I've written before about how really smart people like to make curriculum that focuses on the "basics". Unfortunately, these "basics" tend to be rules and shortcuts that were manufactured by people

*after they*figured things out the hard way. These very smart people make the (mis)assumption that others can just learn the algorithms (the fabricated shortcuts and rules) in an attempt to expedite the whole learning process. And many many teachers are prepared to eat this up because they've got too much to cover anyways - anything to speed up the inconvenient messiness of learning is seen as an asset.But there's a problem.

The only reason these really really smart people understand the algorithms is because they worked their tails off trying to figure things out from scratch. They played around in the mud and got their hands dirty trying to figure stuff out. In other words, they constructed their own meaning and made sense of these ideas; hence why they could make the algorithms.

I've been writing in generalities thus far. It's time for a specific example.

I was reading my students the book Twenty and Ten when we found out that the story was taking place in 1944, and World War II was in full force. I stopped the class and informed them that I was not even born during this time. Even though I'm 31, I still get these shocking reactions - "Really, but you're soooo old." (Despite my strict anti-punishment beliefs, I promptly bend these kids over my knee and... just kidding)

I then explained that my dad (who was born in 1953) was not even born yet. However, I did share with them that my Grandpa Art (here's a hilarious post about my Grandpa and multiple choice tests) was born in 1916 and that he was alive at this time.

I told them he is 94, but then I asked them how old he would have been in 1944.

Cue the debacle.

Kids stared at each other. I'm pretty sure half the class broke out in cold sweats while the other half couldn't avoid eye-contact with me fast enough. Intimidation. Embarrassment. Confusion.

I asked if someone could come up and show how they could figure this out. Jake was kind enough to come up and show his multi-column subtraction skills - but the whole operation went sideways from the get-go because he promptly placed 1916 on top of 1944 and started to carry and borrow.

Gentlemen, start your algorithms.

He ended up getting an answer of 1972. I have no idea how, nor did anyone else in the class understand his explanation. I thanked Jake and he sat down. Student after student tried to tackle this mathematical albatross - time and time again, algorithms went up in dust as they tried to spit rule after rule at these despicable numbers. The class was left in a haze of math mumbo-jumbo.

He ended up getting an answer of 1972. I have no idea how, nor did anyone else in the class understand his explanation. I thanked Jake and he sat down. Student after student tried to tackle this mathematical albatross - time and time again, algorithms went up in dust as they tried to spit rule after rule at these despicable numbers. The class was left in a haze of math mumbo-jumbo.

I was depressed.

After reading Constance Kamii's book Young Children Reinvent Arithmetic, I was all gung-ho to help kids

*think*and*reason*their way through math, but these kids weren't thinking or reasoning; they were spewing shortcuts and riddling rules that resembled the algorithms you and I have come to understand.And there lies the problem.

They were just following the rules.

*You*and*I*have come to understand these algorithms - single and double column addition and subtraction with healthy doses of carrying and borrowing - but they didn't have a sniff of understanding.They were just following the rules.

Based on this experience, I totally understand why Constance Kamii warns educators how teaching students algorithms before they make meaning can undermine a child's understanding for place value:

It took centuries for mathematicians to invent, or construct, “carrying” and “borrowing.” When we teach these algorithms to children without letting them go through a left-to-right process, we are requiring them to skip a step in their development. Babies need to crawl before they walk (although a few walk without crawling). Most say “Ball gone” before they say “The ball is gone.” Teaching them to “carry” and to “borrow” makes children skip a stage of development that took centuries for adult mathematicians to invent. Since 1972, Ashlock (1972, 1976, 1982, 1986, 1990, 1994, 1998, 2002, 2006. 2010) has been publishing an astonishing variety of data showing that children do not understand “carrying,” “borrowing,” and the many other computational rules they have been taught in school.

Even if our goal is to just get kids to simply bark "right" answers to questions on tests, Kamii's research shows that students who make meaning out score children who only know the algorithm. Even if you don't differentiate between encouraging children to make meaning with math and pumping out higher standardized test scores, you're still better off teaching meaning and only introducing algorithms

*after*kids construct their own understanding.
Something I've done is show students this YouTube video http://www.youtube.com/watch?v=omyUncKI7oU&feature=related and get the kids to discuss what just happened. Most kids know that 25 divided by 4 doesn't equal 14, but to get them to prove it mathematically is tricky.

