Teaching Numeracy, Critical Habits 5 & 6

If you’re just joining us, we’re reading and discussing Teaching Numeracy, 9 Critical Habits to Ignite Mathematical Thinking, by Margie Pearse and K. M. Walton.

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Habit 5: Predict, Infer, Recognize Trends, Use Patterns

“Inferring allows students in your math class to make their own discoveries without spoon-feeding the information to them. In this view, understanding cannot be delivered by instructors, no matter how skillful, but must be created by learners in their own minds.”

What I liked most about this chapter (and actually the one that follows) is that it conjures up the image of a classroom of thinkers. The quote above (p 55) sums up a major issue with mathematics instruction in the United States–we don’t allow students to struggle.

Some of the most powerful research I have read relates to the cultural differences in the way we teach math in the United States compared with Japan and Germany. In 1995, as part of the Third International Mathematics and Science Study (TIMSS), Dr. James Stigler studied the instructional techniques of teachers in Japan, Germany, and the United States and documented the findings in his book, The Teaching Gap. He found that teachers in Japan expected their students to be confused and struggle, while teachers in the United States felt that if their students were confused then they hadn’t been clear in their explanations. In other words, struggle is not part of our teaching culture in the United States. I recently heard Dr. Yeap Ban Har speak about math instruction in Singapore, and one piece of information he shared drew gasps from the room. In Singapore, 20% of their standardized assessments are problems of a type students have never experienced before. Now THAT’S problem-solving!

The authors make a great point on page 56 about how important it is that learning is relevant to students. Something as simple as using students’ names in story problems can help draw them into the learning. It’s not that difficult to rewrite word problems using the students’ interests and names, but it can have a big impact. I watched a masterful teacher in my school last year do a Problem of the Day written about one of her students at batting practice (she hit 90 balls, 27 went over the fence and the rest stayed in the field). The students were so engaged and the little girl felt like a star! Not only was it a great math problem, but this teacher also exposed her students to an activity (batting practice) that many were not familiar with.

I was reminded that the purpose of the book is to “ignite mathematical thinking” when I read the passage on page 62 about modeling metacognitive self-talk. It’s so important that we model this type of behavior for students, because it is a learned behavior. Just as reading teachers “think aloud” when they are modeling what good readers do, we should be consistently doing the same in math.


Habit 6: Question for Understanding

“To maximize problem-solving, application, and the development of a variety of thinking skills, it is vital that we pay more attention to improving our questioning in mathematics lessons.”

The research could not be any more clear on the topic of questioning in mathematics, and the authors did a great job of listing and categorizing it (p 69-71). I always appreciate it when a book connects and spirals ideas, so here we see how questioning supports the previously discussed skills of monitoring and repairing and metacognition. The questioning ideas on pages 71-73 are gold! I, for one, will definitely be copying this page and keeping it close by. Dr. Yeap modeled a couple of very powerful questioning techniques that I would like to add to the list. The first technique was asking, “Your friend says that 8 + 6 = 14, because it’s like 10 + 4. Do you agree?”. The other was so simple, yet so effective. After giving him a solution he asked, “Are you sure?”. Right or wrong, he asked the question over and over. Even as an adult who was, I should say, pretty confident with my answers, I immediately looked back and checked over my work. And isn’t that what we want our students to do? I’m anxious to see if it works for them as well as it did on me! Finally, I loved the I Learned/I Wonder two-column notes idea on page 77.  What a great way for students to summarize their thinking in their math journals!

So there’s my recap of the two chapters.  What spoke to you?

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  1. Habit 5: Predict, Infer, Recognize Trends, Use Patterns
    What spoke to me most in this section was the ideas about making inferences. It made me think of a young student I worked with over the last several years. This student has struggled in both math and literacy. I have sat at many team meetings where the literacy teachers have said how he doesn’t seem to be able to make inferences and has a hard time making sense of text. His math learning has been very slow and he has received much support in building his conceptual understanding. After reading this section and thinking about this student, I think he must struggle to make inferences in math as well. “If students don’t infer, they will not grasp the deeper essence of the mathematical concepts being introduced.” I know this student struggles with the deeper essence of the math and I am thinking he needs more work on making inferences! I will be using the tips from the second section of the chapter when I meet with him one on one this summer.

