An interesting set-up of right triangles allows us to prove radical identities.
I was asked by a colleague last week to prove an identity involving radicals. The two expressions arise when considering cosine of the angle pi/12. Normally, one would apply a sum or difference formula
and this would simplify to
However, when one of his students used a calculator, the calculator returned back an unusual expression:
He and the student were able to verify that these expressions evaluated to a similar decimal expansion, so must be equivalent. But then his student asked him how to prove the equivalence of expressions like this. He tried for a bit, unsuccessful – then he tormented me with this problem all Easter weekend. Eventually, I was able to show the equivalence using an old right triangle trick I saw a few years back.
Attach two right triangles together in such a way so that the right leg of the second, and the bottom leg of the first meet at a right angle. On the hypotenuses of the smaller triangles write root 6 and root 2, respectively. This is done so that the hypotenuse of the larger right triangle is root 6 + root 2 – matching up with the numerator of expression (1).
Our goal is to apply the Pythagorean Theorem on the large right triangle, so we need to determine the legs of the larger triangle. To do this, we will determine the legs of the smaller right triangles. For the root 2 triangle, we have the obvious choice of making the legs (1, 1). For the root 6 triangle, we could make the legs (root 2, root 4), (root 3, root 3) or (root 1, root 5). Notice that in expression (2), we have a root 3. This suggests we might want to try the (root 3, root 3) combination for the root 6 triangle. This shows us that the legs of the larger right triangle are both root 3 + 1.
Now we can apply the Pythagorean Theorem on the large right triangle.
Taking the square root of both sides gives
And finally, dividing both sides of the equation by 4 yields the desired result.
I suppose that the moral of the story here, besides seeing some really interesting mathematics, is that I never would have solved this problem unless I had seen the previous problem involving something similar. In general, I believe it is safe to state that in order to be successful solving problems, one should be exposed to many different types of problems (ever wonder how those Math Olympiad contestants get so “smart”?). From a cognitive science perspective this makes sense – it allows us to create problem archetypes (schemata) that we can draw upon to help solve future problems. And the more well-connected these schemata become, the easier it becomes to solve problems.
What a crazy couple months it has been! It began mid-September when I was asked to give a talk about the new mathematics curriculum in BC through the college. From here, I attended researchED New York early in October, where I got to connect with some awesome educators and present about connecting interleaved practice with teaching with the amazing Yana Weinstein. After this, I gave a math workshop to some future BC education assistants. We talked about math anxiety, early numeracy, and cognitive science, all while working on interesting math problems together! Yesterday I was in Saskatoon at the SUM Conference to present on non-routine cognitive tasks – synthesizing some information I gathered from Dan Meyer and Steve Leinwand. Finally, I have a small break before the next two workshops to hopefully eke out a post on some reflections I have had from traversing between these different worlds.
#1) Cumulative Review — Why Isn’t Everyone Doing It?
I have recently read “Accessible Mathematics” by Steve Leinwand, in which he outlines 10 instructional shifts to help raise student achievement. One of those shifts is to shift toward giving ongoing cumulative practice at the beginning of your math lessons. It does not have to be terribly extensive – perhaps just four or five short recall-type questions to ensure that students are not forgetting past concepts. It seems obvious that we should be doing this – but many of us are not!
Why should we be doing it? Well, this was somewhat tied to the presentation that Yana and I gave at researchED. It seems that interleaved and spaced practice are highly effective strategies to increase long-term learning in our students. For instance, I saw a 10% increase in the discrimination of problem type when I used interleaved practice in my integral calculus class last year. However, there are some things that we don’t know about interleaving that warrant future studies – like how many problem-types should we include, or how interleaving affects attention in our students.
Why are we not doing it? Efrat discussed some of the practical limitations of using interleaved and spaced practice at researchED New York. Teachers typically list time investment, lack of support, or an incompatible system as reasons for not utilizing spaced practice. What might change their minds? It seems that teachers are interested in ongoing professional development in cognitive science, and time to work with colleagues in order to help ease them into implementation of such tasks. As this is an area of interest to me, please contact me if you or your school is interested in ongoing professional development in cognitive science – I would be happy to help!
