Archive for the ‘math’ Category

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Shall we stick the Daily Homeschool Report   here?  Yes, we shall.

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Thursday is homeschool class at the Cathedral, with only one more week to go, sadly.

So that means no copywork and no home morning prayer. It’s get him up, feed him, and off we go.

Today his drama class practiced their play and his history of science class talked about Louis Pasteur.

After, we ran to the downtown library branch to pick up an armful of Smurf comics.  (More on that in a bit). Then home for lunch, had him talk to me about Pasteur, finish up Beast Academy 5A, and talk about the Writing and Rhetoric story, followed by several exercises (excerpts from Twain, Anne of Green Gables, etc)  asking him to look for unbelievable, improbable, improper or unclear.

As I said, it is prep for learning how to write refutations in a very ordered, but not at all boring way. It’s about instilling criteria in the mind so that one can give reasons for the case one is making.  I’m impressed with it.


– 3—

By this time, it’s mid-afternoon and rainy, so I pulled out the video of Ken Burns’ program on Lewis and Clark I had checked out of the library and we started watching it.  It’s pretty long – 240 minutes, but he was engaged, so I think we’ll just take it in 45 minute sections and watch it over the next week.

Piano practice, and that’s it.


— 4 —

Honest to pete, as they say, I had never before watched a Ken Burns doc. It’s quality, for sure, but stylistically so repetitive.  Gliding shot of river at sunrise. Voiceover from journal. Talking head. Gliding shot of river at sunset.

I guess there’s really nothing much more to do, right?

And the talking heads – maybe I’m just getting oversensitive as I age, but wow,  I just wanted to say BACK OFF, TALKING HEAD.  Really, pull that camera back even six inches, and I won’t reflexively recoil from you.


— 5 —

Proud that this conference on racial reconciliation is being held in Birmingham right now, held at a local Baptist divinity school and  co-sponsored by the Diocese of Birmingham

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Speakers include Bishop Braxton of Belleville, IL, the Archbishop of Owerri, Nigeria, and today the mayor of Charleston:

Riley recalled gathering the families of survivors together the night of the shooting as the police chief explained what happened.

“There was this choir of sorrow, wailing, crying, that will be with me as long as I live,” Riley said. “We told the community this was a hate crime. He came from 110 miles away. He wasn’t from Charleston. But he was from America. He wasn’t from another planet.”

The city, without a solid foundation in good race relations, could have responded in anger and with riots, Riley said. Instead, people of different races and religious beliefs gathered in front of the church, held hands and prayed.

“He came with hate, and we in this community would respond with love,” Riley said. “We decided we would take care of each other and we would pray. And we did.”

Riley spearheaded and is still working on a plan to build an African-American history museum on the site of the wharf where thousands of Africans were sold as slaves in Charleston, he said. “Forty-four percent of all slaves who came to North America came through Charleston,” Riley said.

Unfortunately, I can’t attend, but it looks really good.  Maybe we will try to sneak over at noon, but no, on second thought I think there is some big music audition/competition going over on that campus right now, besides classes, and a friend of mine was saying parking was impossible on campus, so probably not….

— 6–

Remember that Lent when your early idealism held and you indeed did not have cheese pizza for dinner every Friday?

Yeah, me neither.


— 7 —

Oh, to get back to the Smurfs.

Both of my younger boys, but especially the actual youngest, really like the Asterix-TinTin end of comics/graphic novels.  I’ve mentioned before that the youngest is also a big fan of the Lucky Luke series and occasionally asks if the Gaston series, which he encountered in a cabin in the Pyrenees and gamely tried to “read” in French, has been translated into English yet (nope).

Another short series he likes is Benny Breakiron by Peyo, who was also the author of the Smurfs comics.  I had suggested the latter to him before, but he’d always rejected it because what are Smurfs anyway but something for toddlers, right? (My only real encounter was with the animated series, which I never actually watched, but which made me itchy even just running in the other room. But I had read that the comics were different). The other day, he started reading one in the library, was hooked, and, as I said, asked to return to get like ten more.   I asked him why he liked them and he said he mostly liked how each of the Smurfs had a different personality.

