The purpose of this activity is to receive an introduction to the teaching concepts addressed in TMM. Specifically, what are characteristics of struggling learners that make meaningful mathematics a challenge and what are strategies that are useful in making mathematics meaningful? This will allow the others to get a sense of the major points in the other chapters. We will be returning to this book in the future. It is a very good book.
- In reading your respective chapter you are to address the following:
- The major points (be judicious in what you choose) - provide enough detail so the others get a full sense of the point
- What are the major take-aways you have for each major point listed
- For the chapter on characteristics of struggling learners, address WHY kids are struggling and how this relates to the teacher's perspective (after all the point of this is to inform your teaching).
- For the strategies chapters, address WHY the strategies may work and how these strategies relate to your future teaching. How can you make use of these?
- Again, be judicious and be concise. You don't have to provide a full explanation of the whole chapter.
- In responding to the posts of others, identify a key point made and address how it can or will impact your teaching AND explain your reaction, e.g. are you surprised or is it common sense?
Your initial post is due by Sunday, June 1 at 10PM and your responses to the posts of others is due by Sunday, June 8 10PM.
Here are the chapter assignments:
- Matt - Chapter 4: "The Importance of Valuing Mathematics and Mathematics Instruction for Struggling Learners"
- Pennell - Chapter 5: "Common Learning Characteristics that Make Mathematics Difficult for Struggling Learners"
- Tim - Chapter 6: "Curriculum Barriers to Learning Mathematics"
- Emily - Chapter 7: How Struggling Learners Can Learn Mathematics"
- Noah - Chapter 8: "Making Instructional Decisions: Determining What and How to Teach
- Ted - Chapter 9: "Teaching for Initial Understanding: Using Effective Instructional Practices"
- Kelli - Chapter 10: "Building Proficiency: Using Effective Student Practice Strategies"
- Jennifer - Chapter 11: "Planning Effective Mathematics Instruction in a Variety of Educational Environments"
- Jannine - Chapter 12: "Using Technology to Promote Access to Mathematics"
Chp. 11: Planning Effective Mathematics Instruction in a Variety of Educational Environments
ReplyDeleteThis chapter provides a framework for planning effective math instruction that integrates earlier-discussed techniques for helping struggling learners.
• The Responsive Teaching Framework poses 10 questions teachers should consider when planning instruction that will be responsive to the needs of struggling learners. A few of the most important include: “What did I learn from my mathematics dynamic assessment?”, “How will I differentiate the instructional needs of my students?” and “What authentic context(s) will I use?”
o Takeaway: Don’t just plan “to teach comparing fractions”; consider what the students already know, any error patterns in data, how they can connect it to previous knowledge, where it’s found in real life, and HOW you will do all of those things effectively for 30 individual learners.
• The framework requires teachers to first gather data on what the students know and can do mathematically and then use that data to summarize the students’ understandings and develop an instructional hypothesis. This hypothesis lists what the students can do, cannot do, and why the teacher thinks they can and cannot do these skills.
o Takeaway: Know where the students are at and try to figure out WHY they are there. Don’t just reteach from the top—address what you think is the obstacle. (Very similar to Lemov’s Technique #16, Break It Down.) Targeted teaching is more effective than a broad stroke.
• From there, the teacher can plan how she will differentiate, within either whole-class instruction or in flexible groupings. Either way (or through a combination of both), students’ needs will be served as they are grouped for comprehension and level of teacher support necessary—low, medium, or high.
o Takeaway: Differentiation doesn’t have to mean flexible groupings or just different leveled work for certain students; it can be done in whole-class instruction too. (That was an eye opener! Being able to maintain the classroom community rather than constantly splitting all the kids up but still offering scaffolding and enrichment as needed sounds great.)
• One suggestion for grouping that is offered is to consider the levels as ALL, MOST, and SOME—what ALL students are expected to know, what MOST students are expected to know, and what SOME are expected to know. This allows the teacher to make sure all students are achieving mastery of the targeted skill at appropriate levels of complexity.
o Takeaway: This seems really useful. Not only can you ensure each student is mastering the basic skill, you can also offer enrichment and depth for the students who need it. It also seems less stigmatizing than “High, Medium, and Low”. (Kids inevitably figure out which group they’re in and it can be a real source of pain for them. Avoiding the “low” term can only help destigmatize it.)
• Finally, the chapter focuses on how to make math a “wrap-around” experience for the students, so that they are receiving support at home and make connections to real-world applications.
o Takeaway: The more you include the answer to “when am I ever going to use this in the real world?” in your lessons, the more students will be pushed to learn in a deeper conceptual way and in a more self-motivated manner.
o One suggestion adapted a common reading comprehension model for connections, which is connection to (other) texts, to self, and to world. In math it became connections in my class, in my school, and in my home. Students kept it as a portfolio of real-world applications, listing the pattern and where they found it. This could be done with other skills too. Ex. Volume (in class—storing our manipulatives in different-sized bins, in school—at lunch on our milk containers, at home—fitting groceries in the fridge). I really like this idea of getting the students to notice the math around them all the time; then math is not an isolated, abstract class at school, but part of daily life.
