Book Study: Accessible Mathematics

Watch:  It's Instruction Stupid (5ish minutes)
  • Reflect
  • What are you thinking?

Chapter 1 - We've Got Most of the Answers
  • The author mentions the differences between a typical US classroom and Japanese classroom.  Which are we more like?  What are the pros and cons of each?  Where do the mathematical practices fit into these situations?
Chapter 2 - Ready, Set, Review
  • Would a daily review, although taking time out of class, actually save us time throughout the year?  Should teachers write their own reviews or make an "annual" sequence?  What's more important, the review question or the discussion?
Chapter 3 - It's Not Hard to Figure Out Why Reading Works Better than Math

  • This chapter focuses on the mathematical practices of the CCSS.  Did you know that ELA, Math and the Next Generation Science Standards all have comparable practices.  It keeps questions like "Does that make sense" and "do you comprehend" in the forefront.  
  • How did you react to the comments "homework assignments with too many "practice" problems that cannot possibly all be reviewed the next day are so counterproductive" and "it makes no sense to try to cover so much material during a school year when every professional bone in your body tells you it's not sinking in for many..."
Chapter 4 - Picture It, Draw It
  • I would concur with the number of times I observe a classroom and want to jump and and say "draw it!"  In that spirit, share your experiences with drawing, picturing,or visualizing a mathematical concept.  What was the concept?  What was the outcome? 
Chapter 5 - Language-Rich Classes
  • If you get a chance to read through some of your colleagues responses, there is a definite theme lately.  Depth of instruction and retention.  This chapter is all about depth of instruction.  Increasing our students literacy in math by attacking problems in different ways, letting them increase their background knowledge, and allowing them to help determine the depth of instruction will only increase their retention.
  • After reading this chapter are we doing enough depth through literacy?
Chapter 6 - Building Number Sense
  • What is number sense?  Why should we focus on number sense? What are some important factors to building number sense?  Why is it important to build number sense?
Chapter 7 - Milking the Data
  • Why is the "So?" strategy such a powerful way to introduce a concept?  How could you see this being used in your classroom?  Be specific regarding the content being covered.
  • What are some other major take-aways from this chapter.

Chapter 8 - How big, How Far, How Much?
  • This chapter discusses emphasizing the questions, "How Big, How Far, How Much?"  It is all about measurement and understanding what an amount is (ties to estimation).  No specific question because the skin question is pretty leading on its own.  What are your thoughts?  Where can/do we incorporate this into our instruction?
Chapter 9 - Just Don't Do It!
  • What to remove - valid reasoning for removing it.  What are your thoughts?  This is a potentially dangerous chapter.

Chapter 10 - Putting It All Context

  • This chapter by now should be completely obvious given the focus for the past several years on Mathematical Tasks.  I am putting a collection of them on this blog (under the book study link) over the summer.  If you have ones you find/like, email them my way.
  • For this chapter, I want to know what is stopping us from doing tasks.

Chapter 11 - Just Ask Them "Why?"

  • In my opinion, this is probably the most important chapter.
  • Failing to follow up a student's answer with "why?" or "how did you get that answer?" or "Can you explain your thinking?" is a serious missed opportunity.  
  • What does active discourse reinforce?
  • Why follow up most answers with a request for an explanation?

66 comments:

  1. On page 2 it mentions a common frustration of all teachers. A frustration I experienced on my recent final exam. "If a pitcher throws a ball at 87 mph when it leaves his hand, how fast is it going when it crosses home plate? Answer...264 mph." Really?

    I think I try to offer an engaging activity to increase the background knowledge of the topic, it's just getting the students to think that way is a struggle.

    What are your thoughts?

    ReplyDelete
  2. I wonder if students don't seem to put math problems in context because they don't have to at an early age? We need to teach context for every math skill. If context is used to teach all skills maybe number sense would increase as new skills are introduced.

    ReplyDelete
  3. The eighth grade math PLC gave a midterm that was more task oriented. It was interesting to see that the most current topic we had studied was the part they did poorly on. The older topics seemed to go fairly smoothly. It was the exact opposite of what I would have anticipated. However, the students have background knowledge on linear relationships from seventh grade. Exponential relationships (the part they struggled on) is something new to them this year. Perhaps the lack of exposure may have played a role in this.

    Marla's comment was well said.

    ReplyDelete
  4. I use the ongoing cumulative review every day in all of my Math classes. The upper level students have NO problems with the review questions, but my lower level math classes struggle with them. Quite a few of them just write the problems down and then just sit there until the class discusses the answers and the procedures for solving them.

    I also had the same problem on my final as Mark. I think a worse outcome, is that 50% of my students did NOT put ANYTHING down to try and answer the question.

    ReplyDelete
    Replies
    1. Wow! That's amazing. There is no perseverance anymore to solve problems.

