Head Start Cut 5 Percent

CBS reports on the federal budget effect on Head Start, the children who benefit from it’s services, and the mothers who work in their communities. 

http://m.cbsnews.com/fullstory.rbml?catid=57589532&feed_id=1&videofeed=37

 

Technology Takes Students Further in the Mathematical Thinking

In “Promoting Problem Solving across Geometry and Algebra by Using Technology,” [1] University of Georgia researchers make a case for teachers to use hardware and software in their classrooms (Erbas, 2005). Technology enables students to make connections between representations of numbers and operations in order to find multiple solutions to problems. Students make connections between mathematical concepts involving real life situations by using graphic calculators, geometric drafting software and spreadsheet applications. With a teacher’s encouragement, they may consider multiple answers to solve problems, evaluate plausibility of those answers, and recognize patterns for hypotheses. With spreadsheets, tables of possible answers allow students to check validity of their thinking without becoming discouraged by limited skill in solving algorithms to prove theory. Graphing calculators show relationships of the numbers and free students to consider reasonableness of their answers. Drawing software allows students to sketch accurate geometric representations of problems and connect relationships of numbers with variables. These technological tools do not release the teacher from questioning students. The teacher encourages and challenges students to use the tools to go further in their thinking, beyond one answer. The tools free students to consider their own theories and seek solutions at higher levels of learning.

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[1] Erbas, A.K. (2005). “Promoting Problem Solving Across Geometry and Algebra by Using Technology.” Mathematics Teacher, p. 599-602.

Lift the lesson out of the textbook and into the learner’s life

Teacher Stephen Lee engages the student with a hands-on project of staining pencil boxes. He said the project was successful because the task “lifted the lesson out of the textbook and into the learner’s life.” He engaged by showing a finished pencil box and presents the challenge of how much stain is needed for the project. He said that he was going to go to the hardware store to buy stain and asked the students what Mr. Cunningham at the store would ask him. A student stated the task, “How much stain do you need?” Mr. Lee restated the task to find out the total area of the box so they’ll know how much stain to buy. He provides information on how much area the stain will cover: 1 point covers about 25 square feet. He provides them with wood pieces be measured, calculator, ruler, pencil, and paper. The learners work in groups proposing their ideas. The learners apparently have worked in groups, as they discuss and disagree amiably. Learners have assumed roles in the group:  recorder, calculator, and one who measures. Mr. Lee asks them to tell him what they are doing. He clarifies his understanding and encourages, “Wonderful!” Evidence of their thinking is in their conversation, their recording, and their behavior.  He gives a 5 minute warning by ringing a bell. He advises to estimate what has not been measured and decide on a recommendation for the amount of stain to be bought.  When they met to share information, he recorded their totals for area in square feet, volume of stain, and cost of stain. Group recommendations did not agree. He posed another question, “What in the world am I going to tell Mr. Cunningham?” The students began to estimate the area of 26 boxes in comarison to the size of the room. As they discussed their ideas, they realized their errors. The answers began to converge. He wondered how they could check their answers. A student suggested that the class work together. (Source: http://www.learner.org/resources/series32.html?pop=yes&pid=895#)

Van de Walle Chapter: Geometry Levels of Learning

Task 1

Hmmm…There seems to be a pattern here. We get stuck because we are not presented with problems requiring higher levels of thinking, such as creating theorems, analyzing proofs, writing proofs and reasoning. I would guess that I am at level 1 according to the statistics from a study of high school students,  presented by Usiskin in 1982. Most elementary students are at the level 0, visual classification. Today, preschools are integrating properties of shapes and the statistics may be better for k-1. If this is typical of high school students, at what van Hiele level do most K-5 students fit?  I would think at level 1, because pre-school curriculum is stressing recognizing these shapes in the environment…

Task 2

            In video 1, Mason is asking convergent questions. Convergent questions:

1. How do you know it’s a triangle?
2. How do you tell someone what a triangle looks like?
3. How could you tell someone the description of a square?

