Kindergarten Math

I don’t post as much about what the twins are doing, and I fear this does reflect where my attention tends to run . . . however, there is the trickle-down effect and I hope they are not growing up in complete ignorance.

Dash has been assigned to set the table for breakfast each morning. This is a great chance to deal with concrete quantities. How many people if Papa isn’t here for breakfast? There are three plates in the dishwasher–how many more do you need from the cupboard? If you dropped two forks on the floor, how many would be left in your hand?

Friday nights are our role-playing game nights. The older two are big enough to sit in with the grownups and are even starting to play their own characters, but the twins are sent to play quietly in the other room and then to bed. Last night Dash was playing with Munchkin cards, a game that reduces role-playing to its simplest and silliest aspects. Your character puts on various equipment with bonuses, fights monsters, goes up levels, and gets more treasure. We have played it as a family many times.

Dash can’t read words independently yet, but the cards are vividly illustrated and plainly labeled with numbers that show the size of the monster, the amount of the bonus, etc., and he is very good at reading numbers. (Even 2-digit numbers, which he likes to punch into the scientific calculator and then read off. I have, by the way, no idea when or how he learned this–I guess the trickle-down effect does work.) So he was working it out on his own with a little help.

He got stumped when it came to adding up all the bonuses his character had from equipment to see if he was strong enough to beat a particular monster. Even with small numbers, adding three numbers at once was more than he could do in his head.

Finally I suggested he take the glass gems he was using to keep track of his level and place the appropriate amount on each card–5 counters on the +5 Gatling Gun, 3 counters on the +3 Twenty Gallon Hat. He figured out what I meant from a brief explanation and when I checked on him next, each card was covered with the appropriate number of counters. Then as he battled each monster, he had to compare his total level plus all bonuses with the size of the monster.

It turned out to be the perfect level of challenge for him, as well as keeping him happy through a long evening. And a lot more fun than “Circle the set with more balloons.”

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Math Remix

So, I posted a couple of weeks ago about how math was not working well. We had switched to MEP which I really do like conceptually, but which was bogging us down unbearably. There were too many things to do too quickly, there were some areas we had missed or not spent enough time on to do quickly, more writing than Deux was willing to do, and even doing it on the computer didn’t help much.  Duchess was having many, uhh, dramatic moments. Math had become the most miserable time of day, and since we did it first, it set the tone for the rest of the day. And we were getting nowhere near doing work independently.

Well, I considered several options:

  • Moving backward, to Year 3, which would be *really* easy for them, and expecting them to work alone.
  • Moving forward, to Year 5 (I actually haven’t ruled this out entirely), where the work looks (at short glance) a little more focused on interesting concepts and less on long, tedious calculations to ensure mastery of place value. (Which seems to be dominating Year 4).
  • Scratching MEP altogether and finding something else they could do independently.
  • Muddling through and telling them that this was what math was like, so just deal with it.

None of these seemed quite the thing. I realized that the strength of our math time was the time we spent working through complex and interesting problems together. There were some of these in the materials, but they were usually overshadowed by tedious calculations, or rushing on to the next thing. I also realized that they had not had enough experience with book math to know how to do multi-digit calculations efficiently, and that we needed to master those before going too much farther. . . but that they wouldn’t need a year of repetition to be sure they had grasped those, as math books tend to be written for.

What I finally settled on was sticking with where and what we were doing, but doing it our way. We stopped using the worksheets almost entirely, unless there was one that really caught my eye.  Instead of trying to rush through seven or eight activities, which is how the lesson plans were laid out, I selected one or two that had the potential to be thought-provoking. Or I looked at the area to be covered and used an activity I had seen elsewhere. I’m paying attention to the goals for the year and I will feel free to skip areas I know they have down and spend more time on the others.

To work on the basic calculation skills, I assigned them one long-hand problem to do each day. Just one. But they had to do it all themselves and get it right. I have walked them through these until they got the transition from the mental math they were good at to how it was represented on paper with problems too complex to keep straight in the head.

I also decided we needed to spend some extra time getting math facts automatic, but there was no reason this had to be painful. Right now they are loving DigitWhiz. I love that it’s free and that it uses multiple approaches to reinforce understanding in addition to memorization. Deux especially has trouble with timed things, but so far he is doing OK with Digitwhiz’s approach. (Although I’m not sure he’ll ever be able to pass the “mastery” sections–he thinks deeply about math, not quickly.)

