# PDF Lectures on N_X(p) (Research Notes in Mathematics)

Tao My nationality is Korea I am elementary school student I want to be a mathematician when I grow up Please tell me the secret method of being a good mathematician. Dear Tao, I am very happy to come across you on the website. I am a new maths teacher in a college of southwest China. I major in algebra.

If you have some meaningful references concerning algebra,I ecpect your help by email. I want to do some research on mathematics. James Clark. Organization in writing is a huge key. As time goes on, your writing style will develop and organization changes with that. You have a good informative blog. I am a 8th grade teacher in NC and came across your site while researching some information about writing techniques for my English class this year.

I just wanted to thank you for the great information and articles about writing, and let you know about a site we are putting together to help teachers find trusted resources.

We would love it if you could write a few articles for us, but understand that you are probably busy. In the long run, intellectual speed and the type of traits measured by such statistics as IQ are not actually all that relevant to the ability to do mathematics, which relies instead more on getting the right level of understanding for the subject and on systematically and patiently attacking a problem or range of problems. I have read your advise on writing and career and I would say that all of them were very interesting and smack of reality.

I would liketo hear about teaching from the perspective of a Mathematician of your calibre. Thank you for taking your valuable time to open up a space for hive minds. It reflects great humility for a genius of your caliber to pave opportunities for interaction. However, in the Mathematical Subject Classification in JAMS, there is a section for history of mathematics, logic and foundations, and others. I was wondering if you could clarify on the restrictions as far as philosophy of mathematics is concerned as that section would include logic and foundation.

Do you usually go through your mail before you decide which one to read or someone else goes through your mail first before you receive it? I sent you a letter last year and did not receive a response from you. I was wondering if you had actually read it. Len Shantz. I am extremely impressed with your writing skills and also with the layout on your weblog.

Is this a paid theme or did you modify it yourself? Flavor: Working through a proof.

- BOOK SERIES.
- Step Down Mr President: We need real change.
- 11 Series Titles?
- RUSSIA CALLING: Short Stories.
- Chasing Margaret.
- On writing | What's new.
- Lhistoire de lAtlantide: édition intégrale (Documents) (French Edition).

Season: Pulling together and writing up. Tao is great. He writes with simplicity yet with the complex meanings given by the words. Advice on writing from Terence Tao Science Library. Do you find any results concerning to this situation? Any feedback will be appreciated. Should a.

Or is index a waste of space? The blog of Terence Tao is also an excellent source for advices on writing mathematics and academic careers in this […]. I very confused about the behavior of mathematical community. I have worked 15 years in chaos theory and I have published more than 80 papers and 8 books in several international publishers and in each time I seek comments from other specialist, they answred my positively and they never refuse to read may papers. The main objective is to see the opinions of experts before sending the paper to a journal. Is the mathematical community behaves like the logistic equation chaotic.

More than this, some of them attack me personaly with very bold words in despite they do not know me. I using word WS. I want to know if anyone can recommend a program to write and compile a latex and ArXiv? Katy Smith. Lots of learning stuff here. Thanks for people like you. After reading, I realize that a certain rule is not always applicable to everyone and that includes writing.

I proved hypothesis Legendre wanted to send in arxiv. NT, you would not have helped me? Thank you very much for all the great advice! Unfortunately, since Google Buzz closed down, the following links are broken:. What tools do you use for bibliographies? I find it very time consuming to do it by hand. Typo here: It is also assists readability if you factor the paper into smaller pieces, for instance by making plenty of lemmas.

Which university i can study PhD. Dear Terry Tao — Often I enjoy and learn from your blog. JoAnne Growney. Dear Prof. Thank you very much for your help!! Xu Peng. Thank you very much! On writing Ichigo Ichie. I am writing a research monograph. What also about a set definition laying on its side? This may greatly reduce formula width and this is a great problem for me.

I am an amateur mathematician without official scientific degrees. I am writing a breakthrough research monograph in abstract mathematics. That it needs editing, is quite probable because this is a very new field of research and the book may require changes to make it better and more general. The main issue here is that after the decision there is no way back: If I publish it traditionally I may lose copyright and be not able to distribute my LaTeX files for free, and reversely if I put it online with a free license, this may be an obstacle for publishing it.

This is even despite that publishing under copyleft is better for hunting errors, as in GiHub and similar free Git hosters there are error reporting zillas. Stephen King to share writing tips in new short story collection Ismael Olson. Hi Terry, Some time ago you were an advocate of publishing no further than arxiv. However, with their comments it is more of the same. Is there a time stamping technique you can recommend so that we can move on and publish at our websites?

## Jean-Pierre Serre - Wikipedia

Mathematic Reading futileinfo. In french but easy to understand:. Omitting it would decrease the length of theorem conditions. But without this condition conditions would be warrantenly false. I find interest in maths and equally in philosophy. I have been writing them linking them in the best possible way I can..

I adopt equations and theories from math and duduce explainations for some deep philosophical logics. The fact that my work doesnt incorporate for now a substantial amount of equations and symbols pulls me from submitting my works.. Dear sir, I am a 11 grader. I wanted to know what would be the sum of all the real numbers.