      Delete
  5. Whether it is true or not, math has gotten a stigma about it that is hard to fight past for math teachers. Students haven't seen context to their math in the past, so it seems to lead them to not even bother to look for it in their current courses.

    Even in a very task oriented class (Financial Algebra), I still see that issue. Many students don't even want to try to make that connection between the math and its context. Many of them have gone through so much frustration in math that they just don't care to make that connection anymore.

    ReplyDelete
    Replies
    1. I know this is late... On one of the first days of class I have my students in my AP Stats class take out their calcs. I go through a long button sequence. Add this, multiply, now square, etc etc. When done I say ok what did you get? They all say 343!(or whatever it comes to). To which I say, so what, who cares, what does it mean? They are so excited they got the same thing, they lose focus on the important aspect, what it means. We then discuss how numbers without meaning are meaningless. Thought I would share, even if it happens after a few months....

      Delete
    2. Thanks for sharing. What a neat idea.

      Delete
  6. "The author mentions the differences between a typical US classroom and Japanese classroom. Which are we more like?" With Connected Math and CPM, I feel that we are striving to be more like the Japanese model. It is built into the curriculum that students can discover the theorems and rules instead of just having them handed to them. Modeling is used often which helps students understand the concepts. The top students can learn with either method-but it's the struggling students who need the modeling and applications to their own lives the most. Unfortunately the poorer math students are also the ones who want to sit back and have their classmates do the work when they work in teams which is why it's helpful to have the struggling students in their own group(s) so the teacher can give them more individual attention. It makes it hard though when you have a whole classful of struggling math students!

    ReplyDelete
    Replies
    1. I agree that CPM strives closer, however i feel the "real world" questions are quite contrived and most of the times are not application problems at all. I do agree that our students in the "regular track" are more struggling students than ever before and it is a challenge to get the involved in their groups! However, my job is not boring and a constant struggle to come up with solutions. At least it sounds like we are finally going to talk about the root of the problems: Instruction!

      Delete
  7. I think we agree with and try to incorporate the 10 shifts in instruction on pages 4-5. 8 is a very essential element in this process, what to minimize and cut out of or instruction. We all get caught in that slippery slope of teaching because it is on this ACT or WKCE test or it might be needed for that next level class they are going to, or that is how we learned it, but do the students really need it, and to what level?

    ReplyDelete
  8. After reading the first chapter, I found that the differences that the author mentioned about the US and Japanese classrooms is a generalization that is not necessarily true about the typical DCE math classroom. The fact that he mentioned that the US classroom is "traditional, rule-oriented teaching by telling and devoid of number sense", goes against many of the curriculums and teaching practices that we have been incorporating into our classrooms (pg. 2). Although I do see some benefit of teaching traditionally for certain topics, the benefits of presenting problems in a complex, non-traditional and possibly constructivist manner not only allows students to develop and discover the learning, but develop higher order problem solving skills. One drawback of using only a constructivist approach or non traditional method, is that students sometimes do not get enough practice of the skills that feed into the problems (which sometimes can lead to lack of retention). Thus I feel there needs to be a balance of the two in our instruction and the ability to discern when it is best to use each teaching method. The mathematical practices fall directly in line with the 'Japanese model' of instruction.

    ReplyDelete
    Replies
    1. Very well put Meredith. I agree that there definitely needs to be a balance between "traditional" and "constructive" approaches to teaching. Using a constructivist model to capture the students' attention on a new topic and using it to discover the rule/theorem is very valuable. Reinforcing the topic using more traditional methods helps students master the skill.

      Delete
    2. I also agree with that. A blend of discovery/experience with practice is needed. Can we expect students to "discover or explore" something new or connected but then truly obtain the skill through one or two more "connected" problems in the homework? As for the homework, I feel that in some of the problems there is a smattering of skills that they can get lost in that the student really isn't able to practice your core concept from the lesson. Balance is definitely missing unless implanted.

      Delete
  9. Chapter 2 Response:

    After reading chapter 2, "Ready, Set, Review", I feel that the chapter confirms my beliefs in incorporating some type of review into my weekly, if not daily instruction. When I taught geometry in my first year of teaching, I hardly threw in any review of previous concepts into my instruction and assessments. This was a traditional curriculum, and as a means of surviving my first year, I taught it very traditionally and stuck to how the book approached the material (no review whatsoever). After four years of now teaching from CPM, I see the long term benefits that the spiraling in the assignments provides for students, as well as the spiraling that we include in our assessments. Students don't need a lot of time to review material from early in the year because they keep seeing it here and there. One drawback of relying on the assignments for the spiral is that the low students, the ones that need the spiraling the most, are typically the ones that do not complete the daily homework. Thus, relying on the assignments alone in CPM would not be sufficient enough and I as an instructor need to make sure I try to include more review into my daily warm ups or instruction (something I can improve on).