Hi Jennifer, Thanks for tackling Chapter 11! I really like the concept above about ALL, MOST, SOME instead of low, medium, high level. All kids should know how to determine the length and width of a rectangle. Some kids should know how to calculate the area of a rectangle and the total square footage of a 10 x 12 ft brick patio. SOME can determine how much money you will need to spend at the Home Improvement Store if bricks cost $3 each and you want to build that patio. I'm sure in many cases the differences in level would be more subtle, and that makes me realize I will need a lot of field experience to improve in differentiated instruction using these techniques. I shared your feelings of relief that differentiation can also be done maintaining the class community, through whole-class instruction and not just flexible grouping.
DeleteHi Jen,
DeleteThanks for breaking Chapter down into a concise yet still detailed summary. There's some great stuff in there! I also feel like it's an eye-opener to hear and begin to learn how to differentiate instruction without creating a divisive (even if it's covert) environment. I can't wait to learn more about this.
I really love the idea of making the kids reflect on "my class, my school, my home" I really want to incorporate this, even if it's bringing a weekly journal response as a bellwork question before or immediately following an exam. What a great way to kill two birds with one stone - help the kids to make connections with their math content, and reinforce writing skills to help support interdisciplinary goals and SBAC requirements. Great stuff!!!
I love that idea of making the journal into occasional bell work, Emily! That can be one of the initiators for a day when I haven't had time to make something wonderful and pretty, like Scott was saying. Nice connection!
DeleteThis was obviously a chapter packed with material. One idea that struck me was having students keep a journal of real world applications of math. One thing I have started doing, is to ask professionals when I meet them, how they use math in their jobs. Only once have i not recevied an answer. I wonder if this idea could be incorporated into the concept.
DeleteChapter Six (Curriculum Barriers to Learning Mathematics)
ReplyDeleteThe ‘standard’ curriculum and instructional techniques work for most students, but poses some significant challenges for those that are struggling. This chapter describes five of these especially difficult challenges.
1. The Spiraling Curriculum. This one is what you would expect...as progressive years’ curricula build on top of previous years’ concepts, struggling students get further behind as they progress through school. Math becomes increasingly difficult for the student, which then leads to ‘math anxiety’ and exasperation among both the students and the teachers.
Takeaway: All hope is not lost; astute teachers (who are aware of these curriculum challenges for struggling students) can detect mismatches between curriculum expectations and student learning needs and can/should make appropriate instructional modifications (p.63).
2. Teaching to Mastery. This challenge is linked to the spiraling curriculum in that teachers may convince themselves that students will have ample opportunities to master skills and understanding because of the frequent exposures they will have to the material. Whereas multiple exposures can strengthen neural pathways in memory, learning is frequently uneven, and not consistently retained for struggling students.
Takeaway: Continuous monitoring of students’ progress (rare in today’s schools) will inform teachers immediately of the need for instructional adjustment.
3. Algorithm-driven Instruction versus Teaching Understanding. Many teachers default to teaching computational procedures and algorithms (because that is how they learned it) vice teaching for conceptual understanding. Students may be able to repeat the algorithm, but when questioned, do not understand what they are doing, or more importantly, why they are doing these computations. When students miss the conceptual understanding of math, it is no longer meaningful.
Takeaway: Instruction for all students, especially those struggling in math, should be grounded in conceptual understanding of the material.
4. The Cyclical Reform Process. Curriculum reforms run in cycles. These reforms, in content and instructional techniques, depend on the philosophical and political trends of the day (p.65). Struggling students typically do not adapt well to change.
Takeaway: Not only is a steady diet of both the procedural and conceptual aspects of math beneficial for average (plus) students, it is critical for struggling learners. Incorporating both of these aspects into a strategy matched with the learning needs of struggling students will go a long way in ensuring their success.
5. The Options Not Taken: Lack of Implementation of Effective Teaching Practices for Struggling Learners. For a number of reasons, teachers fall into ‘teacher-centered’ instructional approaches, to the detriment of the struggling student. One of the primary reasons this occurs is because teachers become overly focused on what they can control (the lesson), vice the intent of the lesson itself (aid in students’ conceptual understanding).
Takeaway: Progress can be made if teachers match teaching techniques with student learning preferences. This requires some level of teacher-student relationship, where the teacher understands each student at a deeper level than ‘gets along well with others’, or ‘works well in groups’. This relationship is reminiscent of Nel Noddings’ concept of a ‘caring’ teacher-student relation.
The macro takeaways for me from this chapter are that effective teachers will have enough of a relationship with each student to know what instructional techniques will work for the individual students, and also to have some process(es) in place to continuously monitor student performance. This monitoring will act as a rumble strip for the teacher to make modest (and tailored to the student) adjustments before the student goes off the cliff.
Thanks for the useful summary Tim! I found all the points intriguing. The one that surprised me most was the notion that spiraling in the curriculum can have negative effects on the struggling learner! I have always liked the idea of spiraling because it reinforces prior skills and, I thought, helped students maintain these skills more permanently. However I can see how the learning can be "uneven" for those who struggle and can create greater Math anxiety.