I will be grateful to you,if you provide the answer. More links from math professors — Lucy's World of Technical Communication. Writing Snowdrop's Notebook. Hello Sir, I am graduate student in Mathematics. I have some basic results related to Fibonacci sequence and partition but I am not able to deciding whether it is publishable or not.

Tell me How can I proceed? I want to know how can I write a paper or work under the supervision of a professor? Please help me. Dan Christensen. Can you recommend some book s with rigorous proofs and lots of detail, that I might use as templates for formal proofs?

David J. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email. Blog at WordPress. Ben Eastaugh and Chris Sternal-Johnson.

Subscribe to feed. What's new Updates on my research and expository papers, discussion of open problems, and other maths-related topics. On writing. Somerset Maugham Everyone has to develop their own writing style , based on their own strengths and weaknesses, on the subject matter, on the target audience, and sometimes on the target medium.

One should also invest some effort in both organising and motivating the paper, and in particular in selecting good notation and giving appropriate amounts of detail. But one should not over-optimise the paper. It also assists readability if you factor the paper into smaller pieces, for instance by making plenty of lemmas. To reduce the time needed to write and organise a paper, I recommend writing a rapid prototype first. One should take advantage of the English language , and not just rely purely on mathematical symbols.

Submitting a paper Proofread and double-check your article before submission ; you should be submitting a final draft, not a first draft. Submit to an appropriate journal. On the use of implicit mathematical notational conventions to provide contextual clues when reading. Dick Lipton on an analogy between paper writing and city planning. Like this: Like Loading What is good mathematics? Why global regularity for Navier-Stokes is hard. Top Posts Career advice Does one have to be a genius to do maths? Categories expository tricks 10 guest blog 10 Mathematics math. AC 8 math. AG 41 math. AP math.

AT 17 math. CA math. CO math. CT 7 math. CV 27 math. DG 37 math. DS 78 math. FA 24 math. GM 12 math. GN 21 math. GR 86 math. GT 16 math. HO 10 math. IT 11 math. LO 48 math. MG 43 math. MP 27 math. NA 22 math. NT math. OA 19 math. PR 96 math. QA 5 math. RA 37 math. RT 21 math. SG 4 math. SP 47 math. ST 6 non-technical admin 44 advertising 35 diversions 4 media 12 journals 3 obituary 12 opinion 30 paper book 18 Companion 13 update 19 question polymath 84 talk 65 DLS 20 teaching A — Real analysis 11 B — Real analysis 21 C — Real analysis 6 A — complex analysis 9 C — complex analysis 5 A — analytic prime number theory 16 A — ergodic theory 18 A — Hilbert's fifth problem 12 A — Incompressible fluid equations 5 A — random matrices 14 B — expansion in groups 8 B — Higher order Fourier analysis 9 B — incompressible Euler equations 2 A — probability theory 6 G — poincare conjecture 20 Logic reading seminar 8 travel Assuming self-study: The term Analysis brings books like Rudin to mind.

Hope that helps! Thank you for your opinion, I have been looking for an answer like yours for a while. Tao, I have a question about proofs in mathematics research papers. Is this practice acceptable to the mathematical community? If so, why? Thanks in advance for any insight you or your readers can give me on this topic! This is a wonderful forum! Dear Eno, Id I guess your can publish your paper in the journal of Zeilberger!

Dear prof. Oh my Gosh! Thanks for the update! Dear Prof Tao, I would like you to recomend me a book for studing differential geometry. Thanks a lot for your advices! Dear Prof Tao, Thanks a lot for your answer! Did you know about this quantum calculus subject? Alexandra had a different idea.

- Ignorance Is Bliss!
- Extending the Activity.
- Medea: The Trash TV Show!
- The Simple Secret of Successful Dieting and Weight-Loss That Really Works!;
- Lesbian Passions: X-Rated Lust Volume One (Lesbian Erotica).
- ziwopycaxa.tk: Jean-Pierre Serre eBooks?

So it has to be some dollars and some cents. He explained to the class what he had told me before, that the man had to have five dollars going into the third store in order to have nothing left. Alvin raised a hand, excited. Five plus ten is fifteen, and half of that is seven dollars and fifty cents. Some of the students now nodded in agreement; others were still confused or unsure. I decided that it was time for them to continue working on their own.

Listen to what you should write. The information can help you later, and it can also help me understand how you were thinking. When you figure out the answer, explain in words how you finally got it. And finally, write about whether you think this problem was too easy, just right, or too hard for fifth graders. The students were used to writing titles on their papers. No one else had a question and the students went back to their desks to work. The room was productively noisy, and the students stayed engrossed.

I asked him to explain what he had written, and he did so clearly. As the students worked, I started a T-table on the board, labeling the first column Start and the second column End. I recorded on the table the amounts that the students found as they worked, recording the end amounts as negative numbers when the man was in the hole and positive numbers when the man was ahead. After several entries, the table looked like this:. Several others did, too. I called the class back to the rug for a concluding conversation, and the interchange was lively. The general consensus was that guessing and checking, working backward, acting it out, and looking for a pattern were the most used problem-solving strategies.

