So I share this post for two reasons. One is to allow others to weigh in on the conversation. What are your thoughts on math vs numeracy; 'back to basics' vs real world problems? I know there are other educators across BC and the global community grappling with the same questions debated below.
But the second, and maybe more important reason I am posting this is because I thought it was a shame that these great insights were limited to an audience of Langley Principals only. I know it took me a while to get on to the "blogging stage" but I have always seen the value in smart professionals sharing their thoughts with a broader audience. So, with permission from those involved, I decided to share their dialogue with a broader audience.
It is a long read, but well worth it...
(In reference to the CBC article) Interesting viewpoint on how we have looked at the Math Curriculum. I guess all that is old is not bad after all. Read article down to the section on teaching methodologies.
A contrasting message to the article quoted in Balan's post earlier today. ( thx Chris Kennedy!) We have a lot to celebrate regarding BC education. Let's not lose sight of what we have achieved in terms of excellence and equity. As Chris says, there are things to be learned from the PISA assessment, but let's do that thoughtfully and not react to the 'back to basics' knee-jerk response.
Thank you Kathy for the article. I have shared both articles with my staff, because part of the way we are encouraging looking at curriculum differently with our staff is to look at alternate perspectives. I do encourage you to look at the video by Anna Stokke from the University of Winnipeg, not as a solution but as a perspective. I don't think it is a "knee jerk' reaction, but a cause for reflection.
I do agree that we have a lot to celebrate in the Canadian Education system. I also feel that we need to take a "pluralistic" view of educational change and reform. My focus on sending the article is so that we have a continual dialogue about what and how we should look at Numeracy as leaders in a K-12 system. As Sam has recently highlighted at Principal's meeting it is an area of growth for us as a district. Here is my response to the article that I sent to Chris:
Thank you Chris, for your reflection of the PISA results, however I am more interested in the specifics of the status of Numeracy and Mathematics based programs in Canada. My general observation as someone who has been an Administrator in a Grade 1-12 school is that our students are definitely decreasing in basic Mathematics ability. I am well aware of the greatness of the Canadian Education system and we can be proud of BC, however I think we need to be truly reflective about whether our philosophical ideologies are really making a difference for our students. While the new provincial curriculum is exciting, and project based learning can create more passion for learning, we might also learn from our past and listen to opposing view points. Here is one example of a view point that I read in response to the CBC article below.
Every time a board or region swings the pendulum from inquiry to rote or back again they set the progress of math education back and loose a lot of kids through the cracks. The truth is there should be a balance.
Elementary kids need a lot of repetition to develop foundational numeracy, but the algorithms taught should not be devoid of sense making and reason. Often the standard algorithms for low-level math functions like rounding or 2-digit by 2-digit multiplication are taught in unfathomably abstract, rule-based ways when they could be taught using processes and algorithms that connect to some sense-making or visualization of the concept. Some of the “competent” math students coming into my grade 10 class might be able to invert and multiply to get the answer to a paper/ pencil fraction division problem but can’t tell me how many eighths are in one and a half; or they can calculate 15% of 250 but stare blankly when I ask them what three quarters of eight is.
Inquiry approaches often go too far and loose the demand for mental math and fluency but we have to allow room for kids to figure some things out for themselves — that’s what makes math fun! I think the traditional approach (algorithm, rules-based) is a decent place to start, but sense-making and reasoning definitely belong in math classes.
Ok Balan, you asked me to weigh in so…
I completely agree with your position that there should be balance. However, I also completely disagree with the idea that we need to start with the 'traditional (algorithm, rules-based)' approach. I also am aghast (yes a big word) at the idea of "repetition developing foundational numeracy" (excuse me??)!
