Article Discussion #1
Getting Started
with Open-Ended Assessment (Teaching Children Mathematics, April 2005,
p. 413-419)
This article discussed the benefits
and struggles of using open-ended assessments in a (math) classroom. Overall, the
article showed the value and importance open-ended assessments have for
students and teachers. Open-ended means that there are various strategies that
can be used to solve a problem and various solutions may be considered correct.
The article stressed that we need to show students that math isn’t simply about
equations and formulas but that it can also be conceptual (it requires
reasoning, problem solving, and communication, and involves students to be creative
in their thinking). High quality open-ended assessments should not only focus
on a mathematical concept and elicit multiple strategies/responses, but it
should also be written in a well-balanced manner. In other words, the problem
shouldn’t be too obvious, listing every step the student must do, but the
problem also shouldn’t be too open/broad or too challenging. Students need to
be appropriately challenged and steered in the right direction so they are able
to properly demonstrate their thinking. This article discussed open-ended assessment in a fourth grade classroom. Developing open-ended assessments is
not a piece of cake; it may be challenging and time-consuming for teachers, but
the more teachers practice with these, the more they will see their value and
reward. It is not uncommon for students to struggle or feel uncomfortable with
open-ended problems at first due to the fact that they aren’t used to explaining
their reasoning/thought process (as it was in this classroom). It is hard for them to understand why they
even have to provide explanations, and it can be challenging for them to put into
words why they did what they did. In this fourth grade classroom, they were hesitant when drawing
pictures to go along with their explanations. However, with teacher
encouragement and practice, students eventually began explaining their thinking
and adding pictures. Some ways to help students feel more comfortable with
open-ended assessments are: eliminating the word “assessment,” share student
responses (with no name) on the overhead and discuss the positives/negatives
about those responses, give students problems to solve in class and discuss
them as a whole class, give students a slate or white board to write on and
hold up, share a variety of solutions (to show there may be multiple
strategies), pair students and have students explain their thoughts to someone
else, praise students for participating, and commend students who showed
pictures or other visuals along with their written explanation.
Benefits for Students and Teachers
With assessment problems, students
self-confidence, willingness to participate, pride in their ability to explain
their thinking, and eagerness to help others will likely increase. In addition
to having benefits for students, open-ended assessment problems also have
benefits for teachers which include: gaining a better understanding of how students are
comprehending the material and better recognizing if they need to reteach or review certain
material. Also, teachers benefit from the following: variety of solutions from students (shows how all students think
differently and how a problem can be solved in various ways), engage students
who do not typically like to write or who do not like math, the strong
connection between math and literacy may be more comforting for teachers who
are not as comfortable with math instruction but more comfortable with literacy
instruction, samples of student work on open-ended assessment problems is a
helpful tool for communication (teacher can convey a child’s progress to parents
in a parent-teacher conference; they can show them exactly what their child is
understanding in certain topics), share examples of student work with teaching
colleagues (teachers can communicate how students are learning), and special
education teachers or support teachers might find it valuable to have a
student’s work so they can look at specific examples and areas of
strengths/weaknesses.
Open-Ended Assessment Challenges
Not only
are open-ended assessments time-consuming and require a lot of careful thought
and planning, but also interpreting student responses is challenging. It is
very difficult to try and interpret what you think a student knows based on
their response (this is where a rubric comes in handy).
