Hello all, I will be moving to Word Press over the next few months. I will be sprucing up my past favourite posts and re-posting them at bryanpenfound.wordpress.com. Feel free to follow me over there. Thanks!
-Bryan
Wednesday, October 7, 2015
Sunday, September 13, 2015
Teaching Math in the 21st Century
A few months ago I was fortunate enough to read Barry Garelick's latest book Teaching Math in the 21st Century (yes, it IS available at Amazon.ca here). Aside from having a wonderful illustration of Mr. Garelick himself on the front cover behind the desk, this book is filled to the brim with a unique kind of wit that only Garelick can deliver. Having read Letters from John Dewey/Letters from Huck Finn awhile back, I was genuinely excited to have a go with his new title.
The forward is written by Ze'ev Wurman, an individual who was heavily involved in the 1997 state standards of mathematics in California, a set of standards which I have heard were very strong (unfortunately they have been since replaced by the new, and lacking, Common Core standards). Wurman sets a wonderful tone for the book by looking at the content knowledge, or lack thereof, of many classroom teachers and their administrators. This is not an attack on our educators, but moreso a serious critique of the "professional training" received. It is as though educational training promotes a culture of ignorance, a system where mathematics teaching is devoid of mathematics and those in the system cannot see their lack of skill.
Just as interesting is Garelick's introduction. He speaks of the man in the sailor's cap who uses common trickery to sell the new standards: "How many of you have solved a quadratic equation, who are not math teachers?", "If all we are doing is teaching algorithms, then we are doing our students a disservice", "We want students to 'think like a mathematician' using the eight math practices", "We are moving away from a way of teaching that didn't work". It all seems rather odd to me, considering that the California standards, to my knowledge, were quite good - how was it that they were not working? I highly doubt the man in the sailor's cap actually had any relevant data.
The book describes Garelick's work as a substitute teacher in two different locations. The first he tells us his stories from his time at a high school. Perhaps my favourite story is that of Grant's Tomb being used to describe radical notation. Garelick describes √22 as asking "Who is buried in Grant's Tomb?" to which his daughter replied "Who is Grant?" Being from Canada, and lacking in American history, I have to admit I was a bit like his daughter - I had to do some research to determine that, in fact, it was Ulysses S. Grant buried in Riverside Park in Manhattan.
At the end of the first section, Garelick writes "I showed up for every class period, taught to the best of my ability, and tried to be consistent." He let's the reader ponder over whether we believe this to be success or failure. I think it is fairly easy to see the success he accomplished in a small amount of time. The students appreciated the time they spent with him, and, in my opinion, they learned something of value.
The second portion of the book follows Garelick's long-term substitute appointment at Lawrence Middle School teaching mathematics. Personally, I would have 'run like the wind' while chatting with the principal about his son's 'deep understanding' of numbers. It makes me wonder if all school administrators are under the same oblivious cloak - falling victim to fancy terms like 'deep understanding' to describe mathematics devoid of mathematics. Do all teachers applying for positions have to jump through these hurdles? How does one escape this culture of ignorance?
The most eye-opening portion of the second half of the book was Garelick's discussion of the new Algebra 1 structure. According to Sally (she appears several times throughout the book) only the elite will be permitted into Algebra 1 in California. The explanation given is that "Common Core is very challenging" and only the truly gifted will have access. To me, it sounded like they were trying to keep students behind to beat them to death with 'conceptual understanding'. Of course, there is the typical jargon regarding more 'problem-solving' and 'critical thinking'. How is keeping highly skilled students behind good for their motivation? If America truly needs more students entering the STEM fields, should we not allow the competent students access to higher mathematics rather than holding them back? Hmm.
Garelick continues describing the culture of ignorance with a beautiful example of the quadratic formula:
Overall, Garelick's book is a lovely (albeit scary) adventure into the current state of teaching in America in the 21st century. The text is full of just the right amount of humor to mix with the eye-opening stories of Common Core implementation. He does an excellent job describing the culture of ignorance and allowing the reader to ponder his/her beliefs, despite knowing which side of the fence Garelick aligns with. The book comes highly recommended: the students are charming, the stories are memorable, and Garelick is as witty as ever.
 |
| Inside the front cover. |
The forward is written by Ze'ev Wurman, an individual who was heavily involved in the 1997 state standards of mathematics in California, a set of standards which I have heard were very strong (unfortunately they have been since replaced by the new, and lacking, Common Core standards). Wurman sets a wonderful tone for the book by looking at the content knowledge, or lack thereof, of many classroom teachers and their administrators. This is not an attack on our educators, but moreso a serious critique of the "professional training" received. It is as though educational training promotes a culture of ignorance, a system where mathematics teaching is devoid of mathematics and those in the system cannot see their lack of skill.
Just as interesting is Garelick's introduction. He speaks of the man in the sailor's cap who uses common trickery to sell the new standards: "How many of you have solved a quadratic equation, who are not math teachers?", "If all we are doing is teaching algorithms, then we are doing our students a disservice", "We want students to 'think like a mathematician' using the eight math practices", "We are moving away from a way of teaching that didn't work". It all seems rather odd to me, considering that the California standards, to my knowledge, were quite good - how was it that they were not working? I highly doubt the man in the sailor's cap actually had any relevant data.
The book describes Garelick's work as a substitute teacher in two different locations. The first he tells us his stories from his time at a high school. Perhaps my favourite story is that of Grant's Tomb being used to describe radical notation. Garelick describes √22 as asking "Who is buried in Grant's Tomb?" to which his daughter replied "Who is Grant?" Being from Canada, and lacking in American history, I have to admit I was a bit like his daughter - I had to do some research to determine that, in fact, it was Ulysses S. Grant buried in Riverside Park in Manhattan.
At the end of the first section, Garelick writes "I showed up for every class period, taught to the best of my ability, and tried to be consistent." He let's the reader ponder over whether we believe this to be success or failure. I think it is fairly easy to see the success he accomplished in a small amount of time. The students appreciated the time they spent with him, and, in my opinion, they learned something of value.
 |
| Elisa's wolf drawing on Garelick's final day. |
The most eye-opening portion of the second half of the book was Garelick's discussion of the new Algebra 1 structure. According to Sally (she appears several times throughout the book) only the elite will be permitted into Algebra 1 in California. The explanation given is that "Common Core is very challenging" and only the truly gifted will have access. To me, it sounded like they were trying to keep students behind to beat them to death with 'conceptual understanding'. Of course, there is the typical jargon regarding more 'problem-solving' and 'critical thinking'. How is keeping highly skilled students behind good for their motivation? If America truly needs more students entering the STEM fields, should we not allow the competent students access to higher mathematics rather than holding them back? Hmm.
Garelick continues describing the culture of ignorance with a beautiful example of the quadratic formula:
"Of course, under Common Core, he might not be required to memorize the quadratic formula, but would have to explain how and why it works."
How a student could be deemed to understand the quadratic formula without knowing it was puzzling.This is a very valid point: how can we expect students to show 'understanding' of an operation or procedure if they do not know how to perform the computation? A deeper understanding often comes out of a well-developed schema of the subject matter (students have something to draw on). Trying to force understanding leads to inappropriate and incorrect usage (Daisy Christodoulou has written about this here). Seems to me that those who promote the culture of ignorance are devoid of the idea that procedural knowledge promotes schema acquisition which can be drawn upon to build 'deeper understanding'. Trying to build 'deeper understanding' first seems counter-intuitive.
Overall, Garelick's book is a lovely (albeit scary) adventure into the current state of teaching in America in the 21st century. The text is full of just the right amount of humor to mix with the eye-opening stories of Common Core implementation. He does an excellent job describing the culture of ignorance and allowing the reader to ponder his/her beliefs, despite knowing which side of the fence Garelick aligns with. The book comes highly recommended: the students are charming, the stories are memorable, and Garelick is as witty as ever.
Friday, July 10, 2015
More Poorly-Designed Open-Ended Problems
Barry Garelick brought this article to my attention today, which sparked my thinking of why I dislike open-ended questions all over again.
First, I need to vent about the awful first sentence:
I agree that clear explanations and practice are important to learning maths. I consider myself a more traditional teacher who uses worked-examples and leading questions to develop mathematical concepts. However, my beef comes from the phrase "usually in isolation from other concepts." Why would any teacher agree that maths is best learned through isolated ideas?
For example, in my maths class for pre-service teachers, we recently discussed the number of divisors a number has. We considered two small example first: 12 and 16.
12 = 2^2 x 3 and 16 = 2^4
12 has 6 divisors (1,2,3,4,6,12) and 16 has 5 divisors (1,2,4,8,16)
Then I asked "Is there any connection to the number of divisors and the prime factorization?" A difficult question, so I allowed them to ponder a bit. We eventually connected the number of divisors to the powers of the primes in the factorization through addition of one. For example, 12 has (2+1)(1+1)=6 divisors and 16 has (4+1)=5 divisors. We discussed why this always works by considering the combinations of the powers, then made a generalization so we could find the number of divisors for larger numbers. And voilà, the concept of prime factorization has been connected to the concept of divisors. I don't believe that good mathematics teaching happens inside boxes.
