What is Wrong with an Algorithm?

Nothing! Yes I said nothing! Let me elaborate.

In conversation today I was asked, "What is wrong with an algorithm?"  I get this question a lot. Now this question was asked because like us all we were taught to do an algorithm so we always do what is familiar for us. The question also came about because they were afraid of teaching their child the wrong way.

This inspired me to write this blog post. There is nothing wrong with learning an algorithm; there are some risks in only learning this but there is nothing wrong.

An algorithm was first invented in order to make math easier to do. We have to remember that calculators were not invented at this time and may people were using abacuses and basically counting by ones or a base 16 system. With the use of an algorithm counting became procedural and easier to do.  However, those using it understood mathematical foundations of why the algorithm worked.

This is the problem with introducing the algorithm to early. Let's just reflect on our own understanding of basic addition. We honestly did not learn how to add by using an algorithm first and to be fair our first insistence of remembering an algorithm (at least in my guess) was probably around grade three maybe four. This means that for four-five years, depending if you were in the pre-school or JK error you had a lot of experimentation and exploring in counting, saying numbers, ordering numbers and subitizing.  The problem is that we often forget all of the steps it took us in learning an algorithm.

For the past ten years I have been looking closely at how students learn mathematics. One of the foundation research pieces have been Fosnot and Dolk's landscape of Learning (2000). (full file)

I know that the picture is small but you can see all of the learning that a student has to go through in order to get to an understanding of algorithms.

Now here is my problem with starting with algorithms to early:

1) It doesn't teach proper number sense:

Now you may question me on this but in my honest opinion it doesn't teach proper number sense. The problem is that when a person does an algorithm it forces you to only look at numbers from 1-9. There is no true understanding of our base ten system. Students do not really understand why or how 1 group of ten can be ten things yet 1 ten. This is often seen when you ask a student tell you the value of a two digit number (let's say 24). When you point to the ten's column they will say the value is two. They will even go as far as counting 24 objects then pulling two of them to signify the value. This can also be seen when students use base ten blocks and they count the rods as two instead of twenty.

2) Basic procedural errors:

Now these errors may be dismissed as, they just have to learn the proper procedure. However, these are troubling errors because students not only are procedurally doing the operations wrong they are also struggling with reason-ability of answers or what I like to call no number sense.  

What I propose instead:

In my math class we work hard on developing a number line.  Learning this strategy does a lot of things: 

1) It allows all students to grasp a strategy whether they are counters by one or able to skip count proficiently. 

2) It teachers all of the rudimentary learning of number sense. Students learn one to one tagging, cardinality, subitizing, magnitude and many more options all in one strategy.

3) Eventually they develop more proficient mental strategies because they can conceptually understand what is happening to the numbers.

4) There is no need to regroup and or borrow with a number line, just the use of number sense.  I find this to be the biggest problem with algorithms. For example, let's take 56-49=7. Now for many of us we may not need to use an algorithm but ask your child or a younger student they will start by borrowing from the 5 (not fifty) to make 16 because you cannot take 6 from 9 (which we also no is not true-- ask anyone in debt or who has a mortgage) and then they will realize the answer is five. However, if students are taught proper number sense, they realize that they can just count up to the answer.

A number line also develops efficient mental math strategies that are far faster then whipping out a piece of paper and borrowing or regrouping.  Take a look at some of my grade twos thinking (all done 

Our number talk today. Continuing to build adding and subtracting by ten. #raylearns #engagemath #peelmathchat pic.twitter.com/gIp8HnOnXv
— Jonathan So (@MrSoclassroom) October 31, 2014

Our number talk today. Trying to get Ss to think w/ 10s. http://t.co/Sd45gMG2Do #enagemath #peelmatchat #raylearns pic.twitter.com/pHI7hNd7in
— Jonathan So (@MrSoclassroom) October 29, 2014

By allowing my students to explore, question and develop their number sense they are better mathematicians and yes we have discussed an algorithm, we have developed an understanding of how to use it but my students make less mistakes with a number line then with an algorithm. And to be honest they much prefer the number line then the algorithm, most of the time they do this because that is what they think they need to do.

One of the biggest problems in our math programs today is that we often jump to far into the abstract without thinking about the concrete work that people need to develop in order to understand these concepts.

What do you think? When and where should algorithms be taught? How do you develop number sense? Love to hear your thoughts.

For further reading I recommend reading:

1) Fosnot, C. T., & Dolk, M. L. A. M. (2001). Young mathematicians at work. Portsmouth, NH: Heinemann.

2) Anghileri, J., Beishuizen, M., & Van Putten, K. (2002). From informal strategies to structured procedures: mind the gap!. Educational Studies in Mathematics,49(2), 149-170. 