And then he said he thought he had figured out where Calvin (of Calvin and Hobbes) lives.


Speaking of books…order some from me!  Signed editions of any of the picture books at 8 bucks a title.  Big orders for your entire First Communion class welcome!




For more Quick Takes, visit This Ain’t the Lyceum!

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Time for the Daily Homeschool Report. It’s not fascinating, it’s just a report so you can see how this thing happens in one little corner.

Are you tired of me saying “short day?” Well, here’s another one.  Early dismissal for brother, late start because of the Super Bowl..so yeah. Short day.

  • Scripture reading (we always pray with the daily Mass readings) was 2 Samuel 8 – placing the Ark of the Covenant in the just – built Temple.  I recapped some Solomon stuff that had preceded it.  He recited the list of OT books he knows so far – from Genesis through 2 Chronicles.  Then read the Gospel, prayed the Intentions and an Our Father.
  • Copywork was Scripture, since it’s Monday.  The last part of the Old Testament reading:
  •  Now when the priests came out of the sanctuary, the cloud filled the Temple of the Lord, and because of the cloud the priests could no longer perform their duties: the glory of the Lord filled the Lord’s Temple.
      Then Solomon said:
    ‘The Lord has chosen to dwell in the thick cloud.
    Yes, I have built you a dwelling,
    a place for you to live in for ever.’
  • . Half in manuscript, half in cursive.

  • Math was more simplification of expressions –these worksheets from Beast Academy.  A bit of confusion, but by the end, it was understood.
  • He had had homework of sorts over the weekend – to read the first few chapters in the next volume of Hakim’s Story of US – about the early years of the United States – Washington’s presidency, the Hamilton/Jefferson conflict and the establishment of the District of Columbia.  We talked about that – can’t recall the specific points of interest, but there were some.
  • I then made my big announcement about his next “school” book (remember, we did short stories all last week): The Magician’s Nephew.  
  • “Oh, I’ve already read that. Remember?”
  • No, I didn’t.  I knew he hadn’t read all of the Narnia books, claiming he’d gotten bored,  but I didn’t know what he had read.  I guess I should have asked?
  • Okay, well, let’s think of something else then.
  • His brother read Animal Farm last week, and this one had heard us discussing it, so he asked for clarification of what it was about.  I explained who George Orwell was, what type of writing he did, and then the general point/plot of this novel, defining allegory in the process.  He then asked about Fahrenheit 451 , which had been brother’s summer reading.
  • I handed them both to him and told him that if he was interested, go into his room and read the first few pages of both and then whichever one he wanted to read first, that’s what we’d do. He picked the latter.  So it shall be.
  • Let’s see.  What next….Just a bit of Latin – some translating and parsing.
  • Then, let’s finish up with these invertebrates . Crayfish waiting.
  • Read about Crustaceans in the Animal book, which is become the science “spine” (as they say) of the month.
  • This is our third dissection, preceded by the earthworm and the grasshopper.  This was by far the best. The animal is larger and the organs are easier to see.
  • "amy welborn"
  • I also decided to do the dissection to the tune of a video – it just makes a lot of sense to dissect along with someone who knows what he is doing.  We focused on this one, and then after we were done, watched a bit of this one to get a slightly different perspective. It worked very well. We paused it occasionally so not to rush our cutting.  The most interesting parts were first, the gills – I had never known that the stiff but still sort of feathery things that come off with a lobster’s swimmerettes are gills, but that is what they are. Duh. Very interesting to see those and poke around. Secondly, the crayfish stomach, located practically in the head, has “teeth” in it – you can remove the stomach, see the contents – mud, mostly – and see and feel those little hard protuberances.
  • After we’d finished the dissection, I just named the systems, and as I did, he pointed and explained the course and shape of each in the animal. An exercise like this really emphasized exactly what the nervous system, for example is – you can see that central nerve “cord” and trace it along the body up to the head where it splits to go to the eyes and antennae then meets again for the “brain.”  To see the little heart and the hole in it through which the blood flows to the rest of the body in the animal’s open circulatory system – it simply clarifies the basic functions of all of these systems, not only in these simpler animals, but in all creatures.
  • Finally, he watched a video during lunch and after – the first episode of Egypt – a BBC docudrama about late 19th-early 20th century archaeological work.  I told him I’d just like him to try it out – if he doesn’t want to watch the whole series, that is fine.  He said the first one was good “like that Abbey show Katie watches but more interesting,” but I don’t know if he really wants to watch any more of it. We’ll see tomorrow.
  • Timeframe, including video, 10-2.