The effect after just a couple of weeks? Math is now the *favorite* subject. We are having the fun we used to have with math when they were preschoolers and it was all about divvying out the snacks. On the day that multi-digit multiplication clicked, they were covering the board with difficult problems long after school was over. Deux is now writing out three by three digit multiplications without complaint, where at the beginning of the year he still struggled to remember which way a 2 went. Sometimes we do fun (or just silly) problems during supper with Papa. Duchess came up with a lovely model of multiples by drawing a street with even numbers on one side and odd on the other, then envisioning which houses each number would stop at.

So our new plan works like this:

1) Do one multi-digit basic calculation long hand. (We started with one of each during the week, but addition proved so easy we dropped it and now they are doing one extra of where they think they need work.) When they all are getting too easy, we’ll add fractions and decimals, negative numbers, or order-of-operations challenges to the mix.

2) Explore an interesting math topic. We have drawn factor trees that really looked like trees. (Duchess is so much happier with math that involves art. Or bunnies.) We have played Stake Your Claim. We have calculated the area of the different rooms of the house and figured out how to deal with odd shapes, even triangles. We have worked out complicated word problems from the MEP lessons together, and taken the time to talk it through for full understanding. I look through the MEP plans and choose the activities that are the right challenge level and cover the relevant material without bogging us down on stuff they know.

3) Play on the computer to practice facts.

All told, it’s about half an hour and we are having fun again and they are *getting* it again instead of freezing up. The moral of the story is . . . sometimes you can take curriculum that isn’t working and turn it into something that is. Sometimes less is more. And sometimes the learning is worth the time spent working together instead of rushing to independence.

Planning, Part 3: Subjects

Science: AO Y3 has some nature study readings (Pagoo and Secrets of the Woods) and then some science options, one of which I don’t care for the writing style of and one of which isn’t readily available. So I plan to continue with the next stage after Building Foundations of Scientific Understanding, Elementary Science Education. It just got here in the mail last week and I have started flipping through it. It looks really good, but really challenging–at least for the teacher. Plus, I’m going to need some equipment, most significantly a proper microscope and a triple beam scale, which aren’t cheap but should last us several years. I’m a little intimidated. I’m also not sure how to plan, or how to make sure it actually happens. It slid off too easily or got shortchanged last year. But there’s some really cool stuff in here. We will also need to start keeping notebooks, which means–sigh–more writing for Deux. That AND I want to make Nature Notebooks a weekly thing this year. (We did do many more entries this year than the year before. And this can be an independent activity.)

History: Duchess wants to do her own timeline. I think I’m going to get a small notebook and have her use it just for the time period covered during this school year. I’ll probably need to get one for Deux, too, even if he doesn’t put much in it. I am not certain whether I want to read history out loud or have them read it to themselves. And if they do read it to themselves, how do I provide the appropriate introductions and preparation? The trouble is, the history reads are usually relatively easy to read independently, but they do need context.

History Biographies: For the first term I will have Duchess read the biography of Da Vinci by Hahn, which is fairly lengthy. Deux will read the shorter one by Diane Stanley. In the second term, I’m going to have Deux read the biography of Sir Walter Raleigh recommended for Year 3.5 and Duchess read both Bard of Avon and Good Queen Bess. I haven’t decided what to do for the third term yet–I may have them both read The Landing of the Pilgrims or I may have Deux read Squanto: Friend of the Pilgrims by Bulla.

Geography: An idea I want to try is having them trace their own map of Asia to fill in. I’m going to do one map for the whole year to which they will add the journeys of Marco Polo, plus the weekly blank map to learn to fill in the modern countries. I got the Komroff Marco Polo, which looks good, but I have to schedule it myself–there are only 26 chapters, so I might intersperse with some activities from Marco Polo for Kids, which I will probably just get from the library.

Math: I’m still planning to do MEP–I think it’s the best fit for where they are now. What I really want to do long-term is now the Art of Problem Solving, but they don’t have anything between 3d grade and pre-algebra. (And I don’t care that much for the younger level stuff, anyway.) That means switching curriculum again in a couple of years. Which, of course, is supposed to be bad. However, I think I have good reason for the choices at each stage, and I’m willing and able to help them bridge any gaps. I’m still dreading the writing issue with Deux.

