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Long Division and Constructivism

This is certainly a bizarre piece of scribbles to include on my blog. I might as well post a picture of chicken scratches, or sastrugas in the snow.  But it is an authentic piece of scratch paper from this past weekend when I was recording my son Bruce(6.5)’s thinking out loud while he solved a long division problem.  His current strategy is to think about long division in terms of multiplication, and to keep multiplying until he gets the right answer.

One of the biggest questions people have when they first hear about Constructivism is “How in the heck would you do long division?” The answer to that question is that there are lots of ways to solve long division problems, and that each child will explore and then settle on a strategy that best makes sense to that particular child’s brain.  Maybe that will be choosing to use the traditional algorithm, maybe not. 

In a true Constructivist program, the traditional algorithm would not be introduced until the child had already mastered several other methods.  If a teacher introduces the traditional algorithm too early, thinking and exploration could suddenly halt, which would really crimp the development of true number sense.

This is why it is important for me as a parent to give Bruce lots of opportunity to learn how to do long division at home many months before he learns at school through Houghton Mifflin Math Expressions, which is not “Constructivist enough” in my opinion.  Stay tuned for more examples of creative ways to do long division!

More Math Without Worksheets

My six and a half year old son Bruce has been doing Math Without Worksheets to earn screen-time recently. I write down a number problem, and he writes down a word problem to match. Then he solves the problems numerically on the left-hand side of the page, and in words on the right. This type of Constuctivist activity is the nidus of someday being able to write out proofs in higher level math.

Bruce has been solving problems this way once a week for three weeks in a row now, and you can already see a really big difference in his work. For one thing, he’s writing down his thinking himself now, whereas before I had to write down his answers for him. But there is still a lot of room for improvement. As a former teacher, these are the things I am looking for (rubric style):

  1. The word problem should match the number problem. (+3)
  2. The word problem should be neatly written and in its own space. (+3)
  3. Numerical work should be neat, tidy, and in its own space. (+3)
  4. The numerical work should clearly show the strategy being used. (+3)
  5. The answer should be circled. (+1)
  6. The written explanation should be neat, tidy, and in its own space. (+3)
  7. The written explanation should clearly show the strategy being used. (+3)

If I was a third grade teacher scoring Bruce’s work in the above example, I would give it 12 out of a possible 19 points. The word problem, as well as the written explanation, both need a lot of work. Numerically, Bruce is showing that he knows that since 20 divided by 5 = 4, then 20 divided by 4 is going to equal 5. That’s a solid strategy for solving the problem.

Here’s another example from the same day that shows a lot of improvement. In this next problem, Bruce’s numerical explanation is even stronger, but the written parts are still a bit confusing. I’d give this example 15 out of a possible 19 points.

For the twenty minutes Bruce spent working on these two pages I let him earn half an hour of Lego Ninjago on the computer. Am I a “Tiger Mom”? No. Am I looking for extra ways to boost my son’s academic potential? Yes! The best part of all of this is that it is free and rooted in solid Constructivist pedagogy. No matter what math curriculum your child is using at his or her school, this type of practice can help.

Two Year Olds and the Quantity 3

Last week’s exploration into determining whether or not Jenna(2.5) could visualize the quantity three sent me searching in my Right Start Level C box, to see if I could find any literature about the subject. Bingo! I rediscovered Math and the Young Child, and fell in love with Dr. Joan Cotter and her work all over again. (The link takes a little while to load btw.)

On page 3 of the Transitions Lessons book, Dr. Cotter writes about Dr. K. Wynn’s paper “Addition and Subtraction by Human Infants” and the experiments she did with babies and teddy bears. She would show a baby a teddy bear and then put the bear behind a sheet. Then she would show the baby another teddy bear and put that bear behind the sheet. When she lifted the sheet, the baby expected to see two teddy bears. If the adult added a third bear behind the sheet and the baby saw three teddy bears instead of the expected two, then the baby looked at the bears longer. By tracking infant eye gaze she found that five month old babies could tell the difference between 1, 2, and 3 bears.