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So are you sick of these yet? Well..don’t read them then!

A reminder of why I do this daily update thing, and will for a while: Dissatisfaction with education grows and grows. More folks who once may have said, “I could never…” are contemplating the possibility of homeschooling. There are loads of different ways to do it – here’s ours. No boxed curriculum, no online classes, a touch of unschooling (not as much as I had hoped, I admit), a few texts, a lot of ad hoc. And every great while, the teacher is a little hungover from book group. So.

  • Prayer: We missed the daily readings yesterday because we had to dash out of the house for the Cathedral class, so I caught him up on David today – told him about Absalom, read about the death of Absalom, and then the death of David.
  • Today’s first reading was Ecclesiasticus (Sirach) which was a summary of David’s life, accomplishments and character.  We read it – I read half, he read half – and then after we read the Gospel (death of J the B) and prayed, I swung back around to the structure of the Old Testament – had him recite the books he knows in order (now through 2 Chronicles), then I explained the different types of books: Torah/Pentateuch, historical, Wisdom, Prophets, and in particular the nature of Wisdom books.
  • Usually Friday is illustrate-one-of-your-copywork-entries day. He didn’t feel like drawing, so we just hit the dictation (see Wednesday for explanation).
  • Oh, then we looked at a February calendar and talked about the month’s big days: Chinese New Year, Mardi Gras, Ash Wednesday, Lincoln’s birthday, Washington’s birthday, Presidents Day or whatever it is.
  • 2 word problems for Math revew.
  • Beast Academy  – defining coefficient and variable, combining like terms. A couple of pages of “corralling like terms” puzzles and then simplification with problems like these. 
  • Page of Latin review, which involved once again going over various sentence structures: Subject-Predicate nominative; subject-predicate adjective; subject – direct object. And some vocab and imperfect of sum. 
  • Writing and Rhetoric – summarizing a story – an exercise which builds on the careful method of learning how to summarize that this curriculum teaches. Since it’s a workbook, it involves crossing out inessentials, circling important details and main points, and so on. Very helpful.
  • Look at the material on insects in the Animal book –  keep recommending it to you, and I mean it. If you have a child interested in zoology, this is the book to get. It is chock full of great photos, but more than that is actually a comprehensive presentation of animal life organized according to phyla, with loads of solid information – not just cute pictures.
  • Grasshopper time.  Time to dissect the grasshopper specimen I had purchased from Carolina. It’s big. He studied this worksheet as well as the instructions included with the kit, and went to work.  First examine the exterior of the bug, identify parts – then cut off legs and wings, cut through the exoskeleton to get to the innards. I had watched a couple of videos to prep. 
  • "amy welborn"
  • Next week: crayfish.
  • Story today was “The Lady or the Tiger?” He read it, and then looked over some study questions to prepare for talking about it. His opinion was that the tiger was behind the opened door. The focus of the discussion was on what either possibility would say about the princess, as well as the power of an ambiguous ending. He admitted that he was frustrated by the ending, but on the other hand, if the story did have definite closure – if we did know which door the poor man opened – it wouldn’t be a story anyone would really care about.  It would be just about the princess and what we learn about her through that revelation in a way that puts a lid on further contemplation, rather than encouraging it.
  • (Does none of that make a lick of sense? Well read the story!)
  • So the focus of lit discussions this week has been irony and the impact of a story’s ending. It’s been interesting.
  • And that’s it. Brother had a slightly earlier day and had to be fetched. Piano practice after that, then a couple of hours outside with the neighbor friend.
  • Timeframe 10am-1.