    Habit 6: Question for Understanding
    I think this is the most important habit yet! The difference between a good teacher and a great one can often be the quality of questions they ask. My school district focused on questioning several years ago and it made a huge difference in the quality of teaching and student understanding. The hardest thing for me when I was really looking at asking good questions was waiting before asking a kid to answer it. If you ask a good, deep question, the answer should require some thought. I started watching the clock to see how long I was waiting before calling on someone to give me the answer and it was a very short amount of time. Mostly under 15 seconds. I spent a year working on increasing wait time and asking better questions and it has made the biggest difference in my own teaching practice. I love the list of questions presented in this section as well. There were some questions there I had not thought of before and I am working on getting these into my lessons for this school year.

    Another thing that really struck me while reading these two sections is how well this book would work as part of a course on numeracy. There is such broad appeal to this and applications in such a wide variety of grades. Our school district offers course though the local university that are on site and open to teachers in the surrounding 2 districts. This would be a great book to use with a course on numeracy because it could include such a wide range of math teachers and would be a great place to get teachers of math together and talking about best practices. I think it would be valuable for teachers K-12. I would love to hear what others think about this idea!

    The Math Maniac

    1. Your words tell the story of how this book came to be.

      I literally would have spent $100 for a book that would help my students grow in their mathematical thinking. Sure, I could find plenty of books with terrific activities around a certain math topic. But, as much as I loved and bought so many books, I was beginning to feel limited by them. I needed a way to show my students HOW to think, not just what to think.

      My desire was for students to navigate through the unfamiliar using the critical thinking strategies necessary to fight through the unknown. And that took over 15 years of researching, interviewing, and experimenting.

      I am literally bursting with gratitude over your description of these chapters. You hit right to the heart of my wish for this work. Thank you!

    2. Your book really is fantastic! It is definitely in my top five of all time professional reading materials. I read a lot of professional books so that really says something about your ideas!

      The Math Maniac

    3. Tara, I totally agree with you about the importance of questioning. Effective questioning puts the learning in the hands of the students.

      I also agree with you about the importance of this book. 🙂

  2. As I read chapters 5 & 6 of the book, I realized that the strategies of predicting and inferring are mainly thought of in the area of reading but is also a critical component in mathematics. I am glad that the author covered those important points.

    Chapter 6 for me was my favorite. Like Tara (the Math Maniac), I believe that great questioning is at the heart of great instruction. Great questioning requires one to preplan with intent. As an educator I believe that there is something to be said about wait time. In reflecting, I realize that I should continue to work on wait time in my lesson delivery. This also connects to the following statement, “the individuals who are doing the talking are also the ones doing the learning.” Good questions require students to really think and the answer my require further exploration or research. I am beginning to realize that this is okay because the students are taking ownership of their own learning.

    This chapter also highlighted the importance of students asking questions. I really saw the benefit of this practice as I worked with a small group of 4th graders that needed additional support. I modeled the types of questions that they should ask and the students ran with it. In some of my sessions I would pause so that I could take in the learning that was taking place. I found that I learned as much from my students as I hoped to teach them.

    All of the question stems in this chapter are EXCELLENT. I especially enjoyed the questions that should be asked before, during and after problem solving. I would love to create an anchor chart using some of the questions from this chapter for the classroom (I would probably need the authors permission to do so!!).

    Anyhow, I can go on and on about this chapter but I will not. Just know that chapter 6 is a must read for all educators.

    Greg Coleman

    1. I would be honored if you would make an anchor chart using these questions. Thank you for thinking of it!

    2. I am very excited about including the anchor chart in my teaching. Thanks again!

      I am intentionally planning more anchor moments for my students. I now work with college kids and have found that, even with this age-group, taking the time to first anchor the learning, with a visual, provides that confidence they all need to move forward.

      I also use this strategy at the beginning of every semester to model and think aloud what it looks like and sounds like to be a tenacious person. It is often the tenacity piece that is a struggle for so many of my students.

    3. I found the information from Chapter 5 caused me to shift my thinking when working with students. First of all, I think it is essential that we learn to ” let go” of our thinking (as the teacher and not plan on one set way to show our students how to solve a problem) and we need to allow more time for our students to do the thinking on their own on how and why they solve their mathematics problems the way they do. This chapter gave me a lot of ideas on how I can refocus my lessons to provide more opportunities for students to identify and use patterns, then make predictions on what they “think” is the best way to solve the problem and then give them time to share their thinking out loud. Of course, an accurate mathematical answer is important, but the thinking process that occurs prior and during finding the answer is just as important. The authors offer a lot of insight into this during this and the in Chapter 6 as well. Jeanine K. Brizendine, Math Specialist

    4. Jeanine, I am constantly amazed at the strategies that students have and use. I feel like the learner sometimes as I listen to their thinking!

