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Teaching Numeracy, Critical Habits 1 & 2

Wow! Are we off to a great start or what? What amazing comments you all contributed to our discussion of the Preface and Introduction last week! Each week, I’ll provide links to all previous posts, so any time you happen to wander across the book study you can jump right in.

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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  Let’s dive right in and look at the first two critical habits!

Reading Schedule

Habit 1: Monitor and Repair Understanding

“When we monitor our comprehension in literacy, we pay attention to whether or not we are understanding what we are reading. The same must happen in a mathematics classroom.”Raise your hand if this is what you see happening in your mathematics classroom. What? I don’t see any hands raised, and it’s not just because I can’t actually see you  Ha ha. How could something so simple and common sense have eluded mathematics instruction for so long? I know I hear language arts teachers reference “fix-it” strategies all the time, and anchor charts outlining them are a staple in language arts classrooms. When I saw the list of mathematics “fix-up tools” on pages 11 and 12, I immediately saw the power of teaching these strategies to our mathematicians. With Margie’s permission, I created a Math Fix-Up Tools poster you can download here. There are two versions–one that would be great for a poster and another, smaller B&W one students can glue in their math journals. Just remember it’s not enough to just put up a poster. You have to model, model, model!

Another thing I loved in this chapter was Margie’s narrative about the Working Answer Keys (p 15) and how she used them to improve the homework routine in her classroom. This past year I helped a 3rd-grade teacher implement a new homework process that allowed students to discuss their homework with another mathematician (student) in the class. The teacher then chose one mathematician to share his or her work on the document camera. The mathematical discussions the kiddos had were incredible! We also saw an improvement in the homework completion rate, because students who didn’t do their homework didn’t get to share in the mathematical discussions, and they did not like that at all. I could see that adding the Working Answer Keys would make the process even more effective.

Habit 2: Develop Schema and Activate Background Knowledge

“When a student feels that math is too hard or they can’t do it or they don’t understand it or everyone else gets it but them, you’ve lost him or her. Letting students consciously identify what they already know increases confidence and engagement.”

As soon as the authors made a connection between activating prior knowledge and making mathematics accessible to all students, they had me hooked, and I began looking at it from a whole new perspective. I know that in my own classroom, it was often “…a luxury to be considered only if time allows” (p 20) and not something that I consciously built into every lesson. So I guess I proved their point! Connecting the idea of developing background knowledge to something I’m already passionate about definitely made the learning more meaningful to me. I will be placing more emphasis on building background knowledge from here on out.

I noticed that the Key Ideas (p 20-22) included references to the concrete, representational, abstract (CRA) sequence of instruction, and I was positively giddy to see the authors state that manipulatives are for all grade levels, not just the primary grades. When we rush students through the concrete and pictorial stages, or skip them altogether, we do them a huge disservice. One argument for not using manipulatives more often is lack of time, but how much time does it take to remediate students with no conceptual understanding?

There are so many great ideas in these two chapters, and I can’t wait to read the rich discussion that is certain to take place.  Remember, this is an interactive discussion!  Feel free to participate either by posting your own response to the reading or by replying to the comments of others.

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  1. Excellent points and thank you for sharing the student copy for their journals/INB! Enjoy the conference!

  2. Holy smokes! There is so much good stuff to think about and I will! This is perhaps the BEST professional development ever on math!