Does that really mean that infants understand the quantity 3? Or does it mean that infants understand 1, 2, and “more”? I’ve played the chocolate chip game several times with Jenna now, and she still can’t consistently name the quantity three even though she is 31 months old. But I hadn’t been measuring her eye contact. Maybe she was looking at the quantity three longer and I hadn’t realized.

So I tried my own version Dr. Wynn’s teddy bear experiment. I recreated it as best as I could by myself using a pizza box and some Calico Critters.

It really did seem to startle Jenna when there was an unexpected number of Critters behind the pizza box! I was witnessing the very foundations of addition and subtraction development in her brain! This experiment would seem to indicate that Jenna can visualize and recognize when something is “more than two”. But when it came to articulation, she still hits a brick wall. If I asked Jenna to name a quantity, she could verbalize one and two, but started wildly guessing when I showed her three Critters.

This lack of ability to name the quantity is not immutable; it is only a matter of time before Jenna will be developmentally ready to start quantifying objects. Once she can name quantities up to five, I’m going to be starting her on Right Start Level A. I have no idea when this will be, but I am really excited about using Dr. Cotter’s methods from day one with Jenna, instead of floundering around with other programs like I did with my son Bruce. (And no, I’m not a paid spokesman for Right Start! I’m just a passionate Constructivist.) 🙂

Singapore Math and Constructivism

I am a big believer in teaching mathematics from a Constructivist perspective, which means enabling children to develop their own meaningful strategies for solving problems, instead of just blindly teaching traditional algorithms.  It also means giving children time, space, and materials to explore mathematical concepts and create their own understanding, before you start imposing your own thinking upon them.

One of my favorite at-home curriculums for teaching math is Right Start, but I have always been curious about Singapore Math because it so popular with homeschoolers.  It is also popular with families who are Afterschoolers, even if they have never heard about that term before.

Even just a casual internet search will tell you that many parents who are confused about Constructivism, or unhappy with how it is being implemented in their children’s publics schools, choose Singapore as the at home alternative to help get their children “back on track”. I have even seen a blog that is very much dedicated to how much better Singapore Math is than current Constructivist textbooks.

The problem that I see with all of this public school curriculum bashing is that when I look at the Singapore Primary Mathematics Standards Edition textbooks, I see a lot of Constructivism.  In fact, there is enough of a Constructivist influence in the 4A book, that I have no problem with my son Bruce(6.5) using it on an Afterschooling basis.  Here are the main, Constructivist elements that I like about the 4A book:

  • Metacognition: There is an emphasis on helping students think about thinking.  There is also guidance through the thinking process.  This is very Constructivist!!!
  • Spiraling Curriculum: Most concepts are taught, and then revisited again and again throughout the year.  The Constructivist public school curriculums that I have seen also use this spiraling method.
  • Visualization: There are a lot of pictures and modeling.  A good Constructivist classroom would also encourage pictures, drawing and modeling to help reach visual learners.
  • Multiple Strategies: Again and again I keep seeing examples of more than one way to solve a problem.  Alternatives are shown beyond just traditional algorithms.

This is really surprising to me because some of the most vocal critics of Constructivism I have seen online are also parents who choose Singapore as their children’s math program.  They tend to keelhaul Constructivism and hail Singapore as their mathematical savoir. This is really bizarre, because comparing Singapore with a public school Constructivist curriculum like Dale Seymour Investigations is not like comparing apples and oranges; it’s more like comparing apples and pears.

If you were to think about math as a continuum with Back to Basics “just-teach-her-to-borrow-and-carry” on one hand, and pure Constructivism “she will discover every new bit of knowledge herself” on the other, then in my opinion, Singapore would not be considered a Constructivist program nor would it be considered a Back to Basics program.  It would fall somewhere in the middle, like Houghton Mifflin’s Math Expressions.  Falling closer to the Constructivist end would be programs like Right Start, Dreambox Math, and Hands on EquationsSaxon, Horizon and Life of Fred would be closer to a Back to Basics philosophy.