Languages: I still have to order this, but I’m excited to try using a Spanish curriculum based on the Gouin series. I’m hoping it will help them make the jump from being familiar with the sounds of Spanish to actually *using* it. For Duchess, I’m going to have her try Mango for French–we can get it free through the library.

Art and Music: I’ve changed from the AO 2013-14 selections to ones from various years that will correspond with our current timeframe. Other than that, I think we’ll keep doing what we’ve been doing. It’s not fancy, but it begins an acquaintance.

Twins: I’m making a list of books to work through, not on any particular schedule, though generally rotating through: the “Among the ____ People” books, a book of fairy and folk tales, the “Twin” books.  I’ll try to read a few appropriate history tales when the older kids are doing something similar, and not give up picture books entirely. I’m not going to schedule reading lessons in advance, because I really have no idea how quickly they’ll progress, but we’ll start with doing word building (maybe in some cool little notebooks) and then add in actual CM-style reading lessons when they seem ready. I don’t think I’ll plan math as part of school–I will continue using the recommendations from The Arithmetic Primer, but as part of ordinary life. I also want to work on a calendar of firsts with them. They will continue to participate informally in memory work and in art and composer and nature study. I would like to look through Building Foundations of Scientific Understanding and make notes of specific areas I want to be sure to discuss with them this year–they are asking lots of good questions on their own. Having worked through it once more formally with the older kids, I think I’m doing better at simply integrating it into their own observations rather than making formal lessons out of it.

Gaining Steam

In the interests of finishing all our readings by late June, we have picked up the pace to three readings a day. This is allowing us to do a weeks’ worth of work in three days (we only do memory work and copywork on Wednesdays) and shorten our school year by a few weeks. So far, it doesn’t seem too burdensome. I’m trying to walk the fine line where we are challenged enough but don’t feel like we’re stuffing things in faster than we can absorb them. We are still done by noon every day, though, with plenty of time in the afternoons for enjoying some glorious summer-like weather.

Duchess is thrilled to be studying Joan of Arc for this term. She is something of a “girl power” student. I’m fascinated to finally make some sense of the Hundred-Years’ War, and seeing the Middle Ages fade into the modern era. (On my own, I’m reading The Maid and the Queen by Nancy Goldstone.)

We had the chance to get a guided tour of some local tidepools this past week. Unfortunately we were all still a little woozy after a stomach bug and we didn’t last very long, but we did see some amazing creatures–sea anemones by the thousands, snails and snail eggs and nests, crabs and isopods, and sea cauliflower.

I’m pretty sure I’m going to switch over to MEP math next year–I have been wanting to switch to a modern program eventually, and I think Deux is ready to handle a worksheet’s worth of math. I’m still a little uncertain on placement–Year 4 looks pretty challenging and has a lot of writing and in general a more mature feel, but at the same time Year 3 would be mostly review. What I’m trying is doing the last month of Year 3 in this last month of school–if it goes well, I’ll feel confident that Year 4 will be the right fit for next fall. There will be some transition for the parts that have not been emphasized in what we have been doing (e.g. certain types of puzzles, translating mental math into symbols) but their basic understanding of math seems well up to it. (If all goes well with the twins, I will probably do The Arithmetic Primer with them and start MEP in Year 3.)

The oldest two have been doing a lot more reading–Duchess likes to keep twenty books going at once. Deux seems to be doing a lot of Narnia.

We have grandparents coming next weekend, but I don’t plan to have that stop school. It works well for the twins to spend time with them in the morning, and the older two in the afternoon. Similarly, for DOB’s birthday he took the day off work and spent time with each child in turn–we just did school while the twins had their turns, and then the big kids had theirs.

Year 2, Week 3-5

I did TRY to post this two weeks ago, but the website wasn’t working, then I lost part of my post, and then . . . things got busy again.

Anyway, we did the readings for weeks 3-5 as scheduled, and most of the extra stuff. DOB has asked me to start coming in and helping out at the office a couple of afternoons a week, so we are working to accommodate that.

Deux seems to be gradually growing accustomed to reading to himself; I think by the end of the year he will be doing fine. I am doing a few more of the readings that I had scheduled for them, but we are making progress. Also he is doing much better with very short cursive copywork. Duchess has really taken off with cursive–she is starting to use it as her primary method of writing, even in her free writing. I think she will be ready for dictation in January as I had hoped.