    1. I love, love, love the updated version of my “Math Fix-up Strategies.” Thank you, Donna!

  3. I am a strong advocate both for the use of manipulatives for all grade levels and all ability levels as well as providing answer keys for students. I have found that over the past few years I have adjusted my philosophy about classwork/homework in my fifth grade classroom. I feel like the formative assessments are not only a chance for me to check in with each child to see where they are but also a chance for the student to become aware of whether they are successfully understanding the concept rather than the “gotcha” grading oppportunity. I ran a blended flipped classroom this year with 1-1 iPads and I found the classroom was revolutionized. Kids had ownership of their own learning and it made a huge difference. When in small guided math groups with me, I was able to provide immediate guidance and deep questioning so that I knew how each child was doing. When students were not in group, they were on their iPads using wonderful apps like Front Row (which has virtual manipulatives for the students to use), Learnzillion (teaching videos), and Thinking Blocks (awesome app for the visual method of solving some of the Common Core word problem formats. Students were working collaboratively as well as checking their answers along the way on any written work using answer keys. Students would go grab a bag of manipulatives whenever they needed them rather than on the one day in the past I may have taken them out. The idea of background knowledge is huge with so many topics in math. We are also at each of our gradelevels building the foundation for future grade levels as well. Thanks for the wonderful fix-it poster, Donna!! So looking forward to reading more!
    Ann Elise

  4. Habit 1: Monitor and Repair Understanding
    What most sticks with me from the section about habit 1 is this quote: “If a student is stuck, do not tell her what is wrong, as hard as that may be. Arthur Hyde says that interventions are best done with questions that help a student or group rethink and repair on their own. “ This can be one of the most challenging things for teachers to do but the rewards are huge! I know during my own college experience when I had a super challenging problem in a class and was working with classmates to get it done, the best teachers were ones who would listen to our problems and ask a question or be a sounding board without giving an answer. I strive to do this with my own students but is certainly requires me to do more listening than talking.
    Another thing from this section I love is the examples of how estimation leads to understanding rather than kids memorizing a series of tricks. “Catch phrases are quick fixes without the reasoning. An estimate will automatically bring in reasoning and meaning. “ I find this a lot when teaching things like decimal operations. Kids really don’t need to memorize rules about where the decimal point should go if they have good estimation skills!

    Habit 2: Develop Scheme and Activate Background Knowledge
    I liked the ideas in this section about working through the concrete, pictorial and representational stages of learning. I love using manipulatives in the classroom and have found them so powerful for kids to connect to what they previously know and build new understanding. I find that older kids still like using manipulatives and love using things like algebra tiles with my sixth graders. I also use the National Library of Virtual Manipulatives (nlvm.usu.edu) quite a bit.

    The Math Maniac

    1. Hmmm…I would love to hear some opinions on the pros and cons of concrete manipulatives vs. virtual manipulatives. Is there a difference? Are today’s students changing how they learn with all the virtual options available? Sue

    2. I teach kindergarten, and I was very hesitant to replace the hands on materials for my students with virtual activities. I initially felt that it was not making the learning more powerful. However, after adapting and trying some things, I definitely am of the mind that BOTH are important. Kids need to see, hold, and literally manipulate the objects, but they also must be able to view the same concepts virtually. And of course, some students will connect better with one or the other method.

      In another note, as my school team was reflecting on our standardized assessment scores, we had noted that many students struggled with geometry and spatial tasks on the test. For example, questions where they had to look at an object and consider what it would look like from another angle were very difficult for them on paper, even if they could do it with real objects in their hands. So using the technology to give them exposure to moving and viewing objects as they may look on a flat surface (similar to the page) has the potential to be very helpful.

    3. I am a huge proponent of using manipulatives at all ages, but I also do trainings on integrating technology based manipulatives. So, I am a fan of using both. The world kids live in now, they need that experience with the technology…but there is just something to say about having the real thing in your hands. Recently, I learned about a tool called Osmo for your iPad that lets kids do both. As soon as I watched the video they have on their site, I bought one (but they are backordered until October)…very cool concept. The kids have the physical manipulatives but Osmo lets them interact with the tools through games on the iPad. Amazing, if you want to check them out, here is their site: https://www.playosmo.com/?r=5719088525475840

    4. Osmo looks absolutely incredible! Do you have one already? I think I’m going on the back order list. Sue

  5. These are my two big take-aways from this week’s reading-

    Developing a bank of processing questions to ask students. I jotted down three while I read Habit 1. What does this remind you of? Will the answer be more or less (bigger or smaller)? What should you do first? A few years ago I spent a lot of time writing comprehension questions for reading in my lesson plans until I felt I could ask deeper questions off the top of my head. I think I’ll add that goal for math to my own pd this year.