Thoughts
I thought this
article was very well written. I have learned about the value of open-ended
assessments in various classes and this article clearly laid out the
characteristics, benefits, and challenges of open-ended assessments. As stated
in the paragraph above, it is hard to interpret what students know, which is
why a rubric is a helpful tool. However, as we learned in class, this rubric
must also be created carefully with clear distinctions between scores so it is
easy to determine where students fall. While open-ended assessments are definitely time consuming
on both the teacher part and student part, I think they are extremely important
to implement when teaching mathematics because it helps students dig deeper and
find more meaning in the content. Instead of looking at “What’s the answer?”
students are able to think about “Why is that my answer? How should I go about
solving this? Does this make sense?” Teachers also can better analyze the
students’ thought processes, strategies, and solutions, and gain a better
understanding of what students still struggle with, which helps guide future
instruction. I really liked how the article mentioned, “…goal was to provide
enough detail about both their thinking and the mathematical processes they
used so that another 9-year-old student could follow their reasoning.” I think
teachers should convey this idea to their students because it helps them
understand the importance of why they need to explain their answers (even
creating an imaginary student, such as “Terrible Tommy” gets the students
engaged and more willing to explain their answers). I also appreciated how the
article said, “Assessment should not merely be done to students; rather, it
should also be done for students, to guide and enhance their learning.”
Assessments shouldn’t just be given for no reason, but rather, they should be
given at an appropriate time and further expand student learning. I will use
open-ended assessments in my future classroom because I think it takes student
learning to a whole other level and is a great way for students to learn from
each other and for the teacher to see where students are at.
Discussion Questions:
1.
When do you think is the best time to introduce
an open-ended problem to your class (at the beginning of a lesson so they can
explore with the content or at the middle/end of the lesson where they can deepen
their existing knowledge or at all times)?
2.
How often should you implement open-ended
problems per unit?
3. How will you manage students who get frustrated solving open-ended problems?
Article Discussion #2
A Smorgasbord of
Assessment Options (Teaching Children Mathematics, April 2010, p.
458-469)
Creating inquiry-based,
exploratory, and authentic assessments is an important duty of a teacher in
order to gain the levels of student understanding. There are various assessments we can use in our
classroom. The article described one assessment in particular, student-centered
assessment. In student-centered assessments, students are engaging and
participating in math activities that deepen their knowledge, and the teacher
can gauge student understanding. In other words, in a student-centered
classroom, assessment should focus on helping the students to learn math
content and provide valuable information to teachers and students (they should
help the teacher with future instruction). It is crucial that teachers don’t
give out assessments nonchalantly; as the article stated, they need to
understand why they are giving out the assessment, who will use the assessment,
and how. Students should benefit from assessment opportunities (should help
guide/scaffold students to explain their thinking and gear them towards deeper
reasoning of math content). When teachers create an effective assessment, they
should think about using multiple assessment techniques, focus on specific math
goals, and consider how students will think about/respond to the math. Meaningful
assessments should show what we want students to know/be able to do
(objectives/goals). Assessments are in a sense like a performance; students get
to perform what skills they know and this helps convey their thought
process/strategies. The article mentioned different target categories: mastery
of content knowledge, using knowledge to reason and solve problems, skill
development, and product creations. In the article, a class was working with
various geometric shapes. Despite the broad range of student understanding of
geometric shapes, each student was able to explore, analyze, develop arguments,
and reason about geometric shapes and their relationships. When students are
easily and clearly communicating their knowledge and sharing/explaining to
peers the product they created, they demonstrate mastery of the content. Also,
through communicating, students build their mathematical vocabulary and show
authentic understanding of the learning objectives. Teachers then are able to
see how well students actually mastered the content. Also, there should be both
formative and summative assessments, and assessments should be designed to deepen student
understanding of math concepts.
Thoughts
Designing student-centered
assessments are very valuable for both students and teachers. Students gain a
better understanding of math content and teachers can better gauge students’
thought processes/strategies/understanding, which helps guide future
instruction. I think it is really important to provide many opportunities for
student-centered assessments because I think it is more engaging and motivating
for students and allows them to have a sense of ownership of the content. Also,
communicating is an important life skill, and students get to practice
communicating by conveying their knowledge and reasoning to their peers and
teachers showing their understanding for the concept. There were two quotes
from the article that particularly stood out. First, the article said,
“…assessments to be tools both for
and of learning.” I think this is
important for teachers to realize; assessments are a part of students’ learning
process and help them further explore the content, but they are also used to determine
students’ levels of understanding and see what they know or struggle with.