#2) Depressing — Why Aren’t We Collaborating?
Continuing on the conversation, we could ask why aren’t we collaborating more as a community? Let’s take a look at an example from my life. I had a student come into my calculus class with a TI calculator stating that his teachers at high school said they would absolutely require a TI calculator for college calculus. Literally, what?! With tools like Desmos at our fingertips, why is there a need to drag around a $200 brick? In addition to this, my department doesn’t allow graphical display calculators on major tests anyway. So it looks like I will need to reach out to the local community and try to spread the Desmos love. Why? Let’s look at it form the alternate viewpoint: If I teach Desmos to my students this year, but when they move on, the next teacher doesn’t know how (or doesn’t want to know how) to use Desmos, these students are now potentially disadvantaged. In essence, a teaching tool is greater when we share it with others in the profession and we develop long-term learning goals using similar tools.
#3) Planning — Using Space, Not Time
In a presentation by Nat Banting and Ilona Vashchyshyn, we were asked to consider planning a lesson using quadrants labelled as “Teaching Actions”, “Teaching Spaces”, “Anticipation”, and “Improvisation.” In other words, when it comes to planning, we need to consider our space (the room, manipulatives, desk arrangement) and our actions (modelling, watching, telling). And Nat and Ilona see our actions and spaces situated on a continuum between anticipation and improvisation. In fact, there has to be some improvisation within our classrooms, since it would technically be impossible for us to plan all the possible divergence that may happen in any given lesson.
Of interest to me was their belief that false dichotomies arise when we believe an individual spends all their time within one of the half-planes. For instance, if we believe an educator continuously anticipates and does not improvise in the class, then they are defined as a traditional teacher. On the other hand, those who are thought to improvise all the time are branded as reform or progressive teachers.
This also works for the horizontal half-planes. If an educator is too focused on the teaching spaces, the lesson might be branded as a differentiated instruction type of lesson; and if an educator is too focused on the teaching actions, the lesson might be branded an inquiry type lesson. There is probably more to this conversation, but I am still trying to think more on these two particular diagrams.
#4) Synthesis — Finding Your Balance
In Saskatoon I tried to synthesize some reading that I have been doing as of late. The first bit of information was regarding non-routine cognitive tasks I originally heard of from Dan Meyer at OAME 2017. The main premise is that a mathematical task can either have a real-world context or not. In addition to this, a mathematical task can involve “real work” or “fake work.” There are certain verb choices that we make in a math class that lead to real work (question, predict, analyze, debate) or to fake work (evaluate, simplify). Finally, doing fake work in a real world context is overrated; that is, dressing up a routine task with the air of real worldness is overused in math education. However, pushing students to do real work not in a real world context is underrated; that is, we often fall short of allowing students to use meaningful verbs like question, predict or analyze outside of real world contexts. Think “Calculate when the phone will be charged given the model.” (routine, plug ‘n’ chug, dressed up in real world clothing) versus “Predict the y-value given the data.” (non-routine, analyzing data to predict, non-dressed up mathy question).
In addition to Dan’s thoughts on non-routine tasks, I embedded Steve Leinwands idea to lead lessons with data. My thoughts were that if we are interested in moving toward doing real work, data can help drive questioning, noticing and predicting. Provided things go well with the lesson, we can follow up with verbs that allow us to extend, such as generalize or debate. If you are interested in seeing a bit more, my slides from the conference can be found here.
Realistically, I think it would be quite the challenge to create every lesson as a non-routine cognitive task. To me, it feels unrealistic. Also, I firmly believe that the verbs recall, calculate and simplify have a place in mathematics classes and that they should be respected. For instance, John Mighton of JUMP Mathematics consistently reminds me that cognitive load is important – that is, our students require some skill in order to begin a rich-task such as data analysis. This skill comes with practice, which can easily be acquired via spaced practice involving recalling facts. However, on the other side, Bjork reminds me of desirable difficulties. Could non-routine cognitive tasks be shaped in such a way to support learning and long-term retention?
