Diane said:

“Twice every year, in the week that teachers and parents are meeting for conferences, assistants and specialists get together to plan a special week for the young people — immersed in a study or topic outside of the curriculum plans for that year. The lead teachers envy them that opportunity at times…”

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Jeri describes what our students did during the week this way:

Tammy, Kate and I combined the two 5/6 classes in a large group and we explored the theme of “Peace and Inclusion on Purpose.” The week included a range of activities designed to help the kids better understand the roots of conflict, cooperation, perspective, mediation and peace. We used stories, games, literature, building activities, movement, role playing, art and reflection to connect the kids to these concepts.

“Ours is not to reason why — just invert and multiply.”

Those of you who recall your school math instruction for dividing by a fraction probably were taught the method described above and probably were given no rationale for it. Just learn it and apply it.  What if you forget which fraction to “flip”? What if you forget something else about the process? Can you come up with another strategy and get to the right answer? Probably not. At least, that was my school experience. But at Miquon, our instructional goals include conceptual understanding as well as process mastery. So here is a close look at what a couple of our small groups have been doing recently with division by fractions. It’s an example of our typical approach to computation processes.

We started with a review of the meanings of division and multiplication in order to consolidate prior learning and sort out any misconceptions. For example, many students create a false rule in the younger grades when they have been doing multiplication and division only with whole numbers: Multiplication makes things bigger and division makes things smaller. As we work with fractions, they soon realize that multiplying a number by a number that is less than 1 gives a product that is less than the number being multiplied, and a whole number divided by a number that is less than one will yield a quotient that is greater than the number being divided. For example, 6 x 2/3 = 4, and 6 ÷ 2/3 = 9.

We then moved on to dividing whole numbers by unit fractions. How many thirds are in 15? Always there are students who will quickly say “5.” When that happens, we ask, “Then how many threes are in 15?”  Oh. A sudden insight and an embarrassed chuckle. After a bit of thought, students arrive at 45 thirds in 15, which is the correct answer. We go through many more examples: fourths in 12 (48), halves in 20 (40), fifths in 10 (50), eighths in 6 (48), and so on. Soon students start offering a rule: You just multiply by the denominator — for thirds it’s times-three, for fifths it’s times-five — because that’s how many parts it takes to make one whole. This is thinking that helps to make sense out of that invert-and-multiply algorithm. Math should always make sense. There are no “tricks.” But we don’t start with invert-and-multiply.

Method #1: Manipulatives and visual representations are important to this and most other math instruction. We use fraction sticks and Cuisenaire rods, make diagrams on paper, and use number lines. Let’s set up a sample problem to solve with this and the methods that follow:  3/4 divided by 1/3. We discuss whether the quotient should be greater or less than one.We are dividing a greater amount by a lesser amount, so we should get a quotient that is greater than one. This is an important but simple way of checking our answer for reasonableness.

For the first problem, we’ll use some of the fraction sticks shown at left. (We could just as easily do jumps on a number line or draw a rectangle on graph paper.) We need three of the 1/4 pieces and some of the 1/3 pieces. We line up the fourths end-to-end. Next to them, we line up the thirds the same way. How many thirds does it take to equal the length of the three-fourths train? Two of the third-size pieces fit along with part of another. (It’s a happy coincidence that the image shown here actually illustrates that.) So the answer is greater than 2. But what about that partial third? How much of it fits the three-fourths train? A little investigation with other pieces reveals the part — it’s a fourth of the partial-third piece. So 3/4 ÷  2/3 =2 and 1/4.

Invariably, at least one student says something like But I thought we were dividing by thirds. Isn’t it 2 and one third? This is a pivotal question. Fractions refer to parts of a whole. What is our whole? It’s what we are dividing by. We are dividing by one third. So the quotient is 2 complete thirds plus one-fourth of another third. It’s hard to over-emphasize how important this understanding is. It’s what makes computation with fractions so difficult. The whole is always there but often just implied. Technically, the one-fourth in that quotient is a fourth of a third of a whole. There are really two different “wholes” — the one that defines the third and the one that defines the fourth. If we don’t use manipulatives and visual models, if we just teach a series of computational steps, this may never be questioned and understood. It may take many exposures to this kind of task many times through fifth and sixth grade before it doesn’t wobble into confusion. So we do it many times and in many different contexts.