Not all of the students had the time to write about whether the problem was too easy, just right, or too hard.

### 129 comments

But our class discussion revealed that most of the students thought that the problem was just right for fifth graders. Some students commented on their papers. Travon wrote: This was hard because there is no very helpful pattern that I know of. Kaisha had a different point of view. She wrote: I think it was pretty easy. In this lesson, the marbles and beans, along with the book Great Estimations by Bruce Goldstone, are used to provide students with an opportunity to explore and apply strategies for estimation. Also, collections of grid paper with different-sized grids, measuring cups of various sizes, a few unifix cubes, and balance scales and masses provide engaging estimation tools for students.

Before class, place one cup of kidney beans in each quart-size sandwich bag, filling enough bags for one bag per pair of students. Place the bags of beans in a large paper bag along with a pint-size and quart-size jar of marbles. I think there are more than twenty-three marbles because I could get about ten marbles in a handful and I know there are more than two handfuls in the jar.

Two tens equal twenty so there must be more than twenty-three. As I read the book, listen carefully for strategies you could use to estimate. Yainid shared the first strategy. Continue reading and recording estimation strategies as students notice them. Following is the list of strategies students may suggest:. Ask students to choose strategies from the list to estimate the number of marbles in the jar.

Record their reasoning on the board as appropriate. Gaby thought making rows of ten would be helpful. I asked her to make a row of ten. I think if we took out another ten marbles the jar would be about one-fourth empty. I used twenty because that would be how many marbles were taken out of the jar for it to be one-fourth empty. I multiplied by four because there are four-fourths in the whole jar. That would be a good estimate of how many marbles were in the jar when it was full.

I asked Kaitlin to try her idea for us by taking out ten more marbles to make a total of twenty marbles removed from the jar. The jar looked like it was more than three-fourths full so Kaitlin took out six more marbles for a total of twenty-six marbles. We agreed that it now looked about three-quarters full or one-fourth empty. Twenty-six plus twenty-six equals fifty-two so there are fifty-two marbles in one-half. Fifty-two and fifty-two equal one hundred four marbles.

I know that four twenty-fives is one hundred. Then I added four more because I changed twenty-six to twenty-five. I have to put back the one. Four times one equals four. One hundred plus four equals one hundred four. Then I counted, 20, 40, 60, I put the eighty in my memory. Then I counted 6, 12, 18, I took the eighty out of my memory and added eighty and twenty-four.

In my class, our initial estimates were , 23, , , 1,,, , , and Our class estimate was Finally, tell the students how many marbles you had counted when you filled the jar. To verify your count and to reinforce the value of using a strategy to make an estimation, count the marbles by placing them into groups of ten. I had counted marbles in the jar. My class counted 11 groups of ten with two extra to figure this out. The class strategy for estimating marbles got them very close to the actual total of marbles.

Next, pull the quart-sized jar of marbles from the bag and hold it up. How can you use what you know about the number of marbles in a pint to help you estimate the number of marbles in a quart? So if we know how many in a pint we can just double that to get a good estimate for how many in a quart. Hold up the estimation tools: grid paper, measuring cups, unifix cubes, and balances and masses. Ask: How might these tools help you carry out the strategies listed on the board?

In my class, the students shared that the grid paper could be used with the box-and-count strategy, the balance and masses could help with the weighing strategy, and the measuring cups could help with thinking about how many tens or hundreds fit in a measuring cup or even a unifix cube if the items were small like popcorn or lentils. Katie had a new idea. She suggested that we could use the small geoboard rubber bands as a way to help with the clumping strategy. The class agreed this was a good idea and geoboard rubber bands were added to the estimation tools. Give partners time to work together on recording their estimates and notes about what tool they used to make their estimate.

When students finish, ask them to record their information on a class chart:. Gather students for a discussion. Focus the discussion on the similarity of the estimates. Each bag had approximately the same number of kidney beans. In most cases, the estimates were similar even though partner pairs used different estimation tools. This usually delights the students. The students in my class concluded that differences in the estimations came from varying sizes of beans, how precisely the beans filled an area of measurement, how carefully people measured and counted, and how the number of beans in the bags varied.

I had already prepared bags of objects: quart-size bags of lima beans, cotton balls, teddy bear counters, macaroni, real pennies, and popcorn, along with snack-size bags of plastic counters, cotton swabs, base ten unit cubes, paper clips, and lentils. After a while, Cathy called the class to attention and asked the students to share what they had noticed. You can include drawings if they will help make your directions clear. Before the students began, Cathy wrote protractor and angle on the overhead for their reference and asked them what other words about angles they might use.

She listed all the words the students suggested: acute, right, obtuse, straight, degrees. Students expressed their thinking in different ways. Jenny and Sara, for example, wrote the following directions for measuring an angle: First, you make an angle. Then you place the bottom line of the angle on the line of the protractor. Then you put the dot on the vertex of the angle. Then you get the arrow and take it up to the number and the number that the line hits is the degree of your angle. Cuong and Cheryl wrote: One rule you must always remember is you must always have the bulls eye on the straight line on the bottom.

