Consider the following multiplication problem: 322 x 47. If you’re like me and schooled in a particular era (one that did not include calculators), you might tackle it like this:
But over the phone line from St. John’s, mathematician Sherry Mantyka describes a different way to do the problem. She has me multiply the top numbers by seven – except once I move into the tens column, the two becomes “20” (which it actually represents), and the three becomes “300.” When I return to the lower line to multiply by four, it becomes 40, which is what the four represents. The answer to each individual multiplication operation gets its own row; six in total. What’s more, it works:
The students who come to Dr. Mantyka for remedial help at Memorial University’s Math Learning Centre are often familiar with the second approach. It’s a discovery-based learning technique (sometimes called inquiry-based) for performing two- or three-digit multiplication. It’s transparent, if more complicated.
Trouble is, Dr. Mantyka believes it’s an approach that’s putting too many students on the math casualty list, unable to automatically recall math facts or use quick, efficient algorithms that are the underpinning of more complex mathematical concepts and skills. Students who come to her centre are among the 18-to-19 percent who annually fail a mandatory math test required for entry into any Memorial math course. In some cases students have attempted, unsuccessfully, to solve division questions using dots or successive subtraction, both discovery-based techniques. Most of these students are not math majors, but need at least one math course for their chosen program.
“Thirty years ago this only affected students who wanted to pursue degrees in science,” says Dr. Mantyka, who has been teaching at Memorial since 1978. “Now, it affects degrees in so many disciplines. Even a degree like social work has a fourth-year course in research methods, which is essentially statistics.”
Weak math skills among new students have been a longstanding complaint among Canadian university math departments. But over the last year a more widespread, acute math panic has settled into the popular Canadian imagination, with news stories painting a picture of a generation of students whose mathematical skills and futures are at risk. Last December, the Organisation for Economic Co-operation and Development’s Programme for International Student Assessment, or PISA, showed that while Canadian students were still performing above average in math, the country had fallen out of the top 10 list among 65 jurisdictions. Although some say the change is inconsequential, performance had declined by 14 points, from 532 in 2003, to 518 in 2012. Meanwhile, the OECD’s assessment of adult skills – the Programme for International Assessment of Adult Competencies – released in October, showed Canada was below average in math skills, with the millennial generation showing the worst performance.
The finger-pointing quickly turned to discovery-based teaching approaches in kindergarten to Grade 12 schools. These have grown in dominance across provincial school systems for more than a decade, supported by scholars in faculties of education. While education researchers have pointed to other sources of the mathematics deficit, such as a lack of teacher education, some mathematicians complain the current thrust of discovery-based learning has left students with no firm grasp of math fundamentals.
“You need to have the foundations in place in order for students to do any sophisticated problem-solving or discovery on their own,” says Robert Craigen, a math professor at the University of Manitoba. “It’s not that discovery is a bad thing to do [in elementary classrooms]. But it should not become the main course of your meal.”
The discovery approach, derived from constructivist theories of learning, argues that all students should know how a particular problem-solving approach works, to gain “deep understanding” of mathematics and be able to apply it to new problems. Students memorizing by rote takes a back seat to students constructing their own knowledge.
To develop this skill, students should be given enriched exploration activities, often using concrete models such as blocks or shapes that they can manipulate to discover the underlying mechanisms of mathematics. They are also encouraged to use a variety of strategies to solve a problem. “We want them to think, to develop their own reasoning,” says math education professor Annie Savard at McGill University.
Traditionalists say that unless students have automatic recall of math facts such as simple addition, subtraction and multiplication tables, and second-nature use of standard algorithms, their working memories will be too bogged down by complicated strategies to move to the next phase of deep understanding. Direct, explicit instruction, followed by lots of practice, is the way to go.
However, in her 15 years teaching elementary school in Quebec City before moving into academic research, Dr. Savard noticed that while some students were whizzes at memorizing multiplication tables, they weren’t necessarily good at applying that knowledge. Meanwhile, she had to give zeroes to students who showed sound reasoning in their problem-solving, yet arrived at the wrong answer through calculation errors.
“For students who don’t have a good memory, they’re done,” says Dr. Savard. “And we observe that for students who do have a good memory…they can do it in school, but in their personal life, there’s no [understanding] about it. It doesn’t make any sense.”