    ReplyDelete
  10. Chapter 2 Response:
    I also feel that review is necessary. We try to fit it in through warm-ups and the spiraling in the curriculum, but sometimes it's hard to fit everything into the period! In 7th grade, we probably could add some review problems into the assessments. I thought the junior high examples of review problems on assessments with a separate grade for those problems as a retention grade was a good idea. If the unit tests are already long enough, the review problems could be on a separate assessment a different day.

    ReplyDelete
    Replies
    1. The spiral review portion that we have placed at the end of each summative test have been quite telling. It has been a real eye-opener as to how much or little students do retain. Seeing these results has driven me to find additional was to review and create connections with new and old material. The more concrete and often I can make this, the better results I feel I see. Quick example, in daily conversations (show me with your hand what linear, exponential, quadratic or inverse might look like on a graph, which of those seems to fit our current concept(s)?....

      Delete
    2. Imagine a world where students remembered what they once knew. Oh to dream...

      That said, I like your quick formative check. We need more of these that access that background knowledge they need to retain information.

      Delete
  11. Last week, we (7th grade teachers) included a review/retention section on our summative unit test. I was pleasantly surprised. It seemed that more students answered the questions correctly and not.

    ReplyDelete
  12. I am All for review, I think it is very important. Let me indicate I must admit I am a little negative today!!!!!! I think our practice in CPM does a hell of a job reviewing prior knowledge and a poor job having student practice the objective learned for the day!!!!!I am concerned about having other students correct other students work - I thought that was a big no, no??? I am a little concerned about the time. Yes it should take about 4 minutes , however do you take any questions? Do you re-teach anything? Now your into 8 min. You have to still go over their homework and answer questions on that! Now your into 4 more minutes or more! Time seems always an issue for me if I am concerned about the students understanding vs. covering material. Yes I think "Ready, Set, Review" is a good idea and I usually do it (90% of the time) and I usually run into time issues!!!!

    ReplyDelete
    Replies
    1. I see the exact same problems in algebra 2, Luke. one small solution I've tried (and it only works on day 2 of a topic) is to use any homework review that I can to lead into the lesson. So if a review problem leads itself into the beginning of the new lesson, I'll use it and adjust the rest of the lesson. The problem is that it doesn't often work out that way. It's one of the few ways I've been able to fix some of the time issues though, when it does work out. Otherwise there have been many times I've run into time issues for my regular lesson because I've spent more time on review than I'd like...which then leads to a repeat on the next day...

      Delete
  13. It is chapters like this that make me think about the CCSS. Yes it is smaller than our previous standards but is it really??? Like everyone, I have my favorite topics to teach and have a hard time letting them go. But can we make them fewer and more effective - deeper not broader? More engaging instead of more coverage? Those are my questions that I fight daily.

    ReplyDelete
  14. I agree with you Mark. Daily, I find myself asking: "Does it really matter if students can . . . " There are many times that I decide "no, it doesn't matter."

    Chapter 3 reflects on the idea of going over homework. It should NOT take much time in class. We have really made an effort to change our homework so that it is "low-level" thinking for practice. All students should be able to do the homework independently--in 30 minutes or less. By using the low-level thinking stuff for homework we are able to concentrate on much higher-level thinking problems in the classroom. Students are happy to do the homework because it is short and sweet and they know exactly what to do. The hard stuff is left for in class with discussion and small group efforts. I have noticed that homework completion has improved this year. I enjoy seeing students struggle and complete problems that I know they would have never done on their own. The conversations are great!

    ReplyDelete
    Replies
    1. I agree, Marla. I don't think we spend too much time going over homework in 7th grade math since the homework problems are often a review of what we discussed in class. With the answers on the Smartboard, we can concentrate on the problems that students have questions on. With the format that we have been using this year, most of my students do a good job of completing their homework. Of course it helps that at the Middle School we can keep our students in for commons time Wednesday-Friday (instead of letting them go out for recess) if they have missing assignments. I also don't allow them to take a retest if they have any missing work.

      Delete
  15. I like the idea of focus on the explanation and understanding. Our students really struggle on answering and writing why they know something even though we really focus on it in geometry constantly. It's like it is to much work for them to explain themselves! We also do not give a lot of homework and the way CPM is layed out is not on the lesson taught for the day but reviews prior lessons. Now my bigger concern is when we do get in good discussion of "why" it takes time and then I feel I am rushing to get thought the next lesson (They are (students) rushing because of the constructivist approach we take in geometry in their groups). Does this go against the common Core standards which says we have to "cover" all of this stuff & same thing with the old state standards. (My opinion way to much!!!)

