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# Math

 “Teaching with understanding depends upon building on what the child already knows. Teaching by rote does not care.” (Dr. Joan Cotter)

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.

I believe in giving children time, space, and materials to explore mathematical concepts and create their own understanding, before you start imposing your own thinking upon them.  Here is a great website that explains more about the Constructivist philosophy: http://mathforum.org/mathed/constructivism.html

For many parents, the idea of teaching math from a Constructivist perspective is frightening.  It pushes them so far outside of their comfort zone that they start lashing out at the philosophy itself, or even questioning the sanity and intelligence of the teacher!  This is why, when you teach from a Constructivist perspective, parent education is huge.  Since so many adults learned math through traditional algorithms themselves, the idea of trying to teach math in any other way can seem wrong too them.  Interestingly, the opposite is true if those parents are engineers or work in professions that involve math.

I received over two years of training in Constructivist math as part of a school wide reform movement at a school in which I taught in California.  Once I saw the Constructivist method in action, I was hooked.

Children who learn math from a Constructivist perspective are eventually able to mentally solve equations quickly, efficiently, and without squinting their eyes and mumbling “cross out the four, carry the one”.  They can explain their mathematical thinking and clearly express the strategies they use to solve problems.  They can also use personal strengths in one area of arithmetic (such as addition), to help them solve problems in another area (such as multiplication).  The best example of the Constructivist method in action that I have witnessed is my own son Bruce, who at 5 1/2 could mentally add 3 digit numbers with regrouping faster than I could.

At the upper levels of math Constructivsm is the difference between one students understanding how and why proofs work, vs. another student blindly cranking out formulas to get the right answers.  Unfortunately, I was the second type student growing up!  I still managed to get through college level Calculus with an “A”, but I was not prepared to enter a professional field involving science, math, technology or engineering because my own mathematical understanding was faulty.  This was despite having a wonderful public school experience in the Classical Education style of learning.

Of course there is a big difference from the theory and purity of the Constructivist philosophy and what can actually happen in the classroom setting.  Teachers, who instruct this way, really need to know what they are doing.  They need to be able to highlight and encourage effective student-created strategies for solving equations, while at the same time also appreciating the ineffective ones.  You want children to generate ideas and work through their own mathematical development, but you also want to hone their skills, and let the quick and efficient strategies sift to the top.

Another challenge teachers can sometimes face is parent opposition. (It is sad to say but true.)  If school is teaching in the Constructivist method, but a parent is teaching traditional algorithms at home, then the parents unintentionally put a road-block in their child’s learning process.  Once a child learns an algorithm it is as if mathematical thinking stops.  I saw this over and over again when I was teaching 3rd/4th grade.

Even if a child had the capacity to explain what borrowing and carrying really meant, he would often times cling to the traditional algorithm and cease learning the fastest and most efficient way for his specific brain to work.  As a teacher, I would allow kids to explain their thinking by saying “I used the traditional way.” but some teachers at my school would not.

FAQ

Q: So, do you ever teach traditional algorithms?

A: Yes, but only after a child has developed their own personal mental math skills down cold.  This will happen at a different point for each individual child.  Traditional algorithms are part of our culture and mathematical lexicon.  They need to be learned, but not clung to like life-rafts.

Q: What is the Constructivist order of teaching arithmetic?

A: Addition, Multiplication, Division and then Subtraction.  Subtraction is considered the most difficult concept to teach. Multiplication, as a form of repeated addition, would be introduced in second grade instead of third, under the Constructivist model.

Q: What is a good way to start teaching math with young children?

A: Cooking!  This sounds surprising, but cooking has it all.  Counting, adding, multiplying, fractions, you name it.  Cooking with your children at home is a wonderful way to give them a solid foundation in math.

Q: How have you taught Bruce math?

A: My husband and I did not do any formal math programs with Bruce until he was four and a half.  At that point, he was completing his second year of a Montessori preschool, and was ready for first grade math.  We used the Horizons math program, but I would not recommend that.  Bruce also did Dreambox math, and played a lot of Reader Rabbit.  In his public school Kindergarten, Bruce did the second grade Houghton Mifflin Math Expressions program.  At home, he completed Right Start Level C, and Life of Fred Fractions.  Right now he is working on Right Start Level D.

Q: How are you teaching Jenna math?

A: Right now we are singing Yellow is the Sun, and working on visualization vs. counting.  We are also beginning to explore MEP math.  In the future we will do Right Start Level A, and put her in Montessori preschool when she turns 3.

## Real Life Examples of Constructivism in Action:

Subtraction:

Square Roots:

Fractions:

Math Facts without Drill and Kill:

Cooking:

1. L says:

I have a question I hope you don’t mind me asking.

I’d like to supplement my son’s math at home. He is 4.5 and is attending a Montessori school. I like the idea of RightStart, but haven’t decided if it will conflict or add to his Montessori education (which will hopefully continue into elementary). The abacuses are different, which is my main concern. He loves math games and definitely wants more than I’ve been offering him.