Memo #6 Language and Learning

Mathematics and language have an interesting relationship. Students come into the classroom thinking that numbers and symbols have nothing to do with the way they read, write, listen, or speak. Brenner disagrees and claims that "learning proceeds most effectively in a social context." This means that students understand the most when the problems they read are real, the answers they write are meaningful, the explanations they give are effectively spoken and received by their peers.

The most learning happens in this last state, where student have to think about their thinking, and in turn express to others how they approached a problem. This meta-cognition reinforces the ideas that they are developing, and thus helps students retain the information they are learning. Unfortunately many of our students are suffering in math class because they don't understand the importance of the process and not the product, or because they don't have the means to communicate effectively.

I went to the Teaching for Social Justice conference in San Francisco and had the pleasure of participating in a workshop given by a past classmate, Rick Barlow. He thought it was so important that students start talking about math that he implemented sentence frames and other scaffolds so that they could find power in their voice and start becoming a valuable member of his mathematics classroom. At first they treated it like a joke, but at least they were talking about it. Not only were they talking, but thanks to his scaffolds, they were using the academic language that we often try so hard for students to acquire.

While each group of students is different and should have their unique needs addressed accordingly, I really enjoyed Rick's suggestions and am thinking about incorporating this idea in to my own classroom one day. Until they are comfortable enough to do it on their own, taking baby steps to get to mathematical language competency is sometimes enough. Especially for those who have gone silent for so long...

Dear Future Self

Dear Jacki, I hope you haven't burnt out by now, because I sure am having a hard time staying motivated. I know you will be great because you have always wanted to teach children. Most importantly I hope that you remember WHY you wanted to teach. Because everyone says they aren't good at math, because you felt bad that they were robbed of all the joys that is problem solving, and because if you were really good at it, why can't they be too. This passion has developed in the past few year into something that is more about equity. Not only because you are frustrated that you are surrounded by individuals that don't know how to solve simple problems, but also because these problems would be made easier if they had the problem solving strategies which make you not only a competent mathematician, but a viable citizen, and member of a community (whether it be your cohort, house, or classroom). I want my students to be successful in mathematics, but most importantly I want them to be successful in life. I want to hold them to a higher standard. No, you can't just give up on that problem because it doesn't look like the ones you have already done. No, I will not give you the answers because it's too hard. No, we will not be taking a break, your time here is precious. But I can tell you this, you aren't alone. I will support you. I will give you my best each and every day. I will respect you as a valuable member of our classroom. These things I want my students to know, and know well. Life isn't easy. It isn't even fair sometimes. But if I can help it, I can make a difference to help give the next generation a fighting chance. Fighting chance at what? For some it's a fighting chance for life, others it may one day be the fight to cure cancer, or maybe something in the middle: the skills we need to get by day to day. The curiosity to think of new ideas, and the competence to follow them through. That's all I want. Talk about a long list of expectations. Well... Goodluck then. See you in 10 years. Love, Jacki

Learning and Teaching of School Algebra

Again, algebraic thinking is a student's ability to think abstractly about numbers and symbols, as well as manipulate equations in order to come to conclusions about a once unknown quantity. Kieran suggest that algebra has been taught more simply as a procedure, and teachers crossed their fingers hoping that those that enjoyed working with numbers got it, and those who didn't were able to figure out how to survive. However there is hope for the future as Kieran suggest algebra is more of a generalized, problem solving tool, able to be modeled, and applied to the work at hand. Being a math major and having taken abstract algebra, I think the most important thing about algebraic thinking revolves around the use of functions and variables. The key to algebraic understanding then lies within a student's ability to make sense of the operations they see every day, (+, -, x, /) and how they act on quantities, whether known or unknown. And that understanding comes from a beginning introduction to mathematics and a students notion about what it means to add and multiply (as well as their inverses, subtractions and division). Thus, what I want my students to understand is that algebra is not merely a procedure that can be applied under certain circumstances, but rather a system for which mathematics is constructed, a way of understanding the world rather. I would want my students to develop a habit of thinking about ways in which to interpret ideas mathematically. Because when it comes down to it, most students can do the procedures but fail when given a word problem that expresses the same thing. This tells me that they didn't quite understood the math in the first place, and that they need to develop a deeper understanding of what mathematics, specifically algebra, really means. The biggest problem comes into play when we talk about how to teach our students to think algebraically. Of course it is important to give students multiple representations of the same material. Kieran does an excellent job of laying out a plethora of ways in which students can think about equations and variables, including balancing models, geometric models, arithmetic models, as well as a couple more. Things like algebra tiles, scales, or virtual manipulatives can all aid in acquiring an understanding of what exactly equations are and what it means to manipulate quantities. The other key part of this is that enough time should be allowed where students can really wrap their head around complex ideas like variables and functions. Students really just need plenty of time to master the skills AND understandings that surround algebra. Of course high expectations and emphasis on the process of problem solving, not the product can also help students get on the right track, especially when standardized testing and the state of mathematics education has driven them so far off the path to mathematical competency. Lastly I think it's important to value student backgrounds when teaching algebra, because it is likely they have developed ways of understanding the world that can align with algebraic thinking (like the notebook model Kieran suggests). Not only does involving student's lives show them that you respect who they are and where they come from, but it also helps to scaffold their learning in a really genuine and relevant way.

Memo #5 Generalizations

Generalizations are a key component to the acquisition of algebraic thinking. Driscoll suggests that it is important to not only generalize arithmetic, but also functions and relations between numbers. For example, observing that powers of 2s (ie 1, 2, 4, 8, 16, 32...), 3s (1, 3, 9, 27, 81...) and so on all grow larger, but numbers like 1/2 to positive powers grow smaller and smaller, approaching 0, helps students grasp the concept of the function that is "repeated multiplication." In a way, if we think about fostering algebraic thinking, rather than procedural fluency that is warned about in Kilpatrick and Swafford, then we will students building rules for themselves, and understanding the functions they are applying, or tasks they are following through with.

One of the large hurdles for students is systems of equations, a topic that Driscoll touches on in Chapter four. As in many mathematical situations, sometimes all of the computation is not necessary, and the skillful student will see patterns and develop strategies to take shortcuts while still arriving at the same answer. The hardest part about all of this is inspiring students to pick up on the knowledge for themselves, since problem solving and algebraic thinking is not the easiest to teach.

I would suggest that instead of modeling problems, teaching procedural strategies, and then letting them go to fend for themselves on the homework, that instead we foster a general curiosity and a genuine interest in math by allowing them to come to terms with new ideas on their own. By this I mean that students come into the classroom with so much information already, and (I would like to think) an innate desire to learn. So instead of talking at them for 45 minutes, or an hour, or even two, we should rather entrust them with problems and puzzles that will instead lead to learning. Too often I see students who become frustrated and give up on problems that are not that different from what they know, but they don't see themselves as competent problems solvers. The truth is that they ARE capable of extending their knowledge, like Driscoll suggests. We just need to give our students plenty of time and support.

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.