    ReplyDelete
    Replies
    1. Hoping more than Luke reads this. The CCSS was supposed to be narrower and more focused and I believe it is at the elementary level. It somewhat is at the middle school but at the 8th grade the complexity and broadness of the CCSS are painful.

      This may come as a surprise but I am leaning toward eliminating more standards even though they are in the Core. We know what is vital for our students. Is deriving the area of a triangle of .5(a)(b)(sinC) important? I think there are other things that should be the focus.

      Maybe we can figure out what is the most important. Prioritize from there?

      Delete
    2. As we have been looking over the CCS and long term planning we (8th PLC) have had many of the similar conversations. One of our fears is, if we take something out, is that now to the detriment of the student in the future. Example, we are to cover systems in 8th grade, if we do not do this, will they be missing it in Alg I then...If we were to do some taking out or arranging to make things more appropriate we would then need to coordinate with other levels to make sure things are being covered...
      this perhaps may be a better plan of attach then trying to scurry to cover such a broad spectrum of objectives and rushing to do so...nobody seems to win with that.

      Delete
    3. I agree. I am wondering what other things seem to just be out of place or inconsequential?

      Delete
    4. In response to Mike, this is where collaboration between PLCs is important as if you cover systems in 8th, do we still need to teach them in depth in 9th? Or vice versa...

      In response to Luke, I as well feel the standards in CC cover just as much if not more material. How are we to take time to incorporate such in depth problems when you have the clock against you to get through everything?

      Delete
    5. I agree with everyone above. If we do not teach concepts at an in depth level, what is the point? Only a few of the kids will ever retain the knowledge and it will have to be re-taught at the next level. This is the same broken wheel. I really like the idea of prioritizing the standards and then really teaching and reviewing those standards well.

      Delete
  16. I think it's important to remember to make sure students are truly learning what we do teach because simply covering the material to say we covered it does not do anyone any justice. That having been said, I agree with Mark and his ideas of focusing on the useful concepts of mathematics. I also agree with Mike B that we would need to coordinate across all levels as to what should be eliminated. It would definitely be a topic to discuss when all levels are represented in the room together. It might be a time-consuming discussion but would most likely be worth it as it would benefit students greatly in the long run.

    ReplyDelete
  17. Chapter 3 response

    During this year, I got the pleasure of teaching a regular Algebra 1 class for the first time. Teaching from CPM (and not my self-created Math Matters lessons) was an adjustment as I felt that I had to get through EVERYTHING that was presented in the lessons, where in MM, if I didn't get to something, it was ok because they would be seeing it in their regular class anyway. With that said, my biggest struggle with my algebra lessons was getting through everything in my lesson. I found that when students had questions on homework or when I went through 'feature' problems, I often did not have time to get through the whole lesson which hurt my students in other ways. This focus on breadth vs. depth of knowledge is a struggle for me as I feel that I need to get through all the standards and lessons with my students, keep up with all the other PLC members, but also want students to get the depth of knowledge that the Common Core is looking for. I feel that the suggestions in chapter 3 will help me focus on my instruction on the important parts of the homework and lessons by making sure students are understanding their answers, where they were derived from and if it makes sense. The nightly homework does have an occasional question that will allow me to ask these questions, but again, the tough part is getting students to first off do the homework, and show interest when we are going through it by asking the 'why' questions.

    ReplyDelete
  18. Chapter 5 Response.

    I whole heartily agree with the idea of "language-rich" classrooms. I have always used and taught the appropriate math vocabulary. This school year, I have a had a whole new experience. I have a section of 30 students, 18 of them have "language learning plans." I have never had more of a difficult time with vocabulary as I do now. Having that many students with language deficiencies has forced me to reflect on almost every explanation that I give. I almost always ask students to "define," "describe," "reword" many of my explanations. I had a rude awaking in early October--we were just finishing a unit called Variables & Patterns. The unit teaches about independent and dependent variables, describing relationships, and representing relationships in tables, graphs, and equations. I was working with a student, when she said, "which one is increasing, the one that goes up or goes down." I realized then that some students didn't know what "increase" or "decrease" meant. From that moment on . . . I find myself defining many more words than I ever did before. Most of the time the words are not "math" words, but they still need to be defined in the context of a problem.

    ReplyDelete
  19. Ok, I am a little behind . I just finished reading chapter 4 . which is literally very interesting to me as I am teaching competencies with fractions and I am guilting of teaching the process and not having students picture it!! In geometry we are constantly harping about drawing it, but in Applied math competecies with fractions I am guilty!! The good thing is that I can still help these students and fix it!!!

    ReplyDelete
  20. Chapter 4 response: I think Connected Math does a good job of having students draw diagrams to help them understand the problems. Especially with the lower level students that I have this year, having them draw diagrams helps them visualize what they know and what they are trying to find out. Even students who struggle with integer rules can be successful by using a number line to count right for positive numbers and left for negative numbers. Memorizing rules only helps in the short term. Understanding the "why" behind the math helps them really understand the material for the long term.

