"Summing" it up

Hah. I made a pun. So punny.

Anyways, when I first started thinking about putting the entire course into perspective, one topic instantly came to mind: assessment. It seemed as though assessment came up each and every week, in one form or another, even if it wasn't the main topic of discussion. From the elementary grades to university mathematics and teacher education, the issue of assessment is always present. Many times, we want to forget about it, pretend that it does not exist. But at the end of the day, we have to evaluate our students in some way. How do we do this? How much is too much? Are we accurately representing what our students know? Are students' assessment scores an accurate representation of our work as teachers? How can we assess if we are focusing on problem solving in the classroom? How can we create assessments that are equitable?

Unfortunately, I don't know if there will ever be definitive answers to these questions. They have been long debated over the years, and I am almost sure that they will be debated for years to come. Assessment is a touchy topic for a lot of people; many have very fixed ideas of what assessment should or should not look like. At some point though, there needs to be balance. Just as with the reform movements, where you have groups of people on the completely opposite side wanting traditional drill and kill, there needs to be a happy medium. Yes, having students engage with mathematics at a meaningful level is important. Yes, having students be able to complete computations efficiently is important. One should not take precedence over the other. We should be encouraging our students to be flexible mathematical thinkers, who are good both computationally and conceptually. Assessment should be the same; a balance between concepts and procedures.


For my three burning questions, I have:

1) Jo Boaler speaks of "mathematics for all" in her work. What does "mathematical assessment for all" look like?

2) What role do graduate TAs play in undergraduate's learning of mathematics? What is expected of them? What could be done to help them succeed?

and for my own research

3) Is there a way to extend Ball and colleagues' theoretical construct of MKT at the elementary level to MKT at the secondary level?

Equity and Language

This week, I read "Equity and the Quality of the Language Used in Mathematics Education" by Schutte and Kaiser. In this article, the authors focus on the role that language plays in a German mathematics classroom, focusing in particular on students' whose native tongue is not German. The authors claim that nearly a third of students in German schools have migrated from other countries. PISA results claim that "youths whose vernacular used in the parents' homes is not coherent with the language used during lessons achieve lower competency scores in all domains" of the test (p. 238). By analysing a short episode of teaching "least common multiples," the authors note that subtleties in language are often swept under the rug, leaving students (whose first language is not German) to grapple with the meaning of the "academic language," as well as symbolic notation.

This piece struck a particular chord with me, since I have been teaching international students for the last two years. The majority of my students are from China, and thus spend the majority of their time speaking to their fellow classmates in Chinese. If national representation were more diverse, this would perhaps not be the case. I often wonder what sort of effect that language has on their learning. As an instructor, I try my best to write everything on the board so that if anything is missed verbally, it's at least written on the board. Even still, when working with the formal definition of a limit and using the words "arbitrarily" and "sufficiently," these words (initially) have no meaning to the students outside of the context of the definition. Thankfully, the Vantage program includes courses through the LLED department that focus on the language in their mathematics courses! Although I don't know exactly what happens in these classes, it does seem as though they address the "academic language" the authors in this article were so concerned with.


Unfortunately, this all exists in a program that exists for students who all need intensive English preparation, whereas the authors of this article are concerned with a typical elementary or secondary classroom. They assert that the goal should be to create classrooms that emphasize language regardless of the number of languages spoken. This is certainly a novel goal, but what actions can mathematics educators take to do so?

New Zealand's New Curriculum

In her article "More complex than skills: Rethinking the relationship between competencies and curriculum content," Rosemary Hipkins outlines the 2007 overhaul of the New Zealand Curriculum (NZC). The previous NZC of the 1990s was a detailed-oriented curriculum with prescribed outcomes for each subject and grade. With a desire to move away from skill based learning, the curriculum was transformed to focus on five "key competencies" including (1) managing self, (2) relating to others, (3) participating and contributing, (4) thinking, and (5) using language, symbols, and texts. These key competencies act as a framework for individual schools and teachers to build a more detailed curriculum of their own. Hipkins presents an argument for why this transformed curriculum could be beneficial to the modern student, emphasizing that learning should not be the passive acquisition of skills and facts, but should be active, engaging, and participatory. Finally, Hipkins presents an example of how the curriculum is transformed when a key competency is added. She argues that such a change emphasizes the "big picture" and that learning becomes the process of meaning-making for students, rather than acquiring knowledge from the teacher.