DeleteAnother point I wanted to comment on is the Algorithm vs. Teaching Understanding (computation vs. concepts). In my classroom I hope to put this into practice despite my instinct to use formulas based on memory. When you teach a student, for example, how to figure out the area of a triangle by perhaps doubling it into a rectangle shape (a much easier area to remember, L x W, and dividing in half, you use logic to find the area, instead of memorizing 1/2(b*h). You can do with with many polygons instead of computing all different complicated formulas. Using paper folding, cutting, re-arranging is also a nice hands-on way to approach the same idea. As I said, with all this new information we are getting through Core and Methods, I hope I can remember all these great techniques and put them into practice.
Hi Tim,
DeleteThanks so much for writing this all up! It was filled with lots of detail, I'm impressed at how well you summarized everything. I really see the cyclical/spiraling curriculum happening a lot in my classroom. Even in our school, there is such a push to move students on wards and achieve graduation that students are moving to upper levels of math having not mastered the content from the previous levels. This has presented such a problem, and though I like the idea that "hope is not lost" that is offered in TMM, I am interested to learn how to meet our own grade's curriculum requirements while simultaneously covering the gaps that kids come into the year with. What a challenge!
I know our school will be starting a more intensive Scientific Research Based Intervention (SRBI) program focused specifically on Math this fall. After reading your post, I am realizing this would be a great chance for students with content gaps to catch up and get out of the downward spiral. I'd like to incorporate a "back to the basics" type of teaching in the SRBI class, designing curriculum for each student to go back and prove mastery of their foundation skills.
I really like the basic idea of this chapter and the five specific difficulties you summarized for us. This first year of teaching for me has been all about #2 and #3: teaching for mastery and teaching algorithms vs. understanding.
DeleteSince I've been teaching at a much lower grade level than I had been tutoring--4th and 5th grade vs. high school and college--I really had my eyes opened to teaching for both mastery and understanding. Even in 4th grade, there's a push to get "through" the curriculum and "cover" everything, that can be really detrimental to a struggling math student. Is it really important that a 4th grader see the routine of how to find the least common denominator and add two fractions when she can't even understand equivalent fractions? And how do you teach (and reteach) equivalent fractions when they don't get it the first few times? I'm really hoping we get into how to approach reteaching before we begin our student teaching this summer, since it will mostly be credit recovery.
What I'm interested in figuring out is how to identify those gaps in basic understanding when you're at a higher level and you DO want to teach pre-cal, not just basic algebra. Being forced to teach arithmetic basics got me thinking about how to break down big concepts and how to go from the concept to the algorithm. But when you're in a higher level and you realize that much of what the students do they do out of habit and based on algorithms, not understanding, how far back can and should you go? For example, I had never seen long multiplication of polynomials--when they're stacked on top of each other and you multiple them just as you would multiple two multi-digit integers. This was great! It brings in the concept of place value and relates it to higher order variables. And it eliminates having to gather like term variables in a huge string of numbers. But I had NEVER SEEN it until the calculus class I observed. It wouldn't even occur to me to discuss that with the kids, yet, it gets into a deeper conceptual understanding of how the long multiplication algorithm works--particularly that "magic zero" my third grade teacher taught me about...Finding out when and how to go back to close gaps--and how to identify them--is something I'm very interested in covering during Methods.
Great summary Tim. Another packed chapter with lots of potential; subjects to comment on, but I will confine myself to one, or I will never get though all these posts.
DeleteI was really struck with the last item "Options not Taken". To me it is a real reminder of accountability of the teacher for the outcome. It was a hard lesson for people who I managed in my former life to learn that you may have worked very hard and followed all the book steps but if the outcome wasn't there you have failed. Put another way, I think the author is saying that you can;t be afraid to change strategies if what you are doing isn't working. Also I can a concept of planning from the student outward as a useful metaphor.
Another quick comment, I found the point on the curriculum cycle to be interesting. I have followed the whole core implementation at my daughter's middle school and have to commend the teachers there for by and large implementing it without throwing the students really realizing it.
Chapter 9- Teaching for Initial Understanding.
ReplyDeleteThis chapter discusses four instructional practices that give students an opportunity for initial understanding of a topic. The four strategies are:
1) “Teaching within authentic contexts.”- This technique requires teachers to have some background knowledge on the students they are working with. In this case the instructor is teaching a targeted skill in a context that the student is connected with. (sports, music, theatre, work) It is believed that learning of a topic that is related to an activity that a student is interested in creates better connection and is therefore more effective. The excitement a student feels when they are connected to a lesson is a wonderful motivator.
This technique is one that I have used effectively during instruction.
2) “Building meaningful student connections”- When students are learning a new topic, connecting new ideas to ones that were previously mastered is an effective way to create understanding. This idea also serves to review topics and create a spiral learning effect. In addition, because previous topics were mastered the student would be afforded some success on parts of the new topic and therefore be more motivated to continue even if the topic was challenging.