    ReplyDelete
    Replies
    1. Ann, I totally understand what you are saying about students memorizing 'rules' not making the necessary connections with the material---especially with the Math Matters students. These students won't retain 'rules' and need something visual and deeper to connect with which is why we need to make sure we are continually providing engaging lessons that capture all learning styles.

      Delete
  21. Chapter 5 response: I found word walls and our vocabulary sheets (where students had to give the definition and diagram of the terms for each chapter) very useful last year when I taught geometry. There aren't as many terms for the students to use in 7th grade, but of course it's still important for them to use the correct terminology. Connected Math lessons aren't just a list of practice problems, but often involve real-life problems (word problems) that make students think. Just by using this curriculum, it makes our classrooms more language-rich than the traditional math classroom.

    ReplyDelete
  22. Very interesting thoughts in the blogs. As usual lots of good ideas and concepts, but time, time, time.. What I think the PLC's need to work on, especially High School next year when we see each other everyday, is what were the most important features that we need to improve on. We all do many aspects of the literacy, language rich, pictures etc..., but what would help student the most if we focused on one or two aspects in this book??

    ReplyDelete
    Replies
    1. I would say the focus on being language rich from Chapter 5 would be a huge priority for me. In my Spanish classes in high school, the teacher forced us (for many of us, against our will) to speak almost solely in Spanish, even if some of the things we said were not said perfectly...in even 1st and second year of the language. If a mistake was made that was something that needed to be addressed, she did, but otherwise we were forced to immerse ourselves in the language.

      I see many of the same things in math. I try to make sure that I use the language in the right ways as much as I can with students without overwhelming them with the terminology, but I know I do not hold them to the standards that I should when they are talking to each other and to me in math classes. I think I need to emulate my old Spanish teacher more in that respect. I think that from that language rich classroom, many of the other pieces discussed in the book (including the 10 instructional shifts on the inside cover) will start to come from it.

      Delete
    2. One of my favorite sayings in my classes is, "Yes thats what normal people say, but since we are in math class we say ______." It makes them know they are thinking down the right path, but in math land we have a different name. Example: Students tend to use the word flip instead of reflection.

      Delete
  23. I think the author is a little optimistic on what questions students will give to a set of "open" data in Chap. 7, and think of the time element to make that effective! Also, how often on a WKCE or ACT or the new common core test will their truly be a "open" ended question, it would be impossible to grade it. With that being said they make great higher level thinking questions, and the idea of good number sense and better interpretation are concepts worth spending time on.

    ReplyDelete
  24. OK so reading early in the morning in the corner of the motel room and then stealing my daughter's I-pad and blogging with it did not go as well as expected! For some reason it would not take my writing - I know after 45 minutes of typing I should have checked $#@&*()_+!!!!! So here is the short version: Ch5 I agree that vocabulary is a huge deal in mathematics and it confirms what we are doing with our practice logs that include vocabulary & a writing log!! How ever currently I am struggling with how much thought my students put into their vocabulary, writing log and now task!! the last chapter's logs were pathetic at best and at least 4 students cheated on the task !!!!!Ch.6 building number sense I think is great and constantly needs to be reinforced - geometry students still struggle with rounding and estimation. I think I am going to put a couple of his questions on the next review portion of our next test and ask the students not to use a calculator (to work on estimation skills) Lastly ch. 7I I like the idea of having students make conjectures and conclusions from graphs and this totally relates to our proof section of our curriculum in geometry - However I am a little concerned about the open and depth when I have to try to achieve a specific objective and stay close to my PLC so that we can asses at the same time!! I really struggle with this as I am sure Steven Leinwand would frown upon this practice because what we are doing disrupts the ability to go with a class and their discussion and get depth and still achieve the objective we are suppose to get to each day!

    ReplyDelete
  25. Number Sense in Chapter 6: I have been harping about this for years. Number sense should be emphasized so much more in our elementary curricula. Some students are amazingly clueless about number sense! Sorry . . .off my high horse.

    I try to do everything possible to include number sense in each concept that we teach. I think our text series does a nice job about teaching and using concepts in context. It is much easier to discuss number sense within the context of a problem. In 7th grade right now we are doing a unit on linear relationships. It is refreshing to see slope and y-intercept taught within a linear context. Students are introduced to "rate of change" and "starting point" and become rather proficient at choosing information within a problem to determine slope or "rate of change" and y-intercept or "starting point". It is much easier to discuss the number sense in the context. For example, I love the fact that we can have students make an equation or graph of the relationship between # of t-shirts and cost and then really UNDERSTAND that the ordered pair (4,23) represents 4 t-shirts for $23. What a better way to talk about number sense! Does 4 t-shirts for $23 that make sense in the context of this problem?