The new NZC certainly means well. The intent of building lifelong learners is a novel goal, but I am somewhat worried about the lack of empirical evidence supporting that this sort of change would be beneficial to students. In theory, the new curriculum appears as though there could only be benefits to students' learning. Unfortunately, it seems to be assumed that we live in a world where all teachers and schools are willing and highly capable of engaging with content, students, life-experiences, and the "big picture." How can one assume that all teachers will be able to engage with content at such a deep level? The new NZC puts a great deal of responsibility on individual schools and teachers, and although the author recognizes that schools and teachers should not be expected to work out what is meant by "key competencies" themselves, she does not provide any examples of pathways for schools to do so. Teachers who have been in the classroom for 20+ years may claim that what they have been doing works, and might be unwilling to make such an overhaul. Similarly, new teachers, who have the additional weight of being in a unfamiliar environment, might be overwhelmed with trying to juggle the new experience of working with a classroom full of kids, trying to understand the content they need to teach, as well as how to include these key competencies. Furthermore, if national testing and assessment still aligns with the old, skills-based curriculum, how are schools, teachers, parents, and students supposed to believe that this is actually beneficial to student learning? We've come full circle to the issue of empirical evidence. The goal of the new curriculum foster lifelong learners is of immense value, and is a goal that educators should be striving for in their classrooms. But, it should be noted that transforming theory into practice (particularly at a large scale) does not always turn out the way we hope it might.

Word Problems

This week, I read "Beliefs about word problems" by Greer, Verschaffel, and De Corte. In this article, the authors argue that word problems should not have a bijective relationship with arithmetic computations, but be thought of as exercises in mathematical modeling. The article presents a number of empirical examples of students and student teachers engaging with mathematical word problems that are firmly based in the real world. That is, the problems require the participant to think realistically, rather than simply mapping the problem to an arithmetic computation. They found that only a very small percentage of students and about half of student teachers placed the problems within a real world context. Why it is "ok" to disconnect mathematical word problems from reality, the authors suggest, is due to the beliefs surrounding word problems. They state that although beliefs about word problems are most prevalent in the classroom setting, a broader perspective that includes the school, the education system, and society in general is necessary to understand how beliefs shape mathematical practice. The authors provide the following illustration:

As you can see, I was wondering how and where parents fit into this picture. Just the other day, I came across a friend's post on Facebook which was a video of a mother complaining about the common core standards and advocating for a "just teach the algorithm approach". My friend said something along the lines of "this is stupid and makes me contemplate homeschooling." Of course, I felt that I should at least chime in from a neutral standpoint addressing the pros, cons, and overall intent of the standards. Then, someone replied to me saying "You say the intention is to create independent math thinkers. .... but by frustrating children to the point if tears.... how about we take that money we are spending on this foolish frustrating system.... and we reduce class sizes so teachers can effectively. ... oh I don't know.. TEACH." I gave this woman a well-informed response, but she hasn't said anything back yet. :P

This dialogue was running through my head the entire time that I was reading this article. If parents are so convinced of what constitutes mathematical reasoning and the doing of mathematics, in what ways does this affect the classroom? Moreover, where does the unwillingness to accept that mathematics might be more than plug and chug stem from? How can teachers (who understand and appreciate the new standards) convince parents otherwise?

New SAT Math Questions

Hi all,

Found this article/quiz on the NY Times website and figured I might share it.

http://nyti.ms/23Tqbtg

For a more in-depth article, see http://www.nytimes.com/2016/02/09/us/sat-test-changes.html

New Questions for Old

This week, I read "New Questions for Old" by Prestage and Perks (2001). The authors start out the chapter with stating "take an old question, change it a bit, and hey presto, a new question appears." This was an instant red flag for me and may have influenced my opinion of the chapter. I'll summarize before I state my opinion of the chapter.

The authors recommend four strategies to alter an existing questions: (1) change a bit of an existing question, (2) give the answer, rather than the question, (3) change the resources, and (4) change the format. The authors provide explicit examples of altered questions in each of the categories, why the changes are significant, and how they may have a positive influence on student learning.