  3. Ok, I need to confess that I’m not quite done with Habit 6 yet. I got a little sidetracked with a suspenseful novel that I’m currently reading. 😉

    With that said, I can see myself have students make predictions in math before beginning the actual work. Asking for a prediction and then asking why they think that will help develop the metacognition process. I think that hearing each other vocalize their math thoughts will help internalize that practice to become a habit.

    As I’ve been reading the book, I keep thinking to myself “How can I give more students the opportunity to express their math thoughts?” I plan to show them how they can record their work using iPads. We have a small number to share across the grade level. We have a free app called Show Me that will import a photo and then draw, type, and record about the photo. I can see someone working with manipulatives, snapping a picture, and recording their thoughts or writing a number sentence to go along. These can then be shared in lots of different ways.

    I was excited to read the comments about the thick question samples in Habit 6. I mentioned before that I practiced writing out good questions for my reading groups. Now I have a jump start on thick questions for math! Sue

    1. I like your idea of using the iPads. This would be a great way for the kids to record their work, and to write about it. Thank you for sharing.

    2. Using the iPads will make learning so relevant to your students. It’s probably easier for them to express their thoughts with technology than with paper and pencil!

  4. Donna your thoughts about asking kids “are you sure?” got me thinking about a phrase I have been practising in reading instruction. I find that SO many kids look to me when they are reading after they have struggled through a difficult word. They want me to confirm for them whether they got it right or not. I have been practising saying, “Are you right?” I say that OFTEN, some times when they ARE right, and some times when they are not. I always thought the main purpose was to get kids to be confident with their own thinking. However, the discussion here helps me to see that this type of question (throwing it back in the child’s court) forces them to deepen their thinking. It throws them at first, and for a while they try to figure out whether I am sneakily trying to indicate that they got it wrong. Once they realize that I want them to really THINK about whether they got it right or not, and be able to explain WHY, it really changes the dynamic.

    I loved the list of questions in Habit 6. It is so helpful to have the words ready to use. I find that I need to practise saying certain phrases, questions, etc to make it a habit for me as a teacher. Having a list to use as a reference will be great!

    I also think it is fantastic to use those open questions (that may have more than one answer) to level the playing field in the classroom. When the kids begin to see the teacher as a coach – who may or may not know the answer, but is willing to help them figure it out – it is much less intimidating than a place where they feel the teacher knows all the answers.

    1. Karen, I love the idea of teacher as coach! How powerful.

      So important that we ask Are you sure? Are you right? and Why? regardless of the correctness of their solutions.

  5. Habit 5: I thought the opening quote by L. Cochran was powerful. “…he made me create it for myself. He gave me nothing, and it was more than any other teacher has ever dared to give me.” I think that this connects with what Donna mentioned above about how too often students are not given opportunities to struggle. Struggle in terms of grappling, exploring, and peeling away layers of meaning. I know from my own experience as soon as problems are no longer rote computation, things happen. Too often students prefer that rote computation because it is procedural: follow these steps and get the answer. But is that really thinking when you can do the steps with your eyes closed? Hmm..I don’t think so. I can’t wait to try the problem from pg 60: “Reading the World.” This might be a great way for me to anchor into the idea that just like in reading, we also make inferences in math. If you read the statements too quickly, you can miss important details and not get the gist of what is happening.

    Habit 6: So many great questions and things to think about in this chapter. I really like the idea of the questions stated in the above posts to make students reflect on their own responses and their thinking: Are you right? Are you sure? I agree these questions might throw some students at first because so often students look for affirmation. Once students get the affirmation unfortunately much of the thinking stops. I think these types of questions can definitely help students think twice about their answers. What also stuck out for me was on page 74 when it talks about having students come to the table with questions and having the students arrive at the understanding themselves. The opportunity for deeper thinking is missed when students are given a response or affirmation to their questions too quickly. The statement…when we deliver the answers too quickly, we are not giving them a chance to own the learning is definitely something to think about.

    1. You make a great point that students often prefer rote computation. It definitely requires less effort on their part and they get to experience being “right”. I think that speaks to the environment that is all too common in our mathematics classrooms. Surely we can change that!

  6. Habit 5 : Predict, Infer, Recognize Trends
    When reading this chapter, I was a bit embarrassed to realize that I had, over the past 13 years of teaching grade 5, gotten away from teaching the estimation lessons because I felt that no matter what I did the kids just didn’t get it. It seemed like a waste of a lesson because so many still didn’t see the unreasonableness of their estimated answers – mostly with quotients from dividing by a 2-digit number. Now, as I embark on my new position as Math Specialist K-5, I realize that their estimates were so off because all along their path they haven’t developed true number sense. I am on a mission to change that! It will be so wonderful for me to work with teachers and students from the spectrum of K-5 this year. I will also be teaching a class entitled Math in Our World to students beginning a teaching program at the local college. I will really be able to see the spectrum! I fully intend to emphasis this Habit 5 of predicting and inferring all across the gradelevels. I now see the power in it.