    This nicely leads into Chapter 7--"milking the data." We will often use graphs and table to represent data. I agree with the "So?" idea unfortunately it is often hard to employ this strategy. When your class is 42 minutes long sometimes there is little room for huge variations in the lesson. I admit I could probably be more flexible about allowing such divergence but . . . reality sometimes prevails.

    ReplyDelete
  26. Time is always the thing that pops up in discussions whether we are in our PLCs or on here. We are all constantly fighting that battle, so maybe one of our big focuses needs to be how to fix that issue. I know we have very little say in whether we have 40 minutes in class or 60 minutes. One thing that we maybe need to consider is how to keep more of that time available for those open ended questions and discussions. Some of us have dabbled in "flipping" the classroom, and if there is some success in it, we could potentially free up more time for these open ended discussions.

    I know there are already issues that have popped up with flipping (how do we guarantee the students take responsibility for doing their part, finding resources (or creating them) that accomplish what we want, etc.), but if we can open up even a part of a class period every week for these open ended discussions talked about in chapter 7, we can not only still get the concepts we need to discuss across, but we can take them much deeper.

    ReplyDelete
  27. Chapter 8 is the first chapter I'm in total disagreement with a statement in this book. On page 47 he states how it is a bad policy to state your objective of the day as it is the best way to lose half your class. Really!!! Not anywhere I've ever seen. Now the idea of good measurement is fine, It does not tie into much of the curriculum at the Alg. II level and up, but always a good idea to use when feasible.

    ReplyDelete
    Replies
    1. I don't necessarily think the author is saying that we are supposed to leave the kids in the dark about the topic and make it a mystery to them...although the way he introduced the next topic does have that sort of ring to it. I'll be honest, given the two options for introduction between what the author presented, I'd prefer the mystery option as a student. Even if I did not necessarily know what topic was being presented.

      I do think a better balance between the two can be reached, the objective can often be shared but still not just stated in a way that the students just get that "here we go again" feeling.

      I do very much agree that measurement is not part o f the Algebra 2 curriculum. However, does that mean we be more creative and incorporate it, or assume it should be dealt with earlier?

      Delete
    2. I think at the Alg. 2 level we have to assume that the measuremment topics have been covered. If we get a chance to incorporated here and there in the curriculum fine, but not to add more in. Between getting students covered in the core standards, preparing them for ACT, preparing them for Pre-Cal. this is the most stuffed curriculum we have.

      Delete
    3. Poof, I think all too often every teacher thinks this. For example, note taking. What teacher above grade 6/7/8 doesn't think, my students should know how to do this already. We hold them accountable for something we have yet to identify as a proficient skill. Now granted, sometimes this is fine, assuming kids can measure in Algebra 2 is probably ok, but when asked to find the length of the major axis in an ellipse I got answers that were a full centimeter off from each other.

      Side Question: Where/When/How is note taking taught in our district? Anyone know if this is just teachers do it when they can, or do kids take a semester course??

      Delete
    4. Dustin, I do not know of any note taking class that students take...which is a shame and it shows. I think we all individually give students 'pointers' but this would be a great and beneficial study skill for students to learn.

      Delete
    5. When I taught at Rosholt, I taught a study skills class to all 6th graders, note taking being one of those portions. Could be a valuable class that students are required to take, even if it were only a quarterly or semester class...I agree with the above statements, at times I do explain to them what to take, how to take it and tips to make notes more useful to them, but, not sure it's enough to really make the "big" difference for all...

      Delete
  28. In AP Stats I just had the best discussion I have ever had in math class. I handed out an article on statistical illiteracy. Basically it talks about how professionals know statistics, but don't know the meaning of them. Example, Difference when discussing mortality rates vs 5-year survival rates. Mortality rates are standard across the board, but 5-year survival rates are based upon when you start the 5-years. Bottom line is that this article meant something to them. Much of what was discussed were implications of medical tests and how doctors don't get it. One example showed that if you were diagnosed with a certain cancer there is only a 10% chance you actually have that cancer!

    Kids like to be engaged, the reason this discussion was so good is that it mattered and was relevant to them. And it was relevant on a non-mathematical level, yet without mathematics they wouldn't have understood it. I struggle with Algebra 2 sometimes with the understanding that much of what we teach is tough to model in everyday life.

    As far as measurement, I often times ask my students if answers make sense. Gets them thinking about numbers in context, not just on a number line. For example, if I find a leg in a right triangle across from a 50 degree angle, and it comes out to be the smallest side, I would hope when I asked if it makes sense. Most students recognize that it has to be between the smallest and largest number not the smallest based upon the size of the angle.

