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# Tag Archives: Constructivism

## Putting the sense in Number Sense

The best teaching happens when you make a lesson visual, spatial and auditory. That’s why I love teaching kids number sense with a math balance.

Utilizing a math balance in a whole-class setting of twenty-seven kids would be tricky, but at home with one child it’s easy. The  balance we own came from Right Start and costs \$25.

5 does not equal 8. It’s so easy to see.

Just like it’s easy to figure out that there are many number combinations that equal 5.

In fact, we spent a full ten minutes just figuring out the number 5!

In pedagogy, we call this “Constructivism”.  It means learning a new concept through your own experimentation and discovery. Giving children the full Constructivist experience isn’t always possible, but a math balance makes it a lot easier.

## Math Without Worksheets

One of the best things about teaching math from a Constructivist perspective is the emphasis on metacognition, or “thinking about thinking”. In a good Constructivist classroom the teacher should know, and be constantly assessing, the thinking strategies her students are using to solve problems. One of the ways I accomplished this when I was a teacher was through small group white-board activities.  I would group three to five students according to ability level, and let them create and solve word problems on the white-board for me, while I reordered their thoughts and strategies.

From a Constructivist perspective, listening to how children solve problems is where the true art in teaching comes. You need to know when to ask probing questions, when to lead them on to more efficient ideas, and when to pick apart what they are saying. The idea behind Constructivism is not to let kids wildly flounder, but to gently encourage them towards efficiency and accuracy.

My son Bruce(6.5) attends a school where the district uses Houghton Mifflin Math Expressions, which I like, but would not describe as a Constructivist program. Although there are occasionally times built into the curriculum where Bruce is asked to write out his own math problem as part of his homework, or explain his thinking in words and pictures, there doesn’t seem to be the same heavy focus on metacognition that there was in my own classroom.

Last night after dinner, I decided to do an experiment. How would Bruce handle a third grade Constructivist classroom? How would he do if he was in one of my white-board groups, from long ago? It turns out, not so well, at least not without more practice. And let me tell you, after realizing this type of activity was a struggle for him, we will be working on it at least once a week from now on!

Here is our first example. I asked Bruce to tell me a subtraction story problem about the number 263. That was the first bit of difficulty as you can see!  Then I asked him to solve the problem with numbers on the left side of the paper, and to keep his work neat, and organized within the box. Second bit of difficulty! I also wanted him to write down his final answer and circle it, but I forgot to articulate this very well, so that was “my bad”. Lastly, I asked Bruce to explain his thinking to me while I reordered his strategies. This is when he really started to flounder.

Okay, let’s try again! If you look at this next example, you can see a slight improvement. The story follows the word problem, his number work stays to the left, and he circled his final answer…but his explanation? On paper it looks okay, but I really had to tease this out of him. It did not come clearly or easily even with me doing the writing for him!

In a Constructivist classroom, by the end of fourth grade I would expect a learner to be able to see a number problem, write a story to go with it, and then write out their proof of how to solve it both in numbers and words. I would expect to see clear thinking, organized work, and neatness. None of that, is as easy as it might sound.

Sometimes, these types of Constructivist activities make certain parents turn vinaceous with annoyance and anger that comes from misunderstanding. I would never expect a child to have to write about every problem they solved. Nor do I think it is fair to hold children back mathematically because they might struggle with reading, writing or language. But doing proof work like this once a week in a small group setting is really helpful for everyone. Teachers get insight into how children are thinking, kids learn from the strategies of each other, and mathematical communication skills get practiced.

As an at-home activity, this has the added bonus of being completely free. I am going to continue working on this type of proof work with Bruce in an Afterschooling setting. I like Houghton Mifflin and I am extremely happy with Bruce’s teachers, but I want him to be successful at work like this too.

## It’s the Teacher NOT the Curriculum, that Makes the Difference!

I recently came across a blog called Out in Left Field whose author espouses the polar opposite of all my views on educational theory.  Katharine Beals, PhD, rails against Balanced Literacy Instruction and Constructivist math in particular. Here are links to my own views on why I love Balanced Literacy Instruction and Constructivist math

Katherine Beals takes shots at Bill Gates whom she describes as being misguided, misinformed, and possibly having Asperger’s Syndrome.  She also rails against Stanford University professor Keith Devlin, also known as “The Math Guy” on NPR. I haven’t read any of Professor Devlin’s books, but I now want to read all of them!  I’m not so sure about Dr. Beals’s book Raising a Left-Brain Child in a Right-Brain World: Strategies for Helping Bright, Quirky, Socially Awkward Children to Thrive at Home and at School, but I might try reading it anyway just to fairly consider a point of view so opposite than my own.  (Note: if you have a child on the Autism spectrum, this book would be of a lot more interest to you.  Read the reviews on Amazon, and see why.)