So why does the author set up this idea that individuals believe that concepts are taught in isolation? Well, it is a straw man that leads into the research that he is doing on open-ended questions - the way I think you are teaching maths is incorrect, here is the way I am teaching maths. In case you missed my earlier post on poorly-designed open-ended questions, you can revisit it here. Essentially, the way he is teaching mathematics is by using open-ended questions such as
The main issue I have with open-ended questions is that they are often poorly thought out. For example, if we consider the first question, what time signifies a degree value of zero between the two hands (we need some kind of reference point)? Is it at 12:00? If 12:00 is zero degrees, then technically, any time after 12:15 is obtuse because we need that initial time as a reference point [here, I am assuming that we are measuring angles in a clockwise fashions, rather than the usual anti-clockwise fashion]. However, I have assumed that the hour hand doesn't move (see next paragraph)! So if we assume the hour hand moves discretely, and not continuously, then 12:15 fits the bill for 90 degrees, making any angle bigger then this obtuse. This means any time after 12:30 must be a reflex angle! So there are no acute angles if the hour hand is on 2. This question isn't mathematically sound until more information is added. Questions similar to this may develop misconceptions in your students: when given a reflex angle, they may measure the incorrect portion or fail to add the multiples of 360 degrees.
In our second question, again using 12:00 as a starting point of zero degrees, we might consider that 12:15 must be the 90 degree angle. But again, this technically isn't the case since the hour hand will have moved one-fourth of the way to 1 by the time we get to 12:15, so the hands at 12:15 are actually a little less than 90 degrees. (If interested, you can show that the time the hands will make 90 degrees is actually 12:16 and 4/11ths of a minute). Because of the argument above, this will be the only time we have a 90 degree angle! All others will be 90 + 180k degrees (k is a positive integer). Again, this open-ended question may result in the misconception that the hour hand doesn't move at all.
This article adds to my belief that open-ended questions are poor ways to teach concepts to students who are not well-versed in the content material, since poorly-designed questions may result in mathematical (or real-life) misconceptions. If these misconceptions occur during the vital younger ages and are stored as memories, it can take a significant amount of energy and time to remove these misconceptions from the long-term memory and rework in the correct concepts. The easiest solution: don't use open-ended questions, or use them sparingly.
First, I need to vent about the awful first sentence:
A common view is that students learn maths best when teachers give clear explanations of mathematical concepts, usually in isolation from other concepts, and students are then given opportunities to practise what they have been shown.
I agree that clear explanations and practice are important to learning maths. I consider myself a more traditional teacher who uses worked-examples and leading questions to develop mathematical concepts. However, my beef comes from the phrase "usually in isolation from other concepts." Why would any teacher agree that maths is best learned through isolated ideas?
For example, in my maths class for pre-service teachers, we recently discussed the number of divisors a number has. We considered two small example first: 12 and 16.
12 = 2^2 x 3 and 16 = 2^4
12 has 6 divisors (1,2,3,4,6,12) and 16 has 5 divisors (1,2,4,8,16)
Then I asked "Is there any connection to the number of divisors and the prime factorization?" A difficult question, so I allowed them to ponder a bit. We eventually connected the number of divisors to the powers of the primes in the factorization through addition of one. For example, 12 has (2+1)(1+1)=6 divisors and 16 has (4+1)=5 divisors. We discussed why this always works by considering the combinations of the powers, then made a generalization so we could find the number of divisors for larger numbers. And voilà, the concept of prime factorization has been connected to the concept of divisors. I don't believe that good mathematics teaching happens inside boxes.
So why does the author set up this idea that individuals believe that concepts are taught in isolation? Well, it is a straw man that leads into the research that he is doing on open-ended questions - the way I think you are teaching maths is incorrect, here is the way I am teaching maths. In case you missed my earlier post on poorly-designed open-ended questions, you can revisit it here. Essentially, the way he is teaching mathematics is by using open-ended questions such as
"The minute hand of a clock is on 2 and the hands make an acute angle. What might be the time?"
or
"What are some times for which the hands on a clock make a right angle?"
The main issue I have with open-ended questions is that they are often poorly thought out. For example, if we consider the first question, what time signifies a degree value of zero between the two hands (we need some kind of reference point)? Is it at 12:00? If 12:00 is zero degrees, then technically, any time after 12:15 is obtuse because we need that initial time as a reference point [here, I am assuming that we are measuring angles in a clockwise fashions, rather than the usual anti-clockwise fashion]. However, I have assumed that the hour hand doesn't move (see next paragraph)! So if we assume the hour hand moves discretely, and not continuously, then 12:15 fits the bill for 90 degrees, making any angle bigger then this obtuse. This means any time after 12:30 must be a reflex angle! So there are no acute angles if the hour hand is on 2. This question isn't mathematically sound until more information is added. Questions similar to this may develop misconceptions in your students: when given a reflex angle, they may measure the incorrect portion or fail to add the multiples of 360 degrees.
In our second question, again using 12:00 as a starting point of zero degrees, we might consider that 12:15 must be the 90 degree angle. But again, this technically isn't the case since the hour hand will have moved one-fourth of the way to 1 by the time we get to 12:15, so the hands at 12:15 are actually a little less than 90 degrees. (If interested, you can show that the time the hands will make 90 degrees is actually 12:16 and 4/11ths of a minute). Because of the argument above, this will be the only time we have a 90 degree angle! All others will be 90 + 180k degrees (k is a positive integer). Again, this open-ended question may result in the misconception that the hour hand doesn't move at all.
This article adds to my belief that open-ended questions are poor ways to teach concepts to students who are not well-versed in the content material, since poorly-designed questions may result in mathematical (or real-life) misconceptions. If these misconceptions occur during the vital younger ages and are stored as memories, it can take a significant amount of energy and time to remove these misconceptions from the long-term memory and rework in the correct concepts. The easiest solution: don't use open-ended questions, or use them sparingly.
Monday, May 11, 2015
My Response to an Assignment Question on Discovery Learning
The assignment question:
Write between 1/2 page and 1 page discussing your approaches and feelings towards "discovery" in the mathematics classroom. For example, you could discuss:
Relate this to your learning through the M.M.T. so far.
My response:
Write between 1/2 page and 1 page discussing your approaches and feelings towards "discovery" in the mathematics classroom. For example, you could discuss:
- In what situations do you deliberately use this or not use this approach?
- Where do you find that you have the most success using this or least success using this?
- How do students react to this approach?
Relate this to your learning through the M.M.T. so far.
My response:
Minimally
guided learning, as defined by Kirschner, Sweller and
Clark, is an environment in which learners discover or construct new
information for themselves instead of being presented the new materials. Discovery learning is simply another
term for minimally guided learning. In my position at the University of
Winnipeg, I teach math content courses to pre-service education students and we
rarely - if ever - use discovery learning. I would like to explain further my
reasoning for this, but I must explain briefly the interactions of the long-term
memory and the working memory.
The working memory is quite limited in capacity,
only able to hold 3-7 pieces of information at a time. As certain items are
flagged as important, these ideas move to the hippocampus where memory
formation begins; and later to the neocortex where they are eventually stored
in our long-term memory. The interesting part of the relationship between
working memory and long-term memory, is that new information that has not been
encountered before takes up a lot of space in the working memory; while
information that has been stored in the long-term memory takes up relatively
little space. If a student's working memory is "too full" we often
say that this student is in a state of cognitive
overload.
As a mathematics teacher, I have to be aware of
this cognitive architecture. As I present new material in my course, such as
computations in base-5, I need to respect that my students are novice learners of this material and
support their learning by discussing worked examples. Allowing them to discover
the operations on their own, without any help or base-knowledge (no pun
intended), may put them at risk for cognitive overload.
Carey states that "the harder we have to
work to retrieve a memory, the greater the subsequent spike in ...
learning." This quote, I believe, ties into the current ideology around
discovery learning - if the students undergo some struggle, then this type of
learning is "better" than other learning. However, prominent figures
such as Christodoulou have mentioned that discovery learning often leads to shallow results, especially if the learner does not have a solid foundation of factual knowledge to build
upon. Without a foundation of factual information to build upon, students' working memories may become quickly
overloaded due to the complexity of the task. This, to me, is the main problem
of discovery learning as it pertains to elementary school. All students are
novice learners and most material is new. Asking students to discover their own
way through mathematics without any assistance is not only bad pedagogy, but it
feels unethical in light of current research.
This is not to say that all discovery learning
is bad - I believe that discovery learning has its time and place. Take the
Master of Mathematics for Teachers (M.M.T.) at the University of Waterloo, for
example. Students entering this program have a very strong foundation in
mathematics (a B.Sc. in mathematics, or a B.Math). These students have also
been teaching mathematics at the high school level for several years, which
means their retrieval strength of math
facts and properties from long-term memory is high. With this framework, minimally
guided instruction through the M.M.T. program works well, since students are
able to bypass cognitive overload (for the most part) and make meaningful discovery.
References
Carey, B. How
We Learn, Random House, New York, 2014.
Christodoulou, D. Seven Myths About Education, Routledge, New York, 2014.
