On Thursday night, I read an article in the Globe and Mail that really bothered me. It’s called, Why Alex can’t add (or subtract, multiply or divide). The article discusses this new focus on problem-solving in math, and it says that students aren’t learning basic skills anymore because of this focus on problem-solving.
I disagree. As many people that read my blog know, I’ve been trying to incorporate more problem-solving into my math program. No, I don’t give my students long lists of addition and subtraction questions and get them to solve them. No, we don’t do “Mad Minute” or some other type of math drill activity. No, I don’t have weekly math tests to ensure that my students know all of the addition and subtraction facts off by heart.
Yes, I do teach my students these facts, but by having them understand these facts as well. My students know that 5 + 5 = 10 and 10 – 5 = 5, but they get what this means, and they can articulate their understanding in words too. Yes, I teach them about doubles with dice games and 10 frame activities. Yes, I make learning math more than about paper work, but I record this student learning with photographs and videos too. My students learn how to add and subtract and even multiply and divide, but along with these skills, they learn how to think and talk about math. Why is this a bad thing?
Below is a video of one of my students explaining his solution to a math problem. This student is able to share his thinking while also sharing his knowledge of basic facts:
I think we can have both. What do you think?
Brilliantly said Aviva! Thanks for speaking on behalf of those of us who are helping students learn math instead of drilling them. In the days of skill and drill those students who didn’t have strong short term memories rarely felt successful. Now all students can enter the problem at their level and have their thinking extended by their peers. We are building a stronger foundation for thinking which will result in adults who are confident in solving problems, working with a team and who have positive feelings about math.
Thanks Angie! I think that your comment is brilliantly said too. I hope that others share their thoughts on this post as well. After reading the article in The Globe and Mail, I knew that I needed to blog a response.
Yes, Aviva. We need to expect that students can do it all. I am tired of hearing about ‘traditional’ algorithms and that our kids can’t do math. I do think our students are experiencing a period of transition in how we teach and what it means to think mathematically. I find it interesting that there is concern that ‘our Manitoba professors’ are so concerned that students in grade five can not multiply 12 x 15 using the traditional algorithm. They do not need to! They know they can multiply 10 x 15=120 and 2x 15=30 and add those products in their heads to get 180. And they might also ‘see’ that 6 x 30=12×15. Why can they do this? Because they know what multiplication is and they know the basic properties of mathematics- including the distributive property! It is so sad that many of the students on university and many of our adults have not yet learned this. Thank you for sharing and let us all keep the spirit of thinking in math alive!
Thanks for the comment, Sandi! I love the example that you used. It’s okay to solve a question in various ways, and if students know the WHY of what they’re doing, they might choose a different strategy as a result. The basic skills haven’t been replaced by problem solving; they’ve been enhanced by it! Yes, let’s keep the spirit of math alive!!
It’s vital that we look at these apparent dilemmas at a learner by learner level. I do think that an age of drilling (my grandparents era, incidentally) produced reciters of both poetry and times tables, but not necessarily love or understanding of either literature or maths. On the other hand, many of the learners I come across who struggle with calculation bed one reliable method that will definitely work with any given numbers in order to access problem solving. Traditional algorithms provide this security, even if they are not always the most appropriate or efficient – once learnt they always work. It is the more able learners who have the luxury of choice and playfulness.
Thanks for your comment! I think that you make a very interesting point here. Some students may need more exposure to these traditional algorithms, but once they have them, how do we help move them past always using the same method to the willingness to try something new? How do we scaffold their learning, so that they can be successful with problem solving as well? Maybe some more small group or individual time that allows them to play, but with some more teacher support would help. What do you think? What do others think?
I should have said that I’m really talking about 10-11 year olds who have, for whatever reason, got that far without the kind of understanding you rightly champion. Often these children have experienced considerable failure in maths to date and it is problem solving that they fund most difficult. Teaching them a fail-proof method, and allowing them to achieve a page full of correct calculations can be enough to give them the confidence to tackle problems with a secure calculation strategy. But for the vast majority of children, I’m right with you on building understanding, allowing for experimentation, trialling different methods. It’s just not for all.
Thanks for replying again, @Collospede! I have really only taught primary students before, and so I really don’t know enough about the struggles you’re discussing here facing this group of slightly older students. I hope that others will chime in with what’s worked for them before. Maybe the only option is offering them a “secure calculation strategy,” but I’d like to believe that there are other options. You’ve definitely given me a lot to think about!
I think we must give our students a chance to approach math (and other subjects) – in a variety of ways. I remember as a student teacher learning to use manipulative math materials with students – and suddenly math was fun! And made sense in a more connected way than it ever had before. Giving students real-life problems to solve can engage and give purpose. I do find that many times students “just want to know how” or get to the answer quickly without having to think on many levels at once. I don’t think it’s laziness, I think that at some points a learner needs to know how before they can tackle why and other times it’s the reverse.