If the angle goes to the right you must read the bottom numbers. But if it goes to the left you must read the top numbers. Ron M. The hole is on the bottom of the protractor. What you do is measure and count by the lines on top of the protractor. Will and Pat designed their work as a pamphlet. They titled it The Protractor Manual see Figure 1.

They wrote: There are 2 things a Protractor does for you. It makes new angles. To measure an angle you put the rough side of the protractor down then put the vertex of the angle in the little hole in the middle, shown on pages 3 and 4 and if the angle is acute you use the numbers on the bottom on the right but if it is acute but it is pointing to the left you use the left side and the top shown on pages 3 and 4. To make angles you use the bottom part of the Protractor shown on pages 3 and 4 to make any angle you disire.

To prepare for the lesson, I duplicated for each pair of students the shapes for the activity and a sheet of inch-squared paper. I used yellow paper for the shapes so that there would be contrast when they pasted them on white paper. Also, I made an overhead transparency of each of the sheets I distributed. To begin, I projected an overhead transparency of the inch-squared paper, which was a 9-by-7 grid.

From their study of multiplication, the students knew to multiply 7 by 9 to get the area of 63 square inches. I next placed a 5-byinch index card on the grid, positioning it carefully so its sides were on lines and it covered 40 of the squares completely. I removed the 5-byinch index card, folded it in half the short way, and cut on the fold.

I placed one half on the overhead grid. I then trimmed the 4-byinch card so it was a 4-byinch square. Next I cut the square in half on the diagonal, making two triangles. I placed one of them on the grid. Some students knew immediately that its area had to be 8 square inches. Tracing the triangle on the squared paper and then removing it helped the students see that it was, indeed, 8 square inches. Then two halves make a whole and two more halves make a whole. I then projected a transparency of the shapes the students were going to explore.

I explained what they were to do, also writing the directions on the board:. Students worked in pairs and I circulated, giving help as needed. I suggested to some students that they place a shape on the inch-squared paper and count the squares it covered. If the class had been a regular-length period, I would have collected their work and returned it to them to complete the next day. Then I called the students together to discuss what they had learned.

Nicholas explained what he learned. That was cool. Andrew and Hiroshi arranged their shapes from least area to greatest, as instructed. Roberto and Laura shared the writing of their discoveries.

- Classroom Lessons.
- Robert F. Tichy.
- Lectures on N_X(p).
- Extreme Value Methods with Applications to Finance;

Figure 3. This focus on one-half helps establish it as an important and useful benchmark. The lesson also provides practice with mental computation of whole numbers as students compare numerators and denominators. I used this activity when I had some time left at the end of a periodor as a warmup at the beginning of class. Students who answered also had to explain their thinking. I wrote on the board:. Leslie asked to come to the board. She drew a circle and divided it into three equal-size wedges.

So one-half is one-third plus some more. One day, I chose one and one-quarter for the fraction. Again, there were lots of volunteers. Even for an obvious solution, students may think in different ways. The students seemed to like stretching their thinking to decide if fractions like these were more or less than one-half. I was a little less than halfway through reading K. If each slice sells for one gold coin, and if I can sell 10 pizzas a day. I gave students some quiet think time before asking them to turn to their partners and share their thoughts.

After partners had discussed their ideas, I called them back to attention. Then I finished reading the book. The students enjoyed the colorful illustrations by Giuliano Ferri and were happy when Chris Croc and Ben Bear solved their problem of being hungry by buying food from one another, passing the one gold coin back and forth until there were no pizzas or cakes left. I waited a few seconds and then called on Daniel. Next, I asked the students to estimate which number they thought would be closest to the exact answer.

With a show of hands, we discovered that 10 students thought 2, coins was closest, 10 students thought 30, was closest, 4 estimated 1,,, and no one thought coins could be possible. I then gave students some time to talk with their partners about the reasonableness of the estimates. After a minute, I asked for their ideas. I gave the students a few seconds to think, then called on Jesus. Then I did sixty plus sixty is one hundred and twenty. This time, everyone thought that the exact answer was closest to 30,, except for Henry, who stuck with 2, After Anton explained how he figured the answer using the standard algorithm, we checked his result with Amber.

When Anton and Amber reported the answer—29,—I ended the lesson by asking the students which number was closer to the exact answer: 30, or , Three hundred fifty thousand is way too big! Teaching multiplication of fractions is, in one way, simple—the rule of multiplying across the numerators and the denominators is easy for teachers to teach and for students to learn. However, teaching so that students also develop understanding is more demanding, and Marilyn Burns tackles this in her new book Teaching Arithmetic: Lessons for Multiplying and Dividing Fractions, Grades 5—6 Math Solutions Publications, In the following excerpt from Chapter 2, Marilyn builds on what students know about multiplying whole numbers to begin developing understanding of what occurs when we multiply fractions.

I planned to use these statements as a base for helping the students think about multiplying fractions. To begin, I pointed to the first statement:. Multiplication is the same as repeated addition when you add the same number again and again. After a few minutes, I called on Juanita. I did one-half plus one-half, like that, six times.