Second, the article said, “Assessment should be an integral part of instruction
before, during, and after the unit [and] should focus on children’s
understanding of ideas, problem-solving abilities, and reactions to their
learning.” I believe that sometimes teachers assume that assessments should be
used at the end of a unit, but really assessments can and should be used
throughout the whole unit (before, during, and after). This way teachers are
constantly able to monitor students’ progress. Also, while student-centered
assessments may take a lot of critical planning from the teacher and may be
time-consuming for the students, they are a meaningful tool where students can
show how they are problem solving. I hope to implement student-centered
assessments in my future classroom to engage students, deepen their math
knowledge, and guide my instruction.
Discussion Questions:
1.
Besides requiring careful planning and lots of
time, what could be other potential downfalls of student-centered assessments?
2.
What tools would you use to make meaningful
student-centered assessments and why (i.e. technology, manipulatives, etc.)?
Article Discussion #3
Understanding
Student to Open-Ended Tasks (Mathematics Teaching in the Middle School,
April 2000, Vol. 5. No. 8, p. 500-505)
This article discussed the
value of open-ended assessments in mathematics. Assessments should be able to
show what students can do and understand. They also should help the teacher
evaluate student understanding of the concept. Often, students have trouble
communicating their math knowledge, and teachers often have trouble providing
meaningful opportunities where students can explain their mathematical
ideas/thought process. Open-ended tasks allow students more freedom to use
their own methods/strategies/approaches. This engages and motivates students
because they are able to have ownership and comfort in the strategies they
choose to demonstrate/express a math idea. When students respond to these
open-ended tasks, it provides teachers with evidence of students’ understanding
of problem-solving and communication skills. This article provided student
examples from Ms. Harding’s sixth grade class where students were writing
detailed explanations in response to a written, open-ended geometry task
(irregular area task). Ms. Harding had discussed how to find only the area of
squares, rectangles, and triangles (but no other figures), and she has
conducted open-ended problems with her class before and constantly encourages
her students to write complete solutions when solving these problems. If
students provide an answer (no explanation) and teachers look at that brief answer, the teacher may
assume the students understand the content when in reality they may not. After
evaluating the students’ responses, Ms. Harding seemed surprised. She realized
that some students had good explanations whereas others had an explanation but
it was just basic or poor/weak. Also, she noticed that some students who
appeared to understand the concept did not necessarily understand it, and some
students who didn’t seem to understand the concept may have actually understood
it. For example, one of Harding’s students who isn’t strong in math seemed to
perform well. Additionally, a student who gave an incorrect answer still may
have had greater knowledge of the subject (shown through the explanation) than
someone who gave the correct answer. Even though these students are young, they
are able to use reasoning and knowledge and explain their thinking through
explanations and visuals. The article did mention that it’s important to
realize that it takes time and practice for students to learn to write and
express their thoughts. Through the open-ended task, Ms. Harding was able to
see the range of student understanding. Also, it was noted that using text,
diagrams, and symbols alongside written explanations helps/supports students’
answers and what they are communicating/convincing to the audience.
Thoughts
I
enjoyed reading this article about open-ended tasks. Ms. Harding did seem a
little tough in her expectations and evaluations. I think it’s important to
realize to keep an open mind when evaluating student responses because students
will have various responses, despite their achievement levels. For instance, a
student who is weak in math may do really well with an open-ended math task. Also, it’s important to expect
reasonable responses. I didn’t like how
Ms. Harding thought Becky’s response conveyed a basic understanding but she
wasn’t happy with this and kept comparing her to other students like Ning. Each
student is different and deserves the chance to freely explore and express
their mathematical process (teacher shouldn’t hold them to a certain
standard). It was sort of ironic because
I felt like Ms. Harding wanted them to openly use text, diagrams, and symbols
along with a detailed explanation but then wasn’t always satisfied with how the
students demonstrated their work even if they included those items. Furthermore,
the article discussed how Ms. Harding provided feedback to students who
submitted incomplete or unclear written responses. I believe providing student
feedback is essential so student understand what parts they need to improve.