ReplyDeleteJoe, excellent post! So often we skip right to the "how" without recognizing and teaching students the patterns they need to recognize before choosing the best approach to solving a problem. You've voiced one of my major concerns about much of our math instruction. (And you even mentioned one of my favorite children's books!) Awesome insight & writing!

ReplyDeleteMr. Bower,

ReplyDeleteMy name is Lisa Ferro. I am a student at the University of South Alabama. I am taking the EDM310 class. You can access our class blog by visiting www.edm310.blogspot.com

My blog is available at ferrolisaedm310.blogspot.com

I was assigned to read your latest post.

I enjoyed reading what you had to say about algorithms. I agree that the "tricks" children are learning for the basics of math are causing problems for the ones who do not understand the "why things work the way they do".

I am currently taking a math class for elementary teachers, where we are learning the different algorithms. Still, even at the college level, I see some class mates experiencing trouble understanding some of the different algorithms. How can an elementary teacher expect an elementary student to understand short cuts when some college students can't understand them?

Really excellent post. Earlier this week a through a recipe question into a division lesson. If a jelly recipe needs 450g of sugar for every 600ml of mixture, then how much sugar do you need for 1250ml? (It was a real life problem - the crab apple jelly mixture was sitting at the back of the classroom waiting for the mixture). My top mathematician started dividing everything by twelve so that she could work out the mixture for 50ml of mixture. Somebody else said straight away: "about 900g". I do lots of work trying to develop context for algorithms - see the last minute of this video: http://www.youtube.com/watch?v=UpmSekBywXQ

ReplyDeleteSorry I meant 'throw' not 'through' - shows my strength is math not English!

ReplyDeleteYou would be pleased with our planning this week. We are teaching grid method multiplication, so on the first day the children drew massive 14 x 19 rectangular arrays and I challenged them to divide it up to make it easier to work out how many squares there were. They independently broke it up into smaller 10 x 10, 4 x 9, 10 x 4 and 10 x 9 rectangles, and did really well when the next day we converted that into using numbers in a grid.

ReplyDeleteAnother stellar post. A classic algorithm that is rarely understood is multiplying by the reciprocal when dividing by a fraction. I think one way to try to counteract this type of misapplication is working concrete with physical objects or pictures (digital manipulatives) when first exploring a concept.

ReplyDeleteFor instance, I think my students have a much better understanding of square root then I did when I first learned it because they've constructed squares and identified patterns themselves between side length. These type of base understandings reinforced physically/pictorially let them apply algorithms more effectively.

This might not fit in this topic but yeah

ReplyDeleteIts Allan from last year

I'm at hunting hills school right now and science sucks because all we doing is write notes, and when we are writing the teacher is explaining stuff. So most people don't even understand most things he is talking about, and everyone just guesses. You should really talk to some teachers, because I'm not learning anything from writing notes.

I mean I'm learning NOTHING!

ReplyDeleteWhat you showed is that knowing an algorithm is not sufficient for problem solving. You did not show evidence for when the ideal time to teach an algorithm is. I'm sure some teachers focus too much, too quickly, on teaching algorithms, to the exclusion of other aspects of problem solving. But some probably make the opposite mistake, no?

ReplyDeleteGreat Post Joe,

ReplyDeleteI had a similar experience last year with my "high" math students when we worked on multiplying fractions. I introduced the concept geometrically, essentially having the students color in a fraction of a rectangle, then color another fraction with a different color, showing that the overlapping color was the answer. This involved a good deal of discussion about why it represented say 1/2 of 1/3 and at the end when students went to practice, many simply wanted to fall back into the algorithm of multiplying the numerators and then the denominators with absolutely no understanding of why what they were doing showed 1/2 of 1/3.

I really think its important for kids to learn the "why" of math, the deeper thinking and not just move on to higher level math by knowing the tricks.