So what do I think of Singapore Primary Mathematics Standards Edition?  I think it’s pretty good, but for a complete homeschool program I still prefer Right Start.  For Afterschooling though, I can see the benefit of using a cheaper, more colorful program like Singapore, if your child preferred it.  The pages are smaller and easier to complete, so if your kid gets a “buzz” from completing pages, then Singapore would facilitate that.

If you do use Singapore Standards edition for a homeschooling program, then you really would need to buy the complete program, including the word problem book.  As a former public school teacher I have to say that there are not nearly enough word problems in the textbook alone to adequately prepare kids for state standardized tests.

Our own experience with Singapore remains limited.  I brought the 4A textbook home for my son Bruce in the middle of Christmas break last week and he completed 23 out of 161 pages in about three days.  Part of this is because the Singapore 4A geometry section is much easier than the third grade Houghton Mifflin Math Expressions work Bruce has been doing.  The multiplication section was all review for him too, because he has completed Dreambox 3rd grade.  But Bruce liked the Singapore textbook.  It is a lot more “fun” to look at than Right Start Level D, and he willingly plowed through pages in 4A.

For a more experienced opinion of Singapore, I’d like to include my friend Claire’s earlier comments from my K-1 Summer Bridge page:

For parents who have sticker shock at the price of RS, I would recommend Singapore Primary Mathematics. Note that these are *NOT* the “Singapore Math” workbooks sold at Barnes & Noble (those are by a different publisher and are substantially “dumbed-down” from the original program).  I prefer using a “hands-on” program like RS to a workbook-based one like Singapore in the primary grades, but RS is pricey and very parent-intensive. Also, Singapore is easier to accelerate and/or up the challenge level for a bright student.

I would recommend getting the Primary Math textbook and either the workbook or the Intensive Practice book (depending on whether the student is average or advanced).  The Intensive Practice book is only available in the U.S. edition but it is very easy to match up the topics in the Stds. ed. text with the ones in the IP book. The Stds. ed. text is full-color and has more of a “cartoony” look to it. I actually prefer the “cleaner” look of the US ed. books myself, but overall feel the Stds. ed. is better.  The books are available at Singaporemath.com, Christian Book Distributors, and Rainbow Resource Center.

Thank you Claire!  Finally, I would like to add my own thoughts about Singapore Math and gifted children.  Just because my six year old can burn through 23 pages of 4A in a few days doesn’t mean he should. Gifted children deserve instruction too.  The deserve attention. They deserve good teaching, and they deserve experiences. They do not deserve to have their curiosity or love of learning drowned in busywork, endless workbooks, or too much isolation.

I think that parents of gifted children should have a clear and consistent message. Too often society says: “Oh, that child is so smart.  Just send him off in the corner with an advanced book and he will be fine.”  In my opinion, that does a huge disservice to gifted children.  I would hope that nobody reading this would use a curriculum like Singapore that way.  Instead, here are some of my favorite 4th grade level activities that you might consider combining with the 4A Standards edition, to “jazz it up” a bit:



 More to come!

Hands On Equations is AMAZING!!!

(From Lesson 6)

Can you believe a 6.5 year old can do this? Granted, my son Bruce is very bright for his age and is capable of doing third grade math, but this isn’t a case of “My son is so smart!” or, “I’m such a great teacher!” 😉 This is an example of brilliant, Constructivist curriculum at its finest. Can you imagine how easy Algebra 1 is going to be for my son someday if he can already do problems like this in first grade?

I’m not a representative for Hands On Equations and I don’t have any affiliation with the company. I just think this program is awesome. I really want to write a grant and bring HOE to my son’s school, but I haven’t even broached the subject with his teachers.