However, I hope she also provided feedback to all other students in the class.
As I learned in edTPA, it is important to acknowledge all students’ level of
understanding and comment on what you would like to see them work on or
accomplish in future work. Even if a student mastered the skill, there should
be feedback provided to the student letting him or her know what they should
continue to work on so they are always able to achieve the best work possible
and to their highest potential. Moreover, I appreciated how the article stated that
using visuals (i.e. diagrams and symbols) helps students express and develop
their math knowledge. Many young students are still trying to grow with their
writing skills and have trouble conveying their thoughts in a written
explanation so visuals can be beneficial. I think this may be why Ms. Harding’s
student, Ning, did so well in the open-ended task. Ning didn’t speak English
well and struggled with English, so through visuals she was able to communicate her knowledge/reasoning.
Lastly, open-ended tasks may be complicated and uncomfortable for students in
the beginning. I like how the article mentioned that it takes time and practice
for students to learn to write and express their thoughts. With more
opportunities to participate in open-ended tasks, students will be able to
practice how to effectively explain their thought process.
Discussion Questions:
1.
What are some questions guiding questions
teachers could ask students as they work through open-ended problems?
2.
How do you think Ms. Harding could have improved
her open-ended task?
Article Discussion #4
Assessing
Students’ Understanding through Conversations (Teaching Children
Mathematics, December 2007/January 2008 p. 260-264)
Conversation
is an easy, informal assessment tool that teachers should be using all the
time! Students are constantly talking, and it’s the teacher’s job to purposefully
listen to student conversations so they can learn about students’ thought
processes. Conversations can show faulty reasoning, mistakes in computations,
and misunderstanding of content, all of which can be used by the teacher to
guide future instruction. Through questioning, one-on-one, small
group, or whole class discussions, presentations, or interviews, teachers can
learn about students understanding. With this knowledge, teachers can then ask
follow-up questions, teach impromptu or planned lessons, better plan for future
instruction, revise planned instruction, and adapt instruction to meet the
students’ needs. No matter what the environment of the school or classroom is
like, conversations are a tool that can always be used. Whether teacher-to-class,
teacher-to-student, or student-to-student conversation, tons of valuable
information is expressed through conversation and teachers can use this
informal conversation as an assessment to better grasp what students do and do
not understand.
This
article discussed three situations where conversation proved to be a valuable
tool. In the first situation, a third grade teacher had a conversation with one
of her students about place value to evaluate her level of understanding. When
the student was able to go back and explain her thinking, she was able to
better understand the concept of place value. This shows the importance of
having students explain their thought process, detect their mistakes, correct
their mistakes, and learn from their mistakes, and better understand how to
work through the problem. In the next situation, a fifth grade teacher had a
conversation with some students who made mistakes on a geometry quiz. The
teacher was trying to understand their mistakes and level of understanding. After
the students were able to interact with each other and discuss their mistakes,
they were able to explain their ideas, get feedback on their thoughts, and
understand other students’ perspectives. In the last situation, a teacher
worked with third graders about measurement (measuring long jumps). Many
students made errors in their measurements. This activity showed the value of
planning engaging and interactive activities where students have to collaborate
with their peers. Students can verbalize their thinking, and teachers can
listen to the conversations. It is crucial that the teacher listens to
conversations and monitors student understanding as opposed to assuming
students are proficient at the task (by listening to conversations, the teacher
may realize areas where many students are struggling). All of these examples
demonstrated that conversation is an imperative part of mathematics so teachers
can better gauge students’ understanding.