The student is functioning on Level 1 because she is basing her selection on appearance alone rather than on reasoning. She cannot describe the attributes of a triangle or square, such as the number of sides, the length of the sides in relationship to one another or the number of points. The evidence of that level is the learner said, “Because it looks like this,” several times, tracing the shape in the air with her finger.

            In Video 2, “How can you tell a triangle?” is a higher-level, convergent question. She changed the orientation and asked, “What if I turn it like this. Is it still a triangle? The learner said, “Cos it has a point and is straight at the bottom.” He is still at level one because he didn’t think the same shape was a triangle when it was turned. Why is “because it was upside down.” She built on that understanding and continued to turn the piece. “Is it were still a triangle?”  He made an adjustment to his schema of classification of properties in space when he said, “Upside down triangle.”

In Video 3, the learner didn’t know what to call an obtuse triangle and said, “Because it’s pointed like this. It’s straight across like this and it’ going diagonal to here.” He didn’t know what shape it is. The teacher asked questions to make him describe properties, “How many sides does it have?” He knew the shape had three sides, but he did not associate properties with the definition of a triangle.

In Video 4, the student was asked an open-ended question which allowed the questioner to discern what concept she knows about the meaning of congruent and similar. The next question allowed the questioner to determine what the learner thought was important about the characteristics of the triangle. Her answers show that the student is at Level 0 because she has grasped the concept that there are classes of shapes. She defines attributes and classifies by properties but does not analyze their relationships.

In Video 5, the teacher asked a level 1 questions that indicates classification by properties and 2 question, analysis, about the relationship (comparison) of the degrees to the angle. The learner is beyond visualization because the same angle is cut within circles of different size.

In Video 6, the interviewer asks the learner to sort and classify level 0. The level 1 question, (how you sorted), indicates level of reasoning, analysis by properties. Naming the shapes and defining is also a level 1 question. He knew the term 3-D but did not yet describe the difference between a square (2-D) and a cube (3-D).

In Video 7, the student sorted and classified by appearance, level 0. He related their shapes according to his experience, his name. He classifies visually and not by property. He indicates experience with tandems as he sees the star in the placement of the triangles. He did not recognize the difference between properties of the triangle, sub classification, level 2.

4th-grade math

Direct instruction: my way or the highway

I am observing a fourth-grade class, in the southeastern part of the county. The students are working on multiplying double digit multiplicands. The lessons start with reviews and then direct instruction. Guided practice follows. Then students work alone. The teacher uses an electronic board to explicitly model her strategy and expects students to follow her methods exactly. Last week, she said that she would take off points for skipping a step in her method. The strategies offered make sense and would work if they could be remembered, the first step in Bloom’s  Revised Taxonomy. However, the students will be in trouble if they forget that strategy at the end of the year.

Building experience with concrete objects

 About three bins of manipulative pieces sit above the lockers. I have seen one bin opened for a rainy day recess inside. The child with an IEP ventured inside and brought out some connecting blocks. I didn’t investigate whether he was solving math problems. I opened another bin looking for pieces for my lesson and they were still sealed in bags with the shipping statement on top. I’m only there twice a week; so I missed the array review for multiplication. Arrays with algorithm were posted on the wall.

A problem was posed.

Four students each had some colored pencils. Each student had 3 red and 2  blue pencils. Show 2 different ways to found out the total number of colored pencils.  One student drew an array and then counted the squares one by one.

The problem reminded me of one of  Dr. Piel’s DMI tasks to determine whether children have progressed from pre-operational to  concrete stage of behavior, that indicates whether they have reached the stage of maturation to learn math. So I asked the teacher if I could take a quick poll. “Are there more red pencils or are there more pencils?” About a third of the class of 29 raised their  indicating there were more red pencils. Their answer begs the question, “Are we expecting too much too soon of our students?” Rather, is the state requiring developmentally appropriate curriculum?