    I guess I’m a good case study for activating prior knowledge because the connections to reading strategies really hit home. I want to frame my instruction as more of the Think Alouds I do daily during reading instruction. Asking specific scaffolding questions when a student is stuck. That will be much more meaningful to me as the teacher. I will have a better understanding of where the student is getting off track. Our first grade team talked about the need to better develop our math vocabulary with the students. I’m thinking right now of a personal “math wall” as a section in their math notebooks this year.

    Thanks for the posters Donna! I’ve been reading a lot this week about interactive math notebooks. I’m looking forward to adding notebooks to my math instruction this year. Sue

    1. I was also really struck by the link with reading strategies. It was such a powerful connection for ME to make. We often talk about using consistent language with students (across classrooms, grade levels, etc). I think using the consistent strategy language across the curriculum is just screaming to be done, although it never occurred to me! And, like you, thinking of it that way will help me to wrap my mind around teaching it.

  6. It is a thrill to spend the summer with such passionate, forward-thinking educators. I feel truly blessed!

    After 25 years in the K-8 setting, I am now in higher education. At first, I was hesitant to pull out the manipulatives in college, assuming my students were too old for them. Boy was I wrong! Without peeling the layers back, my students weren’t able to identify their misconceptions. They just knew they couldn’t get it and moved on. But, when I allowed time to concretely wrestle with the concepts using manipulatives, my students really began to blossom in their mathematical thinking. It just goes to show you; good teaching is good teaching.

    1. Margie, Your story about college level students and manipulatives reminds me of a Title 1 conference I attended a few years back. The keynote speaker was a dynamic guy who taught math at the college level. He gave us a story problem involving cows and chickens and heads and legs. I happen to be sitting at a table full of administrators who were all talking about the formulas they might use to figure it out. None of them did though. I quietly started drawing cows and chickens. It took a few tries, but using pictures I figured out the right combination. Sue

    2. A year ago I enrolled my four year old daughter into a Montessori classroom. When we had originally toured, they showed us many of the math materials the children would be able to explore. One of the teachers was explaining to me the way they used the tools to lay the foundations for the students, and I found myself learning math all over again. As a student I was always good at math (I could use the formulas and get the right answers), but I think I still don’t fully understand the concrete representations behind some of the things I was “working out.” By looking at and working with the manipulatives that my little one was going to use, I started to actually build a deeper understanding. I definitely agree that this is so important for ALL ages.

    3. In the last year I’ve learned just how important using manipulatives is. I have very few in my classroom and I am looking to purchase more. What are your suggestions for “must haves” to help my in getting manipulatives for my classroom?

  7. First a big thank you, Donna, for introducing me to this book! I have so many sticky notes in the book already! There are so many takeaways. Here are few of my initial thoughts.

    As I was reading Habit 1, the idea of mathematicians having inner conversations resonated with me (pg 9). I definitely think this is something I am going to explore with my students. I think by making the explicit connection to reading might be a way to make this more “visible” for my students. One activity I definitely plan on incorporating is the Measurement Scavenger Hunt (pg 13). Measurement sense is not something that comes easy to students. I think the way this activity was presented might be a way to engage students and help them to develop a stronger sense of measurement. It also will help students see math as part of their environment. I have used something similar to Working Answer Keys (pg 15) with my middle school students. It was amazing to see students become self-reflective. When students found an answer that was wrong, students were asked to not only reflect on the process but also asked to do what we called a complementary problem to ensure that they understood and could transfer the skill. Working Answer Keys put the responsibility on the students, and that is key!