Out in Left Field bugs me for a variety of reasons; she blasts teachers as sometimes being too stupid to teach math, she thinks academics often don’t know what they are talking about, she implies that many mathematicians are cowardly for not speaking up against Reform Math, and she repeatedly professes a belief that rote learning of traditional algorithms is the best way to create mathematical thinkers.  But what really bothers me, is Dr. Beals’s general thesis that Reform Math and Balanced Literacy Instruction are B-A-D- Bad!

Teaching with a point of view is not bad, not mater which pedagogy you choose.  I could spend the next fifteen minutes telling you why I support the particular education philosophies she hates, but that would be a waste of time.  What I know, is that it’s not the curriculum that helps children learn, it’s the teacher

If you give me a group of well-fed, middle class Kindergarteners from moderately stable homes I will teach them to read.  If you me a Whole Language curriculum.  I will teach them to read!  If you give me a Phonics Based program.  I will teach them to read!  If you give me a Balanced Literacy program.  I will teach them to read!  Magic pedagogy is not creating readers, good teachers are.

Educators have to teach whatever curriculum the school district hands them.  Good teachers deliver the curriculum as instructed, and then use common sense.  They see that little Johnny over there is going to be able to read by seeing a new cereal box in front of his breakfast bowl each morning.  Great!  Let’s make Johnny some patterned books.  Little Suzie over there?  She really needs more phonics.  Bring out the phoneme cards.  The reason I like Balanced Literacy Instruction is that it includes both.  Teachers have to be flexible!

Now, if you give me forty third graders coming back and forth from Mexico, sleeping next to refrigerators, scared by roving pit-bulls on the playground, no working smoke-detector in my classroom, a principal who downloads pornography in the middle of the school office, no support services whatsoever, and then fail to give me my first paycheck, I’ll give you my 110% best but I can’t make any promises.  Even if you give me a phonics based program like Open Court, I might not be able to teach all of those children to read unless you, the community, give me some help.   I’m a teacher not a miracle worker.

When I was teaching math at a Constructivist Charter school and I had kids who said, “I’m going to solve this subtraction problem in the traditional way,” and started to borrow and carry, that was just fine with me.  It wasn’t okay with all of the teachers at my school, but I was fine with certain kids using traditional algorithms when they wanted to.  Kids who have trouble verbalizing, kids who can think faster than they can write, kids who don’t do well in group learning situations… these are students that good teachers make common sense accommodations for.

Good educators teach in ways that accommodate the differentiated learning style of each student.  Some kids are going to need to be taught algorithms as a life-raft they cling to.  Other kids will become the high-schooler in Academic League who can just look at the question and know the answer.  It is unfair to force any one style of learning on all of your students, but it’s not bad to lead into instruction with pedagogy to provide framework and support.  As far as pedagogy goes, I think Balanced Literacy Instruction and Constructivism have a lot to offer.

P.S.  Ironically, both Dr.Beals and I love Story of the World.  🙂

## Hougton Mifflin Math Expressions, A Review

Those of you familiar with my blog will know that I am a former K-4 teacher who believes in teaching math from a Constructivist perspective. This review of Houghton Mifflin’s Math Expression is based on what I have seen as a parent with my “teacher glasses on”, so to say. I do not have any association with Houghton Mifflin whatsoever.

As a parent, I have had the opportunity to help my son complete the second grade Houghton Mifflin Math Expressions curriculum in a home-study way last year when he was in Kindergarten. Now my role has been simplified (yeah!) to only helping him with his third grade homework. I have also volunteered in my son’s Kindergarten classroom and helped lead many of the small group Kindergarten activities. This gave me a chance to see how the teacher’s guide was written and laid out. I also volunteered to tutor a Kindergartener through my church a few years back, so I got to witness firsthand how an English Language Learner responded to this program.