Fraction Models

“Fraction Models” representation on the Illuminations website provides number sense through visual experiences with irrational  fractions. The site also allows children to see that different representations of fractions may be written with different numbers and operations and still mean the same proportion of the whole.  The experience with equivalent fractions will build schema for algebraic equations that require reducing rational numbers.

 The ability to change the squares to show area, length, and width removes the mystery of irrational numbers. Children may see that larger numerators simply means complete wholes and part of a whole. At this time, fourth-grade class is working on multiplication of tens, hundreds and thousands. The wide range option on scale would allow children to explore the concept of decimal places. Seeing the area representations change in proportion to the number line slide would allow my children who are still counting on their fingers to multiply make a connection to skip counting. Playing with the pie models would allow my children to build visual understanding of simple rational fractions. The screen is somewhat busy. I predict that some guidance on looking at different parts of the screen to make comparisons would be required. Also, it was fun to enlarge the pie with the scale and not look at the corresponding percentage and decimals algorithms. (I have trouble focusing sometimes also.)

Our training has taught us that playing with models such as this is an excellent way to build experience before introducing new concepts. The experience also would allow children to make discoveries themselves, without having to teach them directly. We may reinforce their ideas through explanation and scaffold their understanding with questions. As I’ve already mentioned, I also see opportunity for reviewing concepts and clearing errors in thinking with the models.

Equivalent fractions

Okay, maybe it’s late and I shouldn’t admit this; but the Illuminations Equivalent Fractions game was hard!. When I tried to do the math I’d get it wrong. When I let the picture guide me, then I got it right. I guess that is the idea. The concept is that fractions can have different denominators and number of parts, nominators, and still represent the same proportion of the whole. The game builds number sense of fractions. Students may visualize the parts to the whole and then see that the different numbers correspond to the same fraction. Errors in thinking, such as larger numbers mean larger pieces, are disproven with a visual representation that is then associated with the  symbolic equivalent fractions on the right. The game allows helps children see how factors can be used to reduce fractions to the lowest common denominator. The game would build upon an understanding of fairly dividing parts of a whole, like a sheet cake. Good night.

Experience Fractions Game

The Fractions Game on the Illuminations web site had decent instructions. The goal is to slide all the rulers to one by matching the fraction on the card. Individuals may play themselves by beating the previous time required to finish. The sound has to be turned on, which my be best in a classroom.  I was not clear that I had the correct equivalent by the popping sound. A card is turned over and the player has options on how to slide the markers along tracks to reach one. Six tracks are marked with fractions of halves, thirds, fourth, fifths, sixths, and 10ths of one. Players may count one part at a time to reach one, using the same denominator.  As the game progresses,  a player is forced to find an equivalent fraction as the option for using the exact denominator is no longer available. A player may reduce the fraction or multiply by one in the form of a fraction to match an available denominator. When no exact option is not available, the marker may be slid to one if the fraction on the card is larger than the space required to reach one. A weakness is that the tracks became illegible near the end faded and froze up after three rounds. Positive reinforcement was given at the end of each round.

The game builds experience with concepts about fractions. Fractions of different numbers may be equivalent. Different fractions may be added to make one. Fractions are part of one. Larger numbers in fractions do not mean they are larger parts. Numerators tell how many pieces of one. Larger denominators mean smaller pieces of one.

The game may be used with partners or individuals at the computer center. The game may be used in fourth grade after an introduction using concrete manipulative models. The teacher may model using the game on a SMART board. Students may use the representations to practice or extend their understanding about fractions. Fifth graders may need review and practice. with fractions.

Foundation for understanding algebra

Reading the article “Children’s Understanding of Equality: Foundation of Algebra, by Falkner, Levl and Carpenter in Teaching Mathematics, I was surprised that children see the equal sign as an operation rather than a relationship. I recall hearing the words “is equal to” or “the same as” when reading math problems. Also, I was  surprised by that 84 percent of the  145 sixth graders who did not correctly compute 8 + 4 = ? + 5. I found that the small sample of one of three fourth-graders in my clinical class got the addition problem wrong.  Two of three got the multiplication problem 9 x 9 = ? + 63 correct; but they did not make the connection that 63 is a multiple of 9. They just finished factoring. The third child is working on the tens number system,  and number sense. We can help children build number sense by given the equations context and working with a manipulative like a set of 100s blocks.

