In our course we came across this excellent example of planning lessons to help intermediate students succeed in class, socially and individually.
Check it out here-->
http://www.edugains.ca/newsite/math/supporting_classroom_practices.html
We are keeping in mind Marion Small's following BIG ideas.
Measurement Big Ideas (Marian Small):
- A measurement is a comparison of the size of one object with the size of another.
- The same object can be described by using different measurements.
- The numerical value attached to a measurement is relative to the measurement unit.
- Units of different sizes and tools of different types allow us to measure with different levels of precision.
- The use of standard measurement units simplifies communication about the size of objects.
- Knowledge of the size of benchmarks assists in measuring.
- Measurement formulas allow us to rely on measurements that are simpler to access to calculate measurements that are more complicated to access
Questions
1) Let's look at the expectations and big ideas for this lesson.
a) What are the grade 9 expectations for this lesson?
b) Which grade 7 and 8 expectations would support the learning in this grade 9 lesson? To clarify, which expectations in grades 7 and 8 would be foundational for this lesson?
c) Which of Marian Small’s “big ideas” for measurement would encompass the learning in this lesson?
2) Identify learning goal(s) for this Grade 9, 3-Part lesson and list possible success criteria.
3) What was the format of the consolidation for this lesson? What does consolidation do for the learning in this case?
4) What role does student voice play in the learning for this lesson?
5) Group problem solving can look like organized chaos.
a) What classroom management strategies are shown? List all strategies that you observed during this lesson.
b) Choose one classroom management strategy from the lesson. Explain its use and effectiveness.
c) What other management strategy would you try for this lesson or has worked for you during group problem solving situations?
Success Criteria
- I have thoroughly analysed a 3-Part lesson.
- I showed my understanding of module content (e.g., big ideas, learning continuum, learning goals and success criteria, 3-Part Lesson, etc.).
- My assignment is organized. My communication expresses my understanding and analysis clearly.
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In our online course we are asked to break down and discuss a grade nine math class as shown in the "Teaching Through Problem Solving" link on this page shown above.
Write-Up;
The Grade 9 expectations of the Academic level that would be reached are in the
Overall Expectation of
"Determine through investigation, the optimal values of various measurements"; as well as
"solve problems involving the measurements of two-dimensional shapes and the surface areas
and volumes of three-dimensional figures" from the
Measurement and Geometry Strand;
Specifically the expectations being completed/achieved are the following:
-
determine the maximum area of a rectangle
with a given perimeter by constructing
a variety of rectangles, using a variety of
tools (e.g., geoboards, graph paper, toothpicks,
a pre-made dynamic geometry
sketch), and by examining various values
of the area as the side lengths change and
the perimeter remains constant
-
explain the significance of optimal area,
surface area, or volume in various applications
(e.g., the minimum amount of packaging
material; the relationship between
surface area and heat loss);
NOTE: I believe actually, the Curriculum document's overall expectation of "
"solve problems involving the measurements of two-dimensional shapes and the surface areas and volumes of three-dimensional figures" is completed to an extent, but not the specific extent to which the curriculum is actually asking for. In particular the specific expectation seems to be asking for this to be accomplished with irregular shapes as it prepares students for trigonometry and trig functions. So, although in regards to semantics, the expectation is met "overall", its not met specifically.
This understanding has grown from grade 7 and 8 curriculum foundations built in elementary school through the Math curriculum. In Grade 7,the
overall expectation that will eventually lead to the growth of a student's understanding in surface area may be "
determine the relationships among units and measurable attributes, including the area of a trapezoid and the volume of a right prism." Specifically,
Measurement Relationships in which ;"solve problems that require conversion between metric units of area (i.e., square centimetres, square metres) (Sample problem: What is the ratio of the number of square metres to the number of square centimetres for a given area? Use this ratio to convert 6.25 m2 to square centimetres.)"; and "
solve problems involving the estimation and calculation of the area of a trapezoid".
These previous lessons from Grade 7 Mathematics are some of the first examples of exposure to measurements on a 3-dimensional shape to which students are deducing an understanding. The relationships that students develop at this level are crucial because if they are unable to make the connections of a relatively uncommon whole shape such as a trapezoid, to that of a more common whole shape like a square or rectangle, than the student will be trying to develop that relational understanding at a time where they are supposed to be further developing that understanding. I was a student who had "fallen between the cracks" of mathematics when I was going through school and to no fault of mine or my teachers, I just was never really encouraged or had teachers that they themselves were confident in their mathematical skill level enough to try "something different" for students who just were mathematically delayed at particular stages. The hardest part was getting to Grade 10, I was in Grade 9 applied Math and my teacher requested my parents move me to academic grade 10 instead. Big mistake because there was such a disconnect between what the teacher allowed us to learn with in applied grade nine to that of which was expected of us in grade 10 academic. Until the lat three or four years had I actually decided to go back, study and redetermine the importance of mathematics in my life. I am glad I did. Sorry for the personal narrative.
Furthermore in Grade 8,
Overall although students are not looking at a cylinder, cone or pyramid in this particular example (grade 9 classroom example), the students who have completed the act of finding the necessary information to complete the task as instructed for a particularly sized object; meaning that it is correct to say that the overall expectation in which students will be able to/will have
"determine the relationships among units
and measurable attributes, including the
area of a circle and the volume of a
cylinder". The students would have certainly relied on the experience gained from the completion of the
overall expectation,
"research, describe, and report on
applications of volume and capacity
measurement;" because in order to do the research and reporting, students would have worked together and also communicated with not just classmates they are working with but the rest of the class who observes their work.
