Inside the Mommyvan

Homeschooling & Life Inside the Mommyvan - an old dog learning new tricks

math

If you have kids in public school, or if you spend any time on social media, you’ve probably heard, read, or written at least one rant about the new “common core” math (which isn’t really new at all, but that’s a whole ‘nother story).

Parents and teachers are understandably frustrated trying to help young students learn problem-solving techniques that they don’t understand themselevs, and which seem so terribly complex compared to the “carry-the-one” method we all learned in school.

Let me direct your attention to two words in the paragraph above: problem-solving. The object of these convoluted techniques is not to teach kids how to most efficiently find the answer to a particular calculation, but to illustrate and illuminate mathematical concepts and engage students in problem-solving strategies. Both of these are critically important if our kids are to become mathmatically literate – not just cranking out answers by rote, but understanding why those mechanical algorithms work, what’s happening to the numbers under the hood.

solving multiplication conceptually

This is our chalkboard after my kids, as a team and with some guidance from me, produced the answer to 3421 x 54. After we worked through the whole thing, they shrieked when they tapped the numbers into their calculators and verified their answer.

Yes, it took us ten or fifteen minutes to work through. Yes, it involved drawing pictures, invoking “5 times 10 apples” repeatedly to get past the multiplying thousands hurdle, and many more intermediate steps. No, I don’t expect anyone who hasn’t researched and learned this method to understand how we got from Point A to Point B… though I am happy to explain (to the best of my ability) how and why we did it this way to anyone interested.

At the end of it all, my second graders, my 6 and 7 year-olds who are just beginning to soak times tables into their brains, understand that they are able to multiply huge numbers just as easily as they do 2 x 2. They are believers, but better yet, they are understand-ers. They are learning not only how to do multiplcation, but what it means and how to apply it to other situations that involve multiplying numbers (as well as realizing which situations do call for multiplication, because problems in real life don’t come with nice neat vertically-arranged, place-value-aligned numbers to manipulate).

This is what learning math is all about. This is the math that they will use as adults. This is the math that is so much more than memorization and rote calculation, that will live inside them as one of the many problem-solving tools they acquire through their school years, that will make them truly mathmatically literate as adults, even if it’s not a focus of their higher education.

I understand the frustration surrounding “new” math (whatever it is that’s “new” this generation), but there is a baby in that messy bathwater and it’s in the best interest of all of our kids to not toss that gem out with the confusion and misunderstanding.

We use the Life of Fred books from Polka Dot Publishing as a supplement to our regular math curriculum. The kids usually read the Fred stories with Daddy in the evenings or on weekends.

These books follow the adventures of Fred, a five year-old professor at KITTENS University, and his doll Kingie. As we join Fred’s very silly daily life (one day, he watches as the campus bell tower tips over; on another occasion butterflies fly out of his office window), we discover that he uses math everywhere he goes. There’s so much more to Fred, though. The stories are not just silliness and math, they’re full of all sorts of interesting information about topics ranging from astronomy to yurts. There are a handful of questions at the end of each chapter, some about math and some about other topics covered in that chapter or previous ones.

This is fun math. How much fun? One daughter created her very own Life of Fred book:

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(The spelling, yes, I know. We’re doing 2nd grade, I don’t fuss over spelling on just-for-fun projects like this one)

I’m not a fan of the Common Core initiative. It has some good points, but the realities of implementation are overwhelming those with negatives.

You may have seen one of the “Common Core” math homework pages floating around the internet. They show what seems to be an over-complicated exercise to solve a simple addition or subtraction problem that the parent could do in 5 seconds using the “old” way we were taught math.

These purport to show how ridiculous the new standards are, but that’s a short-sighted view. There are many reasons to dislike Common Core, but this method of teaching math is not one of them. This isn’t even a new way of teaching. It’s been used for years in other countries, and most of those places are far ahead of US students in mathematics. Doesn’t that sound like something that might be worth a closer look?

Teaching Student Centered Mathematics

Arithmetic for Parents

I have another guest post at Parentwin:

Last fall, one of my young students began to struggle with a particular math concept. In his case it was adding mubers with sums just beyond the next ten, like 8+7 or 43+9, and doing similar subtractions “across a ten.”. i put that away for a bit and moved on to some different math topics, thinking maybe we just weren’t quite ready to tackle that. The “Asian math” curriculum we’ve been using as our primary is known for being fairly rigorous and fast-paced.