3) “Modeling and Scaffolding instruction using Concrete-Representational- Abstract (CRA)
Sequence”- This method of acquiring initial understanding was one that was demonstrated in the first homework assignment we received. As first, we needed to identify the a,b, and c of the quadratic equation and fill the numbers in the boxes. Eventually the guidance became less and less and we were doing the hole problem on our own. This scaffolding of the problems gives the students guidance and allows the teacher to become a “learning bridge”
4) Teaching problem solving strategies.- In this section the student is taught ideas and strategies on how to solve various problems. This method is closely related to the modeling and scaffolding idea from above. The emphasis of this idea is on the learner to think, analyze, and implement the correct strategy.
Hi TJ, Thanks for the nice clear summary of Chapter 9. I think this is a very useful chapter given the amount of new material we will be dispensing over the years to our students. I wanted to comment on strategy #2 above, " 'Building meaningful student connections'
Delete- When students are learning a new topic, connecting new ideas to ones that were previously mastered." I want to make sure I incorporate this into my classroom because it seems like a perfect way to bring in new material in a way that reduces anxiety and leads to a deeper understanding. The other day my 4 year old son was asking me why my cousin moved to a new house. I was explaining to him in the simplest way I knew how that she sold her house and bought another one. He said (this part is funny), "you don't buy a house!" I guess he thought you just have one, oh to be four again....so now I gave him a quick lesson on borrowing money from the bank. I used a prior concept that he has mastered: borrowing toys from his daycare friends and then giving them back. I said, you borrow money from the bank and then you give it back. The toy parallel seemed to help create understanding of a new and confusing idea. In the Math world, one of my classmates presented a lesson on "completing the square." The next presenter used the completing the square method to illustrate how the quadratic formula can be derived. It was awesome! It took a confusing formula that we all try to memorize and made it easier to understand based on a prior concept.
Hi TJ, and thanks for the great chapter summary. It's nice having the concepts broken down visually as you did, and with concise descriptions.
DeleteMy chapter included a bit on the CRA learning model, and this really stuck out to me as something I want to be successful at implementing. I had heard of scaffolding previously, but the way TMM breaks this concept down using CRA is really helpful. What a great way to empower kids! I think it's great to take the time to have kids reflect, too, on what they were once not able to do, and now have the skills to complete independently. What a confidence booster.
I think utilizing the "authentic contexts" method is also going to be incredibly effective. It does require previous knowledge of the student, but even with a basic awareness of what teens or tweens are excited about (Snapchat? :) Maybe that's too hard to incorporate), we can get kids seeing the world around them mathematically through things that pique their interest.
Hi TJ, thanks for the summary. I am probably borrowing from a post I made elsewhere but as the idea came up gain I decided to repeat myself, since repetition helps to cement an idea.
DeleteThe scaffolding concept (Concrete-Representational-Abstract) makes perfect sense and I can see where it can be an effective technique. I am going to have to work hard at it since it exactly opposite to the way I think. I can see that I may include a little box on my lesson plan somewhere making me check off that I have thought the issue through.
I also like thinking about math connections. Since math as whole tends to have common patterns, the more we can build connections to each subject the easier the next subject may become. Also many problems can solved multiple ways (for example I know an infinite series proof for the Pythagorean Theorem. So the more connections that are made the more tools students have for problems.
Chapter 10, Building Proficiency Using Effective Student Practice Strategies, looks at the stage of instruction after students have reached a level of understanding in a topic (“Acquisition”) and move on to becoming proficient or fluent. Fluency means being able to show knowledge with accuracy and apply it at an appropriate rate.
ReplyDeleteIn short, to build proficiency, practice matters. Struggling learners will require more practice than those not struggling.
Practice should be varied, motivational, and use real-life context whenever possible. Effective student practice for struggling learners uses math concepts that they already “get,” with multiple opportunities to respond. Activities should be designed with students’ learning disabilities in mind. The teacher should first provide directions, followed by continual feedback, positive reinforcement and monitoring. Practice should include a way to measure progress.
There are several Student Practice Instructional Strategies.
1. Structured Language Experiences – Students use their own language to describe their understandings (writing, talking, drawing, etc.).
2. Structured Cooperative Learning Groups/Peer Tutoring – Learners strengthen what they know by practicing with peers.
• Takeaway: make sure peers don’t leave out struggling learners. Assigning roles in advance such as “coach” or “player” can help prevent this. Pairing strugglers with productive learners is also effective.
3. Math Instructional Games/Self-Correcting Materials – Using games (with solutions to self correct) through use of spinners, dice, flashcards, board games, etc. is a creative way to practice and keep learners motivated.
• Takeaway: for strugglers, select games where they have a good chance for success. Keep games age and interest appropriate and spell out behavior expectations first.
4. Continuous Monitoring and Charting of Understanding – Instructors should have students complete a “learning probe” regularly (this is a 5-10 minute exercise in the topic which should be easy to score). Then chart the student results each time in a stimulating visual format, such as using stickers to plot points (pages 164-166 illustrate some good examples worth reviewing).
• Takeaway: be prepared to modify instruction if the graph shows a decline in performance instead of improvement.
5. Maintenance of Mastered Concepts and Skills –Assign practice work to review prior skills already mastered. A daily routine should incorporate this strategy.