    ReplyDelete
  29. After posting Chapter 9 I feel obligated to put the first post. This chapter scares me. With the CCSS, certain things are dictated such as multi-digit multiplication and division. However, what emphasis should we place on this. I just spent too much time researching different ways to incorporate STEM style activities specifically engineering. How do we create time for more long-term concepts without sacrificing something?

    ReplyDelete
    Replies
    1. I almost feel the CCSS in several ways is in opposition to what the author's point. There are several things we had considered trimming out in eighth grade only to discover they are part of the common core. Much of the new common core standards (that haven't been part of the previous standards we were using) seem to be shifting from algebra to pre-algebra. The author mentions at the end about a curriculum of skills, concept, and applications that are reasonable to expect all students to master, not those that have been moved to an earlier grade on the basis of inappropriately raising standards seems to apply to some of the common core. For instance, I do not understand why systems of linear equations should be a focus of pre-algebra. This seems to be rushing the skills. I do not feel that most of my eighth grade minds are ready to fully comprehend this concept. We had to approach it from more of skill work than application because the application would have been too overwhelming for the students.

      Not to mention that some of the examples presented as poor choices are part of the common core. Can we trim out part of the common core? Even if we, too, feel these standards should be addressed in algebra as opposed to pre-algebra?

      Delete
  30. This chapter is the fork in the road we all face as educators. Do we do what is best for students or what policy (administration) demands of us. We clearly have made it the norm in the US (and our own DCE) to make it our primary goal to meet the state and national standards (CCSS in the current year, it was other thinks the last 25 years and will be called something else 5-10 years (or less) in the future) We push curriculum down to levels many students are not ready for, we push them in the advanced level many times because that is what a parent whats. We move them on level to level because they "passed" the previous level B,C,D even F as that school wants them to move on. We currently think all students will reach (and need) and Algebra II level. So clearly we would need to minimize what is taught to have any success with this. But what needs to be covered is dictated also. Seems like an no win, unless we make some unified choices as what to cut (even it is supposed to be covered)

    ReplyDelete
  31. So when and how does any of this change? We talk about it day after day, in every meeting, and we come to the same spot in every discussion. We know what is best for our students and our curriculum, but we decide we can't do anything about it. In reality, can we really make the decision to leave out items in the CCSS in order to improve the depth that we teach? I'm really curious about this. Do we have an option to dig deeper in curriculum, or are we stuck with the wider and wider approach that we have been dealing with? If there is no choice in the matter, then our discussions that we have are just spinning our wheels, and we need to try to focus our attention on making the best of the situation that we have.

    ReplyDelete
    Replies
    1. This is a "catch-22" of our profession. It seems more and more, the decisions being handed down from Federal, State or other levels are not what are considered by our standards what is best for kids. On paper some of these decisions seem like great ideas...in practice, not so much. We DO seem to talk a lot about what can be cut or take off one's plate...answer inevidibly remains...nothing. Is the curriculum a mile wide, inch deep? Should students at earlier ages be taking Algebra...Difficult questions...I do agree with Andy, at some point you look at your situation and say, make the best of it and how can I continue to improve the aspects I have control over...

      Delete
  32. I agree Mark--this is a "dangerous"chapter. Just look at the comments written in this blog. Clearly there is no right answer or no agreement on what should be the right answer. Perhaps we need to do what Andy suggests--make the best of the situation that we have. We all need to teach what the CCSS outline.

    ReplyDelete
  33. Chapter 4 Response:

    I will be playing catch up over the upcoming week...
    After reading chapter 4 I can totally relate with the message that was brought to attention. As a visual and tactile learner, I learn by seeing, drawing and doing. As a result, I try to incorporate these styles into my teaching as I know there are others out there that need to see a visual in order to understand certain concepts (especially with students that struggle as they often need multiple approaches). Although I have a ways to go in illustrating my teaching, there have been a few instances in which I feel my students have benefited by my method of teaching. One skill I use to demonstrate with a picture is when exploring linear equations. By continually making connections between the parts of the linear equation and where these parts are seen on a graph (or a table), students will start to make deeper connections rather than memorizing which is the slope and which is the y-intercept. I also use visuals when reviewing integer rules with the math matters students. I will use two colored counters, number lines, an elevator demonstration and playing cards to help teach the rules of integers. I encourage students to find one of the methods to connect to and remind them that it is a tool they can use and imagine anytime they are faced with such problems.

