CMP - Inquiry

1. How does the CMP curriculum align with the national Common Core and NCTM standards?

In 2000 the Connected Mathematics Project used the Curriculum and Evaluation Standards for School Mathematics developed and issued by the NCTM. It appears as though the CMP has been designed in alignment with the Content Standards set forth by the NCTM, from Number and Operations all the way to Data analysis and probability. Another striking similarity that can be seen as a philosophical alignment are the Six Principles set forth by the NCTM, striking because the language used seems to almost mandate a teaching style that the CMP has been designed to facilitate. Principles such as Equity, which the NCTM says should encourage the provision of extra help to students in need and advocates that teachers not shy away from having high expectations for all of their students; Teaching, where the NCTM suggests teachers not adopt a 'one size fits all' methodology. The NCTM standards and guiding principles go on to basically say that students should not merely be able to perform calculations in vacuum. And that they recognize that helping students improve their problem solving skills is something that all students have the capacity to accomplish.


2. Numerous students are a year or more behind in the basics. How does one address the needs of these students on a daily basis so they can get up to grade level and also experience success in the inquiry to investigation philosophy of the CMP?

Judging on the literature concerning the CMP model it appears as though many students are behind when the begin being instructed in the CMP way, and that the CMP method does not initially raise those students' apparent skill level, based on standardized tests, until they reach their 3rd year of CMP instruction according to a study titled Standards-based Mathematics Curricula and Middle-Grades Students' Performance on Standardized Achievement Tests, published by the Journal for Research in Mathematics Education, 2008, Vol. 39, No. 2, 184-212. Basically they are saying that while the CMP model did not provide immediately measurable positive gains in learning, the learning gains over time were significant when compared to the standard methods of teaching math. "In related investigations, the authors concluded that there is no immediate short-term advantage to CMP, but that the longer view is promising, with CMP students making large gains on a broad range of curriculum topics and processes when compared to non-CMP students."

That being said, CMP model is designed so that overtime those students who need extra attention, are afforded it. With CMP it is imperative that a teacher be instrumental in care-taking each student's needs with regard to their current skill level. Given the instructional material that has been devised to accompany the CMP classroom, students will find that even though they may lack certain mathematical skills initially, they will still be able to contribute to the overall understanding of whatever math concept is being taught, and will come to realize that understanding the concept is more powerful than merely knowing how to do the calculations, which overtime they will gain the skill to perform regardless. So, the CMP student with sub-par skill sets will overtime gain those missing skills by the very nature of learning in the CMP way; that is, through a realistic and relevant problems based curriculum.


3. What is the role of homework (and accountability) in the CMP?

While the role of homework in the CMP is still very much tied to an individual teachers philosophy some of the most common reasons for giving homework in the CMP are:

• To provide additional explanation and practice of the key mathematical ideas in the lesson.
• To grade students' work .
• To assess what students do and do not know in order to plan instruction.
• To connect learning experiences on two consecutive days.
• To instill good study habits.
• To accomplish more mathematical study outside the time limits of the classroom.

4. CMP Investigations are often composed of small-groups (pair-share, teamwork, cooperative learning).

Notebooks and journaling can be on great method for assessing students, while also helping to ensure that all the students are participating. When students are in these small groups some of the ways that teachers can ensure maximum engagement from every students are:

• Have students assign numbers to each student in their group. Then, have them roll a number cube or draw to determine who will present the group's findings.

• Write each student's name on a craft stick, store the sticks in a cup at the front of the room, and choose one stick at random to determine who will present.

• Have the students choose the presenter for their group, but ask each of the other students a question related to the work.

John Geffert - Teaching

For my math methods class I created a video of myself teaching a short lesson about using the Pythagorean Theorem. I used a cloud based presentation software called Prezi, that creates intuitive and smooth transitions between slides for the intro of my lesson, and then, using a standard HD video camera I filmed myself moving step by step through setting up and solving for the hypotenuse of a right triangle. In order to turn the Prezi into usable video I used a cloud based screen casting software called Screenr. Then, when I had all my clips complete, I simple loaded them into iMovie, and spliced them together.

I used the Khan Academy as my template for creating a simple and effective visual for creating and solving math questions. I really enjoyed producing the video, and I truly believe that videos like this, when made available for your students, can have a significant impact on their abilities over time. Being able to review, start and stop a video at home give the student a lot more control of their learning than they might otherwise have during a class period.

