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Saturday, April 2, 2016

Understanding Takes Time

I know math workshop is for me!  Why?  Because I share these beliefs:
1) Students Are Capable of Brilliance
2) Understanding Takes Time
3) There is More Than One Way

Welcome to part 1 of our Minds-On Math Workshop bookstudy.  Here are some of my thoughts from this week! 

Students Are Capable of Brilliance

     My best teaching friend and Kindergarten teacher extraordinaire has this as her mantra.  Her students constantly outperform other Kindergarten students in the district and she is always being asked to share her secret.  Her #1 reason her kids do so well is because she holds them to very high standards.  She truly believes that all kids can learn and in many ways their teacher's attitude about their learning becomes a self-fulfilling prophecy.  Her students learn because she believes they can.  ALL OF THEM. 
     Every time I feel like giving up on a kid and just "Teaching him how to do it" (aka arithmetic without understanding) I remember my friend and how beliefs become her students.  All students can learn and we need to keep expectations high for our students.

Understanding Takes Time

     You can "teach" your students the standard algorithm for subtraction in ten minutes and have them practice it for an hour.  It will look like your students understand subtraction.  Next week or next month you will give them three subtraction problems and they will tell  you that they forgot how to subtract.   Worse yet, they might not tell you that.  They might keep missing a step in the procedure or do a step wrong repeatedly.  Now they are in a position where they think they know how to subtract and they have no idea all their answers are wrong. They don't know how to tell if an answer is reasonable because you didn't "teach" them how.
     Alternatively, you can spend an hour per day for three weeks guiding your students to develop flexible and efficient strategies and giving them opportunities to share these ideas with their classmates.  They will also have a chance to hear their classmates ideas and compare how they are similar or different from their own.  In the process of doing this, they will strengthen their understanding of addition, place value, estimation and inverse relationships.  Next week or next month, you will give them three subtraction problems.  They will solve them all mentally applying a strategy that is efficient for the numbers in each problem.  They may or may not use the same strategy for all three problems.
     The example above illustrates the difference between telling and guiding.  With telling, you are doing most of the talking and learning.  Sure it is faster but your students will lack understanding.  Guiding students to develop and refine strategies takes much more time upfront.  The students do most of the talking, and you get through fewer problems during a class period.   You might in fact spend 20 minutes talking about one problem.  It takes time but it also develops understanding.  Understanding is what will be there for your students next week and next month.  For me, it is worth the investment of time to produce understanding.

There is More Than One Way

     This is one I learned from my students.  There is one student in particular who I will always remember helping me see a new way to look at subtraction.  I was taught that there was one way to solve each math problem and it has taken me years of teaching and learning to undo that thought.  "With faith that each child, given time, has an innate ability to reason out a solution to a problem, even if their initial approach and strategy may differ from how we believe things "ought" to be done, we can begin to turn over the responsibility for learning mathematics to our students." This quote from the book really resonated this change for me and helped me see how my thinking has changed since I started my teaching career.  I know embrace multiple strategies and love that my second graders can currently solve a problem like 17-9 using six different strategies.  Most of them are very efficient and none of them involve counting! 

Do you share these beliefs?  How has your own experience in the classroom enhanced or changed your beliefs? Other thoughts about this weeks reading?  Leave your thoughts below in the comments! 


  1. I think the journey to becoming a better math teacher in the past 5 or so years has a lot to do with believing and practicing these three things. I'm still learning.

  2. Hello! I have been reading some of your posts and have gained ideas I can use in my future classroom! I am currently in school to obtain my teaching license and I found this particular post to be very true and relatable. I believe that all students are capable of brilliance by learning in their own ways. It is our job, as teachers, to set those high expectations and teach to each student's learning needs. I also agree that understanding takes time. When I was in elementary school, I was taught the standard algorithm and that was it. Students now have the opportunity to understand the process of any operation. This understanding comes from the use of multiple strategies. I believe that by building students' tool boxes with several mathematical strategies, students are given the chance to understand the reasoning behind the problems they are solving. There is more than one way to solve a problem and it is definitely worth the time to teach it! I enjoyed reading this post! Thank you so much for sharing!

  3. Hi! I have been reading a lot of your posts and I have enjoyed learning new ways to engage students in math. There are a lot of good ideas in ways to make math interactive and fun for students. I am a student working towards getting a degree in special education and elementary education. I really liked this specific post because I am currently in a math methods class and I think that the three beliefs that you have listed in this post coincide with the themes I am taking away from the math methods class. I think it is important for teachers to remember that their students, no matter what their math ability is, are capable of succeeding in math. I also think it is important to teach students to understand math and not just learn the process. It will take longer to teach understanding but that will help students in the long run. Finally, I have learned this semester in my math methods class and I was reminded in this post that their are many ways to be successful in math. Students use different strategies to get to the same answer. I think it is important for teachers to keep their mind open when students use different strategies and try and learn how they are processing the information. Thank you for sharing! I will be sure to follow this blog more frequently to continue my learning!