• Takeaway: Use a “Problem of the Day,” or Instructional Self Correcting Games similar to those described in the Johnson Motivation series.
Some possible reasons these strategies may work:
With Structured Language Experiences, by explaining a topic verbally in his/her own words, the student is using more than just memory tricks and must rely on understanding the topic. He/she must think it through carefully. This reinforces the material.
Structured Cooperative Learning may work because stronger students can draw out participation of struggling or shy students. Often a team can solve something that can’t be solved alone, so success is more likely. Something may “click” for a learner when shown by a peer.
Math Instructional Games are fun and motivate students. Kids will be willing to practice more if the process is enjoyable. Also the self correction prevents embarrassment and reduces anxiety of struggling learners.
Continuous Monitoring and Charting may work because students can see their learning track, which is great positive reinforcement.
Finally, my thoughts as they relate to my future teaching are that practice strategies require thoughtful planning and a good sense of which students will benefit most from which strategy in order to be effective. In short, Practice, Monitor, and Maintain to help struggling learners retain what they learned.
Hi Kelli! Thanks for the great summary, and your excellent insight and commentary for your take-aways. I really love the idea of charting progress - I never would have thought of this! To have a visual representation of how kids are progressing is pretty cool. Obviously, it'd be important to have this anonymous, but the visual will really help the students appreciate their successes. It'd be great to set a goal, and provide an incentive to reach that goal (ex: homework passes, a "game day", etc).
DeleteAfter reading this entry on games and cooperative learning, I'm excited to incorporate both into the classroom. This goes in hand with our workshop on Friday...better said, this chapter reinforces what we learned on Friday, and gives me even more buy-in to the idea of incorporating games with props to help kids stay excited about their learning. Thanks!!!
Hey Kelli
DeleteAs I was reading your post I started connecting it to Scott Dunn’s presentation. When we practiced the schema activators in core we were using them to activate prior learning. The benefit of starting the day in this way is twofold. It gets the students ready to learn and, more importantly, it allows students who may be struggling with the material to do some extra practice without the fear if getting the example incorrect because the learning takes the form of a game. Over all it’s a less threatening situation and one in which the students feel safe to continue to gain understanding
Scott also gave is some ways he used group work and it was very similar to the ways outlined on your chapter. He suggested making sure groups all had the same general make up with regard to ability. As you mentioned the concept or idea may make sense when explained to the struggling learner from a peer. I hope to incorporate both of these ideas into my classroom this summer. Great post!
Another great chapter with lots of information.
DeleteI am going to comment on the Maintenance of Mastered concepts. I find with a lot of the students I tutor that they can lose prior concepts pretty quickly if not used regularly. With some of my students I began starting each lesson with a five minute review of something previous. I had an instructor in geometry recently who used the Problem of The Day concept. In any event I can see that incorporating review (perhaps cleverly disguised) routinely is important.
Another packed chapter with lots of good infomraiton.
DeleteI will focus on the Maintenance of Mastered Concepts. I found with student I was tutoring, that by the time we got to the mid-terms they had forgotten alot of the prior material (in line with this concept). Toward the end of the year I began a process where I started each lesson with a quick 5 minute review of some previous material (semi-random). A geometry instructor I recently had, started each class with a problem of the day to get at the same idea. I can using some of the initiation activities that were talked about in Fridays class for this purpose.
Chapter 8
ReplyDeleteMaking Instructional Decisions: Determining What and How to Teach
This chapter discusses a two phase process of deciding what needs to be taught and how to teach it in an interesting way. There are three methods discussed, the MSII, the MIDMIDL, and the MDA.
The first part of this process talks about how to makes lessons more interesting to students by incorporating things that they are interested in or can relate to, which is referred to as the Mathematics Student Interest Inventory (MSII). At the beginning of the year, the teacher asks students about their interests. He then picks out different interests and decides how they can be used in different math concepts (such as using video game genres while talking about bar graphs). This method allows students to see real world applications of math instead of just formulas, and it keeps their attention because they enjoy the connection to their life. There have been several blog posts about this technique, and I intend to use this, perhaps as a first day activity. You get to know your students early on, and then later in the year you can use examples that they will be interested in.
The second part of the process is deciding what and how to teach and is composed of two methods. The first is called the Mathematics Instructional Decision-Making Inventory for Diverse Learners (MIDMIDL). It helps to determine the degree of structure and explicitness for different concepts. Using some important learning characteristics (such as number of students receiving special education services, general achievement level of the class, and degree of motivation), a teacher can decide if the class will need a high level of support (this chapter uses a form to tally up points for each category and assigns a score range to each level of support). For new concepts, support level is based on four factors: degree of complexity, degree of accuracy required, available time for the concept, and if it is foundational for future math. This strategy allows a teacher to decide early on how much structure will be needed for his classroom in general, and which concepts will require the most support. These are great things to think about when starting new units that I had previously not given much thought to.