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I am finally getting myself together here for Melanie Bettinelli’s linkup on “Learning Notes.”  Melanie is a homeschooling mother of many who writes a very fine blog. Check out the series she’s doing on Shakespeare with kids, for example.   There are innumerable ways and styles of homeschooling, and if you are curious about this growing phenomenon and want to understand its appeal to families, I think Melanie’s blog is a great place to start.  Conversations, creating, exploring with people who love you? …the best kind of education, for sure.

My first foray into this linkup isn’t going to be a day-by-day account this time because I’ve got math on the mind these days.

For a non-mathematician, I think a lot about math, and this blog post has finally spurred me to put down some thoughts on it.


I come from an academic, humanties-centered household.  There was no mathiness or science or business-type activity to speak of in my parents’ lives or in their parents’ lives. My mother  joked about her wildly contrasting verbal and math scores on the GRE. I did fine in math in high school, took the minimum I needed to in college, and that was it.  I had no opinion of it one way or the other.  I certainly had to work hard and think things through in the higher math (the highest I got was what they called “Advanced Math” in the day – maybe there was a bit of Pre-Calculus in it, and a little trig, but I never even attempted calculus.  I don’t think the school offered it, come to think of it.), but I often had the weird experience of hitting a wall at night when I was doing my homework, then waking up the next morning, saying “Ah-ha!” – my brain evidently having worked it all out when I was sleeping.

As a parent, I’ve had one older kid who needed help in math, but the other two breezed through on their own, doing very well. My second son never studied in high school and made straight A’s, even in Calculus.  Daughter had to study, but still did well, and liked it – “Math is like a puzzle to me, and I love puzzles” is what she’s always said.  And now she’s studying for the LSAT which she was emboldened to do, not just because she took a Civil Rights/Liberties class and really enjoyed doing case analyses, but also because she looked into what the LSAT is and joyfully discovered, “It’s LOGIC!” So.

And then, for the others…. it was time to homeschool.

Math is something that some non-mathy homeschooling parents dread, but I never have, mostly because I picked a program that I found easy and even interesting to work with – from The Art of Problem Solving.  I’ve written about this program before, so I won’t repeat myself.  I’ll just say that Joseph worked through the Pre-Algebra text last year and is making “A’s” in Algebra in 8th grade right now.  He never minded it too much, and neither did I – in fact, in many ways, I found it illuminating.  Plus I love the videos.  There, I admitted it.

Now, I have a theory about teaching.  I actually think that people who are a “natural” at a subject don’t necessarily make the best teachers of that subject.  Think about it – if you have an intuitive grasp of a topic or skill, it might be a challenge for you to communicate the process to someone who doesn’t have a clue.  On the other hand, if you’ve had to work through a process step-by-step and have actually struggled with various aspects of it…you might just be a really effective teacher to the equally clueless.

All that is to not to say that I’m a fabulous math teacher.  But it is to say that I’m not a bad one – at least to my own children –  and I think it’s because I understand their lack of understanding.

Anyway, math is not only on my mind these days, it’s on the mind of many because of Common Core-related issues.  I’ll say straight up that I’m (not surprisingly) opposed to Common Core simply because I’m opposed to all federal standards in educational content, period, without exception and also because I believe that the push for Common Core is primarily profit-driven.  As I’ve said before, no one makes money when teachers are using five-year old textbooks using methods they’re familiar with.  People make money when new textbooks must be written and printed, when workshops on new pedagogies must be paid for, when consultants must be consulted and when – above all – children must be tested.