Generally, I did not like the examples that the authors provided. I found that many of their examples were arbitrary and would not necessarily help students understand a particular concept better, since they didn't necessarily provide anything meaningful for students to hold onto in the future. For example, the problem "What right-angled triangles can you find with an hypotenuse of 17cm" is an extremely general question. What are students supposed to get out of this exercise? The authors claim that since students have to apply the algorithm many times and make decisions about the number and types of solutions. But do they have a more rich understanding about right triangles or do they just know how to compute better?

Ok....enough of a rant for now. Onto my altered problems! I've chosen a problem from differential calculus, integral calculus, and


For integral calculus:

 It is a very common problem for students to find the volume of a cone using the disk method for solids of rotation. As a change, have students extend this to a tetrahedral of side length a, and then to a cone with an arbitrary polygonal cross-section. This could lead students to thinking about some topology, if we want to take it that far. :P This would be changing the question a bit, according to Prestage and Parks.

For differential calculus: 

Original problem: Show that 2x^2 + x -2 has a zero on the interval (-1,1) using the intermediate value theorem. As an extension, we could ask students to consider any odd degree polynomial and have them prove that the function has at least one zero using the intermediate value theorem.This changes the format, since students cannot do the normal trick of finding explicit x values to evaluate the curve at.

For linear algebra:

Original problem: Given u = [5,2] and v = [-2,4], and the linear transformation T(x) = [.5 0 | 0 .5] [x_1, x_2] (that's supposed to be a 2x2 matrix and a vector in R^2). Find the images of u and v under the transformation T.

As an extension, have students plot this into a computer program and discuss geometrically what T does to each vector x in R^2. As a further extension, transformations in R^3 and do the same in a computer program. This could allow students to better visualize linear transformations in three dimensions. This change would be an example of changing the resources.

Teaching for Understanding

Silver et al's article "Teaching Mathematics for Understanding: An Analysis of Lessons Submitted by Teachers Seeking NBPTS Certification" presents an analysis of a random collection of 32 portfolios submitted by mathematics teachers of upper elementary and middle grades who wish to be certified by the National Board for Professional Teaching Standards (NBPTS) in 1998. The authors argue that although there was a variety of mathematics topics covered within the portfolios, there was a lack of cognitively demanding tasks for both assessing and developing understanding. Moreover, the authors found that fewer than half of the portfolios analyzed included student generated explanations. These findings were significant in that compared to previous research, significantly more of the teachers in this study included cognitively demanding mathematical tasks within their lessons. Although the inclusion of such tasks signifies good teaching practices, one must remember that these teachers were most likely submitting what they consider some of their best work to receive the certification. Thus, particularly with the authors' generous definition of "demanding task" in mind, the fact that just under half of the proposals contained no cognitively demanding tasks could be considered disappointing.

One of the most significant points in this paper for me was the authors' claim that teachers need "additional support to learn to solicit mathematical explanations as a tool in developing and assessing students' mathematical understanding." Although the teachers excelled in all of their pedagogical practices, student generated explanations seemed to be lacking. The authors suspect that this may be due to different definitions of explanation, but with mathematical justification being a central feature of the NCTM standards, this is still a surprising finding. Collaborative work and student centered learning was at the forefront for many, if not most, of these teachers. How is it then, that student explanations and justifications fall to the wayside? Why is this so much "harder" to do than integrating technology into the class?

High Stakes Testing

This week, I read Suzanne Lane's 2004 NCME presidential address, which addressed high-stakes testing and questioned whether or not students are participating in complex mathematical thinking. The article, written 3 years after the establishment of the No Child Left Behind Act (NCLB), does not dismiss NCLB, but rather argues that there is a lack of cohesion in the system. Perhaps no child is left behind, but is this keeping some children from moving ahead? Lane contends that high-stakes testing has become more of an accountability system and less of an opportunity for students and teachers to engage with complex mathematical thinking. Lane addresses the misalignment of state standards and assessment, what is meant by proficiency, and the impact of large-scale assessment and instruction.