    ReplyDelete
  34. Chapter 5 response: Language Rich Classes

    This is one area of my instruction that I need to improve on. As one that strongly dislikes reading and writing, and has a hard time comprehending what I read, I do not like to incorporate it into my instruction. I also find that we are rushed so much as it is in our lessons, that it leaves lessons that incorporate literacy on the chopping block simply because they are often the most time consuming as we want to make sure students know how to do something and don't always make sure they know the vocab associated with the concepts. I wish that this chapter gave us solutions to incorporating more language rich lessons besides just the examples of 'word dumps' as I would like to see how others go about incorporating this skill. I have been trying to focus on power words and vocab the past couple of years but do more of telling and discussing the meanings rather than having students make the connections (in which word dumps might help improve this area).

    ReplyDelete
  35. Chapter 6 Response

    Number sense is a term that encompasses quite a lot more than what people typically think. Number sense is not just knowing the place values, the order of numbers, containing the ability to compare numbers, but it is understanding and relating to numbers and tying numbers to situations. We need to focus more on number sense because students often arrive at answers and don't even tie any meaning to them. Students often say the answer is correct because their calculator told them so when in actuality it may be far from the truth and had they given their answer any thought, they would have seen that. This not only will help them in math class but also in real world situations. I feel that I need to do a better job in my instruction in incorporating this skill. This is another skill that often gets left out to dry due to the pressures of getting through lessons and standards that are pressed upon us. We often assume that it is the job of the teachers and grade levels before us to instill this when in fact it should be a constant focus of all levels to continue to focus on it as the difficulty of material increases.

    ReplyDelete
    Replies
    1. Very true Meredith. At all levels a continued effort to increase number sense is important. I believe at each level, with the degree of task, students continue to build on the level or depth of number sense based on the skills and concepts being taught. I also think with number sense there is a close tie to confidence. One perhaps can build on the other, especially for a marginally successful student. Tough part is that many students do not see it as an important skill (I am not tested on it...)but I could be wrong.

      Delete
  36. Chapter 7: Milking the Data

    The "so" strategy is one that intrigues and engages students before being asked a math related question. It is almost a way of buying kids into the learning so that when they are asked the content related questions, they won't even realize that they are in the heart of the learning because they will be engaged in the material. This goes along with strategies such as videos or Dan Meyer related intros. This chapter focuses mostly on charts and tables, but I see the "so" strategy being used with any type of lesson or material. I can see this being used in my classroom and have used it here and there (not necessarily with data, but with other situations and word problems). For example, when completing a lesson on Newton's Revenge in Algebra class, the book layed out the problem and how to solve the problem in a nice step by step manner. This year we decided to instead just give students a small snip it of information they could use and they had to determine the answer to the question "is Newton's Revenge safe for ALL riders" on their own with no help and no data given to them. We engaged them by discussing roller coasters, how they may determine safety regulations and heights for them, watched videos of roller coasters and asked why safety was important. Even though a table of values wasn't given, we were still using the "so" strategy by engaging students for when they did have the data and to get them interested in what they were to do.

    ReplyDelete
  37. Chapter 8 Response

    The skin problem was an interesting approach to the concept of surface area and measurement. After reading this chapter it seems as though the chapter is suggesting the teacher replaces the lesson where they teach the formula for surface area with this problem and allow the kids to attack it and learn it while working on this problem (in a variety of ways). My issue is that what if the students after answering this problem, never learn formulas for surface area because they used a different approach? Thus, are they then missing this 'standard' that they are required to learn? If this is the case, is it then the teacher's responsibility to teach students after the problem how to find surface area using a formula? This them makes a two day concepts essentially double in time in an already packed curriculum. The thought of incorporating these problems is GREAT but I always worry about whether or not students learn the standard that was supposed to be addressed and if there is time to then teach them after the fact. Thoughts? How do you suggest we incorporate problems like this without having to direct instruct the concepts either before or after the fact?

    ReplyDelete
  38. Chapter 10 response. I'll keep it short and simple as to what I think is the problem. Time. Both for us in planning...it takes a lot more time to prep an engaging task that will lead where we need it to and be a productive use of students' time, and time to dedicate in class with all the things we need to discuss according to the CCSS. Not saying it is a legitimate reason to not do tasks, but it is the biggest reason. I also think an unfamiliarity with the approach makes it tough. Just like it is tough for students to get out of their comfort zone in a difficult course, we CNA have a tough time breaking out of our comfort zone in how we teach. At least for myself, I know it is tough to let go of control of the direction of a lesson/activity.

    ReplyDelete
  39. Andy, I agree with your last statement about the comfort zone. I think we all view ourselves as solid instructors with a wide variety of teaching styles. It is/can be tough to step outside our zone/style, nobody wants to feel like they are "wasting time" for if something doesn't go correctly, now I have to back track. I don't do not mind taking time to set up valuable tasks/experiences for students, difficulty for me is the amount of time and accomodations that follow a great task/experience when students are continually missing (mentally or physically). Tho it may only be a few, but the few really create a back log, especially with RTI at every corner. Thus, coming back to the four letter in education...TIME.

    ReplyDelete