But what has gotten folks riled up above all is the content of the standards, especially in math.  I saw a bit of this in the text Joseph was using in his old school, and which we used in the first year of homeschooling (because at that point we weren’t sure if he would be returning to school after our fall in Europe…just in case he was, he needed to be on track.) I rather liked the text because it invited the student to look at problems in a number of different ways and introduced various problem-solving strategies, but I could see how it could be confusing.

(My problem, though, with how this is shaking out in schools is this: I think the various strategies should be introduced.  What I don’t think is right is then tying “success” of a child – and by extension, a teacher and a school – to that child’s mastery of all of the strategies.  It’s terribly confusing and really confounds the purpose of introducing various strategies, doesn’t it?)

So now, to the present. With the Art of Problem Solving and the curriculum which my younger son is using from the same group, Beast Academywe are encountering “new” strategies. That is, they are new to us, all of us having been taught more or less “traditional” math, even if it has been 40-45 years apart.

And here’s the thing.

They’re so much better. 

They make sense.  They are, as far as I can tell from my limited perspective, truer reflections of what is going on with the numbers with more explanatory power than anything I was taught, which was mostly about learning rules and formulas and plugging in the numbers and doing the computations, period.

I’m going to start with a simple example.

(Caveat – I’m only going to say this once, but it applies to every example.  You may have learned this stuff during math.  Maybe I was taught it, too.  But I don’t think I was, and if I was, it didn’t stick.)

When my older son started PreAlgebra with AOPS, he re-learned a lot about basic arithmetic operations.  It seemed, at first glance, kind of silly, but it wasn’t because, as we soon discovered, it really helps to understand exactly what these basic operations are.  So take division.  What is division?  Well, division is a few things, I suppose, but one of the things division is is simply multiplying by the reciprocal of a number.   So…20 divided by five is also 20 times one-fifth.  Right? So there’s your definition of division:  Multiplying by the reciprocal.

Now. Flash back to..I don’t know.  Fourth, fifth grade math.  When you were taught how to multiply and divide fractions.  Multiplying: easy.   Just multiply straight across.  But dividing?  Ooooh…tricky.  You had to remember that weird thing you had to do – you had to flip the divisor and multiply by the resulting reciprocal.  I don’t know about you, but I never understood why you did that.  Why do we have to do that?  Who knows? It’s a rule!

But hey….isn’t that what division is? Isn’t that the definition?

"beast academy"So when you have to divide fractions, you multiply by the reciprocal…because that’s what division is.

My point is – this was taught to me as a rule with no theoretical foundation.  I probably would have had an easier time remembering it if I’d been taught the reasoning behind it in this really very simple way.

Properties were another thing.  Every year we’d be taught those blasted properties, and never did any of them except the Commutative (because that’s easy) make sense to me.  I had to relearn it every year, and barely did so, because the properties were presented as one little section in one chapter and then essentially neglected, probably until Algebra.

In these AOPS books, students are taught the properties early on, and they use them..constantly.  Multiplication and Division are taught within the framework of the Distributive property – basically, they are taught to break the numbers apart in order to both more easily mentally compute, but also to understand, once again, the operations from the ground up.  And really, this is something a lot of us do anyway, right?  I know I do, and always have – if I have to compute, say, 78 times 6 in my head, I do so by breaking it up into 70 times 6 plus 8 times 6.  It’s just that I never knew what I was doing.

SO.  Finally.  Back to this blog post – in which the author says that the way kids are taught to do multiplication  – the algorithm (or system) – undercuts their understanding of place value. 

If you want kids who get right answers without thinking, then go ahead and keep focusing on those steps. Griffin gets right answer with the lattice algorithm, and I have every confidence that I can train him to get right answers with the standard algorithm too.