When i revisited it in December, the results were no better; if anything it was worse. i tried every teaching methid i could think up or read about, but nothing seemed to stick with this child. ALL of the manipulatives came out: the unit blocks, the base-ten set, the abacus, the ten frames. i drew oictures and diagrams. I explained with words and we counted on our fingers. We used online programs and iPad apps to make it more interesting. I offered bribes and made dire threats. He could get to the correct answer by brute force (and, interestingly, he had many of the sums between 10 and 20 already memorized) but i could tell that he just wasn’t gettting the key concept.

(That concept, for those interested, is that the “ones” being added are split into two parts. First enough are “given” to the other addend’s ones digit to complete “the next ten” and then the remainder become the ones digit of the sum. 28+5 becomes, first physically with blocks or abacus and then on paper with little tens-and-ones pictures and finally with numerals, (28 + 2) + 3, and on to 30 + 3, and finally 33. That they learn this before the old “carry the one” vertical addiition algorithm is critical to developing strong mental math skills.)

We’d hit a brick wall. This child was going nowhere, and I had exhausted all of the topics with which I could work around this one. If we were going to progress, I had to find a way to get this idea into his brain. My patience was wearing thin at this point, and i was about ready to throw in the towel and… i don’t even know. We even tried an outside enrichment program, to no avail (it wasn’t a very good one).

Finally, I took a leap and putchased another popular math curriculum. I’d prevoiusly shied away from it because it seemed to have a lot of busywork, drill quesns which looked like duplicates of work we were doing online. it wasn’t cheap for something i wasn’t even sure we’d use, but i was desperate. It devotes a couple dozen pages to slowly building this particular topic up, step by tiny step. Surely the kids would be bored before we were halfway through, going over and over the same material.

I pulled every page relevant to our trouble topic out of both the main text/workbook and the supplement. i reviewed the first baby step with our manioulatives. I took a deep breath, and set the first page in front of him. He breezed through it! We tried two more pages the next day… same result. I could see the light bulb flickering to life! Before long, he’d made it through the entire section. Best of all, he’s gotten a taste of success where previously there had been only frustration, and he’s enjoying it! He is now doing sums in his head that he could previously do only with base ten blocks and lots of coaching.

Often, a failure in the classroom – even a homeschool classroom – is unilaterally placed on the student’s shoulders. It’s inattention, carelessness, laziness or willfull obstinance, even a learning disability. For some students this is accurate, but before slapping one of thise labels on we need to be sure it’s not instead a failure of the teaching. As homeschoolers, we have the luxury of slowing down, even backing up to try a different teaching method or curriculum, but we must remember to take advantage to that and not be slaves to the checkboxes in our lesson planners. In our case, a simple change from one math book to another was the ladder we needed to hop right over that brick wall we’d slammed into a few months back.

 

Graphs are fun. It’s a nice break from the fact-memorizing, the math-sentence-building, answering “Do I do this with plus or minus?” for the 73rd time this morning.

Today was graphs. Embracing the Singapore Math model to its fullest, we built graphs with colored shapes. We identified one problem with using different shapes in our graph, in that they don’t line up with one another very well. We moved on to making math link cubes, in matching colors, to represent each shape.

From there, it was on to picture and symbol graphs on paper, plus a Discovery Education video showing different types of bar graphs. Finally, we did a few SM worksheets, reading graphs printed on paper, into which were snuck a few difference problems. It was fun to watch the kids pull a lesson from last week out of their brains, finding “how many more” and “how many fewer” using subtraction sentences. I’m not sure they even realized they were doing it, but I sure did, and it is reassuring to know that some of it is sinking in!

Lego bricks are a wonderful thing.

Anyone who spends time around kids knows how great they are for young children, encouraging creative play, fine motor development, spatial reasoning (a fancy name for understanding how things work in a 3-D world, a very early precursor to mechanics / engineering / physics concepts), sorting into various different categories (color, size, etc. — this is a cornerstone of both early math and science), even teamwork and following directions (for the kits that come with building plans). Later on, a world of motors, gears, and programming robots awaits to take older kids deeper into the increasingly important STEM (Science, Technology, Engineering, and Mathematics) areas.

I’ve even heard it said that they could be used for home defense, as any parent who has found Lego bricks in the dark with their bare feet can imagine.