As in most intellectual debates, the discovery vs. traditional math-learning controversy does not shake down to either-or arguments. Talk to a pure math academic or an education professor and within a few minutes it’s common to hear that there’s a place for discovery and a place for mastery of hard skills. Disagreement comes over what the mix should look like and where the emphasis should be.
Discovery learning and traditional, direct instruction approaches are “both right,” says Doug McDougall, an associate professor at the University of Toronto’s Ontario Institute for Studies in Education and director of its centre for science, math and technology education.
“We need to have skills instruction because there has to be some memorization. We need to know our math facts – multiplication, addition, subtraction – in order to continue on to do some discovery-based learning,” says Dr. McDougall. “And we can also use discovery-based learning to reinforce and to learn our math facts, to build a better understanding of the big ideas in math.”
Even Quebec, now Canada’s mathematical powerhouse as the provincial leader in OECD student assessments (it ranked among the top eight scores worldwide in the PISA study), uses a discovery-based curriculum. However, it has a clear requirement that students must memorize certain math facts and algorithms by specific grades – unlike other provinces, which may require knowledge or understanding of those things, but not memorization. That is starting to change with public pressure: Manitoba brought back explicit instruction of math facts and standard algorithms last September. Alberta and Ontario have said they are looking at similar moves.
As an example, the standard, traditional algorithm for long division is sometimes left out of discovery-based math classrooms because it is considered too hard to represent through conceptual models. It’s also not always considered crucial for students to know.
Instead, students might be shown alternative strategies for accomplishing the same goal, through estimation or successive subtraction. But that neglects the fact that the standard algorithm is needed when students reach more advanced mathematics, such as dividing polynomials, says Dr. Craigen from University of Manitoba. He belongs to a group of mathematicians, educators and parents called the Western Initiative to Strengthen Education in Math (WISE Math) that helped pressure the Manitoba government into its change.
“Long division is an important systematic process that is mirrored in a lot of higher-level skills,” says Dr. Craigen. “Our entire society is built upon algorithms. People talk [about] 21st-century learning, and then one of the first things they do is they throw out algorithms.”
Robert Dawson, a professor in the department of mathematics and computing science at Saint Mary’s University, says the discovery method does provide a better conceptual framework than “simply learning a technique that someone else shows you.”
On the other hand, a high school student has to learn “a set of techniques that took some very bright people many hundreds of years to put together,” he continues. “To expect everyone to discover this themselves is unrealistic.”
What math and education professors generally agree on is that elementary teachers in particular – typically trained as generalists – need a much stronger foundation in math and mathematical concepts if their students are going to succeed. That’s especially true if teachers are using a discovery-based approach, where activities must be well-designed and where flexibility is key to properly responding to where students are at in their own conceptual understanding and explorations.
In Quebec, pre-service elementary teachers can receive up to 225 hours of in-class instruction in teaching math, whereas in Ontario teachers may receive just 30 hours. Beyond needing more time to learn how to teach the subject, teachers also need grounding in actual math content. Relatively few elementary teachers have any university-level math background, and it is common to encounter elementary teachers who lack comfort with and confidence in the subject, say researchers.
One cross-appointed professor says that faculties of education should require pre-service elementary teachers to have at least one math content course, and math departments need to think hard about how they can serve those students best. Math departments should “make courses accessible and ask questions about what is it future teachers need,” says Donna Kotsopoulos, an associate professor in the faculty of education and in the mathematics department at Wilfrid Laurier University. Some universities provide math-content courses for aspiring teachers, but not all. Dr. Kotsopoulos has advocated publicly for a “radical re-engineering of math education” at all levels, especially at the postsecondary level, where, she says, teaching approaches have remained static over time.
There’s no question that many elementary teachers are uncomfortable with math themselves and are unlikely to have the chance to go back and do a full math-content review. Recognizing that, the highly successful JUMP math instruction program is trying to fill the gap by developing step-by-step teaching resources and lesson plans.
Founded by John Mighton, a fellow at Toronto’s Fields Institute for Research in Mathematical Sciences, JUMP has been described as a “third way,” to teach math. It requires students to become adept with standard mathematical rules and procedures through repeated practice, yet it also embraces concrete modelling and the importance of students becoming creative problem-solvers.
“The things that [math and education professors] are both concerned about are genuine concerns,” says Dr. Mighton. While education profs want to ensure understanding, he says math profs believe facility with rules and procedures needn’t exclude that understanding. “There’s miscommunication between the sides. I think you can have a hybrid or a very effective combination of discovery with guidance that doesn’t draw these false dichotomies.”