An Island Story remains the favorite, especially the Battle of Hastings. It amuses me that the violence is apparently an issue for some. Not the ducklings. Deux was dragging his foot about another reading until he said, “Wait . . . will there be more fighting?” and raced off to get toy soldiers to act it out. The Dover coloring book has several images from the Bayeux Tapestry, which they all colored as we read, and then we looked at a website that showed it piece by piece. Even the twins were enraptured. Our sympathies are all with the Saxons. Who says that history is always written by the winners?

Ambleside Online officially made a change that had been unofficially discussed this past summer . . . stretching out Burgess Animal Book and moving Pagoo to Year 3. So, although I refused to do it this summer because that was the one book I already had planned out, I decided to do it now even though it will mess up all my printed assignment sheets and copywork. 🙂 Two chapters a week is a lot to absorb; this way Deux can do more of it himself, and I think they will get more out of it. They were happy with the change. I will survive waiting for next year for Pagoo, just like I will survive waiting for next term for *Wind in the Willows*.

I have been trying to mix it up with math these past few weeks–sometimes we still do the pages from the arithmetic book as written, but sometimes I will approach the same concepts in a different way. One day we were supposed to be doing calculations with pints and quarts; instead we found appropriate shapes to represent pints and quarts and worked out how many different ways you could measure a given number of pints, using only pint and quarts to measure with. (For instance, if you needed to measure 6 pints, you could do it four ways: 6 pints, 1 quart and 4 pints, 2 quarts and 2 pints, 3 quarts). After we had tried this with increasing quantities their guesses went from close to exact to not being guesses any more as they deduced the pattern. So that was a bit of real mathematics, which is all too rare in elementary school, and I think it sufficiently reinforced the relationship between pints and quarts.

We have also discovered a fun and simple game: Take a deck of cards, two players each hold a card to their head without looking at it, and the third announces what the sum of the two cards is. By looking at each other’s card, they deduce the amount of their own. With slight modifications, we can do this for differences and products, too. (We are practicing the vocabulary in particular, not just the operations.) For next week I have a new variation to try–they each draw a card, then tell me the sum, difference, product, and quotient of the two cards. Then I have to guess BOTH the cards.

Another math activity we did was a more practical one: we have been keeping our main variable budget categories on the refrigerator. At the end of the month, we each guessed the total (discussing some strategies for estimating), then added them up to see who was the closest. I showed Deux how to use the calculator to do the addition. They found this quite interesting and I can see it being a monthly event, to the better mathematical and financial education of all.

Our music and art studies seem pretty minimal to me, but we are doing something. The science lesson was on vertebrates and invertebrates and interested them greatly.We made a very long list of animals on the board in different categories. Then I started looking things up on Wikipedia and had my mind blown by how many and diverse the forms of life are and how complex it all is.

A few things slid this last week, but we carried on pretty well and also squeezed in a harvest carnival (including them designing and assembling their own costumes), a long-awaited visit from the cousins, and me working three half-days. This week I’m only working two half-days and they should both be afternoons; I hope we can just get school done in the mornings like we are supposed to and then they can enjoy their free afternoons with the babysitter, hopefully including playing outside, arts and crafts, and reading stories, which they enjoy with her.

Game Theory

March is here, and I’m ready for spring, as can be easily seen through our memory selections for this month:

Poem: Spring, by William Blake (the twins LOVE this)

Folksong: English Country Garden.

Memory: The Apostle’s Creed (review, it’s being studied in Sunday School)

Hymn: Amazing Grace

I also created a Lenten paper chain with activities for each day until Easter. We are also working on attending the Wednesday evening services at our church, which are beautifully geared for all ages. On Ash Wednesday the pastor helped each of the children plant flower seeds in a purple pot, and talked about the beauty that would grow in the quiet darkness of Lent. Last week there were candles to light and pray, and D1 asked to go up and pray together. They also had the story of Noah (the Wednesday services are going through Hebrews 11) and D1 eagerly told the whole story–that narration is starting to show.