    Habit 6 Questioning for Understanding
    I am a huge fan of Elin Keene and her Mosaic of Thought which is all about reading strategies which look awfully similar to these Habits! I also went to a conference recently by a Math professor who related all this topics to the equivalent in the Reading world. For example, phonics as being the basis for reading being similar to the concept of number using visual cluster cards. For me, it has been wonderful to see how many connections there are between teaching reading and teaching math. For my entire professional career, I have experienced so much emphasis on reading instruction, it is exciting for me to see the same now being given to math instruction in our district and to see that it is not all new – in fact we have been doing these things all along in our reading instruction. I will definitely share the questions on pg 71-73 with my colleagues as I begin my year as the math specialist.

    I, too, think I come to the rescue of students too quickly because I want them to love math like I do. I don’t want them struggling and give up. One suggestion from Rick Wormeli that I love is that once a student answers a question, ask a random person in the class if they agree or disagree and to explain their reasoning. That way, everyone is on their toes and not just the one who answered the question. I look forward to reading more!
    Ann Elise

    1. Congratulations on your new position, Ann Elise! Like you, I love the connections between reading and math instruction. I think for primary teachers, it brings math into their comfort zone.

  7. Chapter 5
    I was thrilled when I read that true meaning comes when students wrestle with what they see and what they know about what they are seeing, and then learn something new from that process. I know that struggling with a concept leads to learning for me. I enjoy challenging students to think about a topic prior to teaching the topic. It confirmed my strategy when I read this chapter.

    Our theme this year is “Look for the Evidence”. This theme will cover many subject areas and will hopefully encourage students to seek information to add to their own thinking and then draw conclusions, make inferences, and thus develop their generalizations. I like the ideas of using riddles and short mysteries to get students to dig deep for clues and then to come to their own conclusions. Students love to work together on puzzles. I plan to purchase the One-Minute Mysteries and Brain Teasers: Good Clean Puzzles for Kids of All Ages and Two-Minutes Mysteries to use in several classrooms this year.

    Teaching metacognitive self-talk is something that sparked my interest. We all need to be reminded that students need to hear what we are thinking. Good conversation of our own struggles is comforting and educational for all students of all ages.

    The “garbage bag: activity reminded me of something that I used to do in my classroom to begin a new year. I collected things from local antique shops to create a family that lived years ago. Students had to identify the tools that were in the bag and decide what they may have been used for during the time period. Pictures of the family added to the excitement. This activity provided great discussion and thinking. Students then brought their own garbage bag to introduce themselves. We all got to know each other while thinking about each other’s memories.

    Chapter 6
    Hyde opens this chapter with a great question himself. “Why ask questions in mathematics?” His answer – To help with understanding, to construct meaning, to discover new information, to clarify what is going on, to check inferences, and to facilitate visual representations. This is a great reminder to take the time to inquire. The chapter does a nice job of reminding all of us of the importance of planning the questions. Because we are all strapped for time, the list of questions will be a nice resource to give to teachers. I plan to create a poster to be a tool that teachers can keep next to their plan books.

    I liked the point that Harvey and Goudvis make – If you ask questions as you learn, you are awake. Amen! Questions indicate thinking. Many times students feel that questions mean they are stupid. I tell them, quite the opposite. They show we are thinking.

    I like the idea of students coming to the classroom with questions. This would be a valuable homework assignment. They would have some idea of the content AND will be ready to interact with the discussion. These questions could be turned into a chapter review activity as well. I like to do an activity that places one student in the hot seat ( a chair in front of a chalkboard or smartboard). Their back faces the board. The students in the class open their books and think of questions that are a review of the chapter. The student in the hot seat gets 3 clues to come up with the answer. Now the student that gave the winning clue, takes the hot seat and more questioning and clues begins. This reminded me of the hot seat activity mentioned in the chapter. Finally, sticky notes are always one of my favorite question tools.

    1. Thank you for sharing so many valuable thoughts, Ellen! You make a great point about students feeling that asking questions is something to be avoided. I think that goes right back to developing a classroom culture that values struggle and questioning. So important!

  8. I’m a little late to this party, Donna, but I’m loving this book study! I started asking, “Are you sure?” this past year and it was amazing! Thanks for the reminder to continue with it this upcoming year.

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