Thanks for the comment, Sarah! You raise an interesting point here. I often struggle with how much to tell my students and how much to let them figure out on their own. I was the student that wanted to know “how,” and I was also the student that was good with “computations” but didn’t really understand what I was doing or why. I just knew the formula. I remember sitting through university math classes and having students ask why are we doing this, and I kept thinking, don’t worry about why; just do it. Then I started teaching, and I need to have my students understand the “why.” I want them to question now. I want them to figure out what these computations really mean. I want them to do more than just figure out the answer. In fact, I’m less concerned with the final answer, as I am with the process. I think that teaching math through problem-solving has taught me this. And, as I said in my post, I still do ensure that the students learn the basic facts, but instead of just regurgitating the facts, I focus on them understanding them. I think that this makes them more meaningful.
I’m loving the discussion here though! There’s so much to consider. Thanks for adding to the conversation!
When I teach addition and subtraction for the very first time (of at least three times) in my grade one class I send my students through rotations looking at the math skill in a variety of ways. I don’t care how much you’ve memorized in the past (or had it drilled into your head) I want you to manipulate items, illustrate your manipulations, and then write a full number sentence. Where I give them choice is in the numbers they chose to do this with. For basic addition and subtraction skills I still (after 20 years) rely on the Math Their Way program. It is one of very few programs that I have seen that allows the math to be open ended, completely hands on, and takes it from concrete (manipulative), to visual (illustrations), to abstract (number sentences). When I taught in Australia the big thing was “show me in pictures, numbers, and words”. If you can’t show me in all three ways, I question whether you really understand the basic concept of math. So yes, I agree process is far more important than product as most (all) of the learning takes place during the process.
Thanks for your comment, Karen! I love the example that you shared. Our students are constantly told to show their thinking using “pictures, numbers, and words.” I think there’s a lot of value in this too. I definitely agree with you that the process is more important than the product. It’s great to see that the students understand what they’re doing and why they’re doing it!
I also feel that when you truly understand how the “basic facts” work your problem solving skills are that much stronger because you really get it. I believe you need a solid understanding of basic skills, and the ability to use those skills to solve real world problems. One is pretty much useless with out the other.
Thanks for the comment, Karen! I think that this “understanding” piece in crucial. With the use of technology, we can all put addition, subtraction, multiplication, or division questions into a calculator and get an answer, but understanding what this answer means and what the question is asking, really makes a difference when it comes to problem solving. I think that it’s through rich problems and good math discussions that we can all push our mathematical thinking forward!
Thank you Aviva on behalf of a teacher who is already tired of justifying the problem solving model to some. It is the power of understanding and articulating ones understanding of a problem that pushes math thinking forward. Students helping students to understand their thinking as opposed to the teacher giving a formula and students testing it out by simply plugging in numbers over and over again. My K’s are amazing the way they attack a problem, looking for the most efficient way to solve it and then explaining their thinking to the group. I do find some parents are still looking for that “textbook” (or now website) fix to guide their child in the ‘right’ direction. What is more powerful and engaging to little mathematicians….figuring out and explaining how to organize and keep track of the different types of food we have collected for our local food bank or counting snowmen on a paper and recording how many? The answer is obvious!
Thanks for the comment, Heather! I completely agree with you too, and I need to thank you. It was our conversation over lunch at ECOO that really pushed me forward in looking at new ways to use problem solving in the classroom. I’m finding that the math discussions my students are having now are so much richer as a result, and the students do truly understand the math that they’re learning. I can’t thank you enough for helping me see the value in problem solving!
P.S. I love the example that you give here! So very true …
I have never believed in “kill & drill” because that’s not math. I don’t care how many problems someone can finish correctly in sixty seconds because that’s not math. Math involves thinking and understanding and using the language of math. It is a subject area that is integrated into our daily lives and involves a good deal of problem solving. One of my friends, who teaches middle school math, says he purposely teaches depth instead of breath and feels as if he’s adding ingredients to a slow cooker for everything to meld together so at the end you have the perfect dish. A group of students who understand the relationships and make the critcial mathematical connections.
As always many thanks,
Thanks JoAnn! I love the comment from your friend about teaching “depth instead of breadth.” How true! There’s such value in having students understand what they’re learning. Completing numerous computations in a given time does not show understanding. As I said in one of my previous comments, a calculator could do that now for you. What a calculator can’t do is the rich discussions that come out of a good math problem. That’s why I love the problem solving model!
Children who can come up with strategies to solve math problems are equipped for life long success. Children who just memorize procedures and facts are doomed.
Thanks for the comment, Gloria! You make a very interesting point. For life-long success, I guess that problem solving really is so necessary!
Hi Aviva. I couldn’t agree with your insights more. I experience the fallout of exclusively procedural math frequently. I eyewitnessed a student trying to find out how much change she would receive from $10 if she were to buy something that cost $9.99. She used the “standard algorithm”. She did it correctly, but it took her about 5 minutes to arrive at a cent. Is this good math? Is this showing reasoning, understanding of numbers, efficient manipulation of numbers? This is what happens when students are told how they must do something. How much more would this student have benefitted from being exposed to the “think addition” strategy for subtraction or from using a numberline to help her visualize the difference between these numbers? Arithmetic is important, but it is not math. It is just one little piece of the puzzle.
Thanks for the comment, Victoria! I loved the example that you shared. It definitely shows that there’s more to math than just procedure.