In general, I am favorably impressed with Math Expressions, but only because as a teacher I have witnessed, and been forced to use, some really awful math programs in the past. (Anyone familiar with Math Their Way?) Houghton Mifflin seems to be trying to strike a balance between Constructivism and traditional algorithm based approaches. This reminds me of the phonics vs. Whole Language debate that resulted in Balanced Literacy Instruction. Math Expressions encourages children to learn to solve arithmetic problems using multiple strategies, but often spoon-feeds the children what these strategies should be.

Pure Constructivists would be very unhappy with Math Expressions because of the way the program directly tells children how to solve problems. This is also a curriculum that usually insists that children “stack” problems (write them vertically) in order to solve equations. Pure Constructivists can sometimes advocate the opposite, and insist that children write problems horizontally. I personally, think children should be able to write problems however they want. On the other hand, Constructivists would be happy with Math Expressions in that it goes beyond borrowing and carrying, as the only way to solve problems.

As a former Kindergarten teacher, I feel very strongly that the Kindergarten curriculum moves way too slowly. The English Language Learner boy I tutored was capable of a lot more than what Math Expressions was having him do. This is also a horrible program for school districts who use the Alternate Day Kindergarten schedule. There is no way you could cover the whole year’s program and only teach math two days a week. On the plus side, the Kindergarten curriculum did include lots of hands-on games and activities to reinforce conceptual learning.

I think  my son’s school district has now been using Math Expressions for three or four years.  If I recall corectly, they were using Dale Seymour’s Investigations before that.  In the past five years, the district’s standardized math scores have gone up 19%, which does not surprise me.  Math Expressions seems to be a very “teacher proof” program (although I had that expression!). The teacher guide tells you exactly what to do. It is up to the intelligence and intuition of the teacher herself however, to make this program really great. As with any subject, good teaching is still essential with Math Expressions. Although if your child was stuck with a “bum teacher” (another expression I detest!), they would probably be better off with that teacher using Math Expressions, then a lot of other programs out there.

Finally, I have to mention one thing that is currently bugging me about the third grade program. My son’s homework keeps requiring him to write down his answer numerically and then make a proof drawing of his work. The problem is, my son often solves three digit problems in his head. This is really just a minor annoyance with the program. It is after all, important for mathematicians to be able to explain their thinking and show their work. Unfortunately for Bruce, a proof drawing has nothing to do whatsoever with how he is solving the problem!

If you are a parent reading this whose school district has adopted Math Expressions I would say “Relax! It could be a lot worse.” But if you are looking for ways to support math learning at home, please take a look around my blog because I have lots of ideas for you.

## Right Start Level D

For the past month and half my six year old son Bruce has been working out of the Right Start Level D workbook, which is at the 3rd-4th grade level. Even though it is summer vacation, he is still doing math each day because the rule in our house is that two pages (or one side front and back) of math earn 30 minutes of screen time. So I’m not making Bruce do math all summer, it is his own personal choice.

We do not own the Right Start Level D Teacher’s Guide, but we do have all of the manipulatives at our disposal. I am also familiar with teaching in the Construcvist method and have read the Level C Teacher’s Guide cover to cover. So we even though we are not following the program exactly, I am still able to deliver a lot of meaningful instruction to Bruce, that is a huge step beyond plunking him down on the dining room table and having him do math worksheets.

I don’t know if you can tell from the blown up picture of what he was working on today, but Bruce has quite a sense of humor! He is also still struggling with number reversals. At this point in his math education, I am having him go back and correct the reversals each time. Today however, I chose to ignore them because Bruce was trying to do extra math to earn watching a movie.

We are about 22 out of 150 pages into the Level D workbook, and so far I’m pretty pleased. I think that the workbook can stand alone as an Afterschooling supplement, but that if you were using this program for Homeschooling, you would definitely want to get the Teachers Edition.

## Review: Right Start Level C

We have been using Right Start Level C as part of Bruce’s afterschooling for about seven or eighth months now.  We switched over to Right Start, after Bruce completed year of Horizons 1st grade math.  I thought Horizons’ first grade program was okay, but that the second grade curriculum focused too much on algorithms.  For more information on our experience with Horizons, please see here: https://teachingmybabytoread.com/2011/05/01/horizons-math/

I believe in teaching children math through the Constructivist method, which is where they discover mathematical concepts and strategy themselves, sometiems through hands on discovery. I was very impressed by the philosophy of the Right Start program which stresses understanding over rote memorization.  Additionally, I was familiar with Right Start, having used components of the program when I taught 3rd/4th grade at a Charter school.