The other method in this part is the Mathematics Dynamic Assessment (MDA). It assesses the students’ understandings and misconceptions of a concept. There are three techniques that are discussed here, the first of which is Concrete-Representational-Abstract assessment (CRA). It assesses students at three levels of mathematical understanding: concrete, representational, and abstract. This allows teachers to see if students truly understand a concept, or if they are just memorizing a procedure. It also gives insight on where a student’s strengths and weaknesses are. The next technique is error pattern analysis, where teachers look for repeated mistakes and then analyze where the student is having trouble. This also allows teachers to determine if there is an underlying misconception that is giving the student trouble. The final technique is flexible interviews. A teacher can ask a student to talk through their work. This can give teachers an idea of how students think and where specifically they are not understanding a concept.
This chapter provides several strategies in determining what concepts are taught and with what level of support, how to keep student interest, and how to determine understanding. The MSII determines students’ interests in order to incorporate them into lessons. The MIDMIDL helps teachers to decide the level of structure they will need in general as well as for specific topics. The MDA uses several techniques to learn about students’ understandings and misconceptions of concepts. Combining these methods will allow teachers to plan what to teach (from the MDA), and how to teach it (from the MSII and MIDMIDL).
Hi Noah, thanks for a clear summary of the chapter! I also like the MSII strategy at the beginning of the year to determine student interests. It seems like such a simple yet effective way to motivate learners. Because we are getting overloaded with instructional strategies and techniques this summer, all useful of course, but still an overload, anything that I take away as a simple approach that is effective gets a gold star in my toolbox.
DeleteI also wanted to comment on the MIDMIDL. I had been thinking more broadly about the degree of structure in a class and the level of support needed, but this Chapter made me realize we have to analyze that for each CONCEPT. I like the four factors to consider: degree of complexity, degree of accuracy required, available time for the concept, and if it is foundational for future math. I think I might want to incorporate those four pieces somewhere on my lesson plan template, just as I would the objective each day and the specific Common Core Standard being addressed.
Hi Noah, thanks for the great summary. I think out of all the material presented, I like the MII and MDA strategies the best. I had read about the CRA scaffolding methods, but I really like the error-pattern analysis. Though this came up before, but I like the way you've described it. It'd be great to chart how/where students are making mistakes, and modify lessons to re-build these skills. It goes back to some of Johnson's books, and an idea to have students find the mistakes in a sample problem that is done wrong, so that they can see the problem from a correcting perspective, not a completing perspective.
DeleteI also agree - I definitely want to bring some sort of survey where students choose or rank in order of preference their out-of-school interests. Doing so early will help in building rapport, but also help in writing or finding problems that connect with their interests. Great idea!
I will echo the previous commenter in applauding the clear and accessible structure of the chapter summary.
DeleteIt has always been a hot button for me that math is often not taught with accessible application in mind. That being said I always struggle with how to put the concept into practice. I thought the concept of a student interest inventory was a good idea for first or second day activity, maybe on a more informal basis than suggested.
I was also struck by the discussion of Dynamic Assessment. I have never really thought about the learning process in this fashion. And for me it is important since I tend to learn in exactly the opposite way moving from the abstract to the concrete. So it is going to be be important for me to deliberately think about this issue as I develop and deliver lesson plans so that I do not confuse students. Also I can see in doing assessment it can be useful to structure thinking about where a student is in the process. I think this idea ties in pretty well with the six stages of learning discussed in chapter 7 (Thanks Emily)
Chapter 7: How Struggling Learners Can Learn Mathematics
ReplyDeleteSIX STAGES OF LEARNING:
1. Initial Acquisition: Develop a beginning level of understanding, 0% to 50% of the concept.
2. Advanced Acquisition: Demonstrate 50% - 95% understanding of a concept.
3. Proficiency: Fluent with the target concept
4. Maintenance: Maintain skill over time (i.e. later in school year)
5. Generalization: Apply the use of the concept in varying contexts.
6. Adaption: Apply the understanding of one concept to another and interpret their relationship.
The ultimate goal is to achieve all 6 stages, but students may not move through stages in order (especially struggling learners). Teachers should always plan instruction with the objective of moving students through these stages successfully.
Takeaway: This framework will help in determining where a student is facing an obstacle in the learning process. It will be helpful to use in developing a plan for improvement.
Two teaching tips are provided to help promote success in struggling learners:
TEACHING TIP 1: Material should move from Concrete to Representational to Abstract.
Concrete: Demonstrate understanding through concrete materials (eg: counting objections)
Representational: Demonstrate understanding through drawing diagrams.
Abstract: Demonstrate understanding through written symbols (eg: operations, variables).
Students may be able to show understanding through the abstract means but lack understanding in the concrete demonstration. (eg. a student may compute 3 X 5, but may not be able to show that this means three groups of five)
Take away: This tip is very helpful, as it reminds the teacher to move material progressively to the abstract. This will help struggling learners because it breaks the learning process down into three manageable and sequential steps, and ensures that the abstract level is not approached until the concrete level is mastered.
TEACHING TIP 2: Material should be both receptive and expressive in format.
When checking for student understanding, provide responses that are both receptive (cued) and expressive (open-ended)
An example of ‘receptive format’ is a multiple choice question.
This is effective as it
- Helps struggling learners build confidence and provides them guidance and cues in place of anxiety.
