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?
Saturday, February 14, 2015
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.
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