I decided to shell out out the big bucks and bought the homeschool deluxe set for level C, and I am really glad I did.  It has tons of math manipulatives that are useful for Bruce, Jenna, and other math activities we do as well.  It also has all of the books, including the Teacher’s Guide.  I used the Right Start manipulatives a lot to help explain concepts in Bruce’s school math program, Houghton Mifflin Math Expressions, and we have also used them to help with Life Of Fred Fractions.

The Right Start program has a teacher’s guide that is very creative, detailed, and goes above and beyond what is just in the worksheet book.  If you were using Right Start as a stand-alone curriculum, you would really need the teacher’s addition.  We were using the program as a supplement to Math Expressions, so we didn’t follow the lessons plans exactly, although Bruce did do almost all of the worksheets.

Since I’m a teacher myself, I read the teacher’s guide almost cover to cover, and then just “winged it”, or taught the lessons intuitively according to all the training I’ve received in teaching mathematics from a Constructivist perspective.  But I would definitely recommend following the teacher’s guide, to anyone who was new to the Constructivst approach, or who was using Right Start as their child’s sole mathematics curriculum.

The hallmark of the Right Start program is the abacus, which I was really excited about when the box arrived.  My husband is an engineer, who has had the opportunity to work with a lot of coworkers from China, many of whom are abacus devotes. (But a different type of abacus, I should point out.)  One of these friends told my husband he could “see the abacus in his mind,” and that’s why he had such exceptional mental math skills.  This is the same claim that the author of Right Start makes.

I am still super excited about teaching with an abacus, and intend to do this with Jenna as soon as she is old enough for Level A.  The problem with Bruce and the abacus, was that he rejected it from the get-go.  You know how there are little babies who reject the pacifier or the bottle?  You think, “Did that mother really try hard enough?  I mean, did she really try?”  Well yes in fact, I did!  Bruce would not have anything to do with the abacus at all.  Probably his math skills were developed enough already that imposing the abacus into his thinking was something his brain just did not want.  He was five years old at the time and had already finished the Horizons first grade workbooks, and half of Hougton Mifflin 2nd grade.  This is also party the reason that we could not follow the Right Start lesson plans exactly, because so much of them are abacus based.

Not using the abacus turned out to be okay for our family, because I am not teaching the traditional algorithms of borrowing or carrying to Bruce at this point.  The way the Right Start lessons set things up, they use the abacus to teach borrowing and carrying in a manipulative way.  So I would have skipped over those parts in the Teacher’s Guide anyway.   (For those of you thinking, “What the heck!  This lady doesn’t teach borrowing or carrying?”, please see my previous post on subtraction: https://teachingmybabytoread.com/2011/03/15/subtraction/.)

A final point of note.  Bruce has almost finished all of Level C, and I’m still stumped by what exact grade level it is.  I’ve thought about it a lot, and reread the 2nd, 3rd, and 4th grade standards in our state.  I’ve also taught 3rd grade for two years, and a 3rd/4th combo for two years.  And yet, I’m still a bit perplexed, because it introduces some concepts that aren’t usually taught in public school until later.  In general though, I’d say Level C seems to be in the 2.5 grade – 3.5 grade range.  They also have a good placement test online to help you determine which book to choose for your child.

## Square Numbers and Square Roots

Bruce is nearing the very end of Life of Fred Fractions, and one of the last lessons was about square numbers and square roots.  He had learned these concepts a while ago, when I taught him the basics of multiplying, but he was unfamiliar with the square root sign.  This lesson was a good review.  To me, it is also a perfect example of how a Constructivist approach to teaching mathematics solidifies conceptual understanding.

First we built out multiples of 2, 3, and 4.  Then we looked to see which arrays were square, and which were rectangle.

Next we labeled all of the numbers.  I then asked Bruce to identify the “square” numbers.

Since we hadn’t built out the fives, I had Bruce square 5 on the whiteboard.

We then moved onto the concept of square roots.  Bruce is still a bit confused on this point.  To be fair, we had just started the square root portion of the lesson when Jenna came running in and scattered all of the blocks!  We’ll have to try again later.

## Too Many Algorithims in Life of Fred

Bruce and I are now on chapter 27 of Life of Fred Fractions, and nearing the end.