    For Habit 2, I also noted the reference to the concrete-representational-abstract sequence of instruction and immediately thought of the posts Donna has here on her blog that are always so enlightening. It has become a habit of mind when planning instruction whether I am working with elementary or middle school students. Sometimes it is a challenge, but definitely exciting to see the progression of learning by students. The Half Snack activity (pg 24) is something I want to further explore with students. This would work perfectly as a follow up to the book, Give Me Half! by Stuart J. Murphy.

  8. Thank you Donna for the great posters!

    I am currently thinking about how to display anchor charts linking the reading and math strategies in my classrooms. I am thinking about showing “Visualizing” with examples of how it is useful in reading, writing, and math. “Connecting,” etc

    One of my big take aways from this reading was the importance of giving students time to discuss with one another their thinking, and defend and adapt their thinking. I have always skipped past estimation (I think because I never liked to do it myself. I always disliked rough drafts in writing as well.) However, thinking about having students estimate and then DEFEND or EXPLAIN their estimation makes so much sense. This forces them to really think about it, rather than guess and wait to be told if they are right or wrong.

    I really loved the idea of using cloze activities for this type of discussion as well. I may sound like a broken record, but again I think of what I have made such a priority in reading, and how to do that with math as well. Partner discussions are a daily routine in our reading workshop, and I see the importance of building that into our math lessons as well. I thought I was using partner work, as I would have students work on solving problems together, but I think what was really happening is the “stronger” partner solved the problems while the other student sat by and watched. By creating more open challenges and by making the conversation their task, it requires both partners to have a richer discussion. “The critical part of this experience lies in the discussion.”

    I also really love the idea of having students develop formulas themselves. Again, so meaningful!

  9. Oh and Donna, I will also be in Vegas this week for the TPT conference (flying from Australia – I may be crazy!), so I hope to be able to meet you in person. Hope you had a good flight!


  10. I wanted to share a tool I have been using lately to promote collaboration and metacognition in math. It is called “Chalkboard Splash.” First, present a problem to the class in small groups. Have groups solve using any strategy on sticky chart paper. Make sure groups are including their process and an explanation of that process. Then, each group sticks their finished work up on the whiteboard simultaneously. At that point you can give students a 3-column “chalkboard splash debriefing form” (headings are similar, different, suprises) and have them brainstorm what they notice is similar/different/surprises about the strategies used. Wow! It really provides some meaningful conversations, especially with teacher-prompted questions. There is always something magical when we challenge students to identify the patterns they notice in the math. I love using this to activate and build background knowledge.

    1. Love this idea!!! I can’t wait to try it with students. Thank you so much for sharing.

    2. Great idea! I am doing a workshop on fractions later this week and will be trying it out.

  11. Hello from Vegas! What a treat to check in on the post and see all the wonderfully thoughtful dialogue going on! You guys rock!

  12. Reading through this week, I was particularly intrigued by the Working Answer Keys. As I was reading about them, Dan Meyer’s video about Khan Academy doing Angry Birds, popped into my head and so I went back and watched it again to gather my thoughts about why it came to mind. It is a very short video, so I encourage you to check it out (http://blog.mrmeyer.com/2012/khan-academy-does-angry-birds/) but here are two important pictures (well, I can’t add pictures so I’ll type out what is on the picture) and a bit of quoted text:

    Angry Birds
    1. Play.
    2. Experiment.
    3. Get feedback.
    4. Get an explanation.
    5. Learn.

    Khan Academy
    1. Get an explanation.
    2. Play.
    3. Experiment.
    4. Get feedback.
    5. Learn.

    “I’m not saying lectures and explanations are never necessary in math and science — or in Angry Birds, for that matter. When I couldn’t get past that one really tricky level, I went online and found a walkthrough. But the walkthrough — the explanation — wasn’t the first thing I did when I experienced Angry Birds. So why does Khan Academy make an explanation the very first thing a student experiences with a new topic in math. When we put the explanation first, we get lousy learning and bored students.”

