Thinking about thinking

Assessment in this clip is portrayed as a way for students' future social status to be determined and if pretty reflective of the ways that students experience school now. In schools now though, testing is much more discrete and though tracking is no longer as emphasized as it used to be, students are still aware of what classes are better to be in, the kids of students who get to take AP classes, and whether or not they will make it to college.

On another note, students' capacity to actually understand concepts and problems is highly undermined when we give them test such as the multiple choice questions like the one seen in this clip. As teachers we must understand that there are always wheels turning inside our students head and gaining access to how they think is one of our most valuable resources.

From student thinking we can then develop strategies to help our students learn. Some students have an easier time when things are worded in terms of money, while others need to visualize the problems like seen in this clip. Giving them the structure in which they will take these strengths and succeed is one of the most important roles of a teacher. One that I hope I will be able to master when I have my own classroom

Memo #4 Algebraic Thinking

Algebraic thinking is a difficult topic to nail down. Some suggest that it's the ability to do and undo mathematics fluidly, building upon rules and patterns in order to logically work through an algebraic problem. Students who possess algebraic thinking also may have acquired the ability to look at the bigger picture and think abstractly about a given problem. While not all students are currently thinking algebraically, it's important for teachers to recognize it's a valuable skill worth developing within our students.

One way teachers may foster algebraic thinking is by taking into account that there is more than one way of thinking about a problem, and certainly more than one strategy to solve it. To incorporate this idea into our practice, we might allow students to demonstrate their ideas to each other, or even the class. This helps students develop a meta-cognition when solving math as they have to explain their answers, and additionally it benefits the class, as individual students may have their own solving strategy validated, or see a connection to a new way of thinking about it. These connections across curriculum are vital, as we have learned from Hiebert and Carpenter. In my placement I have seen my teacher strive for students to voice their thoughts in an attempt to foster critical thought and meta-cognition as suggested, but it is hardly successful. The students even worked on a problem similar to the problem Driscoll posed about trolls, but unfortunately none of the students used algebra, or backwards thinking in order to solve. They picked random numbers and tried again and again until they got the right answer. Even when prompted to describe how exactly they gauged what theyr next guess would be. It magically just comes to them they say.

This leads us to a more important point, being the kinds of questions we ask and the ways in which we prompt for critical thinking. It is a difficult skill to master, but one that is worth paying close attention to. We can often do thi through eliciting algebraic thinking, incorporating wait time, clarifying, encouraging student exploration, as well as various other strategies. The important part is that students have an opportunity to really think, and reflect upon their thinking when building their math skills.

Memo #3 Proportional Reasoning

Proportions & Proportional Reasoning

Questions
1. What is proportional reasoning?

Proportional reasoning is just like what it sounds, it's a student's ability to reason about problems and situations that require them to think about proportions and how quantities directly relate to one another. This can be through fractions, ratios, related rates, even tables and graphs can be a part of this process. A key part of this process is the concept of equivalence. Teaching a young child about proportional reasoning may not be very successful, because they must be cognitively mature enough to think about quantities and how they relate, or are equal to other such quantities. Students need to be able to make predictions and connections and comparisons using abstract representations, such as numbers, graphs, etc. The last key concept students may need to master is the unit rate, which is highlighted in the Catehcart article as being crucial to developing other skills when working with ratios, rates, proportions, percents and fractions.

2. What are the central concepts and connections (between representations, between
procedures and concepts, etc.) for teaching proportional reasoning?

I think that the central concepts for teaching proportional reasoning is wait time and critical thought building. Again what I believe teachers should be emphasizing is student discovery. Students are not going to develop their own understandings by listening to a teacher lecture about certain types of procedures they can use to solve these problems. The unfortunate problem that I keep running into is how exactly do you inspire student initiated discovery, without seeming superficial as well as forced and confusing. The second problem that becomes difficult arises when students come into this from different levels, some more prepared for these ideas, and some needing a little more time. Some students have even been taught procedures already, and with those students, is it even possible to back track and show them the understanding they need to use with these problems, or is it just more difficult?