This
article also discussed the importance of having a safe classroom where students
can freely explore new ideas and share their thoughts. It is essential that
teachers review classroom rules and what is expected of the students in order to have a safe classroom. For
example, students should listen respectfully, accept others ideas/opinions,
take turns, etc. Furthermore, getting students to participate in conversation
is not always an easy task. Therefore, teachers should try to wait an
appropriate amount of time after asking a question before calling on a student
for a response, and they should encourage all students to think/share ideas. [I
also believe teachers should give students praise for participating even if
their answer is incorrect to make students feel safe and more willing to
participate in the future.] If student participation is a struggle in a
classroom, a teacher could have students write their responses on a mini
whiteboard and hold them up so all students are engaged and comfortable and so the teacher can
see students’ levels of understanding. In addition, having students repeat what
other students have shared helps make sure they are listening and
participating. Moreover, students cannot converse if teachers do not provide
students enough opportunities to do so. Having students think, pair and share,
and/or talk in a small group before sharing their ideas aloud to the whole
class allows them adequate time to think about the question, prepare an
answer, and feel more comfortable. Lastly, teachers should make sure their
classroom is set up in a manner that promotes collaboration (i.e. table groups,
open floor space, etc.)
Thoughts
I
agree with the article in that the idea of conversing in a classroom cannot be
stressed enough. Conversation is such a simple assessment tool that is informal
and requires no planning. Conversation in a classroom is a win-win; through conversation,
students are participating and learning from each other, and teachers are listening/assessing as the
students converse. With conversation, teachers can truly assess students' math
understanding and knowledge and what areas students still struggle with. This will
help teachers guide their future instruction, where they need to reteach, and
what parts of their future lessons they need to revise. In addition,
conversation helps students understand their own mistakes! By sharing their
thought process with a peer or teacher, often students catch their own mistakes
and are able to correctly adjust their thought process. I always find it
amazing to listen to the diverse responses in which students share, because
sometimes students share such creative approaches that deepens their own math
knowledge as well as their peers. I think conversation is useful in a classroom because students can
learn from their peers as well as understand the significance of participating
and clearly conveying their thought process.
Discussion Questions:
1.
How would you go about responding to a student
that is enthusiastically participating/sharing their ideas but whose answer is
incorrect?
2.
Besides wait time, encouragement, and praise,
how would you encourage student participation in your math classroom? (i.e.
create lessons that students can connect with, incorporate a real-life
situation, use manipulatives, etc.)?
Article Discussion #5
An Experiment in
Using Portfolios in the Middle School (Mathematics Teaching in the
Middle School, March 2008, Vol. 13, No. 7, p. 404-409)
This article was
fascinating to read about as it discussed an authentic way of assessing
students. Standard tests do not always show what students understand. For
example, if students simply circle an answer for a multiple choice question or
write an equation, they aren’t necessarily demonstrating their thought process
or showing what they understand. Rather than using traditional assessments,
such as quizzes or tests, this article discussed using portfolios as an
assessment tool in an 8th grade prealgebra classroom. The portfolios
were collected after seven weeks. Portfolios are an alternate way to assess
students in math. They show what a student learned over the past few
weeks/months and help students and teachers see the learning growth. The
article mentioned that students gain a better attitude about math and
themselves as well as earn better grades when being assessed with portfolios (I
think this is probably true in part because students have less pressure and
anxiety as they might have when taking a test). Five categories were assessed in students’
portfolios: 1. Mathematical attitude (what a student thought about
mathematics and his or her work in mathematics), 2. Problem-solving (different
ways of solving problems from concrete to more open-ended/abstract), 3.
Mathematical growth (improvement in mathematical knowledge and understanding),
4. Mathematical writing (writing about student’s thinking in mathematics as
well as about mathematics itself), and 5. Mathematical connections (showing
that mathematics exists outside the mathematics classroom). The teacher didn’t
want the students to just get a grade, but rather wanted the students to
understand their strengths and progression. In the portfolios, students collected
pieces of their work (that demonstrated problem-solving, communication, and
connections). Additionally, students had to write a reflection, which explained
the category that their piece was in as well as why they chose that piece.