    I am interested in trying out the Working Answer Keys, but my fear is that it will become like Dan’s list for Khan Academy instead of his list for Angry Birds. I want the students to see math tasks like they see Angry Birds: just jump in and play with the math, experiment and try out solution paths, get feedback by looking at their calculated answer versus their estimated answer as well as my Working Answer Keys, and then get an explanation from their peers and me in class, thus building their learning. But my fear is that the students will see it more like Khan Academy: get an explanation from the Working Answer Sheet, play around with the math to see if they can figure out what I did to get the answer, experiment a little to see if they can discover a different solution path than mine, get feedback based on whether or not they got the same solution path as me, and learn that all math is just about doing the same steps as the teacher.

    Now, I’m being a bit dramatic here for effect…the authors do caution us that there needs to be a lot of discussion, oversight, and modeling when you first start using Working Answer Sheets. So, for those of us that plan on trying it out I think we need to contemplate a lot more about how it will play out and be watching for those kids who are using it like Khan Academy instead of like Angry Birds.

  13. I agree. In fact, since the release of the book, I added a bit of a twist to it. If a student came up with the same answer, but used a different strategy, they could submit it to me on a “strategy addendum document” (a funny form I created asking for clearly written steps). If he/she could explain it to me and better yet, the class, then the addendum would be added to future answer keys with his/her name included.

    The real power in using a working answer key for me was the transference of responsibility for the learning from me to the students. Of course, that took a lot of modeling at first. I literally did a think aloud and pretended I was home doing homework while my students noticed what it looked like and sounded like to do get the most out of practicing math. Whenever necessary, I drew a snowstorm on the whiteboard, with a house and front steps. I challenged my students to never want to get left out in the cold when it comes to understanding. Meaning must be actively pursued. Now, my drawing was terrible, and they giggled every time I repeated it, but the message came through loud and clear. (I hope!)

    On a side note, creating these working keys forced me to complete the homework I assigned each night. Yikes! Doing my own homework was a real eye-opener at times. It was my best lesson in how to create more worthwhile homework.

    Thank you so much for the Khan Academy clip! Love it!

    1. I do love the fact that the Working Keys will force us to do the homework first. I am thinking that I might try showing multiple solution paths for every problem, for a few reasons. First, it might deter the students from trying to ‘figure out what the teacher did’ because there will be so much there for each problem. Second, there won’t be ‘the way the teacher did it’ because I want my students to solve the problem the way that makes since to them and not feel like if they didn’t do it my way that it is wrong (which your idea helps solve that as well). Third, for a kid who is struggling on a problem, if I only have one solution path in the Working Key they might not understand or connect to that path…but if there are multiple paths, hopefully one of them will connect with them.

      Thank you for participating in the book study with us, your insight is invaluable. And you are welcome for the clip…it is one of my favorites.

    2. As I read about the working answers keys (new concept for me), I thought about how helpful they would be for parents also. A big problem that we have found in our district as we move to the common core standards is that parent have no idea how to help their kids with their homework. We had parent meetings each quarter and we sent home a booklet with examples for each standard and how to solve problems but I like the idea of doing it each night with their homework.

      I also love the idea of including an addendum for alternate ways to solve the problem. I learned so much from my students this last year by having them show and explain how they solved a problem. Many times it was different from my way but it worked and it makes their learning more meaningful.

  14. I found some great ideas in these two chapters. I also found confirmation that I am doing the right thing in my professional development. I liked the discussion about inner conversation with oneself. Allington brought out the fact that this is something children do not automatically do – we need to teach it. Students must THINK…do I understand this???

    I loved the Mystery Staff Person idea and hope to implement this as this school year begins. I love to do things that allow the students to know staff in fun ways. Posting the measurements will create a sense of family while creating great number sense.

    The topic of using friendly numbers has been around, but it is really a great way to get students thinking about numbers and strengthening their mental math ability. Too often students are willing to accept an answer without thinking about it logically. Friendly numbers help students KNOW when a number is not reasonable.