Kirschner, P.A. & J. Sweller & R.E.
Clark. (2006). Why Minimal Guidance During Instruction Does Not Work: An
analysis of the failure of constructivist, discovery, problem-based,
experiential, and inquiry-based teaching. Educational
Psychologist, 41(2), 75-86.
Monday, May 4, 2015
ResearchED New York - The Big Apple Welcomes Those form Across the Pond
On May 2, 2015 I had the utmost pleasure of attending the researchED conference in New York. If you haven't heard about researchED, it is an organization of teacher-led individuals who are interested in dispelling myths in education and bringing high-quality research to the table for teachers. If you want to read more about them you can check them out here or, according to Tom Bennett during his welcome introduction, you can wait until he releases his book One Tweet: My Story this fall (please do not hold your breath for the release of said book).
What I would like to do for you is to give you an insight into the sessions that I attended, as well as my thoughts about the conference overall.
The Location: Riverdale Country School, The Bronx
Let me be the first to say that the locale for the event was perfect. Riverdale Country School is a pre-kindergarten to grade 12 independent school located just off the Henry Hudson Parkway in the Fieldston region of the Bronx. I was stunned at how lovely the campus was as I drove up. It also helped that the research gods were shining down on us with beautiful weather and copious amounts of sunshine. The campus was very active for a Saturday with parents and students attending sporting events and writing finals. However, despite the campus being busy, Riverdale Country School invited researchED and all its attendees with open arms - the staff and the school head, Dominic Randolph, were exceptional and deserve many thanks for helping organize such a successful event.
Session 1: Developing Great Teachers - What Works, and How Do We Do It?
The first session I attended was by the well-spoken David Weston (@informed_edu) in which he discussed various way to implement successful professional development sessions for teachers. It did not take him much time to win over the audience (his session, by the way, was so popular, there was standing room only for latecomers) with some shocking statistics:
Session 2: Seven Myths about Education - What are They and Why do They Matter?
I next had the joy of attending Daisy Christodoulou's (@daisychristo) session in which she discussed some key ideas from her book Seven Myths about Education. The main idea she shared with us was the need for strong content knowledge in schools. She shared some hilarious anecdotes and research with the audience, including the thesaurus experiment, in which students were asked to replace certain words with other words from the thesaurus. I believe the audience's favourite was the student who changed "Mrs. Morrow stirred the soup." to "Mrs. Morrow stimulated the soup." This example was meant to illustrate that it takes a vast amount of content knowledge to be able to access a thesaurus in a successful way. In a similar fashion, it can be quite difficult for students to use Google to look up facts, since it takes considerable content knowledge to parse out the incorrect and useless facts, all the while utilizing considerable working memory.
Christodoulou also cautioned for educators to carefully examine their lesson plans, as "students tend to remember what they think about." If your student is being asked to create a PowerPoint presentation on The Great Depression, they may be more focused on searching for pictures, or in making their animations look good - which defeats the purpose of having the student research The Great Depression in the first place. As we develop our lesson plans, lead with the question "What is it that I want my students to think about?" and develop your lesson from there.
Session 3: A New School Centric Definition of Evidence-Informed Practice
The next session I attended was given by Gary Jones (@drgaryjones) in which evidence-based practice was discussed and explored. Jones began by giving the typical definition of the term evidence-based practice, and revamped it to better suit the educational community. Of the points made in this session, I particularly enjoyed the following:
Session 4: How Might Teachers Serve as Advocates for the Science of Learning?
Ben Riley (@benjaminjriley) first got the audience mad - really really mad. He showed us a photo from Texas of a billboard ad promoting a 12 week teacher training program for certification. Yes, you read correctly - after you have completed your Bachelor program, you may enter this after-degree program and be certified to teach in only 12 weeks! A typical after-degree program in Canada is four semesters, not one. In my opinion, it is difficult to call teaching a profession if there are programs like this that treat teacher training in such a simplistic way. How can educators develop the strong content and pedagogical knowledge needed to be effective teachers in one semester?
Among other ideas Riley shared with the audience was a history lesson in which we saw how the business and medical sectors transformed over the years to become professions. Nowadays, people are very proud to say that they got into medical school or business school. We typically don't hear many people proudly exclaim that they got into ed school. I think this says something about the overall value our society puts on the teaching "profession."
Riley also mentioned that it will be a challenging uphill battle, as many educators still believe in debunked theories. For example, he gave a percentage breakdown of teachers in various countries that still believe the three neuromyths (1) students learn better when presented material in their learning style, (2) exercise can help integrate left/right brain functioning, and (3) differences in left/right brain dominance can explain differences in students. These results were shocking indeed.
Session 5: What can 'Top-Performing' Systems Teach Us about Education Policy?
The last session I was able to attend was by the lovely Lucy Crehan (@lucy_crehan) in which she discussed her travels to schools in 'top-performing' systems. She began the session by discussing usual complaints we hear about PISA.
The Conclusion: More ResearchED is Needed on These Topics
All in all, ResearchED NY was an amazing experience and allowed me to reflect upon big ideas in the educational field. Many thanks goes out to all the presenters, the staff of Riverdale, and the organizers.
From what I saw and gathered from the atmosphere of the event is that educators from around the globe are hungry to learn and to share ideas with each other. This is exactly what a professional community should look like, and I am happy to be a part of this community. When it all boils down, we share a common goal: communication. There is a drive in all of us to be a part of the on-going educational conversation. We all understand the implications the conversation has for our communities, how the conversation challenges our ideals and biases, and how open and thoughtful communication is necessary to allowing this conversation to continue. I am looking forward to the next ResearchED event I am able to attend.
PS: If you are looking for future locales for ResearchED, I hear Banff is beautiful this time of year. ;)
 |
| The humorous Tom Bennett during his welcome speech at the introduction of researchED NY. |
The Location: Riverdale Country School, The Bronx
Let me be the first to say that the locale for the event was perfect. Riverdale Country School is a pre-kindergarten to grade 12 independent school located just off the Henry Hudson Parkway in the Fieldston region of the Bronx. I was stunned at how lovely the campus was as I drove up. It also helped that the research gods were shining down on us with beautiful weather and copious amounts of sunshine. The campus was very active for a Saturday with parents and students attending sporting events and writing finals. However, despite the campus being busy, Riverdale Country School invited researchED and all its attendees with open arms - the staff and the school head, Dominic Randolph, were exceptional and deserve many thanks for helping organize such a successful event.
 |
| A view of Mow Hall from Vinik Hall, Riverdal Country School. |
Session 1: Developing Great Teachers - What Works, and How Do We Do It?
The first session I attended was by the well-spoken David Weston (@informed_edu) in which he discussed various way to implement successful professional development sessions for teachers. It did not take him much time to win over the audience (his session, by the way, was so popular, there was standing room only for latecomers) with some shocking statistics:
- Curee (2011) - only 1% of teacher PD sessions were of "high quality"
- Sutton (2011) - the gap between disadvantaged children and others grows exponentially with time; however, disadvantaged children had the potential to make three times as much progress with "highly effective teachers"
Session 2: Seven Myths about Education - What are They and Why do They Matter?
I next had the joy of attending Daisy Christodoulou's (@daisychristo) session in which she discussed some key ideas from her book Seven Myths about Education. The main idea she shared with us was the need for strong content knowledge in schools. She shared some hilarious anecdotes and research with the audience, including the thesaurus experiment, in which students were asked to replace certain words with other words from the thesaurus. I believe the audience's favourite was the student who changed "Mrs. Morrow stirred the soup." to "Mrs. Morrow stimulated the soup." This example was meant to illustrate that it takes a vast amount of content knowledge to be able to access a thesaurus in a successful way. In a similar fashion, it can be quite difficult for students to use Google to look up facts, since it takes considerable content knowledge to parse out the incorrect and useless facts, all the while utilizing considerable working memory.
 |
| Students were asked to replace words using a thesaurus. Some of the 'stimulating' responses are given above. |
Session 3: A New School Centric Definition of Evidence-Informed Practice
The next session I attended was given by Gary Jones (@drgaryjones) in which evidence-based practice was discussed and explored. Jones began by giving the typical definition of the term evidence-based practice, and revamped it to better suit the educational community. Of the points made in this session, I particularly enjoyed the following:
- Research from all sides needs to be reviewed before making decisions on educational policy. Typically, we have seen educational policy come into fruition backed by cherry-picked data.
- We need to challenge our own biases by seeking out non-conforming data. This leads to humble inquiry: the art of asking questions based on interest, rather than asking questions to try to dominate a conversation.
- Policy-makers should develop pre-mortems to analyze potential drawbacks and problems with future educational policies. This will save much wasted time and effort, and allow for more refined approaches to policy-making.
Session 4: How Might Teachers Serve as Advocates for the Science of Learning?