As I continue to navigate the large divide of what feels like a fake world of mathematics and a real world of mathematics education, I can’t help but wonder how we might all be able to help shift the collective from fake work to real work.
“The merit of painting lies in the exactness of reproduction. Painting is a science and all sciences are based on mathematics.” -Da Vinci
Take a moment to read the phrase: “The hungry caterpillar ate the juicy leaf.”
Now quickly complete the word by filling in a missing letter: SO_P.
Out of curiosity, did you complete the word using the letter U to make SOUP? According to Kahneman, author of Thinking Fast and Slow, after processing the words HUNGRY and ATE in a sentence, we are primed to select the letter U in the word above since SOUP is associated with the words HUNGRY and ATE. Let’s explore this a little bit, and see if and how we might think about using this idea in our math classrooms.
What is the Priming Effect?
An idea in our memory is associated with many other ideas. These associations may be categorical, such as connecting the words FRUIT and APPLE, or property-based, such as connecting ADDITION or MULTIPLICATION to COMMUTATIVITY. Ideas may also be associated through effects like how we may connect ALCOHOL to DRUNK, or CIGARETTE to CANCER. When primed with one of the links in an association, our mind has the ability to bring the other familiar and associated words into our working memory.
What Does Priming Look Like?
When priming occurs it is subconscious and Kahneman argues that we are likely not to believe it is occurring due to the way our brain functions (our brain allows us to believe that we are in full control). He mentions several studies in his book, but I will touch on only two to give you a sense of how priming is at work. In the first, participants were primed with images of money. The group that was primed with money images became more individualistic – less likely to help others and less likely to ask for help – on tasks that followed.
In the second group, it was shown that actions can also be primed. In this study, children read sentences involving words associated with the elderly such as FORGETFUL, BALD, GRAY, and WRINKLE. None of the sentences explicitly mentioned mentioned the elderly. When the participants were asked to walk down a hallway, they did so at a much slower pace than normal. The reverse association was true as well: children who were asked to walk slowly for a period of time were more apt to recognize words associated with old age.
Can We Use Priming in Mathematics Class?
I wonder if mathematics teachers have been using this idea already? In most classes and assessments, we tend to be explicit with word choice when we are asking students to perform a task. For example, if I want my students to think in a linear way, I could use an associated word like SLOPE or a similar word like STRAIGHT to help them recall ideas around linear functions. Use of certain cues to aid in recall are most likely beneficial since we know that recall of facts helps with both storage and retrieval strength. I could also see the argument of priming allowing students to access previous knowledge, which may be an appropriate action during the set-up of a teaching task.
On the other hand, we do have to be aware that priming may occur without our knowledge at any given time. That is, if we utilize unnecessary pictures or words to aid in a mathematical task, our students may be thinking about what we don’t want them to think about!
In closing, the priming effect is an interesting process to be aware of in our classrooms. However, Kahneman notes that the effect doesn’t work with all individuals, so we do not have to worry about students becoming zombies to priming effects. In addition to this, it seems that the priming effect has been under scrutiny for robustness, including replicability of certain findings. Perhaps we will have to wait to see what color the first coat is before delving deeper into this theory in our classrooms.
“Diversity if not about how we differ. Diversity is about embracing one another’s uniqueness.” -Ola Joseph
Friday – May 12th:
Any of you who have attended both days at OAME and are out tonight, I am not sure how you are doing it. Here I am, nuzzled in a blanket at 9:02PM with a glass of Moscato contemplating writing a reflection because my internal battery is at 5%, wondering how long it will take to fully charge if I know at 9:22 I will be at 19%. Hint: the relationship is surprisingly non-linear (despite a correlation coefficient close to 1)!