I think the answer is three. The students nodded their agreement, and I wrote OK next to the first statement. The students were familiar with using rectangles for whole number multiplication. I split the rectangle as Kayla suggested, then erased the 1 and replaced it with 1 written twice.

How many squares are there in the unshaded rectangle? Does this still give an answer of three? Three is still the answer. I decided to show the students a way to think about representing the problem with a rectangle. It shows that one times one is one. Now watch as I draw a rectangle inside this one with sides that each measure one-half.

Saul nodded. I planned to develop this idea, and I used the next statement to do so. You can reverse the order of the factors and the product stays the same. What about if we think about the problem as one-half times six? After a few moments, I called on Brendan. You could break the six into twos, and then you do two times one-half three times.

Two times one-half is one. One plus one plus one is three. So it works. Half of four is two and half of two is one and two plus one is three. It works. When you multiply two numbers, the product is larger than the factors unless one of the factors is zero or one. When you multiply two numbers, the product is larger than the factors unless one of the factors is zero or one or a fraction. I thought about how to respond. I posed a problem that had a fraction as one of the factors for which the answer was greater than both of the factors.

## Robert F. Tichy's Homepage

Think about this problem—six times three-halves. After a couple of minutes, I called on Craig. So any number that is zero or one or in between makes an answer that is smaller than the factors. When you multiply two numbers, the product is larger than the factors unless one of the factors is zero or one or a fraction smaller than one. This was just the first lesson I planned to teach about multiplying fractions. After the class had experience comparing two fractions by using one-half as a benchmark, I wrote on the board:.

I waited until every student had raised a hand. While the question was trivial for most students, I planned to build on their understanding and have a class discussion about different ways to determine that a fraction is equivalent to one-half. Before calling on any students to respond, I gave a direction. If you have a different way, then raise your hand again. I knew, or thought that I knew, what Jake was stating. But his response gave me the chance to push for more clarity from him. The symbols are like shortcuts. I turned to the board, explaining as I wrote. Who can explain why this makes sense?

The idea of algebraic variables was new for these students and I tried to explain. If it makes sense to you, it sure will save us some writing energy. He came up and used the notation for whole number division to record. He wrote:. I then returned to the discussion of other ways to see if a fraction were equivalent to one-half. I called on Gena, who now seemed more confident.

George had another idea. I started around the room having students tell me fractions, and I recorded their suggestions on the board. Ali took another direction when it was her turn. Representing your ideas algebraically as I did on the chart is a handy way to refer to many, many fractions. I then gave the class an assignment. On your paper, explain your reasoning for each. See Figures 1 and 2 on the following pages for examples of how students worked on this assignment. Brett used a chart to present the answers.

A few hands sprung up. Raise your hand when you have a fraction in mind. I called on Josh. I called on other students and recorded the fractions as they offered them. I reminded him that everyone could change his mind at any time in math class, as long as he had a reason. Several other students raised their hands. Sam raised a hand. I continued by asking the class for fractions for the other two columns, each time having the student explain her reasoning for the fraction she identified.

Then I repeated the activity using three-eighths and then one-fourth as starting fractions. I continued the lesson until only ten minutes remained in the period. Then I stopped to give the homework assignment. To avoid confusion when they were at home, I duplicated the directions for the homework and distributed them to the class:.

Above the columns draw boxes for the numerator and denominator of the starting fraction. To find the starting fraction, roll a die twice. Use the smaller number for the numerator and the larger number for the denominator. If both numbers you roll are the same, roll again so that the numerator and denominator of your starting fraction are different. Write at least five fractions in each column. The numerator and denominator in each fraction you write must be greater than the numerator and denominator in the starting fraction. Choose one fraction from each column and explain how you know it belongs there.

I explained to the students what they were to do. I emphasized the fifth rule. The next day, I had students report about what they had learned from the assignment. See Figures 1—3. No one reported having this problem. I ruled columns and started the activity.

### Navigation menu

We continued for about fifteen minutes. I then gave each student a Post-it Note to use for his or her initials. Then I asked Maryann to go up to the board and place hers on the Venn diagram. I think I need another Post-it. Aaron had a thought. I nodded my agreement. Everyone thought for a moment. Some children knew exactly where to place their Post-it Notes, and others were hesitant. But animated conversation helped each student decide. I allowed a minute for silent inspection of the results.

We then tallied the information and I recorded it on the board:. However, I wanted the students to reason mentally and talk about their thinking. To help, I wrote on the board what Jason had already said:. Is it more than 50 percent or less than 50 percent? So I needed three 20 percents. Isaac joined in. Sixty percent are. Next I distributed newsprint and markers and asked the students to draw Venn diagrams with three intersecting circles, label each, and guess how many of the class would fall into each category.

At the end of class, I collected the papers. There was a great deal of collaboration, and the class discussion that followed was rich and productive. In order to introduce my students to problems that involve division with fractions, I use problem situations that draw on familiar contexts. I keep the focus of their work on making sense of the situation and explaining their strategies and solutions. Will there be enough for each person in the class?

If not, how much more will I need to buy? After students shared their answers and the methods they used, I gave them other problems to solve, using other amounts for the sizes of the large and small bags. This helped students connect the original situation to the correct mathematical representation. As before, the students were asked to explain the methods they used. The problems they wrote helped me assess their ability to connect an equation involving division with fractions to a real-world context.