Ben Riley (@benjaminjriley) first got the audience mad - really really mad. He showed us a photo from Texas of a billboard ad promoting a 12 week teacher training program for certification. Yes, you read correctly - after you have completed your Bachelor program, you may enter this after-degree program and be certified to teach in only 12 weeks! A typical after-degree program in Canada is four semesters, not one. In my opinion, it is difficult to call teaching a profession if there are programs like this that treat teacher training in such a simplistic way. How can educators develop the strong content and pedagogical knowledge needed to be effective teachers in one semester?
 |
| Billboard ad in Texas: "Want to teach? When can you start?" |
Riley also mentioned that it will be a challenging uphill battle, as many educators still believe in debunked theories. For example, he gave a percentage breakdown of teachers in various countries that still believe the three neuromyths (1) students learn better when presented material in their learning style, (2) exercise can help integrate left/right brain functioning, and (3) differences in left/right brain dominance can explain differences in students. These results were shocking indeed.
 |
| Percentages from UK, The Netherlands, Turkey, Greece and China of teachers who believe certain neuromyths. The lowest percent score was 71% and the highest 97%. |
Session 5: What can 'Top-Performing' Systems Teach Us about Education Policy?
The last session I was able to attend was by the lovely Lucy Crehan (@lucy_crehan) in which she discussed her travels to schools in 'top-performing' systems. She began the session by discussing usual complaints we hear about PISA.
- "PISA is too narrow a measure." While this may be true, this should not be a reason to not look at the top-performing schools!
- "Those countries only do well because they are small and homogeneous." Of the top-performers, only Finland is small and homogeneous. However, it is curious to note that its neighbours are also small and homogeneous and are not doing so well on international testing.
- "Correlation does not imply causation." Of course! However, this phrase is often used to dismiss findings. We need to be mindful of when people are using this statement to disregard potential interesting data.
The Conclusion: More ResearchED is Needed on These Topics
All in all, ResearchED NY was an amazing experience and allowed me to reflect upon big ideas in the educational field. Many thanks goes out to all the presenters, the staff of Riverdale, and the organizers.
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| The animated Tom Bennett posing with me during lunch at ResearchED NY. |
PS: If you are looking for future locales for ResearchED, I hear Banff is beautiful this time of year. ;)
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| Empire State Building as seen from The Rockefeller Center. |
Friday, April 10, 2015
More Time in Mathematics for Pre-Service Teachers
SIEGLER'S STUDY
When we first begin to work computationally with fractions, we are often challenged in our ways of thinking about numbers. Let me illustrate with a few examples. When we multiply two whole numbers bigger than one, the product is larger than our factors (2 x 5 = 10 and 10 is larger than 2 or 5). When we divide two whole numbers bigger than one, the quotient is smaller than the dividend (10 / 2 = 5 and 5 is less than 10).
As novice learners of mathematics, the ideas "when I multiply, the number gets larger" and "when I divide the number gets smaller" rarely get challenged. That is, perhaps, until the novice learner begins computations with rational numbers. Rational numbers are a bit of an odd beast, as I will show you with a few examples. When multiplying two rational numbers between zero and one, our product is smaller than our factors: 1/2 x 1/4 = 1/8, and 1/8 is less than 1/2 or 1/4. If we divide two positive rational numbers between zero and one, the quotient is larger than the dividend: (1/2) / (1/4) = 2, and 2 is larger than 1/2 or 1/4. Notice that this kind of conceptual thinking is counter to how we think about whole number multiplication. When multiplying as novice learners, we expect our numbers to get larger; and when dividing as novice learners, we expect our numbers to get smaller.
When someone incorrectly assumes that multiplying 1/2 x 1/4 yields a larger number than 1/2 or 1/4, we call this a direction of effect error. We use the same term for an individual who incorrectly assumes dividing two rational numbers between zero and one yields a smaller number.
It is here that I would like to point out an interesting paper that was recently published by Robert S. Siegler and Hugues Lortie-Forgues entitled Conceptual Knowledge of Fraction Arithmetic. The full article can be found here, but I would like to give a brief summary of its findings as it relates to the direction of effect error and conceptualization of fractions.
The study can be summarized into three distinct parts. In each part, three areas regarding fractions were studied:
(1) fraction arithmetic (called procedural knowledge), in which the subjects were asked to perform computations involving two positive fractions (adding, subtracting, multiplying or dividing)
(2) the direction of effect error (called conceptual knowledge), in which the subjects were given a statement of the form a/b * c/d > a/b and they were asked to determine if the statement was true or false (here the * represents one of the four standard operations +, -, x, /)
(3) magnitude of whole numbers and fractions, in which subjects were asked to label certain whole numbers and fractions on a number line to determine their accuracy
In the first part, 41 pre-service teachers from Quebec were asked to perform the three tasks listed above. The pre-service teachers had excellent knowledge of fraction magnitude, and excellent procedural/conceptual knowledge of addition and subtraction. However, their conceptual knowledge of multiplication and division was weak, as well as their procedural knowledge of division. Results are given in Table 1:
When we first begin to work computationally with fractions, we are often challenged in our ways of thinking about numbers. Let me illustrate with a few examples. When we multiply two whole numbers bigger than one, the product is larger than our factors (2 x 5 = 10 and 10 is larger than 2 or 5). When we divide two whole numbers bigger than one, the quotient is smaller than the dividend (10 / 2 = 5 and 5 is less than 10).
As novice learners of mathematics, the ideas "when I multiply, the number gets larger" and "when I divide the number gets smaller" rarely get challenged. That is, perhaps, until the novice learner begins computations with rational numbers. Rational numbers are a bit of an odd beast, as I will show you with a few examples. When multiplying two rational numbers between zero and one, our product is smaller than our factors: 1/2 x 1/4 = 1/8, and 1/8 is less than 1/2 or 1/4. If we divide two positive rational numbers between zero and one, the quotient is larger than the dividend: (1/2) / (1/4) = 2, and 2 is larger than 1/2 or 1/4. Notice that this kind of conceptual thinking is counter to how we think about whole number multiplication. When multiplying as novice learners, we expect our numbers to get larger; and when dividing as novice learners, we expect our numbers to get smaller.
When someone incorrectly assumes that multiplying 1/2 x 1/4 yields a larger number than 1/2 or 1/4, we call this a direction of effect error. We use the same term for an individual who incorrectly assumes dividing two rational numbers between zero and one yields a smaller number.
It is here that I would like to point out an interesting paper that was recently published by Robert S. Siegler and Hugues Lortie-Forgues entitled Conceptual Knowledge of Fraction Arithmetic. The full article can be found here, but I would like to give a brief summary of its findings as it relates to the direction of effect error and conceptualization of fractions.
The study can be summarized into three distinct parts. In each part, three areas regarding fractions were studied:
(1) fraction arithmetic (called procedural knowledge), in which the subjects were asked to perform computations involving two positive fractions (adding, subtracting, multiplying or dividing)
(2) the direction of effect error (called conceptual knowledge), in which the subjects were given a statement of the form a/b * c/d > a/b and they were asked to determine if the statement was true or false (here the * represents one of the four standard operations +, -, x, /)
(3) magnitude of whole numbers and fractions, in which subjects were asked to label certain whole numbers and fractions on a number line to determine their accuracy
In the first part, 41 pre-service teachers from Quebec were asked to perform the three tasks listed above. The pre-service teachers had excellent knowledge of fraction magnitude, and excellent procedural/conceptual knowledge of addition and subtraction. However, their conceptual knowledge of multiplication and division was weak, as well as their procedural knowledge of division. Results are given in Table 1:
In the second part, 59 6th and 8th grade students from the greater Pittsburgh area were asked to perform the three tasks listed above. The students had excellent knowledge of fraction magnitude, and excellent procedural knowledge of all four operations with fractions. However, their conceptual knowledge of multiplication and division was weak. Results are given in Table 2:
In the third part, 17 undergraduate students in STEM fields were asked to perform the three tasks listed above. The undergraduate students were fluent in both conceptual and procedural knowledge of fractions in all areas. Results are given in Table 3:
DISCUSSION OF THE STUDY
Now, while the study does suffer from a couple drawbacks, it does raise a few interesting discussion points. The first I want to highlight is that the grade 6 and 8 students had roughly the same procedural knowledge and conceptual knowledge as the pre-service teachers. This to me is interesting and may account for why the general population is so fraction-adverse: perhaps many of our elementary school teachers actually have roughly the same base knowledge about fractions as their students (or less, as was the case in this study). Since the teachers themselves never truly conceptually understand fractions, how can we expect them to pass on conceptual understanding of fractions to their students? In this respect, fraction arithmetic gets placed into its own box in our long-term memory that is mutually exclusive to whole number arithmetic.
The second point that I would like to bring up is that those students who were in STEM fields had exceptional conceptual and procedural understanding of fraction arithmetic. I believe this is due to the fact that arithmetic with rational numbers must be mastered at an early point in their careers. One cannot be successful in calculus and analysis without a firm grasp of rational numbers, as these ideas lead to the more abstract ideas of rational expressions and rational functions. Also, these students see arithmetic with rational expressions every semester of every year - they have had many hours to perfect their knowledge. Compared to our pre-service teachers, I would guess that our STEM undergraduates see several hundreds of more hours working with rational expressions.
