Moving!

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

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.

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 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:

"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.

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:

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.

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: 

• 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.

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).

The humorous Tom Bennett during his welcome speech at the introduction of researchED NY.

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.

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"

Weston stressed that great professional development takes time. A major idea stressed was delivering small chunks of new information to our teachers, allowing them to implement new ideas in the classroom, then having follow-up discussion and revisions. Professional development sessions are then seen as relevant to the teachers, and teachers may believe that it will truly help them and their students. This leads to teacher buy-in - our educators actually want to attend sessions, rather than being strong-armed into attendance by superiors.Throughout the session he stressed that great PD must blend pedagogy with content knowledge - effective PD should not favour one over the other.


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.

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:

• 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?"

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.

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.

1. "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!
2. "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.
3. "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.

She then went into a discussion of her observations on some of the top-performing systems. Of interest here is that all top-performers, except Singapore, have mixed ability classes until the age of 15. That is, students are not filtered into higher or lower academic classes until after grade 10. In addition to this, Asian top-performers tended to have specialized teachers by subject and by grade. This was in addition to the comparatively smaller class sizes and fewer lessons teachers had to give throughout the course of one day. The extra time saved through this system allowed teachers more time for co-planning with colleagues, which, in my opinion, leads to a more cohesive curriculum.


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. 

The animated Tom Bennett posing with me during lunch at ResearchED NY.

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. ;)

Empire State Building as seen from The Rockefeller Center.