Tom was buying wood for his woodshop class. How much wood is left over? Each candy bar has five equal parts. Betty went to the local fabric store for fabric to make curtains. How much fabric is left over? The students shared their word problems, resulting in some very interesting discussions. After hearing the problem about Betty buying fabric for curtains, for example, I pointed out that if I went to buy fabric to make curtains, I would measure and know ahead of time how much fabric to buy and how many curtains I would be making.

Charles makes Pinewood Derby kits from 8-foot stock. How many 8-foot pieces of stock are required to fill an order for kits? After that, no one knew what to do next. I encouraged them to make a model. Then we measured and marked with masking tape 8 feet or 96 inches on the classroom floor.

At this point, the students were off and running. Here is how one student expressed her thinking in writing. Danielle started reading One Riddle, One Answer aloud. Danielle stopped reading at the part where Aziza proposes that she write a number riddle. The princess tells her father that she would prefer to marry the person clever enough to answer her riddle. The sultan agrees. You can use scratch paper to jot down your thoughts.

Then, whisper your ideas about the answer to a partner. Danielle first gave students time to think and take notes. Then, as the students shared their ideas with their partners, she circulated, listening to their guesses. Some thought the answer was one; a few thought that it was zero; many were completely stumped. She stopped at this point and addressed the class. Danielle continued to read. When Ahmed, the farmer in the story, guessed that the answer to the riddle was the number one, Danielle again checked with the students to see if it worked with all the clues, and it did.

Danielle was taken aback. And it works for any number you multiply it by. Amanda nodded and continued with her argument. And when you count, zero comes before one on the number line! Getting students to argue passionately about their ideas in math class is often difficult to achieve. When the short debate was over, Danielle acknowledged that there could be more than one correct answer to the riddle.

Then she finished reading the story, including a section at the back of the book where the author explains how the number one works for each clue. First you have to think of a number to write about. Then you have to write some clues. I want you to practice by brainstorming some possible clues for the numberten.

## Research Notes in Mathematics

Danielle gave the students about five minutes to work in groups. She then called the class together and wrote some of the clues the students came up with on the board:. Then Danielle gave feedback on the clues. When the class was finished brainstorming clues for the number ten, Danielle gave the homework assignment of writing one riddle sentence for the number one-half. The next day, Danielle asked for volunteers to share their riddles.

Some students revealed some very sophisticated thinking about one-half, while others exposed misconceptions. Soon, the class began to hum with conversations as students shared their ideas with one another. As Danielle circulated, she encouraged students to think of subtle rather than obvious clues. After about thirty minutes, she pulled the class back together. In this lesson, excerpted from A Month-to-Month Guide: Fourth-Grade Math Math Solutions, , Lainie Schuster has her fourth graders start the school year with an investigation that offers them the opportunity to work in pairs to collect, represent, and analyze data.

In this book, Martha, a dog who is able to talk as a result of eating a bowl of alphabet soup each day, finds herself in a pickle when the owner of the soup company reduces the number of letters in each can of alphabet soup. After listening to the story, students pair up to investigate the frequency with which each letter in the English alphabet is used. They gather and organize their information, then summarize in writing what they notice about the letter frequencies.

They create posters to show their findings and participate in whole-class discussions to share their thinking. To extend the activity, Lainie gives students pasta letters from which they form as many words as possible. I began the lesson with a read-aloud of Martha Blah Blah. Following the reading, we had a class discussion about letter distribution in words. Why might that be? Do you think consonants are used more than vowels or vowels more than consonants?

How could we collect the data? How could we chart the collected data? If Granny Flo, the owner of the soup company, was determined to eliminate the least-used letters from her soup, how could we help her make an informed decision at least for talking dogs? I directed the children to discuss these questions with their neighbor and come up with ideas for carrying out a letter-frequency investigation.

As the children talked, I circulated around the room, listening and handing out a record sheet containing writing prompts. See end of lesson for blackline master. When I felt everyone was ready, I called the class back together so students could share their ideas and discuss the writing assignment. In response, I asked the children if it was necessary to write down all the letters in the alphabet or if they could write down just the ones that were in their reading selection.

This led to a rich discussion about the importance of being able to quickly assess the holes in the collected data. Then I described the task. I explained that, working in pairs, they were going to determine how often each letter of the alphabet was used. I gave each pair a piece of newsprint on which to record all their work. I told them to post the data they collected in whatever format they were comfortable with and to leave room on their newsprint for writing summary statements and posting the results of another task, which I would give them the next day.

I also told them that they would have an opportunity to decorate their posters at the end of the investigation. Once we agreed on what they were being asked to do, the children set off to collect and represent their data. Many chose a paragraph from a book they were presently reading. Others picked one of the books in the classroom book display.

I had asked that the paragraphs be relatively short because it was the beginning of the school year, and I was more concerned about the manageability of the task than the length of the paragraph. Before all the students were finished collecting their data, I asked for their attention, and we discussed the written part of the task. I asked the children to write two summary statements on their sheet of newsprint.