3. What are recommendations for teaching this topic for understanding?

a) What should I emphasize when teaching proportional reasoning?
Teaching this topic for understanding is actually very difficult to do, especially with middle school students who are just beginning to think abstractly about mathematics, and manipulate quantities using multiplicatives, instead of addatives. For certain, procedural based learning will not benefit students as well as allowing them to honestly work through the ideas for themselves. Teachers should let students discover for themselves what it means to invert and multiply, giving them the tools they need to do so.

b) How can I teach proportional reasoning using objects, pictures, and word
problems?
Another tricky problem here is that proportional reasoning is abstract, but it also involves real world ideas. Bringing in pictures that show students how things can be grouped in order to compare quantities (ie: for every two kids there is an adult watching them, for every frog in the pond there are two fish, and so on). I have actually seen this at work through a program called "Success Maker." In "Success Maker" they see gumballs in a machine and have a plethora of questions relating to proportions, probability, ratios and fractions.

c) How can instruction address common student difficulties?
Instruction addressing common student difficulties is another hard subject, especially when we don't necessarily think the same way that our students do. So I would say the first step in addressing common student difficulties is to begin thinking like a student and start noticing things that might give them trouble. I really do believe that it's the small things, and the frequent reminders that help students, and to do that we need to get inside their head. The last way that we can address common student difficulties is by involving students in the teaching. If a student comes up to the board to solve a problem and demonstrates a misunderstanding, not only does it illuminate misunderstandings, it allows for opportunities for inprovement in a very real way.

Memo #2: Presentation

Lamon: Proportions, relativity, and the unit. on Prezi

Memo #1 Understanding

Part I-

Understanding is key when teaching mathematics, and too often we have turned our classrooms over entirely to procedural fluency. While solely relying upon procedural fluency is detrimental to a child’s learning, procedural fluency does have its place in the classroom. Hiebert outlines the fact that each of us have a limited amount of mental effort that can be expended at any given time. This means that “the more efficiently a procedure is executed, the less mental effort is required." With less of our mental capacity being tied up doing mundane tasks, the more we can use our brains to focus on meaningful mathematical knowledge. If students become bogged down by difficult computations or frustrated at the first step, they will never move on to grasp the larger scheme of things. In this way, procedural fluency is crucial in building conceptual understanding.

In alignment with the idea that the strands of learning should be taught en masse, procedural fluency and conceptual understanding are inseparable. For example, if two students want to add the numbers 1458 and 267, one may proceed with procedural fluency and begin adding the numbers like so:
            1 4¹9¹8             where the other may “borrow” 2 from                 1 5 0 0
            + 2 6 7             267 in order to round 1498 to 1500,                       + 2 6 5

             1 7 6 5             then add the remaining 265 like so:                      1 7 6 5

The second student may use his conceptual understanding of how to add, but he or she is still relying on the knowledge of how to compute 1500+265, a procedure simpler than the former, yet still a procedure.

How can we teach mathematics for understanding?  

Mathematics is difficult to teach as it is, but to teach for understanding is a whole new ballgame. In order to teach for understanding, we would need to shift our focus from procedures and practices to concepts and content. The most difficult part of this process is that the design of education places 25-40 students in every class, which makes encouraging critical thinking in each and every students mind a difficult task. Even more difficult is providing feedback to these students, whether in the form of informal or formal assessment. I personally think that another element of teaching for understanding involves genuine student inquiry, which is difficult to plan, especially when each student comes from a different level. Math is often taught in a superficial sense, emphasizing procedure over concepts and hurting our students in the long run. 

Part II:

Questions:

1.  What is an external versus internal representation?

External and internal representations are just like they sound. An internal representation is something that one might think about, like the concept of gravity, namely that things fall and we know and have experienced this over our lifetime. An external representation is something more concrete that expresses out internal representation. In this case, it may be anything from a graphic organizer detailing the things that we frequently think about falling, like balls, or people sky diving from planes, maybe even apples from a tree to a verbal explanation about what happens when you drop . An external representation could also be a diagram, or the equation for an object accelerating towards the ground. Representations are different for every individual, depending on their experience with the concept called to mind. (The graphic organizer maybe for small children, and the velocity/acceleration equation for a skilled high school or college student).