3) Kami and Dominck: The Harmful effects of Algorithms in 4-9

Why do we Need to Argue over Math? --> A Call for a Balanced Approach

About a month ago a colleague of mine Kyle Pearce wrote a post "Does memorizing multiplication facts hurt more than help." It was a very interesting read and I happen to agree with Kyle's point of view.  As many of my frequent readers of this blog know, I prescribe to the constructivist approach to learning mathematics.  I believe that students through discovery and proper guidance will be able to understand a wide load of big ideas and theories.  Not only do I believe this but I have witnessed this first hand with my students in every grade that I have taught.

However, this is not so for many people.  In fact it was a discussion on Kyle's blog post (feel free to read the thread) that has me thinking more and more about this topic. And not only thinking about it but trying to fix and insight thoughtful discussion around the ways in which we are teaching math.

Maybe a little background first.  Math has been a hot topic for the past year, if not for the last century.  For many countries, provinces, and states, math curriculum has undergone a significant change from what we grew up with as children.  Some. like myself, believe that these changes are for the better, some have not.  I was recently at the OAME and listening to Brent Davis a professor at University of Calgary.  In his lecture he shared that the reason math curriculum was introduced was so we could have a work force to crunch numbers, nothing more and nothing less.  As we have evolved beyond that (not saying fact crunching is not important) our skills have also changed and I think this is what we need to remember; we have evolved.

For this reason I and many others are proposing a  more balanced approach to mathematics.  Lets stop this war and needless debates and get to teaching good mathematical practises.  One in which our students will push their thinking and really think about the numbers.

If it was up to me this is what I would include:

1) Math should be linked to Big Mathematical Ideas:

I think this is the first step to thinking about our students as mathematicians.  Catherine Fosnot (2002) has some very interesting work around making our students mathematicians.  One of the most interesting facts is there was a study done with so many mathematicians and they were asked to solve a problem.  Not one of those mathematicians solved it the same way.  I found this interesting because that is what I see math.  Math is about the mathematics and there is not one way of doing things.  We have to teach our students the understanding, the flexibility and the patience to be mathematicians.

2) Math is about real numbers:

Students need authentic experiences to learn.  Let's think about ourselves and how we learn.  Now some do learn through reading and replication but if you honestly think about how you learn a concept the best; it is through trail and error and than guidance from a mentor.  This is the same for our students.  They need real experiences so that they can play and discover the mathematical concepts.  In my personal experience both in tutoring high school students and teaching mathematics in the primary and junior divisions it's the contexts that allow students to really understand what they are doing.  It's the context that helps them build models of representation.  In my classroom, these are often done through social justice problems and real life contexts.

3) Students need time to explore:

This goes hand in hand with the above comment. As much as we need instruction, we also need exploration. Students need time to make mistakes, reflect, debate and discuss. These experiences allow students to make connections between concrete and abstract thinking. I was reminded at the OAME that every mistake makes a new synapse in the brain. We need these mistakes I order to solidify  our learning.

4) Students need Mentors:

My most recent research in understanding teachers questions has shown me the importance of teachers, not that I didn't believe that before.  With a shift towards discovery learning, we as educators have forgotten the importance of our role, or what even our role is.  I truly believe that we should not be the dispenser of knowledge but the mentor of that knowledge.  Though through exploration students will learn (many studies to show this) they may not have tools to reflect or pull together the big ideas.  I think this is where much of the back lash has come from discovery math, reform math or whatever you want to call it.  As students explore and discover there needs to be some sort of guidance.  This is where a teacher can shine and help students with the mathematics.  However, my research shows that for this to happen, a teacher needs to 1) have a good understanding of mathematics, 2) have a good understanding of how children learn mathematics and 3) plan. I know that all teachers plan but this planning involves thinking of big ideas, landscapes and possible questions.  It is these questions carefully placed that can allow students to figure out and make mathematical connections.

Take a look at this video of three of my students thinking about fractions.

They have never been taught fractions from me before this and in fact as a class we haven't even started the unit. However, that being said think about their learning and the role they play and the role that I play as a teacher.  Where do my questions come from? Why did I ask them at the time I did?

5) Time for debating, conjecturing, discussing and proving:

For me this is the time for a teacher to shine; however not in the traditional sense of standing in front of the class and lecture or tell students how it should be done. Just like students need time to explore they also need time to debate as a community. It is through this debate that students defend their thinking, conjecture, question and solidify their learning. Moreover, it goes beyond just showing and telling.  As a teacher the types of strategies that you show matter. How are you building the learning up, what questions are you modeling? How are you focusing the talk? How are you fostering talk? These are all questions that a teacher needs to be asking.