Can we get beyond the current math impasse and move on, to a new era of excellence for students? Dr. Mighton thinks so. But ultimately, he says, teachers must be freed to experiment with what works best, along with a rigorous tracking of student results and appropriate adjustment of approach in response. Not only can students do better in math, they can go much farther than they’re being asked to now. “Even the advanced countries of the world who do better than [Canada] aren’t even coming close to realizing the potential of children,” says Dr. Mighton. “We are in a crisis – but every country is in a crisis.
I was a bit disappointed with the article on “Professors debate the best way to teach math” by Moira MacDonald.
In reference to the “discovery” method of teaching math she quotes “It’s not that discovery is a bad thing to do in elementary school, but it should not become the main course of your meal.”
She cites an example comparing the traditional method for multiplying two long integers [I hope this formats okay]
with a different method which she refers to as illustarting the discovery method, namely
I think the second way is wonderful. It shows how multiplication works by repeated addition.
The issue here is NOT discovery, but understanding. The student is not “discovering” a new method at all. The second method is presented so that the student can understand long multiplication and then understand the first method as an optimization of the new method.
The point is that if we teach the student how to “do multiplication” but not explain how it works, we have failed. That is the reason (I believe) for the new “discovery syllabus”. It’s not primarily about discovery, it’s primarily about understanding.
The article then goes on to say “As an example, the standard, traditional algorithm for long division is sometimes left out of discovery-based math classrooms because it is considered too hard to represent through conceptual models. It’s also not always considered crucial for students to know.”
It’s not true that it’s conceptually hard.
Division is just repeated subtraction!
The problem here is that our teachers don’t understand division!
Because when we all learned division our teachers didn’t understand it and we had difficulty with it.
Here is how you could divide 15134 by 47 using repeated subtraction quotient
47 | 15134
– 4700 100
– 9400 + 200
– 940 + 20
– 2×47 + 2
The division algorithm we learned in school is just an optimization of this to reduce the number of steps which involves estimating each digit of the quotient.
I believe it is helpful to teach why methods work and a method for hand calculation.
That’s the reason for the “discovery method”.
It’s not primarily about discovery, it’s about understanding why methods work.
Do we need to know the optimized long division algorithm. Not really, the one I just gave is good enough.
Checking with a calculator would be wise.
Thank you for providing a balanced discussion on this topic. The only thing I’d like to add, is this. It seems every time followers of the constructivist/reform math (i.e. “Discovery based math”) enter into this discussion, they are quick to point out that arithmetic needs to be taught using a variety of strategies in order to promote a deeper understanding of this subject, yet they quickly dismiss the more traditional methods to gain automatic recall of math facts. In essence, they are completely close minded to realizing that some tried and true methods, are still legitimate to teach children math facts. Why is that? Whereas, I have found that the more traditional instructors/professors/teachers, have a higher success rate in teaching arithmetic, using effective methods and creating a strong foundation, first. I don’t know why this whole discussion came about regarding “understanding vs. rote”. Learning arithmetic has always included a deep understanding of math facts, but somehow this has been forgotten in these math wars which are now taking place in our classrooms and in our homes.
I would also like to find out why we now have 30-50% of school aged children attending outside tutorial centres to learn their math fundamentals. I am sure teacher training is part of the reason for this, but then why are parents left to teach their children multiplication tables and standard algorithms at home, or pay for someone outside of school, to do it for them? Not every child deserves to be a math scholar, or to even like math. But ALL of them deserve to possess confidence in reciting their multiplication tables and understand long division. It’s sad to see that even that, is now under scrutiny.
I can’t teach by rote if I were to try. Throughout my career in mathematics education, ranging from kindergarten to university, I have followed Socrates and Archimedes. Given any mathematical concept to be presented to a group of students (however large the group), the process is in two parts:
Part 1 should consist, not in discovery but, in guided discovery. The former ends in frustration while the latter ends in excitement, clear understanding, and a desire to know more.
Part 2 Given any mathematical concept introduced this way, it is then essential to have the students practise (I) in recognizing when to apply it and (2) to do so with ease. For any students who, during a test, forget a needed mathematical rule, my experience starting with myself has been that a minute or two spent in deriving the rule from first principles (e.g. during a test) is all it takes. Depending very much on the nature of a given test, I have at times given the students a set of formulae and left it to each of them to decide which to apply when, if at all.