Method #2: Students learned how to change two or more fractions to equivalent ones that had a common denominator when they were learning to add, subtract, and compare them. It may not be secure learning yet, but it’s a familiar process. So let’s do that with our sample problem. We can transform 3/4 divided by 1/3 into 9/12 divided by 4/12. At this point, the fact that we are working with twelfths is unimportant. We have a set of 9 things, and we want to break it into groups of 4 things. We will end up with 2 complete groups of 4 and 1/4 of another group of 4.

Method #3: When we multiply fractions, we multiply across. That is, the numerators are multiplied to tell us how many pieces we have. The denominators are multiplied to tell us what has changed about the size of the pieces. So why can’t we divide across? Isn’t division the opposite of multiplication? It is, and we can. It’s just that some problems are easier to do this way than others. Let’s set aside our example problem for a moment and look at one that is easier to understand. How about 4/9 divided by 2/3. Two goes into four 2 times, so the numerator is 2. Three goes into nine three times, so the denominator is 3. The quotient is 2/3. It works. (We can check it with multiplication: 2/3 times 2/3 is 4/9.)  Now let’s look at our sample problem: 3/4 divided by 1/3. One goes into 3 three times, so our numerator is 3. Three goes into 4 one and one-third times, so our denominator is one and one third. That’s mathematically valid, but it’s also needlessly complicated. To simplify it, we’ll multiply the numerator and denominator by 3 to get an equivalent fraction and wipe out that inconvenient fraction (thirds) in the denominator. Here a student says, “Why are we multiplying by three?” So we go back to the number of parts it takes to make a whole — three thirds, five fifths, eight eighths — and the student constructs a new rule for herself. We can simplify a stacked fraction by multiplication, just as we simplify other fractions by dividing by a common factor. Back to the problem with the awkward-looking quotient. Three times the numerator is 9.  Three times the denominator is 4. And nine fourths resolves to 2 and 1/4. Same answer yet again.

Method #4: Yes, it’s time for invert-and-multiply. We have explored division by a fraction through three different methods that all yield the same answer, and we have had a number-sense discussion that has led students to come up with a rule for dividing whole numbers by unit fractions. So it’s time to take on the method most parents know (but most can’t justify or explain). How many thirds are there in 6? Eighteen. Yes —  so how many groups of two-thirds are there in 6? A few students suggest 36. Others say 9. Which is it? Out come the fraction sticks again. Oh. It’s nine. Why is it less? Because two thirds are a bigger piece than one third. So let’s keep working on that rule. Eventually, we have one that works. To divide by one third is the same as multiplying by three. To divide by two thirds means multiplying by three and then cutting it in half. So one third goes into 6 eighteen times, and two thirds goes into 6 nine times.

Can we come up with an algorithm? Let’s learn about reciprocals. The word reciprocal describes a relationship, and that’s not just in math class. (We take a moment to talk about reciprocating when someone gives you a compliment or a birthday present, and we also find out how a reciprocating saw works. Then back to math.) Two numbers are reciprocals if, when you multiply them by each other, the product is 1.  So 3 times 1/3 = 1.  Five fourths times four fifths equals one. Try out a lot of fractions like those on your calculator. Got a rule for fractions? Yes — just turn it upside down. Can we write any number as a fraction? Yes. Seven is just 7 over 1. Three and a half is 7 over 2. So the reciprocal of three and a half is 2 over 7. One way to divide by a fraction is to multiply by its reciprocal. That is, we can use the opposite operation (multiplication instead of division) and the opposite fraction (3/2 instead of 2/3). What we are dividing cannot and does not change. But the operation and the divisor can. So we can solve our sample problem this way, too. We will rewrite  3/4 divided by 1/3 into 3/4 times 3/1. Three times three is 9. That’s the numerator. Four times one is 4. That’s the denominator. And we already know that 9/4 = 2 and 1/4 because we’ve solved this problem three other ways.