As a former teacher myself, I know that I really have to watch myself. I don’t want to be pushy, I don’t want to be the problem parent, and I just want to let Bruce’s wonderful teachers continue doing a wonderful job as they see fit. I know as well as anyone that public school teachers can sometimes be under enormous pressure to strictly implement district-approved curriculum, without any room for creativity. So for now, HOE is just a fun activity that we are doing at home. But oh my goodness! The teacher in me sooooo wants to run a HOE small group during centers time. I know those first and second graders and they would eat this up.

3-4-5 Triangles

This weekend Bruce(6.5) and I had fun building real life triangles. I was inspired to try this activity after reading Keith Devlin’s thought provoking book, The Language of Mathematics. Unlike The Math Instinct, The Language of Mathematics focuses more on the history of mathematics rather than best practices in teaching mathematics.

Dr. Devlin wrote The Language of Mathematics for the general reader, but I found it quite challenging even though I was an A- student in mathematics all the way through college level Calculus. I wish I had read this book back in high school, because it might have helped me knock off those minuses!  On page 44, he discusses 3-4-5 triangles and ancient builders.

A 3-4-5 triangle is also mentioned in a children’s book Bruce and I read recently by Avi called The Barn. That too, was an excellent book although I would not recommend it for children under ten (oops) because it dealt with a really depressing storyline. In The Barn, the children use 3-4-5 triangles to create the right angle of the barn they are building for their father who is dying after suffering from a stroke. Here is how we built our own 3-4-5 triangle:

First, Bruce measured out 12 segments on masking tape that was folded on itself so it wasn’t sticky. He used Legos as his nonstandard unit of measurement. It would have been better to tie knots in rope or yarn at 12 equal lengths, but I knew that tying knots with a six year old would have added about ten minutes of utter frustration to the activity, and fine motor skill improvement was not my learning objective. Plus, I couldn’t find any rope. 🙂

At the 3 and 7 marks, we pulled the triangle tight. This picture doesn’t do the triangle justice, because I was also trying to hold the camera, but we did end up forming a right triangle with sides that were equal to 3-4-5 Legos. This triangle was also scalene because all three sides were different lengths.

After building our right triangle, we built a giant Isosolese triangle on the ground for fun. We also got out our protractor and measured angles, but as you can probably tell from the picture, since we had built our triangle with masking tape instead of rope, the angles were off by a few degrees. Bruce had previously had trouble remembering what an Isosclese triangle was, so hopefully this helps him remember. You can fit an “I” in an Isosceles triangle. Hopefully he also remembers what a right triangle is too.

Does Your Two Year Old Know the Quantity 3?

In case you are wondering, I’d offer a hesitant yes to this question regarding Jenna(29 months). She correctly identifies quantities of three about 80% of the time. There is no way she is ready to move on to four yet, despite her ability to “count”, i.e. rattle off numbers without correspondence. I got a pretty accurate understanding of where my daughter’s thinking currently resides, by trying out lesson 1 from Right Start Level A. You can download the first few lesson plans yourself for free and give it a try with your own two year old.

I wish I had known to try this experiment with Bruce (6.5) when he was two, because of course I’m now uber-curious what his thinking was back then. When did the quantities three and four really solidify for Bruce? When will they solidify for Jenna? The Psychology major in me is going to be periodically checking to figure this out. Looking forward in lessons 2 and 3 of Right Start Level A, the other two abilities to monitor are A-B-A patterns, and sorting.

The reason why I decided to investigate all of this to begin with, was that I have just finished reading Keith Devlin’s wonderful book The Math Instinct: Why You’re a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs). This book was well written, meticulously researched, and thought provoking. The final chapters reaffirmed everything I learned about Constructivist math in my professional development as a teacher. The research he presented about young infants understanding the quantity of two was especially fascinating. I had previously been proud of Jenna understanding the quantity two. Now I realize that’s no big deal!

I’m going to continue on reading more of Keith Devlin’s books. I’m not a “mathy” person myself, but he writes in a way that is easy to understand… even for someone who hasn’t studied Calculus since 12th grade.  🙂