In conclusion, (a) present a problem with the attitude summarized as “I wonder … if … .”, (b) guide the students and always look out for any sign of frustration in the process, (c) as a class, as distinct from the effort in groups, the steps leading to the discovery are recorded, and (d) practice in both recognition of its use and in the mechanical aspect.
Part 3 With regard to test results, the students have a chance to redeem loss of marks suffered through an erroneous solution: the next time that a concept in question occurs again in a later test, the mark obtained the second time erases the first one.
Although not every student makes the hundred percent mark at the end of a term, students who have been labelled as math failures prior to joining my class have ended the year with a mark of between 60 and 70 percent, and they are thrilled with the change.
A few things. First, thanks for the article. There is always a danger when presenting different sides to a story that people will assume the sides are equal in weight or accuracy.
The expanded algorithm described above for multiplication is NOT a “discovery math” technique. It has been around in some form for six or seven CENTURIES.
For teaching students it is useful for reinforcing the place value concepts inherent in multiplication though this is achieved even better if the steps are clearly explained:
15 (5 x 3) (or better 5 ones x 3 ones)
100 (5 x 20) (or 5 ones x 2 tens)
>and so on.
This is NOT as the post above states, repeated addition. This would be 123 +123 + 123 + 123… 45 times.
The biggest problem with the current movement is the false belief that there are no BEST ways of doing math. Much of the techniques (especially use of manipulatives) we’d like to show students (or have them ‘discover’) are very good for helping develop an understanding, but they are NOT effective for DOING math. An end goal of better and best ways of doing things is crucial.
The so-called standard algorithm for multiplication can be used, as is, to teach the understanding behind the multiplication. it should not be frowned upon by applying a misguided belief that it is barren of understanding. can someone use it without understanding? Certainly. Does it inherently create a lack of understanding? No.
As to the issue of basic facts. They are essential. Don’t let anyone tell you otherwise.
A student who does not have a solid grasp of fundamentals has serious limitations put on their working memory while trying to deal with more complex problems. This is not solved with the “use a calculator for that stuff” approach.
Michael Monagan, The current methods of math education are breeding dependance on computers rather than developing math skills.
My personal understanding increases with practice and improved skill. This applies from basic skills like tying my shoes, through the most complex math included in my Master’s of Mechanical Engineering. It even applies to the pure fun stuff like rolling a kayak in grade 5 whitewater, and martial arts.
I use the word skill on purpose. The entire point of getting an education is to gain the ability to do things. Why else would I invest any effort into getting an education? Being able to do things requires skill, and skill means doing things quickly, consistently, at a high level of quality. Having the optimized techniques mastered helped me on this path a whole lot.
Your comments to the effect that the traditional division algorithm is just an optimization that is no longer necessary astounds me. Why on earth would you not want your students to learn the most effective “optimized” techniques? I find this attitude completely astounding, and quite honestly I think it shows great negligence towards the future well being of our students. Why do you want your students to learn “good enough” instead of “the best”? How does “good enough” help them achieve “their best”?
Let’s apply your mentality to reading, I don’t really need the ability to read do I. Understanding that it can be done is good enough correct? I certainly don’t need any skill at it do I? I know there are electronic readers, I’ll just get one of those and it will do all the work for me.
That brings me to another problem with how math is being taught today. Practice and memorization are discouraged, pretty much treated as evil. This does not teach the need for effort, and does not present any real challenge with respect to being able to complete assignments by a dead line, or rise to any challenge like getting something completed. I guess begin able to rise to a challenge is of no concern either right?
Every successful person I know credits effort for a significant part of their success. Doing “what ever it takes” to get what ever completed keeps coming up when I ask what it took to be successful. We can’t find anything like this in Alberta’s math curriculum can we?
No wonder people refer to the current math programs as Dumbing us Down.
My final comment is that division is not just repetitive subtraction. Subtraction is when I sneak my hand into the cookie jar, take a cookie and eat it. It will never be seen again as a cookie. I’ve just subtracted one cookie from the cookie jar. Division is when I divide those cookies into equal groups and put them into plastic bags so I can share them with my friends. Stealing a cookie from the cookie jar and eating it is just not the same thing as dividing up the cookie jar to share them equally with a group of my friends now is it?
How about getting back to building skill at the foundation things like reading, writing, and math? With these mastered, students stand a chance at figuring out a lot more on their own,