Surprise is not a Surprise with Desmos
If you did not get the reference above, then I am not mad… just disappointed that I didn’t see you in Dan’s presentation this morning regarding the functionality of Desmos. I feel like I have grown so much over the past year through using the graphing calculator and the activity builder in my calculus classes, but am still learning more as the months progress. I have started to realize that with Desmos, I am consistently amazed, but never surprised (anymore). Two awesome functions of Desmos we got a peek at today were the geometry beta, and Desmos for the visually impaired. This was one of the few times in my life I got to hear what graph sounded like (the other being when I studied Fourier Analysis).
By the way, if you want the animation from the cell phone 3-Act, Dan tweeted it out to us. Play away.
One of my afternoon sessions looked at the five practices for facilitating effective discussion in classrooms. This was a great connection to to Deborah Ball’s Domains of Mathematical Knowledge for Teaching (below). Regarding a lesson on perimeter of connected hexagons, we observed student work and strategies during two points in time – near the beginning of their pondering, and much later during the process. We had the opportunity to ask questions of the students to elicit how they were thinking about the problem, connecting them and us to Common Content (generalized math knowledge not specific to teaching) and Specialized Content Knowledge (knowledge specific to teaching) present in the problem.
One interesting aspect of this presentation that was not present in others was time near the beginning to discuss possible misconceptions and strategies of students as the problem progressed. Thinking about student misconceptions would fall under what Ball calls Knowledge of Content and Students, and thinking about different strategies to tackle the same problem falls under Knowledge of Content and Teaching. At the end of this questioning period, we had the opportunity to decide which three student solutions we would present to the class and in what order. We opted to choose a visual strategy to solve the problem first, followed by a tabular strategy, then finally a solution containing both a table and a picture. Interestingly, no students opted for a graphical strategy; although I argued that perhaps it was unnecessary to get the information required for this task. Understanding the progression from visual to table to graphical would be an instance of Horizon Content Knowledge, or knowing how one idea/topic connects to another. All in all, a very well-laid out execution of the pedagogy of classroom discussion!
In the alternate assessment session, we saw non-typical ways to assess students. I thought these assignments were beneficial to have ready to go for when needed (use for students on vacation or sick for example). I was fond of the interview, mostly due to the fact that I have used it before with my elementary teachers. Giving onus to the student to develop and justify their mark is awesome. However, a good interview with prompts does take a lot of time to prepare for properly, especially if you want to mark it objectively.
One thing that caught my eye was the use of an alternate test format. Students could choose the alternate test, which was more open in the sense that a typical question involved elaboration (or explaining all he/she could on a particular topic.) which is a well-known strategy for learning. Of interest, was that the students didn’t necessarily see a decrease in stress/anxiety levels when comparing a typical test to a non-standard test. That is, students writing the more elaborative test had roughly equal nervousness as a student writing a standard test.
Igniting my Heart
Fermi Problem: How many objects can Matthew throw on the stage during his Ignite?
Jimmy and Jon reminded us that we need to be our own teacher. We are not, and should not be, 40% Dan, 30% Marian and 30% Cathy with a dash of basil and sauce. We need to be thoughtful about which strategies and philosophies work for us and our students. This was a refreshing thing to hear, especially after the Twitter conversations I had had earlier in the day. Bouncing off of this, Kyle brought up the idea that the debate is not about automaticity, but how we get students toward automaticity. Often those arguing on Twitter forget that automaticity is definitely and end-goal of many teachers attending OAME, and to state otherwise is rude and uncalled for. I was reminded of this while having dinner with the teachers who inspired me last year to develop my interleaved project at Okanagan College. And to me, that’s exactly what OAME is: a place to gather with, and learn from, math educators of varying walks. Not everything that you see will resonate with you, and it does not have to.
As I finalize this post the morning of Saturday the 13th, I look back and realize how lucky I am to work with and be friends with such amazing educators. Here is to an amazing 2016-17 school year, and to many many more together.
“Developers who understand the whole stack are going to build better applications.” -Mike Loukides
What a whirldwind of a week so far. Monday – interview for a recurring role at the college in BC. Tuesday – an affirmative nod that I will be back again (and again and again… hopefully) for a few more years with the college. Wednesday – fly to Ontario and commute to Kingston, all while ceremonies for the faculty awards I was a finalist for are occurring.