  4. Hi, I was reading through your blog as an assignment for my math methods course, and I was very pleased with your posts. One in specific that stood out to me was your section titled "There is More Than One Way." An insight I have taken from my class was how important it is as future educators to understand this and have it reflect in our classrooms. I really enjoyed the quote you posted as well, and that made me think even more about this topic. I believe that teachers too often use only one way to teach a math concept, and this does not benefit their students whatsoever. Many students learn differently, and they should be given as many options as possible that will maximize their learning and success!

  5. Hello! I really enjoyed reading your blog and I gained lots of new knowledge that I can apply to my teaching. Currently, I am attending a university and going for a degree in Special Education and Elementary Education. I can relate to your blog because I am learning about different math strategies to use in the classroom INSTEAD of the standard algorithm. I agree 100% with your Kindergarten teacher friend that ALL students are capable of learning and can learn in many ways. We as teachers should not settle by just teaching our students how to do a math problem. We need to actually teach our students so they understand the material. Like I previously mentioned, I have recently learned multiple new efficient strategies to add, subtract, multiply and divide. The new strategies I have learned are a lot easier to do and I actually understand why they work. When I was in elementary school I was only taught the standard algorithm and now that I know all of these new strategies I wish I would have learned them when I was younger! I strongly agree with you and think we should not be restricting our students to only one strategy (that they will most likely forget the next day). By showing students multiple strategies, they are able to pick which strategy they like and understand the best. I really liked your post about why it is important to give students more time to learn different strategies. Spending 30 minutes explaining and discussing one problem, is way more effective than spending 10 minutes "teaching" students how to solve a problem. I also agree that it is very powerful for our students to be able to share and compare their answers with their peers. That way they are learning from the teacher and their peers. It takes time for students to learn these strategies, but it also helps develop understanding. It is worth the amount of time it takes for students to actually understand the material. Thank you so much for sharing your knowledge with me! I greatly appreciate it.

  6. Hi there,

    I absolutely love your stressed importance for taking time for each student to understanding the process and not to simply go through the procedure. When teachers teach students procedures, the students may succeed at solving the procedure and finding the correct answer, but later in their math career, they will most likely struggle with an understanding of advanced math problems. It amazes me on how many different ways students learn math through the use of different math strategies. We have included our students in a math number talk, where students are encouraged to explain how they found their answer to a given problem. This not only increases their understanding, but it also provides classmates to hear different math strategies that they may use in the future to solve similar problems. Overall, it is of high importance. to encourage inventive strategies of students to increase the likelihood of students understanding the process.

    Thanks for the post, I hope these ideas continue to spread to other educators!

    Heather Thiel

  7. Thanks for sharing your great ideas and beliefs! I too am currently in my teacher preparation program and am learning a lot in my current methods courses about the most effective techniques to use in the classroom. In response to your first belief I completely agree that all students are capable of learning and holding each and every one of your students to high standards will increase their own math confidence. From my experiences thus far, I think sometimes teachers forget to hold similar high standards for those students who may be at lower levels however how does that give those students the opportunity to reach maximum success? Comparable to your other two beliefs, in our math methods course we have been forced to think about math problems in a completely different way than what we learned throughout our K-12 years. So much of our experience in math class has been revolved around the standard algorithm and memorizing math facts. Although sometimes challenging, working through math problems in this new way has really given me insight into how students should actually be taught math. Your idea about telling vs guided instruction also reminded me of work done in our methods course in which we are given the problem, asked to use our own strategies to solve it, and then we also are given the chance to share our strategies with our peers. By doing it this way we are forced to put real work and understanding into what we do, making math more meangingful. Thanks for all of your great ideas and I hope to take some of these same beliefs into my future classroom!

  8. I enjoyed reading your reflections on the quality versus quantity of instruction here. I completely agree that although there are some quick and "easy" ways to teach students a math concept using plug-and-chug methods and meaningless algorithms, these strategies will not benefit students (or teachers) in the long-run. Instead, I appreciated your insight into the idea that investing more time and energy into one topic and teaching toward understanding is far more beneficial for students; yes, it takes far more time, but students gain a rich understanding of the content that can only come from exploration, discussion, collaboration, and time. Instead of pushing toward fitting in the maximum amount of content in the shortest amount of time, I see the value in what you explained about digging deep for longer periods of time in order to establish a foundation that students can build upon for weeks, months, and years to come.