CONCLUDING REMARKS
So how can we use the ideas put forth in this study to inform our practice? First, faculties of education should recruit undergraduate students who have strong procedural and conceptual abilities, as they will make very strong teachers. Faculties of education often have very low entrance requirements, which runs counter to my personal beliefs. If we want good educators, shouldn't we set our bar higher? I also often I hear the argument that students who have higher-level education in mathematics do not make good teachers. This is clearly a false statement. Those who have spent more time studying mathematics have much more consolidation of conceptual knowledge when compared to those who have spent less time studying mathematics.
Secondly, we should encourage our pre-service teachers to spend more time studying mathematics with mathematicians - not by math educators in faculties of education. Since our math educators in faculties of education have not spent as much time studying mathematics, they will likely suffer from the same drawbacks as the pre-service teachers in the above-mentioned study. However, graduates of faculties of science will have spent many hours perfecting conceptual ideas related to elementary school mathematics. If we want our pre-service teachers to be better math teachers, faculties of education have to work more closely with their colleagues in mathematics to develop math content courses that are relevant and designed to deepen the knowledge of our future teachers. And these courses need to be delivered by well-trained individuals with a strong mathematics background.
Tuesday, March 31, 2015
The Fallacy of the 'Multiple Strategies' Approach
It has recently come to my attention that many the proponents of the multiple strategies approach to mathematics education are completely hypocritical. Allow me to explain.
Let's take an example to discuss exactly what is meant by the multiple strategies approach. Consider the question 24 x 3. A student approaching this question might use base-ten blocks to model the question as an area, might try to use a 'doubling-halving' approach to alter the question to a basic fact that they already know 24 x 3 = 12 x 6 = 72, or might try finding three groups of 20 and three groups of 4 using the distributive property 24 x 3 = 20 x 3 + 4 x 3 = 60 + 12 = 72. All of these strategies to tackle this problem are taken as equally valid solutions to the posed question.
The strategies above seem fine; however, there is one glaring problem. I haven't included the use of the standard algorithm as an acceptable way of solving the posed question above! What I see in practice is that proponents of multiple strategies dismiss the standard algorithms under the guise that students do not "understand" the mathematics if the algorithm is used. They instead favour visual and tactile models, and mental calculations - all of which are cognitively heavy on our students, and only work in contrived scenarios. Try to use the doubling-halving method on the multiplication 15 x 15. It won't work because one of the numbers must be even to use this "trick"! I have written about this particular "trick" before, stating that it would be more beneficial for students to learn prime factorization (at least prime factorization comes up often in practice and works in more situations than doubling-halving does). Even better than prime factorization is to teach the standard algorithm! When performed correctly, it is much more reliable than mental calculations, and much more efficient than visual or tactile models. The amount of time spent on the inefficient multiple strategies listed above could be more efficiently used to teach the understanding of the mathematics behind the algorithm. Problem solved!
Despite what I have mentioned above, proponents of the multiple strategies approach continue to word assessment questions in such a way so that students are forced not to utilize the standard algorithms. Or if the student does utilize the standard algorithm where appropriate, they are severely punished. So much for being able to use "multiple" strategies to solve a question!
To give some illustrative examples, I have taken some screen shots of an assessment for Grade 7 teachers.
The first figures show the probing question "Write everything you know about fractions." (Aside: What an AWFUL question. How can you determine what is a good answer and what is not a good answer? There are no guidelines for the student to follow!)
The first student shows that he/she knows some basic fractions and that he/she knows what these fractions look like as a part-whole diagram. Yet this student is "not meeting" the intended grade level objective. How can the student know that by looking at the question? The student answered the questions just perfectly! He/she put down knowledge of fractions.
The second student, who can model fractions using diagrams and algebra, and who even knows the ideas of mixed/improper fractions doesn't even meet the standard! The student must connect fractions to decimals to get a "meeting" grade level objective. But was that clear in the posed question?
The third student here demonstrates that he/she can clearly represent fractions in about 2000 ways, so he/she must be meeting the standard. This is exactly what the proponents of multiple strategies want: a student doesn't "understand" fractions unless he/she can model them in many different ways. However, how can our students be expected to recall ALL of this information with such a poorly designed open question? Would it not be better to have them recall specific ideas, rather than overload them cognitively with such a daunting probing question?
Continuing on, let's see another goodie. Here students are asked to order the following fractions, decimals and percents:
But students might use the standard algorithm to convert fractions to decimals. So, to make sure students don't use it, we will insist that students have to use multiple strategies to get to the answer! Again, a multiple strategies approach was not asked for in the question, so now they are simply punishing students for utilizing a perfectly good strategy.
And, finally, here is my favourite clip from that document. In this question, students are asked to calculate 36% of 25.
This student calculated the correct answer using the standard algorithm for multiplication. Very nicely done! But I am sorry to say that the algorithm is not allowed, so you are "not meeting" expectations. What we actually wanted from you is a contrived mental math strategy given below:
So, I only hope that this gives you some insight into why I am very wary of proponents of multiple strategies. These individuals are secretly removing the standard algorithms from our curricula under the guise that their "methods" (pictures, mental strategies, etc.) promote "understanding." When, in actuality, their methods are archaic, inefficient and non-transferable to many situations.
I leave you with two links. One is to the Heinemann blog on the Standards for Mathematical Practice (SMP) for the US Common Core. In particular you should read standard #5, in which John SanGiovanni clearly states that "In many situations, pencil and paper are inefficient and using them is not strategic." He is promoting use of multiple tools to solve problems, yet you can infer that he would not be happy about students using pencil and paper to solve the question using the algorithm for subtraction (he doesn't even mention the standard algorithm as a viable strategy).
The other link I wish to share is a piece written by Michael Zwaagstra inThe Chronicle Herald, a Nova Scotia newspaper. In this piece, Zwaagstra comments on how Marian Small, a prominent figure for the multiple strategies approach, hypocritically states that different strategies to solve questions are good, but to leave out the standard algorithms from the curriculum.
Sunday, March 1, 2015
Do Educational Organizations Have Any Clue?
Recently a post by the Canadian Education Association (CEA) came across my desk. It was titled "The Facts on Education: Is Inquiry-Based Learning Effective?" so of course I was curious to see what the CEA had to say. The full post can be found here. And the sources referenced can be found here.
What I want to focus on is how the educational organizations have no clue about how to properly teach children. Let's analyse this passage:
According to the CEA About Us page, their mission is to "transform public education by ... disseminating research that can impact practice." Seems to me as if they are preaching the usual "fuzzy" ideals we see from many educationalists. Do we really need "professional" organizations like this distributing biased research to our teachers and government?
What I want to focus on is how the educational organizations have no clue about how to properly teach children. Let's analyse this passage:
So they begin by stating that inquiry-based activities can boost "learning" yet they have decided not to define what they mean by "learning." I admit that I did take the time to search Sharon Friesen's literature review on inquiry-based learning since she was listed as a reference (She submitted this review to the Alberta government by the way, completely ignoring any research that didn't support her viewpoint surprise surprise.). This literature review had 254 instances of the word "learning," yet Dr. Friesen failed to define "learning" once. How can one discuss inquiry-based learning if one doesn't know what "learning" even means? Very peculiar. Seems like the CEA is doing this in the above paragraph as well. How can we trust the CEA that inquiry-based activities can boost "learning" if they have failed to define it?
The CEA continues down the rabbit hole by stating that unguided or minimally-guided inquiry may not work with students who have less previous knowledge. They even mention that "learning" may be blocked, or misconceptions may arise, if inquiry-based methods are used. In these couple sentences they are definitely referring to the referenced work of Kirschner, Sweller and Clark. This seems very odd to me - why go out of your way to claim that inquiry methods are excellent, then reference a paper that argues against what you are trying to state? Kirschner, Sweller and Clark give a very good argument as to why student lessons should be scaffolded with many worked examples - which tends to be the opposite to inquiry-based methods. In fact there is a large body of knowledge (not just from cognitive psychology) that has found explicit instruction to be superior to non-explicit instruction (like inquiry-based learning). See this post by @greg_ashman for a bit more to this picture. So it seems to me as if the CEA really doesn't understand the research properly. It is even more apparent that they are clueless when they make the following absurd claim:
According to the CEA About Us page, their mission is to "transform public education by ... disseminating research that can impact practice." Seems to me as if they are preaching the usual "fuzzy" ideals we see from many educationalists. Do we really need "professional" organizations like this distributing biased research to our teachers and government?
Saturday, February 14, 2015
Parallel Thinking
In my most recent post, I discussed Marian Small's book Good Questions. I really wasn't a huge fan of the book, but there were some redeeming features that I did enjoy. Her view of parallel tasks is one that deserves some mention.
What do we mean by a "parallel task"? The idea of a parallel task is taking a question that a student might be facing and "scaling it back" to a question that is more familiar to the student. I tend to do this a lot in precalculus, especially when working with rational expressions. Let me give you some examples.
Suppose we have to add 1/(x-2) + 1/(x-3). We might write:
1/(x-2) + 1/(x-3)
= (x-3)/(x-2)(x-3) + (x-2)/(x-2)(x-3) [find LCM]
= (2x-5)/(x-2)(x-3) [add the numerators]
To motivate this, we can think of the simpler problem of 1/2 + 1/3.