Thursday – Day #1 OAME.
The Humanness and Non-Linearity of Teaching
Matthew consistently reminds me that learning is non-linear and messy and that the process of learning is interesting. As teachers we need to accept and be aware of the non-linearity of our learners. One thing to remember is that we have all used mathematics so many times that the act of going through the consolidation process is already complete in our minds, so we tend to forget (as humans do with information that is not being used semi-consistently) what this process feels like. Oddly enough, I was reminded of the joys of being a student at dinner tonight with a good friend of mine taking the MMT program through UWaterloo. What an interesting role to be in – one where you are once again the learner. What an excellent way to gain perspective, remind ourselves to be humble and to accept the human element of being a teacher.
How Many Fermi Problems Can One Find in a Calculus Class?
I also began wondering what Fermi problems would look like in a Calculus class? Can one realistically develop a Fermi problem to discuss estimation with derivatives – or is there a certain magnitude component to a Fermi problem that allows it to escape more complex mathematics? I feel like making a Fermi problem related to derivatives would be awkward, but I’m open to suggestions and thoughts.
So interesting and fitting that I attend my next session on transitioning through high school mathematics to college / university mathematics – as I recently transitioned out of a fairly precarious work environment into a very accepting one. It has been very interesting to see the amount of freedom and flexibility (within certain constraints, of course) that I am able to bring to the table at the college. For example some things that I have tried, that definitely would have been a no-go in my previous position, are: open book assessments, collaborative assessments (groups and pairs), an Amazing Race (calculus-style), take-home assessments, and a final exam consisting of mathematical stations. One day I will get around to blogging about the latter two, but until then I will leave you to ponder about the possibilities.
One thing that caught my attention is how sessions like this often boil down to both sides (secondary teachers and post-secondary teachers) complaining about the lack of content knowledge or skill sets of students. However, recall that students (because they are humans) are naturally going to forget mathematical information because they are not like us and are not using this information on a semi-regular basis. So, to me, the real question that needs to be asked is what methods are we putting into place to help students decrease the amount of forgetting that is happening as they transition from high school to university, and how can both sides contribute?
An Interleaved Approach to Interleaving
What can I say. Jamie and I work probably too well together. But in all honesty, I was extremely happy with what we were able to bring together considering that we were several provinces away from each other (how cool is a cross-country collaboration? It’s pretty cool, not gonna lie.). Many thanks to Doug Rohrer for his insight into interleaving and mathematics, as well as Yana & Fabian from the Learning Scientists for all of their hard work in making interleaving sexy and making a very accessible spreadsheet resource. If you want to know more about my interleaved project at the college, you can read my blog posts here, here and here, view our slides from our presentation, or connect with me via email. I’d be happy to discuss interleaving anytime.
There’s Something about Meyer
What is it about Dan “full-stack” Meyer that hits you at your core? There is definitely something about the way he thinks about the process of teaching and learning that leaves you walking away thinking “Yeah, that makes a lot of sense.” In Kingston, he spoke of lessons that should involve more action words than simply recall and compute. However, it does leave me wanting. I wonder if it is realistic for all knowledge to be consolidated this way? If yes, there is no argument to be had. If no, I wonder if there exists an optimal strategy that involves both the process he explained mixed in with something else?
My intuition tells me that cognitive science plays an important bit here, and it feels connected to Lucy’s book, where we learn that one of the Asian educational systems she visited had an interesting strategy toward mathematics learning. First, the teacher would ensure that students had appropriate background knowledge, typically done through a direct instruction method. After this, the teacher would break students up into groups, each tackling a challenging problem that hey had never seen before, and might contain the topics learned at the beginning of class. Discussion of the problems followed at the end. This feels about right to me: (1) introduce students to the tools they may need, perhaps done in an interleaved fashion, (2) work through a more complicated non-routine problem involving some of the concepts we wish for our students to recall within this process that Dan describes. Anyone looking to co-create a study involving these aspects, hit me up.