The Pythagorean Theorem from John Geffert on Vimeo.

Inquiry & CMP research

The guiding principles behind the CMP INstructional model are designed to help students generate meaning and relevancy for what they are learning, in their own terms, by working through problems that have been devised to incorporate common skills as the student explores problem solving methodology. Thus the intent of CMP is to help students create a deeper understanding of a subject or key idea through the direct and open investigation of a well crafted problem situation.

The CMP model of instruction does appear to be inquiry based, in that it relies on open ended inquiry and investigation. Where the student's interest in a subject or idea may be piqued by the Teacher, but ultimately the Teacher is relying on the process of the activity to impart knowledge and to further a Student's understanding of that subject or Idea.

True inquiry based models of teaching require that the Teacher is ultimately encouraging the Student to think for themselves, by not looking for constant validation. The Teacher is reinforcing the Students behavior, to continue to be curious and to continue to work on a project despite hardship, and without giving the information directly to the Student, if it can be helped. CMP seems to follow some of these basic tenants as well.

Anticipatory Set and Closure

The Anticipatory Set (AS) is used by the Teacher at the outset of leading the class through new material. I looked at a couple of definitions, one from the Oklahoma Baptist University and another from About.com. These two definitions agree that the AS should whet the students' appetite for the new material, be used to assess what level of understanding about the subject the student's may already possess and give the students a clear indication of what the lesson will cover and what is expected.

Some of the methods that can be utilized during the AS are direct questioning, role playing or skits, quiz, modeling and even stories or anecdotes. If at all possible I really like role playing with math, because if you can get the body involved in the math principle then it really has a chance to stick. With measurement you could have the students role play the transition from 1 gallon, played by one student, to 4 quarts played by 4 other students and so on. This method also taps into the visual learner as well because now the concept has lept to life and is more visually tangible.

The Closure Set (CS) is more than just a summary (Oklahoma Baptist University). I like OKBU definition because it puts the responsibility for carrying on the information in the hands of the students. Through the closure the students are empowered to reflect on what they've learned as the Teacher puts the new information in a larger context or even better, leads the Students into forming their own context or framework with the new information attached in order to increase retention and to facilitate transfer. The closure is also an opportunity for the Teacher to assess whether the students really got the subject matter.

Some of the methods that can be utilized during the CS it that the Teacher could lead a class discussion, have the students make short presentations, do project in class, or require writing in the students' reflection journal.

Practicum - Sharing a Lesson

Recently at my practicum site I had the opportunity to work with the flute section on a J.P. Sousa March that features a very well known melody (well known maybe 20 years ago) but none of these students had ever heard the march let alone this solo flute part featured in it. My task was to introduce it to them and begin working with them on it. I had about 15 minutes and we were relegated to a room where there was standing room only, and where I could not move around easily, also with the lay out of the room (a storage locker really) I could not easily get the attention of all the students because some students had to stand behind other students to make room for all of us. My strategy was to briefly play it for them at a very slow tempo and then have them all play together as many times as I could in 15 minutes, while selecting various students from time to time to play a certain section by themselves.

From the moment we entered the room I was sidetracked by comments ranging from, "I don't like this, why do we have to play it?" to "flutes aren't supposed to play that high, why are we playing what should really be a piccolo part?" Out of the ten students, 4 of them made comments along these lines. There are no quick satisfying answers to these questions, and so I could only push forward and at least get them focused on trying to play the piece at the moment.

What worked for me was to have a student that I knew could play it, do so. That got everyone's attention, and even the students that were resistant at first gave it a greater effort. We played through it twice without any comments from me. Afterward, I noticed that there were 3 students of the 10 that clearly did not have the fingerings under their control and would not have been able to play it at that moment at any speed, so I enlisted the help of my 'complainers' to take the next 5 minutes to work with those students to write in fingerings so they would have the correct material to practice at home.

At this point we are about 10 minutes into the short session.

After this I had the group play together again. Then I asked someone I hadn't engaged yet to play a particular 4 measure section. This served as one of my checks on learning because then I asked the two students next to him to play together.

