teacher bought a lollipop for three cents, but gave the little girl four pennies. The little girl gave the teacher back one penny.
The teacher bought 2 lollipops and gave the little girl a dime. The little girl gave the teacher back the entire dime saying that it only cost 6 cents. When asked how much a dime was worth, the little girl said “ten cents.” She knew the value of the dime, but even with prompting from the teacher could not figure out how to use the pennies she had collected from previous transactions in her money box to make change for the dime.
To correctly solve this problem, it requires her to think about several things simultaneously: a dime is worth ten cents, subtracting 10 cents – 6 cents to make change for a dime, and giving the correct change back to her teacher.

Another similar situation, counting change by groups of tens and ones, also causes students to think about more than one concept simultaneously. Students may count this as:
As teachers, we know to transfer the counting from groups of tens to ones when we count it. We are able to think of both concepts simultaneously as we are solving the problem.

How can we correct this in our students? By using the same concepts as working on elapsed time.
1. Allow students to create their own understanding
2. Use meaningful examples.

NCTM: Divide and Conquer: Making Sense of Long Division

Presentation by Dr. Juli Dixon, 
University of Central Florida

Well, this session was just what I needed. For two years, I have had discussion after discussion with fellow mentors and colleagues on how to conceptually teach the long division algorithm and, today, I got the answers that we have been looking for! I will offer an overview of what I learned here,  but I really can not wait to get back to school to engage in some collegial conversation over what I learned today with fellow math geeks Rick, Suzanne, Tom,  Patti,  Sherrie, Tammi, Bridget,  Heather, Katie, and Bobbi! It’s sure to be a lively conversation. Are you ready for some enlightenment? Well, here it is! 

I learned why it has been so difficult for us to try to bridge our students’ understanding from the partial quotients method (or Russian Peasant as we like to call it J), to the procedural method of long division and now it all makes perfect sense. 

After reviewing the differences between measurement and sharing/partitive types of division problems, Dr. Dixon taught us that we need to represent long division with a SHARING (not MEASUREMENT) approach. (This means that there is not a natural or easy transition from Russian Peasant Division to Long Division.) Instead, the use of base ten blocks (hundreds, tens, ones) and/or quick draws of base ten blocks, along with “sharing language”, needs to be reinforced.  


As an example, for the problem 536 divided by 4, you would need 5 hundreds flats, 3 tens rods, and 6 cubes.  Starting with hundreds, you would equally share “1” hundred flat among four piles (notice the recording method in the example shown below). The one “left over” would then be decomposed into ten tens rods to make 13. Sharing the 13 tens rods equally in the 4 groups would put 3 in each group with one left over, which would again be decomposed (exchanged) into 10 cubes for a total of 16. The 16 cubes shared equally into the four piles would place 4 in each pile with none left over. 

The end result would show 4 EQUAL PILES OF 134.


            4  )  536

                   -4          (hundreds)

                    13        (tens)

                  -12        (tens)

                    16       (ones)

                  -16       (ones)


Once students make sense of this, the efficiency of the long division 

algorithm can be introduced.


Dr. Dixon also shared a new division model with us that I have not used before which connects to Partial Quotients Division. This involves the decomposition of the dividend (ex. 8,652 = 8,000 + 600 + 50 +2) and left-to-right division (starting with the largest place value first).  See the example below:

                   1,000        200        30        6

          7  )  8,000     1,600     250       42

                 -7,000    -1,400   -210     -42

                  1,000        200        40         0   

Dr. Dixon also reinforced several ideas that we certainly understand at Chets in our own mathematical communities:

1)   1) “Does McDonalds Sell Cheese Burgers?” is A PROBLEM! (Divide, Multiply…) Where's the math in that?

2)   2) Students need a context when working with division so they can model and connect with it.

3)   3) Use language that supports the meaning of division.

4)   4) Students need to see many division strategies, which happen with the Measurement/Partial Quotients/Russian Peasant Method (not with the Sharing/Partitioning or Long Division Methods) so they can see the efficiency in various approaches.

5)   5) Students need lots of experience with both measurement and sharing division types. They need to understand BOTH!

6)   6)Procedurally based algorithms don’t make sense of the math- they make the math quick.

7)   7) Florida’s New Generation Sunshine State Standards have 5th grade as teaching long division. It’s not a bad thing if we don’t “teach it badly.