2. What are the benefits or results of mathematics understanding?

Students benefit from mathematical understanding rather than procedural fluency because information that is learned with understanding, in the sense that it is tied to other concepts and beliefs and has a firm foundation in a students mind, is called to memory quickly and more readily accessible to the student. The result of this is that a student with understanding is more able to reason and problem solve on their own, specifically when coming up against problems they might have not seen before. Even problems that are more abstract can be better understood by those with a true understanding of the concepts, instead of knowledge of procedures which are of no help if a student does not know how to use them. Both understanding and procedures are important, but they must work together to reach the outcome. 

3. What is ‘conceptual knowledge’ and ‘procedural knowledge’ and how are they connected in terms of math understanding?

In mathematics education, we describe two different approaches to solving an equation as either demonstrating procedural fluency or conceptual understanding. Procedural fluency is the ability of a student to apply a procedure, that is to say, the student has mastered an algorithm for solving a given type of problem, and is now applying it to a similar problem. An example of procedural fluency is when a student uses long division on 152/80928 to change it into decimal form. The student will setup the problem and begin chugging away at the answer in the manner that he or she was recently taught. The student will most likely arrive at the correct answer, but they may take some time and if they make any mistakes along the way, they might not be able to catch them.

Conceptual understanding is the converse of procedural fluency. Conceptual understanding involves a student being comfortable with the problem at hand and their ability to think critically about its meaning, applications, and the best strategy to solve it. The same example of 152/80,928 being changed into decimal form can be tackled using conceptual understanding. A student using conceptual understanding may reduce the fraction first in order to find a simpler solution, they may also obtain an answer such as .937 but quickly realize that this cannot be so since (roughly rounding up) 200/81,000 is nowhere near .937. A student using conceptual understanding also uses procedures, yet is able to think critically about which ones to apply in order to best obtain a practical answer.

There are key differences between conceptual understand and procedural fluency. Both aim to prepare students for the tasks ahead of them, but they tackle the issue in fundamentally dissimilar ways. Conceptual understanding puts at the forefront the student’s ability to think creatively and use logic in order to comprehend, interpret, and solve a problem. Procedural fluency is centered on the student’s ability to produce the correct answer, regardless of how they arrived at their response or they understood its significance. By Kilpatrick and Swafford’s definition, a student using procedural fluency has “skill in carrying out [a] procedure flexibly, accurately, efficiently and appropriately,” and a student demonstrating conceptual understanding will show “comprehension of mathematical concepts, operations, and relations”

While students using conceptual understanding or solely procedural fluency might show no difference in their answers, they may vary in their ability to explain how they obtained the answer and why it is significant. If the answer was wrong, students using understanding are will likely diagnose what went wrong because they can logically justify their steps in obtaining the answer. Students who possess procedural fluency may get frustrated at incorrect answers because they are simply following a rule and cannot provide the reasons why their formula or algorithm worked or not. As educators we strive to teach the former, as understanding also leads to a better grasp of the concept and its significance. 

4. Why is it difficult to assess mathematics understanding?

Understanding, specifically mathematical understanding is difficult to assess, because "understanding usually cannot be inferred from a single response on a single task; any individual task can be performed correctly without understanding" (89). Besides this, in classrooms we see today, it is difficult to assess each student's understanding specifically because they may express it in a variety of ways. For example, if you see a student has solved a problem wrong with no work, you might ask them how they got that answer, and they might tell you they did everything right but at the last moment made a simple mistake with subtraction, giving them the wrong answer. There is no way to know by simply looking at an answer to tell what the student was thinking. If a student shows work and you can tell by mistakes their misconceptions that they had the correct understanding but made a mistake, it is still hard to capture this in a multiple choice test that teachers use so often. The last scenario is where a student has the right answer, but no work showing how they came to that conclusion. They could have used the process of elimination, they could have guessed, used the wrong procedure, or even worse, they might have copied another student. In this case, it is next to impossible to tell if the student understands the content unless they showed their work or verbally explained it to you.