- Gives the teacher the chance to encourage through mistakes. Even though they selected the wrong answer, they made several good choices by eliminating some of the other incorrect answers.
Take away: Ultimately, the goal is for struggling learners to do math independently. By having questions in a receptive format, students feel more confident to do the problem because they have a better chance at getting it right. Students who are struggling likely lack confidence. If their choice options are infinite, as in expressive format, they are likely to give up as the probability of succeeding is low. To know one of the answers already in front of them is correct, as in receptive format, they will feel more confident to attempt the problem and determine how to make their mathematical process result in one of the answers provided.
To apply this to teaching, I would want to ask the same exact question in different formats (at different times). This may show that the student has a good understanding of part of a concept, and thus was able to answer the question correctly in the receptive format, but is actually still struggling with a more fundamental part of the concept.
Concluding the chapter is a helpful table outlining teaching strategies for struggling learners.
These include:
- Linking new concepts to previous knowledge.
- Builds confidence as the math is already learned
- Modeling using multiple modalities, (eg: visual, auditory, tactile)
- Gives you a better chance of reaching every type of learner
- Teaching strategies to help students learn independently, (eg. mnemonics or rhymes)
- Allows for future cuing by the teacher without giving the answer, promotes confidence through independent achievement
Thanks for the summary of this chapter Emily! I was thinking a lot about teaching tip #1, moving from Concrete material, to Representational, to Abstract and how struggling learners must progress to Abstract gradually. This made me reflect on a comment made by Paul Vicinus during the Core sessions this week. He made a funny statement that teaching K-2 graders "how to add" is much more difficult than teaching Calculus! Because you have to start with explaining the concept of combining sets to mean "adding" and this is hard for children to grasp. Once that becomes a skill, you move on. What I'm getting at is the breaking down into pieces in order to build up. I'm wondering if as teachers we will automatically know to do that. I can't see an educator throwing a kid on a bike with no training wheels and sending him/her onto the Rails to Trails the first time they are on a bike. I'm wondering if anybody else thought some of that teaching tip #1 was common sense even though it is definitely important and useful.
DeleteThanks Emily for a great chapter summary.
DeleteI am not going to pick out a specific strategy here but instead comment on the theme of developing independent thinking. The overall process reminds me a technique taught in business leadership. You assess where a person is on two scales, motivation and knowledge. Based on that analysis you start with very regimented approach, graduate (hopefully) to a more coaching process and end up with independence. The movies Twelve O'clock High and Hoosiers are often use to illustrate this technique. I have always structured my mentoring of employees this way and I wonder if, in the context of this chapter, if it can be adapted.
Chapter 5 – Common Learning Characteristics that Make Mathematics Difficult for Struggling Learners (Because Google is giving me trouble I cam going to have to publish this in two parts)
ReplyDeleteIn this chapter the authors describe eight different issues that are common to struggling math learners along with some strategies to address these issues. This chapter relates directly back to their universal Features model. I think these characteristics can be usefully divided into two sets of four. The first four relates to issues that arise because of previous instructional issues (Learned Helplessness, Passive Learning, Low Level of Academic Achievement and Math Anxiety). The second four related to clinical issues (Memory Difficulties, Attention Difficulties, Cognitive/Metacognitive Thinking Deficits, and Processing Deficits). The information below summarizes the main issues.
1) Learned Helplessness
Description - Characterized by students over dependence on the teacher for help, resistance to trying new strategies
Cause- Low rates of success in past situations, teachers focusing on only one way of solving problems.
Takeaways - Can impact learning because the student is unwilling to try problem strategies that may work best for them. Can impacts student’s ability to make connections
2) Passive Learning
Description - Characterized by avoidance, students tend to stay from activities that require exploration, that may result in a wrong answer or that require making connections with what they already know.
Cause- Similar to Learned Helpless this characteristic appears to be a result of past failures and the resulting consequences.
Takeaways - Can seriously impair ability to develop broader numbers sense. Using examples relevant to student s life/experience can help
3) Low Level of Academic Achievement
Description - Low level of past Achievement in both math and other subjects can impact current learning. Unless correctly identified, gaps in math knowledge can impact the student ability to learn the current concepts. Also gaps in language and other skills can impact the student’s ability to understand what is being communicated or asked in problems.
Cause Traditional methods of teaching basic math skills may not work for students with delayed numerical awareness.
Takeaways - Students may end up using cue words to solve problems which can be ineffective. Going back to skill development, which may have been the cause of the issue, may be ineffective. The Authors discuss focusing on the development of concepts and their connections to what the student knows.
4) Math Anxiety
Description - Anxiety resulting from repeated failures.
Cause- Often the result of the other issues discussed, math anxiety can intensify the impact of the other learning issues.
Takeaways – The Authors point out that success is the best remedy. From a teacher’s point of view, celebrating small success and breaking problems up into manageable parts can help.
5) Memory Difficulties
Description - Students have difficulty remembering basic math fundamentals, have problems with multistep problems that require multiple problem solving strategies and may have problems with generalizing from the concrete to the abstract.