Specifically, students who have succeed in "
research, describe, and report on
applications of volume and capacity
measurement; determine, through investigation using a
variety of tools and strategies;" Students in the video were able to use charts, and manipulatives to describe and understand the question that the teacher had posed to the class in the beginning. Along with the idea of investigation in general, it was noticed that a student was able to take the idea of the surface areas of all the cubes and utilize the fact that although the cubes together made a large surface area; could it be that the shape changes to maximize the "SA"? This thinking that the student exemplifies could be argued as nourished in his past experiences of completing the specific expectation of, "the relationship between the
area of the base and height and the
volume of a cylinder, and generalize to
develop the formula (i.e., Volume = area
of base x height)". Realistically, the skills that students are applying to the specific expectations are related to the growth of their particular understandings of tasks they they will face in the future, for example teachers can note that within the following expectation students are utilizing the problem solving skills (math process) to "determine, through investigation using
concrete materials, the surface area of a
cylinder; solve problems involving the surface area
and the volume of cylinders, using a
variety of strategies." Students who completed "solve problems involving the estimation
and calculation of the circumference and
the area of a circle" will have found the experiences helpful because of the simple acquirement of skills needed to estimate and hypothesize in the creation of numbers to manipulate.
From this longitudinal perspective its recognized that Marion Small's BIG ideas to which teachers can apply to this particular lesson is, "
The same object can be described by using different measurements." as well as "Measurement formulas allow us to rely on measurements that are simpler to access to calculate measurements that are more complicated to access". These BIG ideas of Marion Small are represented in this particular lesson through the investigation that the students participate in when determining the Surface area of the Cube. The apprehension of ideas through individual manipulation of numbers is important to note as well. The teacher in the lesson specifically mentions that regardless of the numbers of cubes used, there could be a connection to the overall optimization. He encourages students to manipulate the numbers that he originally shares with the students. Through the success of specific students, the class becomes successful when sharing ideas at the front of the class. These are the two BIG ideas that best resonate with me in regards to this lesson.
The learning goals of this lesson as I might described them to my students are:
1. Expressing mathematical ideas in different ways
2. Measuring the maximum area of an object with some similar size sides and some different size faces
The role of consolidation in the lesson as presented, is done in a number of ways. The Lesson consolidation is done through the designing of chart with findings after the experimentation and exploration process. Students as a group are to go to the front of the class and present findings along with methods of process. Choosing only a select few is a productive method for the teacher's lesson and student's time, however it does send a negative message to the groups who were not able to present. If there are ways of consolidating through more accessible means to all students with a given amount of time (much like how we do in the ABQ through discussion posting), this may assist in a stronger consolidation for students. This idea of more time on "accessible" consolidations, could also factor into differentiation of student learning in which students who are ahead can continue ahead while the teacher reviews with specific students who need it.
The role of the student voice in this lesson is layered. The students are communicating which is not only a Math Process as discussed by the ministry as an essential tool; but also a skill utilized to consolidate the learning that has occurred in the class. The students who are able to use their voice have participated and represent the more vocal and confident side of the class. As noticed in the video, there are students who are not documenting the process or discussion, it is sometimes the voice that can also motivate students to share their ideas in order to avoid the act of writing when tired or not interested enough to invest in the written documentation aspect of the lesson. For the teacher the role of the student voice is also to assist the students within a group support each other.
Classroom management strategies that were used by the classroom teacher are the use of an "Exit Ticket"; "Gallery Walk"; "Think Pair Share"; to which I believe that the most effective one used was the Exit Ticket. In my opinion the Chart Paper was an exit ticket for the group. This was effective because the students are most likely eager to share their ideas on the paper, this way the teacher and students can see their work on display which also signals that "Hey, I did this too!" The chart paper is an accumulation of the groups ideas (in most cases) and this is also important because if there are students in groups who are unsure of what is happening in the class, it is a good time for them to sit back and observe. It might actually be a stronger tool if the students were given a checklist of things that thy need to do to succeed in this lesson. for example, if the classroom teacher had given a page that has a series of "I statements" where students can check as "completed" and "not yet" Checkrubric; the teacher list some of the following things,
1. I have shared an idea; Write idea shared below
2. I have written something on the chart paper; Write your contribution below
3. I have Participated in the presentation of our data and experience to someone else
4. I have asked a question to our group members to see if they understand
5. I have asked a group member if my understanding of the topic is correct
However for this lesson I think I much rather would have done a bit of a "Socrative Circle", in which two groups are working together, to prod for questions to possible answers. Instead of just speaking and sharing (which some students did not do because there could only be one speaker at a time), they would be expected to think about what they could ask to get a complete and clear answer from another "visiting" group. This could even be done and different groups could be given different shapes? Different manipulatives? This is a basic classroom and seems to be an applied mathematics 9 class which is also worth considering when developing lessons. Students who are not engaged may benefit from having access to different resources or being placed in a group with people they are comfortable with.
Sources:
http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf
http://www.edugains.ca/newsite/math/supporting_classroom_practices.html
https://vimeo.com/193163993