There are many ways that Lego pieces can be used as math manipulatives. Just do a web search for Lego and math, and you’ll discover tips and lesson plans for everything from preschool-level color identification, sorting, and pattern-making to fractions, graphing, statistics, and more.

One concept that popped up here during Lego play one day was multiplication. Although it will be a while before my own kids are memorizing their times tables, they are already “getting” the concept of multiplication while working on addition: they know that 2 + 2 = 4, and that 4 + 2 = 6, so it’s a short Lego bricks logical jump to 2 + 2 + 2, or three twos, is also equal to 6. Did you get that? Three twos make six. 3 x 2 = 6.

They didn’t quite get it at first, but one of the most common Lego bricks illustrated the concept perfectly. See the red one there in the middle? Three rows of two studs each. 3 x 2. I pulled a few more bricks from the bin and we looked at their “multiplication stories” — 2 x 6, 2 x 2, 4 x 4, 1 x 4 — which happen to be very similar to the names commonly used to refer to the size of a brick or plate (“one by four” or “two by two”).

Suddenly I saw light bulbs flicker to life above my little students’ heads. Multiplying is almost the same as adding, which they already understand. In fact, multiplication is adding, just a shortcut to adding the same number together many times. As is true for other math ideas, it’s a lot easier to see in some concrete way for the first time, or the first several times. When the manipulatives are something as fun as Lego, the learning sometimes seems to happen all by itself!

We’re working on basic math facts right now, and as any parent of a grade-schooler knows, this part — as with most things that require rote memorization — isn’t much fun. It’s essential, though, for them to have the basics drilled into their heads until they come as naturally as walking, because these are the tools they will use when they’re doing more advanced math later on.

The basics right now are the “number bonds” (as the Singapore Math curriculum calls them) up to ten. That means that they’re working on all the ways to add two numbers to make any sum up to and including 10. I especially like the “number bonds” concept because it encompasses subtraction and the beginnings of algebraic thinking using single-digit numbers, without much more work on the part of the student.

My younger readers (or those with older kids) may already know this, but for those of us who attended grade school in the stone age, it’s a new way of thinking. Here’s the old way: addition facts (1 + 1 = 2, 1 + 2 = 3, etc.) now, subtraction next year, word problems after we’d memorized the number facts, and algebra much later on.

The new way is to learn number bonds or number families, and the whole & part operations related to a number family all at once: for example, we worked on 5s today. 5 has three number bonds, one for each set of addends: 0 + 5, 1 + 4, and 2 + 3. For each “family”, they wrote down the addition and subtraction facts, plus some missing number problems — this is the algebraic thinking that will serve them well in years to come — e.g. 2 plus what equals 5?

There are also “number stories”, where pictures and words are used to help the kids grasp the idea of part and whole — a concept as important as the number facts themselves. The picture might show a group of 6 animals with various distinguishing characteristics: standing vs. lying down, different colors, babies vs. adult, and so on. The student is to come up with as many different “stories” as they can about the picture, such as “Three dogs are sitting and four are standing up. How many dogs are there altogether?” This reinforces the part-and-whole thinking that gives meaning to the addition and subtraction facts they are memorizing at the same time: 3 (sitting) + 4 (standing) = 7 (all of the dogs). For a subtraction-based story: of the seven dogs, four are standing up; how many are sitting (translation: 7 – 4 = __ )? This is sure to take some of the pain out of deciphering word problems later on too!

At the end of the day, though, they still have to memorize those basic number facts, and repetition is the way to get there. Nothing says the repetition has to be bland worksheet after worksheet, though, so I have been on the hunt for ways to make these drills fun. Ihit on some real winners yesterday, activities from The School Bell’s Number Family area. On the Worksheet Packet page, the “T-Bar & Puzzle” worksheets were a big hit! After writing all of the addition facts in the T-Bar section, they got to color, cut, and paste the puzzle pieces… and then write the numbers again underneath each completed block. The Number Family Booklets are also turning out to be fun — I’ve skipped the circle mat and counters, and just let them draw little X’s or spots instead of writing the numbers in each half of the circle on the booklet pages.

I hope these resources help, and please share anything you’ve found to make this math memorization process more fun!

In my desperation to do some Christmas shopping that didn’t involve a web browser and a UPS truck, I packed up the kids and took them to the mall. Christmas was still a couple of weeks away, so this is not quite as insane as it might otherwise sound.