A side note:  Every fraction is a division statement. Any division problem can be expressed as a fraction. Sometimes we can solve the problem just by writing it in that format. The numerator is what is being divided. The fraction bar is the division symbol. The denominator is the divisor.  What is 11 divided by 17? Write 11 as the numerator, 17 as the denominator . . . and you have the answer: 11/17. How many times does 2 go into 3? Write it with 3 as the numerator and 2 as the denominator. You have 3/2, which simplifies to 1 and 1/2. One of the standard symbols for division is this one: ÷. It’s actually a picture of a fraction with a dot for the numerator and the denominator. Another symbol for division is this: /. That’s a fraction bar, the line that separates the numerator from the denominator. Every fraction is a division statement.

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How long does it take to teach all this? A lot of time, and it’s not accomplished all at once. The stronger a student’s background is, the less time it takes, but we can get everyone there.

We have several important learning goals that we want students to embrace:

1) Answers should be reasonable and should be checked for reasonableness.

2) Answers can and should be justified in conversation with peers.

3) Problems should be solvable in both visual and computational ways.

4) Mathematics work should be based on number sense and conceptual understanding, not on memory of rote processes.

5) Mastery of number facts makes many things easier — estimation and computation, especially.

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So we hope you have gained some understanding through this detailed example of what we do in math classes all year long, for every topic. And we want to remind you that it’s quite all right for students’ homework to come back with mistakes and questions as well as right answers. All of this tells your child and the teachers what does and does not need to be done in math class the next day. Homework is a tool for planning, not for judgment.

Wildlife Research: As our wagon trains are proceeding toward Hacker’s Valley, the families have a lot of chances to observe the environment through which they’re passing. Most people traveled the vast distance on foot. Only the very old or very young were likely to ride in the wagon on the journey. Even the wagon “driver” often led the wagon, especially if it was being pulled by docile oxen. Oxen typically walk at a rate of 2 or 3 miles per hour, and they would do best if the wagon didn’t have the added weight of unnecessary passengers. As a result of this slow pace,many kinds of animals and plant life would all be easy to observe. In order to enable our students to learn more about all this, we divided the group into 3 small teams and asked them to research and prepare a Google presentation on one of three categories of plains and prairie life: birds, reptiles and amphibians, and mammals. Then we stepped back to let them figure out the details.

It was impressive to watch. It’s not easy for 4 or 5 people to collaborate on a single presentation. All three teams had some organizational rough spots but solved their problems with good communication and compromise. They realized that there were too many animals in their assigned category to do much about all of them, and they also decided (rightly) that a comprehensive list would not be very interesting. So they made some selections that represented a good variety, found attractive images and, in some cases, video clips, and worked on composing their slides.

I did some coaching and advising about what makes a good presentation slide. Text should be large, brief, and note-like, perhaps using bullets. People generally don’t like having an information-dense slide read to them. The words on the slides should be key ideas that are expanded by the presenter’s talk. Images should be large and attention-grabbing. Animations are fun, but too many distract the audience from the content as words and pictures twirl and spin and change size. Practice is important. You should know what you are going to say and be ready to deliver it with confidence. Finally, since this is a kind of publication, it needs to be edited carefully for spelling, punctuation, and wording. One bit of information that was new to many of the group was that animal names are not capitalized, as a rule. Only the ones whose names come from proper nouns should have capitals on some of the words, such as Rocky Mountain bighorn sheep or Woodhouse’s toad. They nodded and went back to take off the extraneous caps.

The resulting presentations were wonderful. Students had good information, attractive slides, lengthy source lists, and some of the most accurate editing we’ve ever seen. We encourage you to ask your child to show you his/her team’s project. It’s something to be proud of, not only as a product but as a collaborative process.

Book Groups: Our 3 book groups met for the first time on Tuesday. There was some confusion about what the full assignment was. Everyone had read the section of their book that was due, but many did not understand what the extent of the written part of the work was, even though we had gone over it both verbally and in writing multiple times. We are hoping for a better result when we meet again on Nov. 18th. Everything they need to do is available in their homework binder and also on the homework blog. A little parental reminding on this would be appreciated. The absence of other homework during Conference Week should make it easy for students to be ready for the next meeting, even though the reading assignment is long.