But we should not kid ourselves that we are teaching mathematical thinking along the way. Griffin turned off part of his brain (the part that gets 37 times 2 quickly) in order to follow a set of steps that didn’t make sense to him.

Ding-ding-ding.  As a non-mathematician, I am in total agreement.

So now back to my issue.  Have you ever tried to explain 2 and 3 (and more) digit multiplication to a kid?  And what “carrying the ones” means?  And keep any sense of place value?  I mean..try it.  Right now.   Explain why you do those things to an imaginary (or real) nine year old.

It works, sure. You get the right answer.  And there’s a reason it works.  But now….let me tell you about how Michael is learning multiplication of 2, 3 and more digit numbers..  It may not be new or radical to you…perhaps it’s being incorporated in some of the other new math materials out there.   But it’s new to me, and I’ll admit when he first started, I got nervous.  I was thinking, “Wait. This isn’t the way I was taught.  I mean, I don’t really understand the way I was taught..but this is different! I don’t think it’s what they’re doing in regular school.  WILL HE BE AN OUTCAST?”

Well, not really on that last part. So let’s go to the photos:

"amy welborn"

That’s how I learned it.  You, too, probably.  Again, imagine explaining to a kid why you carry the 1 and then the 3 and why you put a zero in the units place on that second line. Try.

Now here how Michael’s learning.

"amy welborn"

Do you see? It’s the Distributive property, in action.

In case you  don’t – it’s (3X5) + (3X40) + (70 X 5) + (70 X 40). It’s an accurate, clearly laid-out expression of what is happening in the act of “multiplying” these numbers.

The beauty of it is that if you can do part of it your head –  if you know that 45 X 3 is 135 right off the bat – feel free to just put it down that way. Doesn’t mess anything up.

"amy welborn"

This makes so much more sense.  To me, a non-math person. Yes, it takes up a bit more space on the paper, but it preserves the sense of what the numbers are and what is going on in the act of multiplication.  In other words, it’s not just a “rule” but a clear process.

Has this been the dullest blog post ever on my blog(s)?  Probably.  But at least I got it out of my system.

The examples of “Common Core” math that I have seen do, indeed seem unnecessarily complicated and frankly convoluted.  I think the intention is to encourage a deeper “number sense,” but they end up confusing instead.  My point is that what I have encountered in the AOPS programs has certainly been new to me, but as not-mathy person I haven’t found them confusing at all,but rather illuminating and quite interesting.  There is a way of teaching a way of doing math that is a more accurate expression of what is going on and which doesn’t seem so random, especially to the non-mathy person. The tragedy is that a worthy end is being massively screwed up and, as a consequence, raising suspicions against any attempt to develop better ways to teach our children math, better ways that are out there and that are not crazy or needlessly confusing – in fact, are the opposite.

Below are some of 9-year old Michael’s math pages from last week. You’ll probably have to click on them to get a better view.

"amy welborn"This was the first workbook page on which he had to work with this new algorithm.  Robots optional.

DSCN4447On this page, he was given just a few numbers of each problem and had to work out the rest.  So, for example in #144, he would have to work out what do you multiply 6 by that gives you a number with 8 in the units digit..well, it could be 3 or it could be 8..so you have to go from there and figure it out.  We left the last one to do as review later.

"amy welborn"

He started exponents late last week.  On this page he had to work out where to put parentheses so the equation would work.  If it worked without parentheses, circle it.  (Obviously it was also an exercise in understanding Order of Operations.)

The way that Beast Academy is planned (they haven’t finished all the books yet…) the student will be ready for Pre-Algebra after competing level 5 (this is 4C, with one more to go in the 4th level) – I had my doubts when I heard that, but as we go on…I can see it.  Michael is going to have a completely different, deeper understanding of math than any of his siblings..and it will be better, I have no doubt.

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