A great article to read is: https://drive.google.com/file/d/0B4245QONE7HaSExrSUlWTjBrNEU/edit?usp=sharing 

Also take a look at my most recent grade two conversation of multiplication.

6) Repeated Practise:

Yes I said repeated practice! Students need it, but it's not just doing procedural learning over and over again.  When I say repeated practise I mean a similar problem for students to continue their exploration.  Students, well most students, cannot solidify their learning through one experience.  In a typical unit my students will solve about 7-8 problems that could take three to four weeks to learn.  These problems build their learning and knowledge from day to day.

7) Skills:

Yes skills are important. They are needed but they are not needed like we use to think about it. For me it's how are we introducing facts. Do we make students think through facts? Are they taught in isolation or allong with concepts?  In my classroom, students practise facts at home, they play math games in the classroom (here is a file of my math games:  https://drive.google.com/folderview?id=0B4245QONE7HaaHl3M3ZKNWd3SUk&usp=drive_web). In addition before my problems start I often use string lessons which builds on mental math strategies and learning how to be flexible thinkers and playing with numbers. This to me is more important then struck memorization. It teaches students that numbers are not confined facts but that you can pull apart numbers and use known facts to solve other facts. Through this process students often learn all of their math facts, can recall them and use them in problems, which to me is way more important.

These are just a few of my thoughts on what I am calling a balanced math approach. I have a few more  to hash out around integrating and  implementing a center approach within my problem solving approach.

What are your thoughts?  Don't you think it is better to discuss and fix our problems rather than lay blame about which is better?  Shouldn't we think about our students first and their needs in their 21st century world?  Love to here your thoughts.

Problem based in learning in the context of math wars. Thoughts are myp.o.v.

I was recently given an article from Suril Shah (@thrilsuril), a colleague of mine in the peel Board, (http://news.nationalpost.com/2014/02/28/does-discovery-learning-prepare-alberta-students-for-the-21st-century-or-will-it-toss-out-a-top-tier-education-system/) and then later on another article from another colleague Aviva Dunsiger, a teacher in the Hamilton School board (http://www.theglobeandmail.com/globe-debate/canadas-math-woes-are-adding-up/article17226537/ ).  Both articles discuss (or rather reprimand) the notion of “Discovery Math” needless to say I had to respond.

As many of you know from my blog posts, math is a very important passion of mine.  I have in a way devoted my educational career to learning about math education and how it can help transform student learning.  This has gone on for me for the last 9 years of my teaching career and five years before volunteering at an amazing school in Peel.  Over the course of these 14 years the arguments in the above articles have always been happening; so I think it is funny that when Ms Wente mentions that this is a “new faddish fuzzy notion.”  Since mathematics was first introduced into the curriculum in the fifteenth century it has always been a debate over skill versus conceptual understanding.  This debate will always be there all I can give you is fact from experience and from the classroom (which I will say many who write articles in the Newspaper or make policy cannot).

Let me first start of with my own evolution.  Like many of you I was taught with very traditional methods.  My father drilled in me from a young age that fact recall was the most important thing.  I still remember practising for hours on hours flash cards and being randomly asked multiplication questions to see if I knew these facts.  I also remember that my Math class was all in a work book and my teacher sat at the front of the room and wrote many things on the board and then we did questions to practise and show what we learned.  This continued all the way through school and as I got into the high school and eventually University this is what I remember of my Math class.  Did it help?  No, I don’t think it did.  Don’t get me wrong, I did learn math.  In fact Math has never been a hard subject for me (except problem solving).  I was able to work through and memorize what was needed and then when the test came I was able to retell those facts and get an A.  My problems never came until University Calculus.  Here I because I didn’t have a good foundation in Calculus I struggled, in fact I failed. 

Sorry I digress here.  This method of teaching stuck with me, more so because this was all I knew.  During University I changed majors and decided to become a teacher.  I was able to volunteer at an Amazing school in Peel and soon learned Reform Mathematics (what discovery math was called at that time).  I was also fortunate enough to have an Amazing principal who let me question her and learn what reform mathematics was all about.  At first I said the same things that many of these article, and our parents say when they see problem based learning. You have probably heard these before (I know they are in the article):

1)      What is wrong with Rote, it worked for me?
2)      What about facts? There not learning them like I did?
3)      I memorized and got good grades?
4)      They can’t possible learn this on their own?
5)      What do you mean discovery? What is your job then?
6)      You’re the teacher so teach?
7)      This look chaotic, there is no order, how can they learn?
8)      What about the language, seems like more reading than math?