1/2 + 1/3 = 3/6 + 2/6 [find LCM]
= 5/6 [add the numerators]
Sometimes the rational expressions get trickier with common factors like
1/(x^2 - 25) + 1/(x^2 + 8x +15)
= 1/(x+5)(x-5) + 1/(x+5)(x+3) [factor]
= (x+3)/(x+5)(x-5)(x+3) + (x-5)/(x+5)(x-5)(x+3) [find LCM]
= (2x-2)/(x+5)(x-5)(x+3) [add numerators]
To motivate this, we can think of the simpler problem of 1/6 + 1/9 (since 6 and 9 share a common factor, but do not share their other factors).
1/6 + 1/9 = 1/(2)(3) + 1/(3)(3) [prime factor]
= 3/(2)(3)(3) + 2/(2)(3)(3) [find LCM]
= 5/(2)(3)(3) [add numerators]
= 5/18
So this is the idea of parallel tasking. Where my students tend to have troubles is with the fact that they are not strong with fractions. Perhaps teachers are not showing the importance of using the LCM when finding common denominators (ie. 1/6 + 1/9 = 9/54 + 6/54... WHY WOULD YOU TEACH THIS?!), or not dedicating enough time to prime factorization (this second fault might be due to the awful curriculum we have that forces "mental math" strategies that only work in contrived situations).
In my opinion, parallel tasking is something that an excellent teacher will do and use effectively without thinking. However, it requires a lot of knowledge in the subject matter, and the ability to see how everything is interconnected (something most teachers do not attain from education faculties). But that is another post for another day.
Have you had any success with parallel tasks?
What do we mean by a "parallel task"? The idea of a parallel task is taking a question that a student might be facing and "scaling it back" to a question that is more familiar to the student. I tend to do this a lot in precalculus, especially when working with rational expressions. Let me give you some examples.
Suppose we have to add 1/(x-2) + 1/(x-3). We might write:
1/(x-2) + 1/(x-3)
= (x-3)/(x-2)(x-3) + (x-2)/(x-2)(x-3) [find LCM]
= (2x-5)/(x-2)(x-3) [add the numerators]
To motivate this, we can think of the simpler problem of 1/2 + 1/3.
1/2 + 1/3 = 3/6 + 2/6 [find LCM]
= 5/6 [add the numerators]
Sometimes the rational expressions get trickier with common factors like
1/(x^2 - 25) + 1/(x^2 + 8x +15)
= 1/(x+5)(x-5) + 1/(x+5)(x+3) [factor]
= (x+3)/(x+5)(x-5)(x+3) + (x-5)/(x+5)(x-5)(x+3) [find LCM]
= (2x-2)/(x+5)(x-5)(x+3) [add numerators]
To motivate this, we can think of the simpler problem of 1/6 + 1/9 (since 6 and 9 share a common factor, but do not share their other factors).
1/6 + 1/9 = 1/(2)(3) + 1/(3)(3) [prime factor]
= 3/(2)(3)(3) + 2/(2)(3)(3) [find LCM]
= 5/(2)(3)(3) [add numerators]
= 5/18
So this is the idea of parallel tasking. Where my students tend to have troubles is with the fact that they are not strong with fractions. Perhaps teachers are not showing the importance of using the LCM when finding common denominators (ie. 1/6 + 1/9 = 9/54 + 6/54... WHY WOULD YOU TEACH THIS?!), or not dedicating enough time to prime factorization (this second fault might be due to the awful curriculum we have that forces "mental math" strategies that only work in contrived situations).
In my opinion, parallel tasking is something that an excellent teacher will do and use effectively without thinking. However, it requires a lot of knowledge in the subject matter, and the ability to see how everything is interconnected (something most teachers do not attain from education faculties). But that is another post for another day.
Have you had any success with parallel tasks?
Tuesday, February 3, 2015
Bad Questions: A Great Way to Build False Confidence and Remove Mathematics from Math Class
Rather than reflecting on a particular curriculum outcome or the way a particular lesson was taught, I thought that this week I would put in my two cents about the following book: Good Questions: Great Ways to Differentiate Mathematics Instruction by Marian Small.
Now, without knowing anything about teaching mathematics or Marian Small, one might be led to believe that this is quite an excellent book. It is rated 4.7/5 stars on Amazon at the time of this blog
So it seems there might be enough information here to sway someone who is a novice teacher into thinking that this book was worth the money. Let's hold on to that thought because I want to talk a bit more about what this book has to offer first.
Small opens with the following example (pg. 2)
So Small certainly appeals to the "typical" teacher who has students at varying levels of understanding. And I can see why a typical teacher might side with Small here - most, if not all, teachers would have spent a lot of time learning about differentiation in college. However, what they would not have been told in college is that there is very little scientific research supporting differentiation. Differentiation is a bit like an amorphous blob to the scientific community - we can't quite seem to get a handle on what it looks like and how to measure it properly. As such, any claims from studies looking at differentiation, and any source claiming they can help with classroom differentiation, must be taken with a grain of salt.
Continuing on, we see that Small tells us that to effectively differentiate instruction, we need the three elements: Big Ideas, Choice, and Preassessment (pg. 4). So, if we didn't get it from the title, by page four, we see exactly what Marian is trying to do to us - sell us differentiation techniques. We have reason to be worried - if there is no rigorous research promoting differentiation techniques in the scientific community, then why are they being promoted in this book? The answer comes in the next few pages where she discusses her two core strategies for differentiating mathematics instruction: Open Questions and Parallel Tasks (pg. 6). I will leave the idea of Parallel Tasks for another time, but for now I want to discuss Open Questions.
According to Small (pg. 6) "the ultimate goal of differentiation is to meet the needs of the varied students in a classroom... [and it] becomes manageable if the teacher can create a single question... that is inclusive..." Some might say that the question has to have "breadth," or be wide enough so that all students can access it. For example, rather than ask "What is 6 x 7?" Small (pg. 24) suggests asking "The answer is 42. What is the question?" What kind of value does the second question have over the first? Very little if my students do not know the basic fact 6 x 7 = 42. What we generally see with open questions is that the relevant mathematics is removed entirely. A student might answer "What is my dad's age?" and be rewarded for this non-mathematical answer.
I do believe that there is some value in asking open questions, but they need to be carefully planned out, and come after the basic ideas have been well-developed. Take, for example, this graphic that I recently came across on Twitter
Now, without knowing anything about teaching mathematics or Marian Small, one might be led to believe that this is quite an excellent book. It is rated 4.7/5 stars on Amazon at the time of this blog
and it is one of Google's top recommendations when looking at the string "great ways to d" (I am slightly appalled at the top choice, but that is for another conversation altogether).
Small opens with the following example (pg. 2)
In one cupboard, you have three shelves with five boxes
on each shelf. There are three of those cupboards in the
room. How many boxes are stored in all three cupboards?
To which she has various students answering the question in different ways. For example, Liam looks immediately for the teacher, Angelita uses a drawing to help her, and the others use some kind of adding or multiplication to help them. Small then goes on to explain the role of the teacher during the down time: to promote symbolic use for Angelita, discuss the advantage of using multiplication instead of addition for the others, and to possibly scale back the example for Liam. She then stresses the importance of the "need for a teacher to know where his or her students are developmentally to be able to meet each one's educational needs" (pg. 3). From here, Small begins her discussion of differentiation.
So Small certainly appeals to the "typical" teacher who has students at varying levels of understanding. And I can see why a typical teacher might side with Small here - most, if not all, teachers would have spent a lot of time learning about differentiation in college. However, what they would not have been told in college is that there is very little scientific research supporting differentiation. Differentiation is a bit like an amorphous blob to the scientific community - we can't quite seem to get a handle on what it looks like and how to measure it properly. As such, any claims from studies looking at differentiation, and any source claiming they can help with classroom differentiation, must be taken with a grain of salt.
Continuing on, we see that Small tells us that to effectively differentiate instruction, we need the three elements: Big Ideas, Choice, and Preassessment (pg. 4). So, if we didn't get it from the title, by page four, we see exactly what Marian is trying to do to us - sell us differentiation techniques. We have reason to be worried - if there is no rigorous research promoting differentiation techniques in the scientific community, then why are they being promoted in this book? The answer comes in the next few pages where she discusses her two core strategies for differentiating mathematics instruction: Open Questions and Parallel Tasks (pg. 6). I will leave the idea of Parallel Tasks for another time, but for now I want to discuss Open Questions.
According to Small (pg. 6) "the ultimate goal of differentiation is to meet the needs of the varied students in a classroom... [and it] becomes manageable if the teacher can create a single question... that is inclusive..." Some might say that the question has to have "breadth," or be wide enough so that all students can access it. For example, rather than ask "What is 6 x 7?" Small (pg. 24) suggests asking "The answer is 42. What is the question?" What kind of value does the second question have over the first? Very little if my students do not know the basic fact 6 x 7 = 42. What we generally see with open questions is that the relevant mathematics is removed entirely. A student might answer "What is my dad's age?" and be rewarded for this non-mathematical answer.