Moreover, parents were involved in the portfolio process. Parents had to sign
after seeing the portfolio and provide feedback about how useful they felt the
portfolio was in helping them understand what their child was learning in math.
The portfolio was assessed based on
the organization and on the students’ learning and mathematical understanding.
The portfolio weighed just as much as a traditional assessment. However, the
goal of the portfolio was to understand the level at which students
comprehended the concepts and how well they were able to succeed. The teacher
designed a portfolio checklist to help her appropriately assess the students.
The
benefits of using the portfolios is that the teacher provided more
opportunities for her students to write, problem solve, and complete projects
in which both the teacher and the students can analyze their thinking. Also,
the teacher realized the importance of having students reflect on their
learning and solving more realistic problems that require reasoning. Lastly,
the teacher implemented more meaningful (often open-ended) activities that
students could use in their portfolios.
Students
also had to keep a learning log, write a math autobiography, and write
introductory reflections for each portfolio piece. Learning logs helped
students show what they did and did not understand about the assignment. It
also helped the teacher understand what challenges the students were having and
where they excelled, which helps the teacher plan for future instruction. The
autobiographies helped students think about math. They got the chance to show their math
experiences (i.e. what they liked/disliked, excelled at/struggled with), which
made the teacher more aware of their abilities. The introductory reflections
helped the teacher learn what the students thought about their work.
Some
of the students said that the portfolios took more time than a regular
assessment. Many students said they appreciated being able to see their
strengths in math. Also, parents enjoyed that they got to be a part of their
child’s math learning process by seeing their work and looking at their child’s
strengths, weaknesses, and feelings about math.
Portfolios
are time consuming to put together and time consuming to grade. However, it is
an authentic and effective way for the teacher to be able to assess student
learning and see where they (the teacher) can adapt future instruction so
students can be more successful in math. Portfolios allow students to analyze
their own learning, growth, and understanding of math.
Thoughts
Portfolios
are excellent assessment tools that stray away from the “typical” and often
ineffective traditional tests. I think students can learn more from a portfolio
that they put together than from a test. In addition to teachers assessing
student understanding, the students themselves can evaluate their own level of
understanding and progress. I really
like how students are able to have the freedom to pick what pieces they put in
their portfolio as well as write a reflection on those pieces that they chose
(makes students think about what is good about that piece of work/why they are
proud of it). I believe the portfolio is like a book that the student has
written; it contains various work samples created by the student. I think
portfolios create a sense of ownership and confidence in students and help them
evaluate their own learning and understanding of math. Moreover, I think
portfolios are an amazing concrete example that teachers can use in
parent-teacher conferences to show the student’s progress, level of understanding,
strengths, and weaknesses. The teacher in the article mentioned that she wanted
to use portfolios three or four times a year. I think this may be a little
excessive and time-consuming. However, I do think it is valuable for both
students and teachers to implement a math portfolio at least once a year. Also,
although this portfolio is for a middle school classroom, I could adapt it to make
it appropriate for younger students.
Discussion Questions:
1.
How would you adapt the portfolio assessment to
make it more appropriate for younger students?
2.
In addition to portfolios, what are other
authentic assessment tools teachers can use in the math classroom?
References:
Bacon, K.A. (2010). A smorgasbord of assessment options. Teaching Children
Mathematics. 16(8).
Leatham, K.R., Lawrence, K., Mewborn, D.S. (2005). Getting started with
open-ended assessment. Teaching Children Mathematics. 11(8).
Maxwell, V.L., Lassak, M.B. (2008). An experiment in using portfolios in
the middle school. Mathematics Teaching in the Middle School. 13(7).
Moskal, B.M. (2000). Understanding student to open-ended tasks. Teaching
Mathematics in the Middle School. 5(8).
Vanderhye, C.M., Zmijewski Demers, C.M. (2007/2008). Assessing students'
understanding through conversations. Teaching Children Mathematics. 14(5).