Cause- While this problem may be one of information storage it may also come from how students store information and what strategies they employ to remember specific information.
Takeaways -It is important to distinguish between the two types of memory problems in devising strategies to deal with the issue. This issue may be characterized by an overreliance on cue/keywords in solving problems. This problem may also impact student’s ability to focus on the key features of a mathematical concept. The authors address mitigation strategies later in the book but suggest focusing on other types of cues (visual, auditory,…)
This chapter has some great points that difficulties in math can sometimes be a teacher's fault. I have had several students who don't want to participate in class, and when you ask why they say they don't want to look stupid. It's important that we can foster an environment where students understand that getting wrong answers is a part of the process in math.
DeleteI've also had students whose previous teachers hold their hands too much, and don't try to push them past the remembering/understanding levels. When they get to the next class, they struggle significantly when expectations are higher.
A good deal of my tutoring goes toward trying to undo some of these mindsets. It is highly rewarding though when students start to see that it's ok to make some mistakes, and they eventually build up some confidence about their math skills.
Chapter 5 – Common Learning Characteristics that Make Mathematics Difficult for Struggling Learners (Here is the second part)
ReplyDelete6) Cognitive/Meta-cognitive Thinking Deficits
Description - Difficulty in monitoring and communicating the learning process.
Cause- Really a form of communication deficit, an inability to monitor and repair breakdowns in learning/communications..
Takeaways - -The authors point out that students with this difficulty need to be explicitly taught how to be metacognitive learners. Strategies to address the problem include identifying information structure and self-monitoring.
7) Processing Deficits
Description - Difficulty with processing information inputs. Examples can be an inability to repeat back information or a much slower take-up rate of information.
Cause- This is not an issue a physical problems (i.e. poor eyesight) but rather with how the brain process information. It can relate to just one sense (visual) or multiple senses.
Takeaways - -My impression is that these can one of the most difficult issues to diagnose.
The authors recommend instruction in many different modes (visual, audible, tactile…) to help address.
8) Attention Difficulties
Description - Students miss important information and/or are unable to focus on the key features of a mathematical problem often picking out small or irrelevant details.
Cause- An inability to filter out stimulus. The authors put it very succinctly by saying “students with true attention deficits actually attend too much!”
Takeaways - Teaching students how to use self-cuing methods to identify important aspects of the problems and using non-verbal cues are strategies to address the issues.
Good summary of 8 important items Pennell. Regarding Passive Learning, where "students tend to stay from activities that require exploration, that may result in a wrong answer or that require making connections with what they already know" one strategy you listed advised "using examples relevant to students' life/experience can help." I was drawn to this characteristic that makes learning Math difficult because so many from our generation were products of passive learning environments. Based on our methods books so far and one full week of Core sessions, you can see you vastly different the classroom is today. I was surprised during field observations how much more "hands on" instruction has become, and how much differentiated instruction occurs. But mostly I am now keenly aware of how real-life examples in math problems and projects bring out the interest and enthusiasm of students. If you have heard of "MathTown," it is a project given to 6th graders to construct houses out of cardstock to a particular scale from a real blue print. This is done at Renbrook school in West Hartford (or Avon? It is on the line). The project results are amazing. Kids use proportion, measurement, ratios, etc. and work diligently in teams. Those who typically struggle are teamed with stronger students and/or select a simpler blueprint from which to work. I think projects like this are exciting and engaging. I hope to utilize similar ideas in my own classroom. Incidentally I had to make one of these home for my last class. You find yourself wanting to work on it on a Saturday night, as do the kids! Anything that excites kids to do Math on the weekend must be completely opposite of Passive Learning!
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ReplyDeleteThis is my post in response to Matt's summary of Chapter 4 which he emailed to the group -
ReplyDeleteThanks for the synopsis of Chapter 4 Matt. I really want to take a look at the self evaluation test in Appendix A to assess my own attitudes towards instruction. My thoughts on this chapter are mainly a reaction that many math teachers’ focus is knowledge, not pedagogy. I feel that at least with the ARC program we are taking great strides to learn instructional strategies, but given math shortages I can see why perhaps requirements leaned more towards math knowledge. Math historically has a bad “reputation” by not just kids but adults who “hated Math.” Hence anyone with the know-how of doing Math perhaps got a free pass on the pedagogy more than other subjects in the past. I’m glad this has changed. It is so important for kids who struggle with Math to learn it for their own survival skills and life skills. Today’s teachers must meet the needs of those kids who fall through the Math cracks.
Hi Jannine,
ReplyDeleteSounds like Chapter 12 is a great reference guide for technology uses that we might refer to when teaching especially if we encounter specific areas where a student is struggling. I will want to check out the WebQuest website, since the authentic context theme is prevalent in all our teaching books. My only experience using technology in Math has been (besides the graphing calculator) Geometer's Sketchpad and most recently MyMathLab on the Pearson website for an online Algebra class. I will want to review pages 210-11 with the guidelines on When and How to Use Technology because I don't want to use it in my classroom "just to use it" because it's cool. I want it to be purposeful and case specific. Another example of how much pre-planning we will be doing for lessons!