We did succeed in picking out a few gifts for the kids to give Daddy and their big sister, and I even managed to work in a brief lesson about money, price tags, receipts, and credit cards over lunch. This was a powerful reminder that I really can find those teachable moments anywhere, if I slow down and look around. I am certainly guilty of rushing ahead toward a goal or destination, completely missing all the things my children could be learning along the way!

We’d pretty much finished our shopping when we walked past the new movie theater, with its Wreck-It Ralph posters. I decided that 1pm on a Monday would be a great time to take them to their second movie ever (on the big screen, anyway; we watch plenty at home). In the empty theater, I was able to point out to them that most other kids were in school while we were shopping and going to the movies — those little nuggets I like to tuck away in their brains for the times they think that they want to go to regular school instead because I’m working them too hard that day.

20121120-073954.jpg“Making tens” is a key concept in early math leaarning, and as usual I am always on the lookout for ways to get my kids to practice math that don’t involve those boring worksheets.

This game is one I have read about in a few different places, and now I can’t recall any of them. The idea is simple: give the child a single-digit number, and have them come up with the number needed to “make ten” — that is, the difference between the given number and 10.

My version is equally simple. Supplies needed are an egg carton and a dozen beads in each of two different colors. Of course, there are many things that could fill in for either or both of these items, as long as they fulfill two requirements: the “board” needs to have exactly ten spaces, and the counters need to be of two distinct types — color, shape, material, anything that makes them visually distinct from each other.20121120-074006.jpg For the egg-carton version, just cut off the lid and flap, plus two compartments from one end. Making and decorating the board could be a craft project as well, and might make playing even more fun.

When my kids played, I had them take turns giving the first number and answering with the corresponding “make ten” number. The first player counts out beads of one color into the egg carton to make the starting number, the next then states their answer out loud. The answerer then counts out beads of the other color into the remaining compartments to check their answer. Because the board has exactly ten spaces, the game is self-correcting and can be played without an adult verifying answers. The visually different counters let the kids see the two distinct sets put together to make ten, and I had them repeat the two numbers after they found the answer to help cement it in their mnids.

This isn’t a game that will keep kids entertained for hours, but it beats worksheets, doesn’t make a mess, and can be brought out for a little while every day. You can run through all of the combinations that make ten in a few minutes, and with the addition of a jelly bean or other small reward for correct answers there is some added motivation to think about the numbers instead of just tossing out a guess.

20121003-111730.jpgNumber bond? What’s a number bond? Simple, it’s a number (often inside a circle) with two smaller (circled) numbers connected to it. The top number is the sum of the two bottom numbers. From a number bond diagram, one can derive two addition and two subtraction facts, or ‘number sentences’: for 7, the two smaller numbers might be 3 and 4. From there you get 3 + 4 = 7, 4 + 3 = 7, 7 – 4 = 3, and 7 – 3 = 4.

The Singapore Math workbooks we use (Essential Math: Kindergarten B currently) use the concept of ‘Number Bonds’ to help students understand the relationship between operands and sums or differences. A book I am reading, Arithmetic for Parents, discusses using names and terms for everything, emphasizing that children love special names and explicit wording. “They are proud of their ability to use them,” it says, and I find this to be true with my children. 20121003-111956.jpgThey stand a little taller, speak more seriously, when they talk about ‘number bonds’ so I’m going with it.

Now that my kids have a pretty good understanding of how to do addition and subtraction a few different ways, it’s time to start memorizing. I have decided to post the number bond diagrams for sums 0 through 10 on the walls to help. Since we have a rainy morning with nothing else going on, let’s turn it into a craft! (I was going to make it a craft anyway, but a rainy day makes a nice excuse, doesn’t it?)

What I used: cardstock in several different colors, tacky glue, assorted embellishments (beads, sequins, jewels, buttons, etc.), plus a ruler and pencil to mark the numbers and dividing lines. I’ll use a Sharpie marker to go over the numbers after the glue is dry.20121003-111829.jpg
I made one page myself as an example, and the kids are taking turns making the rest. The only rules I’ve given them is to keep the beads away from the numbers & lines, and each number gets its own shape — all of the markers for each number are of the same shape, and each number on a page has a unique shape. Colors and sizes may vary a bit, since our bead collection is not large enough to find many exactly the same. You might decide that each number will have a color, or an item cut from magazine pages… anything that makes each number group cohesive when you look at it. Once the glue is dry, I will hang the finished products around the room to help with math lessons and memorization of addition facts up to ten.