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Saturday, April 9, 2016

Shallow Versus Deep Math

Welcome to our second week of looking closely at math workshop.  Get more details about my math workshop book study here.  

Deep Versus Shallow Math

In this week's reading, I was struck by the difference between deep and shallow math.  Here are some characteristics of each type of math.

Shallow Math

- Memorizing algorithms
- Applying an algorithm (usually a word problem found on the bottom of a page full of practice for that algorithm.
- Hunt & copy exercises
- Plug and chug numbers
- Not considering what the numbers mean
- About covering the content
- Teacher gives out knowledge

Deep Math

- Engaging, exciting, exhausting & inspiring
- Pushes learners out of their comfort zone
- Mental models
- An understanding of a concept that can be built upon later
- Discourse
- Challenging tasks
- Students wrestling to make sense
- Content understanding
- Teacher as a facilitator of learning

When I was in elementary and middle school 99% of the math I did would be classified as shallow math.  I was the queen of the plug and chug.  I thrived on algorithms and hated "word problems".  When I was in high school, it was more of the same until I got to Algebra 2 and was faced with new and challenging problems that no one had "taught" me how to solve.  This took my enthusiasm for and understanding of math to an entirely new level.  Math class became exciting and invigorating and for the first time I got to invent my own strategies for solving problems and compare them to my classmates.  It was such a dramatic and marked change for me that it really is what sparked my interest in becoming a teacher.

Now when I teach math, I try my best to keep most of what I do with my students at the deep level.  Math workshop provides me with a vehicle for giving kids support solving challenging tasks.  

Your turn!  Can you think of anything that is missing from these lists of shallow and deep math?  Where did most of your own learning take place? Please respond in the comments below!

Come back next week for part 3 of our Minds on Mathematics book study! 


  1. I am an education major at UW-Oshkosh and really enjoyed reading this blog because I could relate to you as well. During my schooling, most of what I was taught in math was shallow math. By teaching lie this, students don't really get a deep understanding of what they are really learning about. They might get how to do the problem or solve an equation, but they really won't know what the reasoning to solving the problem is. As a future teacher, I need to make sure I am using the deep method of teaching math. It might be easier to use shallow, but teaching is to make sure your students get a deep understanding of the content you are teaching. If that means putting in a little extra effort, then that is fine. We need to do what is best for our students and what will help them learn the content with understanding.

  2. I got great grades in math, but I'd say it was pretty shallow. I was a rule follower and didn't even realize I could invent my own strategies to solve problems until recently and I'm 44 year old. I've been missing out. I hope I can change that for my students.

  3. I am also a education major at University of Wisconsin Oshkosh and can relate to your blog as well. All through elementary school and middle school, the math instruction I learned was very shallow and there was one right way to solve every problem. We did many problems out of the textbooks that we could check our answers in the back of. Problems were also laid out for us and pretty much spoon fed to us on how to solve them. I am in a math methods course right now and we have talked about how now with common core, students are learning addition and subtraction and some multiplication and division well before they learn the standard algorithm for them. I believe the standard algorithm should always come last in teaching math concepts. Teaching the strategies of math concepts and having students talk about the strategies they are using out loud to their classmates is super beneficial for all students. We can then push our students to try other classmates strategies and those students can then use those strategies if they work for them. Standard algorithms can be spoon fed to our students but we aren't helping our students understand the concept or how the concept works by doing that and we are not benefiting our students for the future.

  4. My name is Rachel Weisensel and I am also a student at the University of Wisconsin-Oshkosh, studying to become a teacher. I enjoyed reading this blog post a lot! The tables you made to compare shallow versus deep math really spell out how I learned math versus how I will teach math in the future. Until coming to college to learn how to teach math, there was not much, if any meaning behind the numbers I used in math. I was good at it though, so I liked it. While taking three math courses that were on how to teach algebra, geometry, and probability and statistics, I learned an amazing amount! Math was actually exciting because I learned HOW and WHY it worked. I've gone on to tutor all of those math courses because I know how challenging it can be for us to wrap our brains around learning math this way because we're used to the complete opposite (shallow math). Now I am in a math methods course learning more of the hows and whys, how to plan engaging and meaningful instruction, and how to assess and grade math concept understanding. When reading your blog post, I thought about a video we watched on the first day of this class. It was a collection of random interviews taken of a ton of people asking how they felt about math. People made comments about math like “I can’t understand it unless it relates to something physical,” “There’s no room for interpretation,” and “It’s repetitious and tedious.” If everyone could experience math on a deeper level, rather than just drill practice and memorization like these people probably encountered, their brains would rewire themselves to make them stronger at math. I wish that all students could have this experience with math so that they would be engaged and inventive like you talked about feeling after taking Algebra 2. Jo Boaler said in the video that there is no such thing as a math brain and that everyone can be a "math person." It is our job to convince students of this by letting them dive into math that is "engaging, exciting, exhausting & inspiring."

  5. Brilliant observations. I've been an enthusiast for deep and creative math for awhile now. It is refreshing to find other like minded math facilitators!

  6. Thank you for this insightful reminder that, too often, we lean toward "shallow" teaching by distributing algorithms and handing out answers instead of pushing students beyond their comfort zones and engaging them in challenging tasks. We often focus on the speed of learning as opposed to the depth of learning, and this post made it clear that investing time in "deep" mathematics instruction is far more beneficial for student learning! I love that you stated that math instruction should be "engaging, exciting, exhausting, and inspiring." What an incredible reminder that math is not meant to be quick and easy! Math is challenging and frustrating and tiring at times, but it is a wonderfully complex puzzle that students will make meaning from when given the time and freedom to explore and engage in discourse. This post was also a great reminder that our role, as teachers, is not to simply give out knowledge, but rather to facilitate learning and encourage students to become patient problem-solvers. Thank you for providing these insightful distinctions for teachers to learn from and reflect on.

  7. I was great at math until high school! Then I was introduced to geometry and proofs and theroms and struggled with both of them! I had a horrible experience with my geometry teacher and literally quit trying at math. I did not go to college and joined the Naval Reserves because I was not good at math! Now I look back and realize I had the shallow understanding of math and hated the word problems! Now I love teaching kids math...yes, I teach math (although my certification is 1-8 self contained with a reading specialty!). I live and breathe math! I am a math nerd (at least on the elementary side!). My kids love the word problems and how to solve them (or at least they let me think so!