DOB sets aside two evenings each week to play alone with one of the children. They all anticipate their turn eagerly, and he likes the chance to get to know them individually instead of as a collective chaos. He’d had a bit of a struggle finding what to do with D1, though–they each have very different playing styles and both want to do things their own way. Finally he struck on the idea of teaching her chess. It’s been a huge hit. She is studying it closely, reading books, practicing. She challenges D2 (who is a little more tentative) and her oldest cousin.

Meanwhile D2 has discovered one of those jumping-peg boards, as well as a chinese checkers board. Between them, he’s been absorbed creating, modifying, and studying different patterns, the effect of different rules, etc. I also found an amazing book in my sister’s homeschool library, Anno’s Math Games. It’s exactly what I’ve been wanting–a puzzle and game approach to math that works at a conceptually high level but that is appropriate for early elementary. We have had fun working through the first chapter together and I look forward to doing more. And now I see there are two sequels! Anno’s math books are some of the best out there for this age.

Robin Hood maintains his appeal. I tried reading from one of the illustrated simpler editions we had borrowed from the library–very thin and insipid compared to the Howard Pyle audiobook they listen to every night. What’s Robin Hood without varlets and forsooth?

In writing, D1 wrote up a page of instructions for chess and a lengthy letter to her cousin asking him to play chess. She usually has me write out words she thinks she needs help with. I notice a lot of copying poetry has led her to settle on the rule of starting each line with a capital–we probably need a basic lesson on sentences here soon. D2 is sticking to single words, but they are getting longer, like “bobolink” and “cukoo” from “English Country Garden.”

D1 is devouring Tintin books; she also has been reading some of the Flicka, Ricka, Dicka and Snipp, Snapp, Snurr books. D2 picked up some easy-reader joke books and has been browsing picture books, Frog and Toad, Amelia Bedelia, etc.

Everyone has been drawing a lot, mostly on paper. The big kids added on to some maps of imaginary worlds they created several months ago. The twins are showing more variety of strokes. D3 likes to color things in and sticks to the lines pretty well; D4 likes to draw his own things and sometimes verges on recognizable shapes.

Read-alouds have mostly been Heidi and various tales from the Brothers Grimm in various forms. I reserved The Apple and the Arrow from the library to supplement the Switzerland study; I hope D2 likes it.

The kids are becoming more pronounced in their different selections–D1 usually wants to choose stories and art; D2 usually wants to choose science and math. They are starting to notice this, and I wonder if they even differentiate for the sake of being different. Regardless, it ensures we get a good variety of activities every week.

In addition to all the games and the wonderful Anno’s Math Book, D2 has been asking to do some math worksheets. I’ve been having a bit of trouble with this. The ones from MEP math have a nice variety of problems, and they stick to smaller, more concrete numbers in Grade 1, but they still are not a perfect fit. Some of the problems really need extensive instruction to do (and I’m just not ready to start in on math lessons), and some just seem too abstract in approach for what they need right now. But no other free pages I have found have the nice mix of problems–most math pages are all-of-a-kind, which is not only boring but I suspect dulls the brain from paying attention. DOB finally suggested the rather obvious solution of continuing to use MEP, but marking any sections that I didn’t think they should be doing as “Play” and letting them write whatever they want on them. Duh.

I’m thinking we may wrap up studying Europe within a couple more weeks–true, we’ve only done a few countries, but we have learned a lot about their neighbors. We have some anchor points. And I still kind of would like to move on to Asia and Africa through the rest of the spring and summer and start a definite chronological history in the fall. We’ll see. I’d also like to be in our own house by then. And have the twins potty trained. I can dream. 🙂

Math: A Third Approach

Yesterday I had one of those spasms of panic that occasionally afflict those with a relaxed schooling approach, concentrated on our decision to delay formal math. I had been reminded of the placement tests for Singapore Math, and decided to print out the first grade tests to see how the ducklings did with them. (D1 is 6, D2 is 5.)

I presented them merely as a page of math puzzles for them to figure out; I explained that some were easy, some were tricky, and some would use ways of asking questions and writing things down they hadn’t seen yet. I let them ask me for explanations, although I tried to make sure they each thought through each question for themselves.

They did fine–certainly they’re not any behind–although it did highlight a few areas I want to work more on (mostly correctly reading and writing numbers, since we’ve been almost entirely oral).

But working through the tests helped me see a significant divergence of approach in how mathematics can be taught, and I’ve realized that the approach I have instinctively been taking doesn’t really fit the two common math approaches.