NCCTM conference awesome!

The North Carolina Council of Teachers of Mathematics 40th Annual State Conference in Greensboro Friday, Oct. 29, was awesome! Presenters gave evidence that constructivist instruction is happening in elementary schools in the mountains, from the piedmont, and to the coast.

Leigh Hutchins and LaShay Jennings of Buncombe County Schools , which uses the Pearson Investigations curriculum, challenged teachers to differentiate for students’ strengths and weaknesses by modifying games. They had developed involving dominoes and money chips. They suggested send concept modifications to Chutes and Ladders home with students to make a home-school connection with parents. They stressed giving children experience with numbers to build concepts of their relationships and operations.

Amy Scrinzi of the NC Department of Public Instruction, Office of Early Learning, set up centers of games for players to internalize facts. She gave a thumbs down to flash cards and timed tests. She said that as a child she thought there were hundreds of facts to be learned and that she would never memorize them.  Instead, help children internalize facts and number relationships by playing games in centers. She offered inexpensive games made of discount store pieces, like pompoms, cups, paper plates and markers. Players focus on one number at a time. Place the number of pompoms under focus in a cup. Spill the pieces on the plate halved with a marker line. Children record the number of pieces that fall on the  left half of the line and the number on the right side. They continue spilling and recording to build hands-on experience with ways to build a concept of that one number. Then, children discuss what they noticed about the number and how many ways to show it. Building number knowledge with games using one-to-one, hands-on objects like connecting cubes, plastic jewels, dice, and counters. The pieces were used to correspond to numbers on number lines, spinners, dice and balances.  Try “Grab Bag Subtraction” for two players. One player fills the bag with pieces of the number under focus. The other pulls out some counters and shows them. They work together to predict how many are still in the bag. The exchange may be, “I put in 6 and you took out 3.” How many left? They record a number sentence and guess before counting the pieces left in the bag.

Sonya Gregory of Charlotte-Mecklenburg Schools challenged teachers to offer experience solving problems and using reasoning skills. She recommended identifying a child’s strategies for solving a problem and building upon their models. Teach children to ask themselves, “What do I know? What do I need to find out?  Are there any roadblocks?” She provided these mind hooks for steps to solve problems: “Do it,” concrete; “Explain it,” verbal reasoning; “Picture it,” visual representation; and “Write it,” number sentence or expression.  She also shared “Box the Operator,” a strategy for solving word problems. Highlight operation words with a box. Underline numbers needed to solve. Circle unknown references. Draw arrows around equality words. Then put the highlighted data in an equation.

Tammy VanCleef, also of Charlotte-Mecklenburg Schools, showed card games can be used to differentiate among students in the classroom. Pose problems of comparing numbers with cards. Whoever has the larger or smaller number takes the cards. The player with the most cards has the greater number. Ramp it up by challenging to compare pairs of cards, which will build understanding for addition.  Add a third player for even more comparisons She showed video of two girls who challenged themselves without prompting. She also suggested making video, with proper permission, for observation, assessment and documentation.

From Cumberland County, the district math resource coordinator  Dawne Coker showed how to turn textbook problems into higher-order thinking exercises. Replace the numbers in a word problem with blanks. Scramble answers in boxes to be filled into the blanks. Challenge players to replace the numbers in the proper order. The method can be used from first to fifth grades. Another game, “Error Detectives,” makes the classroom a safe place to learn from errors. In teams with captains, players receive problems with errors. They are challenged to find the error and then show the others. They solve the problem correctly and show the others. Teachers often see students making the same error repeatedly. Ms. Coker suggested going home to write a problem and make the same error while modeling the next day.  See how quickly students catch the error. Students will learn to feel safe to engage, even if a mistake is made.