Have you looked into the mathematical practice standards? These are standards that students should master to become mathematical thinkers. They are skills student’s can use to access mathematical content in a way that makes deep conceptual sense to them.

Here are the practices students should be practicing each day:

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Traditionally, these practices have not been emphasized in our math classrooms. It seems easier and less time consuming to just tell students steps and drill them to practice. But research has shown that students do not retain knowledge this way, they develop a bad attitude towards math (that has permeated all of society at this point), and they never develop the skills that mathematicians/scientists use to discover new things, solve big problems, and think critically.

But if we have accepted the fact that students need to be practicing these skills, where do we start?

With discussion! Students should be speaking about problems, suggesting possible strategies and solutions, and refining their arguments. Through this process, students come into contact with the practices! And along the way, they pick up content as well. Content they access through deep discussion, problem-solving, and reflection is much more likely to stick and be used in future learning. Plus students are much more engaged when involved in a challenge or problem than reciting material or copying steps that are hard to remember. A problem-centered classroom banks on students’ natural curiosity and humans’ natural desire to solve a puzzle (just look at the profitability of escape rooms!) The human brain is so amazing because it has great capacity to make connections and reason through new experiences.

One way I love to engage students in mathematical discussion with me and more importantly, each other, is using pictures. Students don’t often think about how mathematical ideas are all around them and don’t look deeply to find ways to apply problem-solving to their every day lives. Pictures are low-stakes, not intimidating, highly usually engaging, and bring out students’ creativity.

Armed with some good questioning and discussion techniques and a single photo, you can have some of your deepest mathematical experiences in the classroom. All you need to do is show a photo and some simple questions.

The important part of this process is that every notice is validated, every thought is valued, and every avenue is explored. One of the reason so many students are intimidated by math class, and turn into adults intimidated by math, is because it has long been the “right answer” club. If you don’t have the capacity to remember the steps, recite the answer, or regurgitate the information because you don’t have any true understanding, you feel ostracized.

The beauty with this process and the math practices is students can start anywhere in the process and they can always get better.

The only “rule” I give my students is to try to incorporate math into the discussion/picture. At first I tell them to try this with numbers, but it is important to know that math concepts don’t always use our number system. More, less, length, width, comparisons, ratios, etc. are all ways students can talk about math without necessarily assigning numbers. But if students are apprehensive or need a starting point, dealing with numbers can help. I also tell them to either find a problem, create a problem, tell a story, or turn the picture into a question. Students love doing this and they naturally get better at it, especially with their peers to work off of.

Once you have found a good central question from your picture, your students can get to work solving. The great thing about the math pictures is that many good questions can come out so you can revisit pictures again and again, especially as students refine their skills and learn to ask deeper questions or create more complex problems. Students should be working together and trying many things in their quest to solve the central question. The teacher should not be the wise sage that holds the answer - all students can obtain a solution they all agree on and can defend together. Students should have access to manipulatives, drawing tools, talking time, and anything else they need to go on this journey. And teachers should be asking questions, furthering students curiosity and encouraging them to try new things.

If students come to a central question that needs some starting point (usually a number) you have some options. You can either have the students come up with a number they think is probably right, give the students a number you know they can work with, or figure it out (if possible). For example, if students want to solve how much money is in some stacks of cash in a photo and they can see there are 17 stacks but don’t know how much is in each stack, it may be impossible to count that stack (it’s just a photo!). You can encourage students to estimate (a very important skill) and go with that number or give them a number (they’ll never know!). I always like to open it up as much as possible so I usually tell my kids if they literally cannot solve an amount, they should estimate and go with it! This could mean different groups of students will be working with different numbers and amounts - but that is great! The beauty if in the thinking process, not the finish line. That isn’t to say students should just come to a wrong conclusion, but through questions, refinement, and reflection students should be able to accurately defend their final solutions and strategies. If they cannot, they will most likely realize they missed something and will go back to work again. The right answer is the rightfully justified and defended argument, not the number or “answer”.

Pictures are a great place to start if you’ve never run a problem-centered classroom before. They are also great for students who have never experienced a problem-centered classroom before (wonderful back to school activity to spark discussion and show students how math class will run!).

That is why I have compiled all my favorite photos I use for Math in Pictures with my 3rd graders. Each photo has the 3 central questions, the important little hint, and a page for students to record. We use them digitally so students type and annotate on an iPad but they can, of course, be printed out or just displayed. The beauty is the discussion and thinking that will fill the classroom! So, if you are interested, click below and happy mathing!