I still really like the book a lot, but I’m less impressed by it as we near the end.  I think the author  relies too much on traditional algorithms to teacher mathematical concepts.  This is completely contrary to my approach to teaching math, which is Constructivist in philosophy.  (For more information on Constructivism, please see my post at: http://teachingmybabytoread.blog.com/2011/03/15/subtraction/)

Earlier in the book, it was easy to do a lot of Constructivist activities and explanations side by side with Life of Fred.  To compare, order, reduce, add and subtract fractions for example, Bruce experimented with the Right Start Fraction strips in addition to working with the algorithms taught in Life of Fred.  So when the book talked about 2/6 being the same as 1/3, Bruce really understood that well, because he could build those fractions himself.

The fraction strips were also helpful for learning about mixed numbers, and the rules about 0/7 = 0 and 7/7 = 1.  In all honesty though, Bruce had a really good understanding of fractions before we even started reading Life of Fred, because of all the times he played Reader Rabbit 2nd Grade math.  When he was four and five, we also played lots of games with my homemade fraction cards, such as Fraction War, Capture the Fraction, etc.

I had originally taught him how to reduce fractions, using M’nMs and peanut butter.  I’ll have to save the explanation for that one for another time!

Fast forward to now, when we are in the higher chapters of Life of Fred and it is now talking about multiplying fractions, and rules about canceling.  Now I can really tell that I never taught past fourth grade, or I probably would have had some tricks up my sleeve to teach these concepts from a Constructivist perspective.

In multiplying fractions, “of” meaning the same thing as “times” makes sense.  I’ll be able to explain this if I drag out some M’nMs and peanut butter again.  But explaining how canceling works, in a hands-on lesson, still eludes me.  If anyone has any ideas please let me know!  I’m definitely not satisfied with relying on the algorithm alone, which is what is offered in Life of Fred.

## Subtraction

Bruce finished off his major chapter on subtraction last month, and he wasn’t too keen on it at first.  I teach math from a Constructivist perspective, which means enabling children to develop their own meaningful strategies for solving problems, instead of just blindly teaching traditional algorithms.  Here’s a nice website that explains more about the Constructivist philosophy: http://mathforum.org/mathed/constructivism.html

Bruce, myself, his classroom teacher, and my husband were all in the trenches there for a few weeks as Bruce explored different strategies for solving triple digit, minus double digit problems with and without regrouping.  In normal classroom situations, Bruce would have been exposed to his classmate’s strategies and thinking, which would spur his own ideas, creativity (and showmanship).  But since he is working independently through the second grade book, I had to introduce some ideas to get him started.

The first strategy I introduced was solving equations on two abacuses, flipped over on side A, so that they represented 200.  (We were starting with problems no greater than 200.)  This went on for about a week, and Bruce was not impressed.  He does not like using an abacus at all, and so I never got to teach him the side B methods.  I was really bummed about this, because my husband worked with an engineer from China, who was just brilliant at math, and Chang said that when he solved problems in his head he could picture the little beads on the abacus sliding.  I wanted this for Bruce, but maybe I didn’t introduce the abacus early enough.  I’ll know better next time for Jenna.

The second strategy I introduced to Bruce was thinking about the problems in terms of money, and then counting out the equations in pretend change.  236-89 would become \$2.36- .89.  Bruce was not too thrilled with this strategy either.  He could easily solve problems this way, but under protest.

Finally I broke down and taught him one of the methods the book demonstrated, ungrouping, which uses pictorial representations of hundreds tens and ones.  I had resisted showing him this method, because Houghton Mifflin uses it as a precursor to borrowing, which I do not want Bruce to learn until later.  But I’m eating a big slice of humble pie right now because this is the method Bruce absolutely loved.

All of a sudden Bruce started tearing through his subtraction work at top speed.  He called this strategy “Busting open the hundreds”.  I’d usually draw out the squares, lines and dots for him on the white board, and he would erase them and then redraw things as he started working.  Pretty soon, he was solving problems in his head, by just looking a the picture and not even erasing anything.  Then he didn’t even need the pictures at all.

The way Bruce would solve the above problem is that he would think: “100-89 = 11.    11+36=47  100+47=147”  He doesn’t need to do any carrying or traditional algorithms at all, which is a testament to the Constructivist method.  Constructivism encourages children to think and understand what numbers really mean, instead of just blinding computing an algorithm, which can stunt their development of number sense.  Bruce’s accuracy isn’t 100% yet, but it is on par with your typical second grader.  I’ll teach him the traditional methods of borrowing and carrying someday, but not until Bruce can do even harder problems intuitively, in his head.