Let me start with the second item first. Numeracy is the ability to understand and work with numbers. It is not only the ability to do the basic operations (add/sub/mult/div) but rather to make sense of them as well. The second part is of huge importance in order for students to be successfully numerate individuals. A comprehensive definition is provided in the BC Math curriculum document (p.11 WCP 8/9). Foundational numeracy (for it to be foundational) must have a degree of sense making included. I think you are referring to the basic arithmetic operations and the ability to do them accurately and efficiently. That is important (critical even) and that's where your idea of balance is important. There needs to be a balance between the emphasis on arithmetic operations and understanding of use/application of these operations. The 'new' curriculum, that has been with us for a period of time (since my IS days if we can recall back that far), emphasizes this balance by ensuring that teachers also focus on communication, connections, mental math and estimation, problem solving, reasoning, technology and visualization. However, it does not say that these should be done at the expense of the basic operations or in lieu of them. Students need to know their times tables and how to manipulate numbers! Yes. They should! However, this also should not be at the expense of the processes! How often have you heard a teacher interrupt a student from doing their 142 identical timed practice questions because they weren't visualizing correctly? How many actually spend time on Mental math which is an essential skill in my opinion? BTW, mental math is addressed in the curriculum as one of the mathematical processes and it is not a part of the traditional approaches you were advocating for. Just sayin'. :)
The two things you mentioned that I think are true is that we need to balance the two and there is a pendulum shifting in the wind. For far too long it was all about the operations and basic arithmetic. Now, in some classes, it's too far the other way. Balance is the key. Implementation is also key. In many classrooms, the implementation is disjointed and frustrating. Why are we forcing the use of algebra tiles? Sure they can be helpful, but sometimes it's just as helpful to pull out a piece of paper and write!! Sometimes we need to use common sense and not just the newest fad. I hear your pain. However, that doesn't mean go back to the tried and true! Anyone still have Journeys in your school? Shame on you!! (This from Tom O'Shea so don't shoot the messenger).
My second area of disagreement lies in the notion that we should be starting with a traditional approach. This is the whole issue! Starting with this focus leads us to right back to squeezing the motivation, relevance, creativity and originality, not to mention understanding, right out of a student's math learning experience. We are trying to fight the notion that there is one way to do things in math and build in students the idea that they can use numbers to creatively problem solve and represent the world around them. Using patterns to teach operations is a good thing. Using patterns and relationships to teach algorithms is also a good thing. But to start with algorithms and only then look at making sense of math is narrow and self-defeating!
With as massive a change in mathematics education as the one we've undergone, I'd expect a dip in performance as teachers and parents figure it out. But have we really dipped significantly? Ask Chris Kennedy. Despite significant changes we are still in the top performing group, English speaking and otherwise, with a potentially stronger base of numerate individuals to come!
Finally, we have always maintained that schools needed to teach students and parents the importance of a balanced math curriculum. Contrary to Anna Stokke, parents must play a significant role in public education or at least allowed the opportunity to! If they can't then we need to look at feasible interventions that don't just promote rote learning.
“Kids spend six hours a day [at school]—I think the schools should be able to teach math to children themselves,” says Stokke. “It’s completely wrong-headed. And the moment you say parents should play a significant role in public education, you have a two-tiered system.”
- Anna Stokke (Macleans article)
The strategy of teaching students a basic operation and then drilling it into them using programs like successmaker, kumon, …etc. and then leaving them to figure out how it applies is outdated and backward. Lessons learned from research projects like the one of Brazillian candy sellers shows that there's a discrepancy between school mathematics and the ability to understand and use math effectively in everyday life. It's the latter that is the definition of numeracy and should be our goal.
Bottom line is that the more students take music, the better they will be at math. It’s not good enough to be able to visualize math, you need to be able to feel math. There is a rhythm to it. That’s why so many drummers are closet computer programmers.
More music – better math results.
These are my random/abstract thoughts for the day :)
Balan wanted more long winded thoughts… Overall I agree with the balanced mathematics program approach. But balance means balance! I think we need drill and repetition in addition to solid teaching for understanding and problem solving. I’ve been teaching math since 1986. I’ve always aimed for understanding, appreciation, critical thinking, and application AS WELL AS familiarity through repetition and exposure to a variety of problems. I have taught quite a number of courses and students in my time in the math classroom. Remedial courses, applied courses, honours, etc. - many students considered themselves strong math students, others felt they were not good at math, and a full range in between. The vast majority understood what I was teaching them and some didn’t… I did teach in rows with me at the front much of the time, but I was always trying to get them to inquire and engage in a very structured way.
Of the many, many successful/strong math students who went on to finish university degrees in mathematics, science, medicine, engineering, education, arts, business/commerce, etc. I don’t think I can recall any who did not know his/her math facts. At the high school level, where most of my time was spent, I didn’t have much class time to drill basic math facts. They needed to know their facts coming in from elementary. I do recall many students who would clearly understand the high school level concepts being taught to them – only to find themselves “stumble” when they made basic (primary) calculation errors that would interfere with their further application or cause them frustration in their mathematical comprehension/literacy. I do recall some students who knew their facts, but did not necessarily grow to be strong higher level math students.