So 15 minutes is not a lot of time to work with 10 flute players who are learning a short solo section they've never really seen before, but as the minutes wore on I was able to assess the ability and motivation level of the individuals within the group. Collecting bits of data in this way helped me focus on the strengths of the students by asking them to be examples for the group, and to deflect/diffuse the poor attitude of a number of the other students by asking them to coach/mentor the few students that were struggling with simple things like the fingerings for common notes.

Had I not been paying attention, or had I let the initial comments from some dissatisfied students corrupt my intentions for the instructional period then I might've spent the entire time lecturing about why the piece was important to learn instead of letting the music speak for itself.

Were I to relive this session, I would have identified some key notes that have multiple fingering possibilities to avoid confusion right off the bat for my inexperienced players. I would have brought in my Ipad with a really awesome rendition of the piece cued up to the flute part so the students could have the full experience of what they are working towards, and I would not have had the full group playing all at once. I would have had students play in pairs, and would have kept things moving more quickly. Pacing is very important and I would like to get better at keeping the momentum.

My site supervisor asked how the session went afterwards, and just as I was about to tell him that I really didn't think we accomplished that much, one of the students that had said she didn't like the piece because it reminder her too much of circus music, came around the corner and said that the session was really helpful and that she felt like she got a lot out of it. Kinda made my day!

Warm-ups in Math Education

Warm-ups serve to set the atmosphere for doing math and should help students begin to shed the mindset of the previous period and begin shape their minds for the lessons they are about to embark on. And, this may be a radical thought, but in math class, or any class for that matter, warm-ups don’t necessarily have to be related to the subject. The class could be led through a follow the leader clapping exercise, or any activity that could get the students attitudes and minds ready to be engaged in the class. When using math problems as a warm up exercise the teacher’s role is to quickly evaluate the quality of the answers produced and decide if more problems of that type are need for certain students. Also warm-ups can be a great way to help give the teacher an idea of where their class is with the fundamentals.

Appropriate Use of Technology

Appropriate Use of Technology

Salman Khan was a hedge fund analyst who was helping his younger cousins with their math lesson. What makes him special is the unique and effective way he used technology. Khan, after giving his cousins a lesson in person, would then record a short Youtube video covering the same material, which was meant to be used as supplemental to the face to face lesson. However, what is remarkable is that Khan discovered that his cousins' preferred his video posts more than the face to face lesson. The idea here is that because everyone learns at a different pace and because we all are differently abled when it comes to attention span, a well crafted lesson that each of us have access to, can start, stop and rewind at will, means we all have our own personal tutor. This subtle but powerful notion is not meant to replace the traditional class room, but it is obvious that technology in this case is more than merely a conduit for supplemental information. With web 2.0 tools available, the Khan Academy has created a virtual learning environment where students are held to high standards of excellence, but are given the opportunity to fail as many times as necessary to achieve that excellence. This is something that only the most charismatic and talented teacher can do for a class of 40 or 50 students.


I found a tutorial covering the concept of indirect measurement using a given scale to measure the size of an object. This video used a simple picture of a helicopter and a small length scale a literal inch represents 7 feet in the image. The teacher walks the student through the thought process of answering various length question as he self talks through the problem. Then the video moves on to measuring the height of the Empire State Building based the relationship between its shadow and the the shadow thrown by a yardstick at the same time of day. I felt like this type of tutorial was very effective, because the visual aspect of online learning brings to bear the full capability of computers and technology with regard to resources that are rich and vibrant (in contrast to a piece of yellow or white chalk), combined with the audio of someone who is doing more than just lecturing, the teacher's in these lessons are also modeling their own thought process and methodology. Very powerful. As I mentioned earlier technology allows the use of more polished, interesting and thus engaging material than a simple chalk/white board can. In addition to this benefit of technology, with the use of streaming videos I can pause for a moment to take in a point the speaker has made without fear of missing out on the material that follows. This serves to reduce my stress level during the lesson. There are, or course, other methods in which this lesson could have been taught, but for the medium - the internet - this was very effective. I can see this lesson being taught by taking a class outside to get a more realistic view of scale and measurements, maybe with taking a model car that has been built in 1:25 scale and comparing it with the original. I honestly can't think of any fundamental way I would have delivered this lesson any differently. I thought the teacher in this case did a great job. Any differences would lie in my own style of delivery, that is my speaking mannerisms, not in the sequence or manner in which he delivered the material versus how I would deliver the material. I thought he was spot on.