I explained that a summary statement should explain what they noticed about the data they had collected. I had pairs work together to create summary statements but asked them to individually complete the writing prompts before discussing and comparing their opinions and answers. As the children went back to work, I circulated around the room to offer assistance. Talking about the mathematics is one thing—writing about it can be quite another. Sometimes it was necessary to remind students to refer to their collected data as they wrote.

Writing Prompts for Martha Blah-Blah We would suggest that Granny Flo take out the following 7 letters: Without these 7 letters, it would be difficult for Martha to say the following words: Without these 7 letters, it would be easy for Martha to say : Why? Since math time was almost over, I continued the lesson the next day. After students retrieved their newsprint, we began a class discussion of their findings and decisions. Most agreed that more consonants should be removed than vowels.

I concluded this investigation with a word search made from a cupful of alphabet pasta. I gave each pair of students a paper cup containing twenty-five pieces of uncooked alphabet pasta. I then gave the following directions:. They were especially curious about the frequency findings, the conclusions that were drawn, and the words that were made from the pasta letters. This lesson gives a four-step plan, including a 3—5 sample task and corresponding authentic student responses.

Open-ended tasks allow students to control some of the difficulty level themselves. In the example task below, students may limit their consideration to only a few shapes or by focusing exclusively on two-dimensional shapes. Similarly, students may choose to use drawings, charts, or diagrams to communicate their ideas, or they may rely more on prose. After a brief discussion about the task expectations, students are normally eager to begin their task.

Some students might think for a minute or so before beginning to record their ideas, but most begin immediately. Following are examples, including authentic student work, of how students responded to the above task. Some students used shape templates, while others preferred drawing freehand. Most students began by drawing a shape on their paper and then writing some words above or below it.

A few students began by writing an idea or the name of a shape, which they then illustrated. She classified shapes by their number of sides and provided the correct name for three-, four-, five-, six-, and eight-sided shapes. Though she did not name the shapes that she drew within her quadrilateral category, she did include a trapezoid, a square, and a parallelogram. She provided one example of a triangle, a pentagon, and a hexagon. Fourth grader Tai included references to concave shapes and polygons, and made connections between two-dimensional and three-dimensional figures.

He also introduced pyramids, right angles, and the term parallel. He was excited as he worked. He recorded one idea and then his eyes lit up as he thought of another. As these ideas were not necessarily related, he often recorded a thought and then drew a ring around it to separate it from his other recordings. Rafael was a strong visual learner. He often made diagrams to summarize the events in a story and his language often reflected his visual preference. Just that morning the teacher was listening to another student explain to Rafael why they should play soccer that afternoon instead of going on a bike ride.

Note his two representations that connect metric and English units along with the geoboard, compass rose, and protractor. Discover what each student chose to include; perhaps it is what he knows best, or what she believes is most important, or what he finds most interesting. Also note what concepts students did not provide evidence for, or for which the evidence is. Share your findings with other teachers. Look at the similarities and differences across grade levels.

Following are observations and plans that teachers made upon reviewing responses to the sample task. Teachers were amazed at the differences across the grade levels. Sample 1 , the sides within her triangle, pentagon, and hexagon had approximately the same length and the figures were drawn with a base parallel to the bottom of the page. Such orientations are common; in fact, many students do not identify some of these figures when their sides are not congruent or when they are not placed in traditional positions.

Perhaps they could repeat the assignment later in the year, and the next year they might use it as both a pre- and post-assessment. Playing the Factor Game provides an engaging format in which students can become familiar with the factors of numbers from two to thirty by playing a two-person board game. To play Factor Game, each player chooses a number while the other player finds the sum of the available factors of that number.

My method of introducing the Factor Game to my class is not standard. Many texts suggest that you discuss the term factor and how it relates to this game. The title of the game board was available to the children, but they paid no attention to it. I told the children that I would go first.

I told them that I was choosing 29 and earned those 29 points. I crossed 29 off on the game board. The, I told them that they would receive 1 point as the result of my choice and crossed off 1. Now it was their turn to choose. The class worked together to choose a number—and invariably chose After all, it was the largest number on the board! I crossed off 30 and posted it on their side of the T-chart while keeping a running total. The class now had 31 points. I deliberately thought out loud as I calculated my points. That gives me a total of forty-one points!

As I was thinking out loud, the children were buzzing about how I was earning my points—and how they were losing theirs! Animated mathematical conversation erupted. Some children were aware of some of the rules of the game at this point. They realized that when one player chose a number, the other player earned points related to the numbers multiplied together to get that particular number.

The language of factors, multiples, and products was not yet being used, but that was fine at this point in the game. When I first began to play the game in this manner, I was astounded at the inefficiency of the discussion accompanying the game without the availability of this terminology. What a great lesson to learn about the power of mathematical language! As we played one or two more rounds, I began to share a few of the rules—the first being that when you choose a number, the other player must be able to earn points.

If the other player can earn no points from your choice, you lose your turn. The language of prime and composite had not yet been introduced, but the children quickly learned that they needed to stay away from prime numbers after that first move because they could not earn any points on the resulting move. After several rounds, I introduced language that would be helpful as the children discussed potential moves. As words were discussed, I wrote them on the board for accessibility. The class was familiar with the term product, but not at ease with its application in their casual mathematical conversations.