    The discussion about assimilation vs accommodation reminds us to prepare our lessons for all students. Creating backbone knowledge means letting students struggle BUT giving them the tools to make the understanding attainable. Using manipulatives, providing opportunities for discussion, creating representations of their learning are all necessary to develop strong mathematicians.

  15. One thing that stood out for me was the importance of doing “think alouds” with my students. I often skip this step due to lack of time and this section made me realize that students need to “see” how I think so that I can help them learn to think. I also like the idea of teaching my students to ask questions throughout the problem. I think this will help to keep them on track.

    Making connections and building background knowledge are crucial to our students’ learning. “We need to provide lots of opportunities for students to either activate or build their background knowledge” (Kindle Location 537). After participating in the Intel Math program last year I realized how important it is for students to use manipulatives and pictures when working math problems, What I used to think took to much time has now become an important part of my teaching.

  16. I have already learned so much just from these first two habits, I don’t even know where to start. I can say that at first as a kindergarten teacher I wasn’t sure if this book study was going to be beneficial to me, but wow, I see that I was wrong and I am glad to be apart of this.

    The first thing that really caught me was the idea of repairing our understanding. I have tried in the past to not just give the students the answer when they can’t figure it out, but now I see that when they are struggling I need to have a conversation with them to see if they can figure out where they went wrong. I am excited to use the “fix-up” tools, which I know I will have to model. I loved the early elementary ideas that were in the book, made me want to try them right now!

    I also loved Margie’s story about the math key that was sent home with her son. I think back to my own experience with math when I was in high school. If this had been an option for me, I think that I would have been more successful in math.

    I also thought that the number surgery was a cute idea and a good one to do before we start addition.

    My head is swimming with all kinds of good ideas, I can’t wait to read more.

  17. I love the ‘Fix-up Tools’ Posters and will definitely use them next year. My students are young (JK & SK) but they still apply. I am going to tape them to all my educator math clipboards so we can use them as prompts when observing and interacting with students. I will also post a larger one in the class for educators and older students to refer to.
    Habit 1: Monitor and Repair Understanding
    This past year our school focus was on Reasoning and Proving in mathematics. This started because so many students never considered how reasonable their answers were. The reading states that students should begin with estimating and then to try repairing. I highlighted the quote: “that inner voice should begin with a reasonable answer”. I also really like the plan: “Have students record their predictions and reasons behind their thinking before they start to calculate. Give students an opportunity to talk about their estimates, thus allowing them to take responsibility for their learning.” In this chapter it became very apparant to me that I am going to love this book. In many professional books I read I find it very frustrating when you are inundated with theory but leave with very little idea of how to implement. The end of the chapter (and every chapter) gives a number of great ideas on what you can ‘do tomorrow’ to help develop this habit.

    Habit 2: Develop Schema and Activate Background Knowledge
    “Letting students consciously identify what they already know increases confidence and engagement”. In my mind I am often struggling with the very different backgrounds my students come from. They have very different experiences and levels of knowledge. I had never thought of this in numeracy before reading this chapter. It is suggested that teachers “craft every lesson so that it provides equal access to the learning for all students, regardless of how weak or strong their schema may be.” At first this seems a bit daunting but after reading the key ideas and lesson suggestions it is more practical. The best part is thatI am already doing most of these things anyways it is just a new way to look at them.

    As I am reading these habits I and trying to think of a practical way I can focus on them this upcoming year. I realize that the goal is to use all of them all the time but to start that way would be very overwhelming. Maybe in September I can start but really focusing on habit 1 and then each month adding a habit. As a new one is added it is not that the first is dropped it but is a practical way to get my mind around things and integrate the focus in my classroom. Just a thought.

    1. All the new learning can feel overwhelming, and I agree with your strategic approach to implementation. I think that’s why the How Can I Use This in My Math Class…Tomorrow section of each chapter is so helpful. By choosing several of the activities and incorporating them into your instruction, you can bring in several habits. I love the way the activities are tied to the habits!

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