I do believe that there is some value in asking open questions, but they need to be carefully planned out, and come after the basic ideas have been well-developed. Take, for example, this graphic that I recently came across on Twitter
@rcraigen came up with some interesting discussion. Let's say we don't ask any closed questions regarding the area of a triangle and we simply pose the open or probing question.
What if the student answers "(-16cm) and (-10cm)" to the open question? Are you able to justify why this is not an acceptable answer? Will this lead to classroom/student confusion?
What is preventing every student from answering 80cm and 2cm or 160cm and 1cm (the "easiest" answers) when the point is to have a more abundant array of answers?
Do you see how the scale of the triangles could cause confusion, and students will think that the two sides have to be the same length? Are your students prepared enough to handle the spatial reasoning of varying sized triangles in their working memory as well as the area formula, or will they shut down because their working memories have been overloaded?
If we have not asked enough closed questions, the student will not have committed the area formula for a triangle to long term memory, so how can he/she answer "How do you know?" in the probing question? Again this will be too overwhelming to handle and the student is likely to shut down.
So we see that a poorly designed open question can lead to a few things: (1) cognitive overload due to students having to keep too much information in their working memories, (2) a significant possibility of students selecting the easy answer or incorrect answers (by the way this makes more work for you as the teacher because now you have to go around and fix all of these misconceptions), and (3) students shutting down because they have become overloaded.
What if the student answers "(-16cm) and (-10cm)" to the open question? Are you able to justify why this is not an acceptable answer? Will this lead to classroom/student confusion?
What is preventing every student from answering 80cm and 2cm or 160cm and 1cm (the "easiest" answers) when the point is to have a more abundant array of answers?
Do you see how the scale of the triangles could cause confusion, and students will think that the two sides have to be the same length? Are your students prepared enough to handle the spatial reasoning of varying sized triangles in their working memory as well as the area formula, or will they shut down because their working memories have been overloaded?
If we have not asked enough closed questions, the student will not have committed the area formula for a triangle to long term memory, so how can he/she answer "How do you know?" in the probing question? Again this will be too overwhelming to handle and the student is likely to shut down.
So we see that a poorly designed open question can lead to a few things: (1) cognitive overload due to students having to keep too much information in their working memories, (2) a significant possibility of students selecting the easy answer or incorrect answers (by the way this makes more work for you as the teacher because now you have to go around and fix all of these misconceptions), and (3) students shutting down because they have become overloaded.
This is why I tend to advocate for well-planned open questions being spread out among the closed questions. This gives students more time to commit the tools/formulas/facts needed to long term memory, preventing the cognitive overload and ego-trouncing associated with an abundance of open questions. By the way, there are certainly better problem-solving questions (like the one Small starts off with above about the cupboards) that can promote discussion without losing the mathematics.
Now back to the topic at hand. Small (pg. 7) does give her view on the psychological aspects of open-ended
questions, but to me, she gets it all backwards: she deems that asking
an open-ended question psychologically "is a much more positive situation"
compared to asking a closed-question. She then notes that "many
students and adults view mathematics as a difficult" and because we are
not encouraged to "express different points of view" once a student
fails they will simply shut down. Research in cognitive science has actually shown us that over-use of open questions quickly wears out the working memory of younger students and leads to cognitive overload, especially if the students do not have the basic tools to access the question. It is quite likely that this then leads to the psychological effect we call "math anxiety."
We see that Small's solution to making students less afraid of mathematics, is to remove the mathematics from the question. If we keep removing the mathematics from our questions, then these students will never get the basic skills committed to long term memory, and will continue struggling with open-ended questions! It just doesn't make logical sense. Why not put more focus on teaching the students the basic skills, rather than insisting on removing more and more mathematics from your open questions? This gives both the teacher and the students false confidence in mathematics.
I believe that Small is trying to implement Dweck's Growth Mindset: a strategy in which we instill confidence in our young learners and show them that they have the ability to learn mathematics. Here is how I view Dweck's Growth Mindset:
(1) Teach the student the basic skills required, and put that student in a situation where he/she can show off their skills. This first step is the most challenging since it takes the most time and students may have to be shown failure (remember Growth Mindset is not about "babying" our students, it is about teaching the skills then instilling the confidence).
(2) Continue with repeated practice so that the student builds more confidence and automaticity of fact recall. Automaticity is important since it frees up space in the working memory.
(3) Then, after much practice, you can expose your student to a more open-ended problem.
I feel like Small is on the Growth Mindset bandwagon without really knowing how it works at its core: give the skills and build the confidence, rather than "babying" and giving false confidence.
As we come to the conclusion of this blog, I want to sum up what we have seen:
(1) Open questions need to be carefully implemented sparingly and not be the core of your mathematics curriculum. It is much more vital to teach the facts first, and the open questions later after much repeated practice.
(2) Open questions can lead to cognitive overload. If students don't have basic skills, how can we expect them to be able to process an open question? Only after repeated practice will students have enough available working memory that they can focus on a more open problem.
(3) Removing mathematics from your math class does not equate to Growth Mindset. In fact, it is the exact opposite of what Dweck is trying to tell us! Removal of mathematics from you class tends to give false confidence to students, when what is really needed are the basic skills.
So, overall, how do I feel about Smalls book? Let's phrase is as a poorly-planned open question with a sassy answer.
Open Question: The answer is (-25.22). What is the question?
Answer: The amount of money I would be willing to pay for Marian Small's Good Questions.
We see that Small's solution to making students less afraid of mathematics, is to remove the mathematics from the question. If we keep removing the mathematics from our questions, then these students will never get the basic skills committed to long term memory, and will continue struggling with open-ended questions! It just doesn't make logical sense. Why not put more focus on teaching the students the basic skills, rather than insisting on removing more and more mathematics from your open questions? This gives both the teacher and the students false confidence in mathematics.
I believe that Small is trying to implement Dweck's Growth Mindset: a strategy in which we instill confidence in our young learners and show them that they have the ability to learn mathematics. Here is how I view Dweck's Growth Mindset:
(1) Teach the student the basic skills required, and put that student in a situation where he/she can show off their skills. This first step is the most challenging since it takes the most time and students may have to be shown failure (remember Growth Mindset is not about "babying" our students, it is about teaching the skills then instilling the confidence).
(2) Continue with repeated practice so that the student builds more confidence and automaticity of fact recall. Automaticity is important since it frees up space in the working memory.
(3) Then, after much practice, you can expose your student to a more open-ended problem.
I feel like Small is on the Growth Mindset bandwagon without really knowing how it works at its core: give the skills and build the confidence, rather than "babying" and giving false confidence.
As we come to the conclusion of this blog, I want to sum up what we have seen:
(1) Open questions need to be carefully implemented sparingly and not be the core of your mathematics curriculum. It is much more vital to teach the facts first, and the open questions later after much repeated practice.
(2) Open questions can lead to cognitive overload. If students don't have basic skills, how can we expect them to be able to process an open question? Only after repeated practice will students have enough available working memory that they can focus on a more open problem.
(3) Removing mathematics from your math class does not equate to Growth Mindset. In fact, it is the exact opposite of what Dweck is trying to tell us! Removal of mathematics from you class tends to give false confidence to students, when what is really needed are the basic skills.
So, overall, how do I feel about Smalls book? Let's phrase is as a poorly-planned open question with a sassy answer.
Open Question: The answer is (-25.22). What is the question?
Answer: The amount of money I would be willing to pay for Marian Small's Good Questions.
Saturday, January 24, 2015
Thoughts on Fractions
I had some great discussion last week around the two ideas below. I will tackle them one at a time so as not to overload our working memories.
The Improper Fraction Dilemma
When adding the fractions 4 1/2 - 2 1/9, is it better to show Grade 8 students the "fail-safe" method of using improper fractions
4 1/2 - 2 1/9 = 9/2 - 19/9
= 81/18 - 38/18
= (81 - 38)/18
= 43/18
= 2 7/18
or to show them that we can subtract the whole parts and then the fractional parts, but noting that sometimes a "borrow" is necessary?
4 1/2 - 2 1/9 = 4 - 2 + 1/2 - 1/9
= 2 + 9/18 - 2/18
= 2 + (9 - 2)/18
= 2 7/18
Note that if we were to switch the place of the 1/2 and 1/9 then a borrow from the whole part (indicated with **) is necessary since we obtain a (-7) as the numerator of our fractional part:
4 1/9 - 2 1/2 = 4 - 2 + 1/9 - 1/2
= 2 + 2/18 - 9/18
= 2 + (2 - 9)/18
= 2 + (-7)/18
= 1 + 18/18 + (-7)/18 **
= 1 + (18 - 7)/18
= 1 11/18
This is an interesting question, since the biggest concern I was hearing from the Grade 8 teacher I was working with was that the fractional skills of the students were not solid enough to handle teaching them the borrowing technique. So the teacher opted to show them only the improper fraction technique method, which works in all instances and students do not have to deal with borrowing if a negative numerator occurs.