Our Lunch Sale: We had a perfect order and delivery this time, as well as a very enthusiastic response to our new menu. This was an interesting lesson in good business practices and in re-evaluating your product. We started with a “taste test” of an assortment of sandwiches from Subway last week. On the whole, our group liked the new rolls and also the new fillings. One of the ongoing problems we have had is soggy rolls — the shredded lettuce in every sandwich contained a lot of water (along with very little nutritional value). But Subway offers fresh baby spinach leaves instead, and they are dry. So we found the things we liked (not the all-veggie sandwich, which was sort of a salad in a roll and which a couple of kids said they thought “the teachers might like”) and decided that the teriyaki chicken would be a good addition to our existing choices, all of which would be reconfigured with spinach instead of lettuce and no onions on anything. The post-lunchtime feedback from several teachers of younger students was just about entirely positive. Most kids actually ate the spinach instead of discarding it, as they had been doing with the lettuce. Even the few students who preferred the rolls that Giant used said that they liked the sandwich filling better. And about that totally accurate order and delivery? That’s a wonderful thing. Details are difficult for any of us to manage. To record every lunch order correctly and then to pack exactly what was ordered is a heavy demand on our students’ ability to focus and work with care. We had no phone calls this past Friday, no requests for a missing item, no need to trade a chocolate pudding for a vanilla. They were rightly proud of themselves.

Looks like a theme — reading in 3 contexts . . .

We finally found a time in which we could get together with our first grade reading buddies. In truth, we won’t just read together. We’ll play games, do some math activities, perhaps have a block-building day or two, and more. But this first meeting was all about books. Some of our class brought in favorite titles from home, I lent out three of my beloved childhood books, and there were stacks and stacks of interesting reading materials in Rich and Elisa’s classroom. It was a delightful way to spend the first part of Thursday morning. Before we went to their classroom, we had a discussion about reading with younger children. Our students shared lots of advice — let them interrupt the story with comments and questions, be sure they can see the text and the pictures while you read to them, let them read to you, be sure to read with expression, and make sure they are interested in the story or choose a different one.

We started book groups this week. Students were given lots of titles to choose from, and we used their preferences to organize 3 groups: one reading Un Lun Dun with Amy (our librarian), one reading Truckers with Jeri, and the third group reading Because of Mr. Terupt with me.  The groups will meet weekly to discuss the story and characters, make predictions, and use the “Notice and Note” strategies to think about the stories more deeply. This approach comes from a book of the same name. Essentially, it points out 6 different “signposts” that writers often employ to engage the reader more deeply. For example, a signpost called “An Aha! Moment” is one in which a character has a sudden insight about something. This should prompt the reader to stop and think about that realization and consider how this might change things. We hope that focusing on our books through this lens will enable students to slow down a bit and do more thinking as they read, especially those who tend to zoom along on the action of the plot. A copy of the full set of signposts is here: The-Notice-and-Note-Signposts . We introduced this metacognitive element by sharing that page with students. Then we read and discussed “Thank you, Ma’am” — a short story by Langston Hughes that contains many contrasts and contradictions (one of the signposts). Finally we asked students to think back over stories and films they enjoyed and try to come up with an example of each of the signposts from those sources. Although not everyone came up with examples of all six categories, they were mostly able to identify at least four.

Something that we have been working on with everyone is improving the appearance, accuracy, and completeness of their work throughout the day. One important element is reading the directions. Our constant litany is that we need to read the directions and/or question at least twice: once before we start, and once when we think we are finished. Some students skip over the directions entirely. Some get so absorbed in a problem or response that they forget there was a second part to the work or that the answer had to be given in a particular form. Reading the directions or the question after as well as before will prevent that much of the time. We’re also working on how students organize their handwritten work on the page. Start at the top left. Leave some space between responses instead of separating them with frames. Pay attention to the margins of the paper. Choose appropriate paper for the work — do you need plain paper, lined paper, or graph paper? Number your responses to match the worksheet or textbook page. This is needed not only for mathematics work but also for other kinds of writing. Making it easy for someone to read and understand what you have written is always a good idea.

Friday ended with our traditional Halloween celebration. It was a delightful way to wrap up the week. Thanks to all of the family members who were able to join us for the afternoon.