Lane's mention of the imbalance of classroom assessment and high-stakes assessment was the most significant point in the article for me. Even at the university, we often speak of the imbalance between homework assignments and what students encounter exams. Instructors remark that they want students to see conceptually heavy content and problems in their homework, but at the same time, produce exams and midterms primarily testing procedural knowledge. Is it reasonable for instructors to expect that all students will spend the time to learn particular material if their "most important" form of assessment does not consider such knowledge? Within the elementary and secondary system, Lane expresses that a "balanced assessment system is needed with a focus not only on quality large-scale assessments, but also on quality classroom assessments that reflect the content standards and are designed to enhance student learning" (p. 13). She emphasizes that assessments can be opportunities for learning, not simply an analysis of.

If classroom assessment is based off of content standards, why aren't large scale assessments? Why is there such an imbalance between two educational practices, which presumably, were developed by the same group of people? As a teacher in such a system, how might you navigate such unsteady waters?

Mathematics-For-Teaching, Davis and Simmt

This week, I read the article "Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) know" by Davis and Simmt. The article develops a theoretical description of what mathematics teachers (need to) know from various workshops they conducted with in-service mathematics teachers. The teacher workshops were conducted to help teachers gain a more nuanced understanding of the mathematical knowledge at stake within the mathematics curriculum, but also acted as a research opportunity for the authors, to see how the teachers learned "new" mathematics. The authors used complexity theory to interpret their results, although the discussion of this theory as a method of interpretation in research would require a completely different discussion.

The authors identified four major aspects of teachers' mathematics-for teaching. Namely, "mathematical objects," "curriculum structures," "classroom collectivity," and "subjective understanding." (Sidenote: It was unclear to me how these aspects were uncovered and why there were not any others. I was generally unsatisfied in the authors' description of how these themes emerged in their research.) In the discussion surrounding "mathematical objects" as an aspect of mathematics-for-teaching, the authors note that many of the teachers' were unaware of a figurative notion of multiplication. That is, multiplication as something besides "repeated addition" or "groups of." They were simply unaware that alternative images and metaphors for a core concept of the mathematics curriculum existed.

My question is, how are teachers expected to come to know this? Is making such connections not a key component of building mathematical understanding? Should teacher education programs include deep, mathematical discussions of core concepts?

Teahers as Mentors by David Tall

David Tall's article "Teachers as Mentors to encourage both power and simplicity in active mathematical learning" argues that mathematics teachers should mentor their students into becoming independent mathematical thinkers. He argues that teaching mathematics sequentially, as is traditionally done, leaves students doing mathematics, but not necessarily thinking about mathematics. The majority of the article is spent explaining what is meant by procedure, process, concept, and procept, and how a teacher might introduce particular topics in a manner that allows students to gain meaning from symbol manipulation.

After reading Tall's article, I was left wondering what he meant by mentor. He begins by stating that "teachers need to act as mentors to encourage their students to build thinkable concepts that link together in coherent ways," but it is unclear to me why he chose the word "mentor." Are good teachers necessarily mentors? Are mentors necessarily good teachers? There was obviously some importance for teachers as mentors, but I'm unsure as to how Tall intended to use it.

Tall's recommendations for teaching the various concepts mentioned within the article are very engaging ways of interacting with the concepts. A great deal of creativity went into some of the examples, particularly the example of a difference of squares, where he uses a rearrangement of geometric shapes. Tall is a well known mathematics education researcher with a strong background in mathematics, and although I see great benefit in utilizing the teaching practices he mentions, I question how the majority of teachers would react to non-sequential mathematics. What if the teachers themselves think of mathematics sequentially? How could one expect to teach mathematics in terms of "thinkable concepts" when they do not think of mathematics in this way? What if the teacher needs to be a mentee?


There is an enormous amount of research regarding the insufficient practices of teachers in mathematics classrooms around the world, as mentioned by Jo Boaler in the video last night. If we want to change classroom practice, it is my opinion that we need to start with teachers who have not yet entered a classroom. True, one could provide future teachers with a list of neat, embodied ways to understand factorization, but isn't this just as bad as having one way to factor it? Although there is more variety, are we not still encouraging teachers to have a toolbox of ways to do and understand things? How might we encourage teachers and students to come up with their own concept images, rather than relying on ones predetermined by someone on the outside?

Hello World!

Red 5, standing by!