Walk-by interventions, as I call them, are crucial early in the school year. The game title identified the new term factor and its meaning in reference to the game being played. You can also introduce multiple, but be prepared for its misuse. Because of the newness of the language, many fifth graders will interchange factor and multiple.

They will often use factor correctly in isolation, but run into difficulty when asked to construct a sentence with both factor and multiple. An entire class period was devoted to this introduction of the Factor Game. I was delighted with the mathematical observations, insights, and discussions that occurred within this format. The language of factors, multiples, and products was immediately meaningful because it supported the children as they discussed and analyzed their number choices. They also learned an important lesson about the importance of implementing appropriate mathematical language.

Once the initial games are played, the children can set off with partners to play a game or two on their own.

As the children play the game, circulate through the room, making note of interesting strategies. You may also want to note who continues to struggle with the recall of their basic multiplication facts. A lack of fluency with the multiplication tables can make playing this game difficult and tedious.

As you move around the room, you may wish to visit some of the pairs and ask them the following questions:. Pulling the class together for a processing session is important and necessary after the children have had the opportunity to play several rounds of the game. Processing the game gives mathematical meaning to the activity. The children need to realize that although games can be great fun, as this one certainly is, good mathematical games also have purpose. Crafting, asking, and answering good questions can further the mathematical understanding of just about any activity.

Good questions can set the stage for meaningful classroom discussion and learning. Students are no longer passive receivers of information when they asked questions that deepen and challenge their mathematical understandings and convictions. Good questions. Questions such as those that follow can help to scaffold and articulate new understandings that have come about as a result of playing the Factor Game. Processing questions in a whole-class format also gives you the opportunity to implement talk moves.

You can help to establish respectful discourse by asking for agreement or disagreement. Revoicing can emphasize important mathematics, insights, or strategies. You can have follow-up lessons that draw upon the understandings constructed from the Factor Game. My class explores perfect, abundant, and deficient numbers as well because of the connections they can make to number choices on the Factor Game game board.

Exploring and applying divisibility rules also now have a place and purpose in the curriculum. Being mathematically proficient goes far beyond being able to compute accurately and proficiently. It involves understanding and applying various relationships, properties, and procedures associated with number concepts Math Matters, Chapin and Johnson The Factor Game and the lessons that it subsequently supports can do just that. The Factor Game Game Board for Materials A collection of coins dated before , placed in a clear plastic bag Overview of Lesson Marilyn is always on the lookout for ways to provide students experience with computing mentally.

Her colleague Jane Crawford gave her the idea of presenting older students with the problem of figuring out the ages of coins. To prepare for the lesson, Marilyn collected loose change for several days, choosing coins that were made before Marilyn planned to ask the students to figure in their heads rather than use paper and pencil. Her goal was for them to focus on making sense of the numbers and to discuss the different strategies they used for figuring. Show the class the plastic bag of coins. List on the board how many of each coin are in the bag.

For example, my bag contained the following:. Record their answers on the board as they report. In my class, as Dylan reported, I wrote:. Now tell the students that you have another problem for them to solve mentally. Choose one of the pennies, and show them where, for example, appears on it. Ask one student in each pair to raise his or her hand and explain how they figured out their answer. When I taught this lesson to a class in the year , students reported several different methods of figuring that the penny was twenty-two years old.

Then I subtracted eight from thirty to get twenty-two. I still had two more years because nineteen seventy-eight is two years from nineteen eighty. So twenty plus two is twenty-two. Pose the same question for another coin, again recording while students report. I posed the same question for a nickel. Then have students each take one of the coins and, working individually, calculate its age. In a classroom in , after explaining how she knew that her penny was thirteen years old, Rachel showed her calculations.

This activity is good to repeat from time to time. Before class began, I drew on the board six 4-by-4 square grids like the ones on the worksheet the students would be using. Talk at your tables and see what you can come up with. The noise level in the class rose as students began to talk. Some got out pencils and paper to sketch.

After a few moments, I called the class back to attention. Lots of hands were raised. I called on Andrew. Can I come up and draw it? I turned to the class. She came up with her paper and drew on the last grid. Everyone had a thumb up. I then asked the class to look again at the brownies cut with diagonal lines, as Katia and Sophia had suggested. I wanted the students to think about counting and combining halves of squares.

A buzz broke out in the class and I waited a moment before asking the students for their attention. Then I called on Claudia. Carolyn was satisfied and there were no other questions. I then told the students what they were to do next. Look for ways to divide each brownie on the worksheet in half in a different way.

For each brownie you divide, be sure that you can explain how you know that the two pieces really are halves. The students got to work. As I circulated, I noticed that some students relied on counting squares before drawing while others drew, then counted, and made corrections if necessary. As they worked, I erased the brownies we had divided on the board and drew six blank grids for a class discussion. When about ten minutes remained in the period, I called the class to attention. No one had completed the entire page, but they had done enough to make me feel confident that they understood the assignment.