One of the biggest roadblocks I find at the university level when I am teaching fractions, is my student's reluctance to accept that improper fractions are not necessary. I often get solutions that look like
23 25/113 - 20 12/113 = 2624/113 - 2272/113
= (2624 - 2272)/113
= 352/113
= 3 13/113
rather than
23 25/113 - 20 12/113 = 23 - 20 + 25/113 - 12/113
= 3 (25 - 12)/113
= 3 13/113.
We can see that the second approach is favourable since there are no unusually large numbers in the numerators, and there is also no borrowing. Most students don't get to a correct final answer if they use the first approach due to the heavy computations involved.
So to me, the real trouble here is that students are likely never shown the borrowing technique at all. So students are not really thinking about the best or most efficient way of answering a mixed number subtraction question. I do agree with the teacher I was working with: it is important for students to see one, and only one, method for addition/subtraction of fractions when it is first introduced. After students have mastered that method, then we can branch out and explore other, more efficient methods. While the improper fraction method is not necessarily the most efficient, it does reinforce ideas like mixed number conversions and simplification of fractions. Time is also another factor to consider: is it better to only show the improper fraction approach and have the students very good with one method (saves time), than to show both approaches and have students weak with two different methods (takes more time)?
I thought a bit about this question and came up with the following idea. Why not introduce basic fraction arithmetic earlier in the curriculum? Why does it happen so late (in Grade 7 and Grade 8)? If we introduced it around Grade 5, then students will have had a few years to grasp the abstractness of fractional addition and subtraction. Then by Grade 7 or 8, we can have them decide which strategy to use when solving a mixed number question like this.
The Cancelling Common Factors Dilemma
When multiplying fractions, we can either decide to multiply the numerators together, then multiply the denominators together
8/5 x 5/2 = (8 x 5)/(5 x 2)
= 40/10
= 4
or we can decide to cancel any common factors, then multiply
8/5 x 5/2 = 4/1 x 1/1
= (4 x 1)/(1 x 1)
= 4.
You can clearly see that if we decide to cancel off common factors first, then the multiplications we do in the second step are much simpler. I was a bit shocked to see the first method being used over the second one. Although, I was ensured that the second method is where the students would get to by the end of the chapter. The Grade 8 teacher told me that the multiplication skills of the students were very weak, so this was a way for the students to review their multiplication facts.
This brought up two thoughts in me. The first is why we care about the idea of cancelling off common factors. When working with rational expressions, we are often concerned with making our expressions simpler by cancelling off "prime" linear or quadratic expressions. Students who are not shown how to do this with rational numbers tend to not grasp the idea when it becomes more abstract. So I will always encourage teachers to eventually have students master cancelling off common factors when multiplying or dividing rational numbers - this way the leap to multiplying and dividing rational expressions is not so large.
The second thought was regarding the statement "the multiplication skills of my students are very weak". This suggests that curriculum is not sufficiently doing its job. According to the curriculum "recall of the multiplication ... facts up to 5 x 5 is expected by the end of Grade 4" and "recall of multiplication facts to 81 ... is expected by the end of Grade 5". However, students are entering Grade 7 and 8 without the basic facts committed to long term memory. Without this, how can we hope for them to be able to work with integers, or with fractions?
Why is it that students do not have a good grasp of the basic facts? One major reason may be due to the fact that these outcomes occur so late in the curriculum. The earlier we can introduce the basic facts and allow students to play with them, the more our students are going to encounter them. The more their working memories encounter the basic facts, the more likely it is that the basic facts will be stored long term. To be having our students only have the facts up to 9 x 9 memorized by the end of Grade 5 is really tragic. Since there is no mention of multiplication facts before the Grade 4 curriculum, our Grade 2 or 3 teachers looking at the curriculum document outcomes may not think about introducing multiplication facts at a more age-appropriate time.
Sorry for a long post, but I just had to get some of these opinions off my mind. Have any of you had similar experiences with fractions? I would love to hear your stories.
The Improper Fraction Dilemma
When adding the fractions 4 1/2 - 2 1/9, is it better to show Grade 8 students the "fail-safe" method of using improper fractions
4 1/2 - 2 1/9 = 9/2 - 19/9
= 81/18 - 38/18
= (81 - 38)/18
= 43/18
= 2 7/18
or to show them that we can subtract the whole parts and then the fractional parts, but noting that sometimes a "borrow" is necessary?
4 1/2 - 2 1/9 = 4 - 2 + 1/2 - 1/9
= 2 + 9/18 - 2/18
= 2 + (9 - 2)/18
= 2 7/18
Note that if we were to switch the place of the 1/2 and 1/9 then a borrow from the whole part (indicated with **) is necessary since we obtain a (-7) as the numerator of our fractional part:
4 1/9 - 2 1/2 = 4 - 2 + 1/9 - 1/2
= 2 + 2/18 - 9/18
= 2 + (2 - 9)/18
= 2 + (-7)/18
= 1 + 18/18 + (-7)/18 **
= 1 + (18 - 7)/18
= 1 11/18
This is an interesting question, since the biggest concern I was hearing from the Grade 8 teacher I was working with was that the fractional skills of the students were not solid enough to handle teaching them the borrowing technique. So the teacher opted to show them only the improper fraction technique method, which works in all instances and students do not have to deal with borrowing if a negative numerator occurs.
One of the biggest roadblocks I find at the university level when I am teaching fractions, is my student's reluctance to accept that improper fractions are not necessary. I often get solutions that look like
23 25/113 - 20 12/113 = 2624/113 - 2272/113
= (2624 - 2272)/113
= 352/113
= 3 13/113
rather than
23 25/113 - 20 12/113 = 23 - 20 + 25/113 - 12/113
= 3 (25 - 12)/113
= 3 13/113.
We can see that the second approach is favourable since there are no unusually large numbers in the numerators, and there is also no borrowing. Most students don't get to a correct final answer if they use the first approach due to the heavy computations involved.
So to me, the real trouble here is that students are likely never shown the borrowing technique at all. So students are not really thinking about the best or most efficient way of answering a mixed number subtraction question. I do agree with the teacher I was working with: it is important for students to see one, and only one, method for addition/subtraction of fractions when it is first introduced. After students have mastered that method, then we can branch out and explore other, more efficient methods. While the improper fraction method is not necessarily the most efficient, it does reinforce ideas like mixed number conversions and simplification of fractions. Time is also another factor to consider: is it better to only show the improper fraction approach and have the students very good with one method (saves time), than to show both approaches and have students weak with two different methods (takes more time)?
I thought a bit about this question and came up with the following idea. Why not introduce basic fraction arithmetic earlier in the curriculum? Why does it happen so late (in Grade 7 and Grade 8)? If we introduced it around Grade 5, then students will have had a few years to grasp the abstractness of fractional addition and subtraction. Then by Grade 7 or 8, we can have them decide which strategy to use when solving a mixed number question like this.
The Cancelling Common Factors Dilemma
When multiplying fractions, we can either decide to multiply the numerators together, then multiply the denominators together
8/5 x 5/2 = (8 x 5)/(5 x 2)
= 40/10
= 4
or we can decide to cancel any common factors, then multiply
8/5 x 5/2 = 4/1 x 1/1
= (4 x 1)/(1 x 1)
= 4.
You can clearly see that if we decide to cancel off common factors first, then the multiplications we do in the second step are much simpler. I was a bit shocked to see the first method being used over the second one. Although, I was ensured that the second method is where the students would get to by the end of the chapter. The Grade 8 teacher told me that the multiplication skills of the students were very weak, so this was a way for the students to review their multiplication facts.
This brought up two thoughts in me. The first is why we care about the idea of cancelling off common factors. When working with rational expressions, we are often concerned with making our expressions simpler by cancelling off "prime" linear or quadratic expressions. Students who are not shown how to do this with rational numbers tend to not grasp the idea when it becomes more abstract. So I will always encourage teachers to eventually have students master cancelling off common factors when multiplying or dividing rational numbers - this way the leap to multiplying and dividing rational expressions is not so large.
The second thought was regarding the statement "the multiplication skills of my students are very weak". This suggests that curriculum is not sufficiently doing its job. According to the curriculum "recall of the multiplication ... facts up to 5 x 5 is expected by the end of Grade 4" and "recall of multiplication facts to 81 ... is expected by the end of Grade 5". However, students are entering Grade 7 and 8 without the basic facts committed to long term memory. Without this, how can we hope for them to be able to work with integers, or with fractions?
Why is it that students do not have a good grasp of the basic facts? One major reason may be due to the fact that these outcomes occur so late in the curriculum. The earlier we can introduce the basic facts and allow students to play with them, the more our students are going to encounter them. The more their working memories encounter the basic facts, the more likely it is that the basic facts will be stored long term. To be having our students only have the facts up to 9 x 9 memorized by the end of Grade 5 is really tragic. Since there is no mention of multiplication facts before the Grade 4 curriculum, our Grade 2 or 3 teachers looking at the curriculum document outcomes may not think about introducing multiplication facts at a more age-appropriate time.
Sorry for a long post, but I just had to get some of these opinions off my mind. Have any of you had similar experiences with fractions? I would love to hear your stories.
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