The big division between mathematical programs that everyone talks about is mastery vs. spiral–do they stay focused on one thing until it’s finished, or do they jump around and then come back to stuff. Mastery is all the rage right now: first you learn addition, then subtraction, then multiplication, then division. (This is the approach of Math-U-See, and RightStart Math, which I had formerly been very interested in. Four-digit addition before you even move on to subtraction.)

But even spiral programs think in this operational way, they just jump about a bit more. Singapore at least combines addition and subtraction and multiplication and division, but you still have to get through two-digit addition before you move on to multiplication. There is always a rapid move from concrete to abstract, and then back to concrete once you move on to a new topic.

But when I gave D1 the test on the second half of first grade, she skipped ahead to the very end where there was a pictorial depiction of multiplication and division and effortlessly filled out the correct answers. Then she went back and sweated over the multi-digit addition. Neither of them had any trouble with understanding what to do in a word problem, which I understand is a source of terror and loathing for many–they found these easier and more appealing than the bare calculations.

This makes sense to me, because what we have always done is stick with small, concrete numbers, and real-life situations primarily. Our math time is usually around the lunch table and usually consists of questions like, “If you ate five bites and you have seven bites left, how many did you start with?” or “If I gave each of the four of you three slices of apple, how many slices are there?”

Or even, “If I am going to give each of the four of you half of an apple, how many apples will I need?” Technically that’s a question calling for division with fractions–a late elementary concept that most schoolteachers haven’t figured out–but because it’s a real object and an easy to visualize amount, it’s easy for them to do. I don’t have to explain it–they work it out for themselves. And although they have a decent aptitude for numbers, they are not child prodigies.

I never consciously set out to do it this way, but the more I think about it, the more I think the approach I am taking is the right one. After all, 5-7 year old children are still fairly concrete thinkers. And large numbers are very abstract. You cannot meaningfully visualize a four-digit number. (Even with the best manipulatives–which are very pricey–it’s hard to play with effectively.)

But twelve? You can do anything with twelve. You can get out twelve beans or glass beads or poker chips and split them up in unequal groups (addition and subtraction) and in equal groups (multiplication and division). You can test whether it is odd or even, prime or composite. You can apply fractions to it. You can tell which numbers it is larger than and smaller than.

All this time you are gaining a deep instinctive understanding of the relationships numbers have with each other, of the patterns and just of the fun of playing with a number.

And isn’t that what they will most need? After all, if you understand how the operations work on numbers under twenty, and if you understand place value thoroughly, all those dreaded algorithms and lengthy calculations are easy to grasp, once you are developmentally ready to handle numbers as pure abstractions.

So I started thinking through math curriculum because I was sure somewhere I’d seen an approach that stuck with small numbers and used all four operations. Sure enough, Ray’s Primary Arithmetic, that long-dress-and-bonnet companion to McGuffey’s Readers, uses just that approach. Primary children learn to add, subtract, multiply and divide combinations making up every number under twenty. And they do it with manipulatives (the basic kind: counters of whatever is on hand) and with real-life problems. Writing the operations comes after they are thoroughly understood, and larger numbers are saved for later years.

Since this is from the era when algebra and geometry were part of an eighth-grade education and people calculated compound percentage longhand, I think it’s pretty evident that this approach is not going to shortchange anyone in solid mathematical skill. It just surprises me it is not more common anymore.

Then I also remembered to check the free MEP curriculum, an experimental program developed in the UK based on Hungary’s mathematics curriculum. Sure enough, this one also concentrates on multiple operations with small numbers first, one number at a time in the first grade. It also seems to keep things fairly concrete at first, although there are the workbook pages so essential nowadays and more use of symbols early on than I would do.

But I will be reexamining it closely to see if that’s the program I would like us to transition into when we’re ready to start doing more bookwork. (The one big downside I see is that it only teaches metric measurements–but then, measurements are one of those real-life skills that can be learned on the side.)

I’m surprised that this approach is so rare (is there even a modern commercial program using it?) when it makes so much sense–to me, at least. I don’t even know a general name for it. It gets lumped with the “spiral” programs, but there’s a huge difference between deliberately mastering the combinations and relationships that make up each number in turn, and hopping about between topics for the sake of variety.