My practical take away on this is: Ensuring that a student knows her math facts does not guarantee a strong higher level math student. They eventually “peak out” somewhere along the line. BUT, letting students slip through without solid knowledge and recall of their math facts almost certainly severely limits virtually all those students later on.
Theoretical thoughts (much longer…):
I question some of our use of “novel” problems to assess students in mathematics. In my Masters program at UBC, the professor, who was near the end of a well-respected, long tenured career at UBC, was teaching the “Problem Analysis for Administrators” course. He said he had given up on the idea of teaching transference in problem solving, i.e., if he wanted his students to know how to do something, he taught them that specific set of (complex) skills because he found that most people didn’t do very well with taking what they learned in one circumstance/problem and applying those skills in new ways to solve a novel problem in different conditions. I have no difficulty giving a comprehensive assessment and having some small portion of the overall questions test the higher level creative/novel skills. I do question giving all students a 1 or 2 hour assessment that has three or four questions on it, and two (or all) of them be questions the kids have never seen before. We know that grit/perseverance will be a big determinant of success on this type of assessment. The question is, are we assessing their math skills/knowledge or their ability to persevere? How valid will that data be in terms of informing us on their mathematical learning?
Teaching the understanding and application of an algorithm means trying the get the students to know when they should/can apply a process, when to alter the process, when the question being asked doesn’t apply to a particular process, etc. If transference of skill/knowledge is already hard, and wanting them to understand an algorithm is “easier” (but still hard), then how can we expect them to be able to do either of those things without fluent and quick knowledge of basic math facts?
I don’t see it being much different from language literacy. If a child has to spend lots of their processing power on figuring out letter-sound relationships, or sounding out sight words while reading, I can almost guarantee it will interfere with reading comprehension.
Likewise, if children have to interrupt their thought process on a two, three, or four step problem so that they can count out, or process via doubles + 2, or reach for a calculator to find the answer to "6 + 8", I can’t help but think it will interfere with their mathematical comprehension in the overall problem solving process.
My gut is telling me that we are facing the same challenges in mathematics education as we have in literacy education. When whole language became the way to teach literacy, I believe some (many) people erred in only hearing part of the message. They mistakenly thought we should let invented spelling, making their own meaning, not paying attention to form or conventions, etc. be everything and throw out all the “old school” phonics and structured resources. Nowadays, I think we are well aware that we need all of it and more! Not only that, we need to carefully manage when to keep moving students along in their development so that they don’t stagnate in their language skills. How long do we let a child stay at reading level 18 for? Best answer – we don’t – we move them along to keep them growing.
In math, I believe there is a danger to letting students settle onto one strategy that “works for them” as they can also stagnate or find out days or years later on that the process that works for them, won’t work for them when problems become more complex. Ultimately, when we understand something, if we want to make significant growth in that skill, sport, or subject we can’t be consciously processing fundamental basic components any longer, we need to just know it or be able to do it without thinking so that we can keep our main focus on the “bigger picture or process.” How do we do levelled, guided instruction in math? I don’t think we’re there yet.
How often are we hearing now that humans really aren’t great multi-taskers? I feel that for most people the same is true when it comes to solving mathematical problems.
Calvin, I can't help but respond to your comment about math drills being a return to basics. I do not argue at all that students need to know math facts. However, if we are drilling math facts before students have acquired math understanding, that is not only not helping, its counter-productive. When students have gained a strong number sense, we can use games or drills to help improve their speed of recall. I agree with John , its about implementation; what are we doing when to help students learn to reason mathematically in developmentally appropriate ways? Recently, one of our teachers posted a math question on a bulletin board in the hallway, attached a pencil on a string and allowed students to respond. The prompt was "The answer is 11; what is the question?" Classrooms at all grade levels had great conversations about the possible questions. These kinds of questions help students see the patterns and relationships in math, which needs to happen before we ask them to commit facts to memory. To me basics versus math understanding is not an 'either or' question, but what, how, when and why?
Thanks all for sharing your thoughts, this has been a rich and engaging dialogue.