Now, that was time well spent! Thank you, Dr. Dixon, for all of this enlightenment! This is exactly what we have been searching for! J

NCTM: Importance of Words: Powerful Strategies for Building Math Vocabulary

Importance of Words: Powerful Strategies for Building Math Vocabulary
Presenters: Brenda Spencer and Andrea Guillaume of California State University

This session began with a sample problem “A stop sign has 8 sides of equal length. Ryan knows that the length of each side is 10 inches. What is the perimeter of the sign?” We were given about 10 seconds to solve this problem.
Why could we solve it so quickly?
Because we have background knowledge of the different cultural (example stop sign) and mathematical vocabulary (example equal length, perimeter) used, allowing us to focus our attention and energy on finding the perimeter, rather than determining what the vocabulary is.

Why should we bother with vocabulary development?
-Math has a high concept and linguistic load.
-Math vocabulary presents special problems. First, most vocabulary is caught, not taught. We “catch” the vocabulary in our everyday life, we do not usually explicitly learn the vocabulary. Second, math vocabulary appears in outside context infrequently, so students bring less background knowledge with them to the classroom. Finally, math vocabulary represents abstract concepts.
-Students need support. Early exposure to math words and home experiences vary from student to student. Students need to learn the math vocabulary in order to catch up.

The research-based framework used to support this session came from Spencer & Guillaume’s book 35 Vocabulary Strategies for Content Area Vocabulary. According to the research, there are four stages of vocabulary development and practice.
1. Prepare: get kids interested in words we want them to learn, pre-assess and self-assess, connect to prior knowledge, learn concepts first, set goals, increase motivation to learn.
2. Build: use strategies rich in context, multiple exposures, depth of word meaning, connecting words to concepts. Use a variety of meaning-based strategies (see below). Make sure to still have students check the actual definitions. (this does need to still be part of the process, however only a small part.)
3. Apply: language rich, use words in new context, engage in writing and speaking opportunities with the new words.
4. Independent Word Learning: Students need to master strategies to learn new vocabulary.

Ideas for preparing vocabulary include:
1. Using vocabulary cards (small cards with a different vocabulary word on each such as equation, odd, even, sum, difference, quadrilateral, etc.) Have students sort the cards into two stacks: familiar and unfamiliar or much background knowledge and little background knowledge, etc. This is valuable because it becomes a quick pre-assessment of student knowledge before you check for true understanding of the words.
2. Vocabulary knowledge rating is another way to prepare as pre-assess student knowledge. For this activity, students should think of the vocabulary words as being on a continuum. How much knowledge do you have of the word? No clue, I know a little, I’ve got it down. A great product to use with this strategy is the Computer Response System pads. (We use them daily in my classroom in both math and reading. Stop by my room for a demo. I think KK has a set available for check out.)

Ideas for building vocabulary include:
1. Focus on multiple meaning words. (odd, even, product, formula) and discuss the different meanings of each. For example, most students will think of formula as something to feed a baby rather than a math word if not already exposed to it.
2. Focus on words with teachable word parts. For example, in the words quadrilateral and quadrant, teach that quad means four. Also, look at cognates. In French, four is quatre and in Spanish, quarto.
3. Use word histories to help build meaning. Teach students were the word came from and teach them how to look up the etymology of the word.
4. Use content links. Have a different word preprinted on pieces of paper. Have several students go to the front of the room holding their cards. Students seated try to find ways to link the vocabulary words together to a partner. They have to justify their reasons because words have the possibility of being linked to more than just one word. (see word chains below) For example, quadrilateral-polygon.
5. Use word chains to stretch student thinking. This strategy is similar to content link, however students are linking a chain of words, not just pairing them together. For example, the words decomposing-strategy-equation-equal-balance.

Ideas for applying word knowledge:
1. Use word posters to demonstrate knowledge. Have students create word posters with pictures, etymologies, synonyms, etc. Use a flip video to record and share the student’s work. (Check with Melanie to check out a Flip if you do not have one of your own!)
2. Allow students to make books. Have students many any number of book types (petal books, ABC books, strategy books) and publish them on the web using the Web 2.0 tools we have been learning about through our book study.
3. Play a math version of Apples to Apples. Before play, make two sets of cards. Set A with terms like mud, plastic, Dr. Seuss, taco, etc. Set B with the math vocabulary you have been using. Students should play in small groups. The object of the game is to gain the most number of points during the entire game. One player is the judge. All of the other players draw 3 cards from set A and keep them a secret. The judge puts one vocabulary card face up and the rest of the players put down the card they think matches the vocabulary word the best. The card that is chosen by the judge as the closest math gets the point. For example, the vocabulary word is fraction. One student might say taco because a taco shell is usually broken into pieces or fractions of the whole. Another student might say ocean because there are different layers in the ocean. The judge would then decide who wins that round.

Tuesday, April 21, 2009

NCTM 2009

Blogging coming to you live from the National Council of Teachers of Mathematics conference in Washington, DC. (Wednesday though Saturday) We can't wait to share what we learn! 
Angela & Melissa