I can go on but they start to sound the same.  During this process I was able to see students truly excel and showcase their learning.  In fact, looking at scores (which is not the end all to be all), the school went from 42% to 93% in that first year in mathematics.  I was also able to reflect on my own learning and how I learn.  This started the ball rolling and has helped me to ask questions back.  Here are a few to think about:

1)      How do you truly learn as an adult learner? 
2)      Do you memorize things and then succeed? Or did you have to make mistakes, go back and relearn or have someone help you through it?
3)      When you are learning do you like to ask questions? Or just sit and receive information?
4)      (my favourite one) As a successful adult how did you become successful? What traits do you like in your employers?

Here are my thoughts to these questions:

I personal learn by doing, struggling, asking questions and then going back to relearn it.  True mastery comes from doing something over and over and over again.  Yes I can see how this backs learning facts, and I am not saying facts are not important, but my learning is in context to the concept not in isolation.  Memorization only works with some things but I still make mistakes no matter what I am doing and then I learn from them.  As for success to me I value students who are free thinkers, creative, adaptable and able to see past just simple direction.  This has been the case even when I was managing people in the private sector in my University jobs.  I don’t think the world can evolve from people who can only follow direction and not think beyond what is on the paper.

With this in mind I began my teaching career.  Here I too continued to question but now I also had to field questions from the general population about my style of teaching.  Here are my responses.

Q: Why is this better than traditional learning?

 A: I hope that I may have answered this above but most students, and adults do not learn through traditional learning.  There are a very few who do and we also have to consider that style but many don’t.  Learning is developmental.  It doesn’t happen in a linear fashion and PBL allows for this to happen.  Learning in PBL also doesn’t happen in isolation from the world, or other subjects.  It is always connected to a context, which helps all students to hold on to something and work with it.  Furthermore, all learners can access PBL, whether gifted or with a learning disability all students can do the problem.  Also, personally, it makes the day go by a lot faster, I enjoy it and so do my students.  Check out this video: http://curriculum.org/secretariat/justice/insights.html for student reflection on what context can do.

Q: You know my kids don’t know facts, why aren’t you teaching them?

A: First and foremost, I want this to be said, “FACTS ARE IMPORTANT!” they must be taught and learned; however, how are we learning them.  Let’s go back to my question back to you.  Can you recall something where pure fact learning has help you be successful?  If yes, no think was it just fact memorization or was it in a context?  Fact knowledge is important and needs to be done.  I prefer to do this through games and mini-lessons.  This allows me to talk about a strategy and have students discuss the pros and cons of the strategies.  The talk focuses the learning.  Check on my previous blog post on it.

Q: “Teachers and Students are learning together” Great so now we have the blind leading the blind!

A: This is the one that bothers me the most.  It bothers me because PBL actually takes more understanding, more planning and a lot more patience then traditionally teaching.  I have almost completed my thesis, in where I researched the impact of my questions on students learning of fractions.  It was interesting to see where I had moments of direct teaching that my students stopped talking.  In fact, they just sat there.  Which is exactly what traditional teaching does, students sit and listen then do.  PBL takes planning.  In another of my posts I talk about five practises that teachers should be following for PBL implementation (http://mrsoclassroom.blogspot.ca/2013/11/blog-post.html ).  Teachers actually need to learn the mathematics and it is through critically placed questions that the learning is brought out.  Students develop at a faster rate through this proper questioning style and can achieve a higher level of understanding.  I have grade twos right now who are learning about equivalent fractions, ratios, division and adding three digit numbers in their head.  It is truly amazing to see what they can do.  But this takes planning on my part.  It takes understanding of learning trajectories and  understanding what students are doing (so you can redirect or push beyond) and understanding the math to be effective in PBL.

Q: Test scores are falling?

A: this might be so but I would caution you on this.  First of all tests are a snap shot of learning at a particular moment in time.  They have a place in assessment.  In my personal opinion a very far place but a place nonetheless.  There are many factors to low test scores: 1) poverty, parents education, home life, social problems that day, being sick, stress, reading level, context, etc. The list is endless.  When we put all emphasis on test we are taking away so many other factors of learning.  I know more about a student from a problem that they solve then by what they can retell me on a test, just a matter of fact.

 I am going to stop here for now as I think I have written more than I ever have in a blog.  This topic is very dear to me and I have heard a lot of the questions in this “Math War.” It will not go away but please don’t take this as a discouragement to stop PBL or even start.  To me PBL is the best way for ALL students to learn.  It gives the teacher the most time for true assessment and understanding of their students needs and next steps.  It allows you to meet all levels of students and be able to get to all of them.  I have and will continue to only teach through PBL (problem based learning).  Love to hear your personal stories, questions or answers to this lovely debate.