I recently finished Jo Boaler’s new book ‘Mathematical Mindsets’ and was, as always, blown away by her work. I learned a lot from the book and as a result am going to create some add-ons to our lesson plans and videos. Firstly, if you haven’t already been using the differentiated tasks that are in pdf format with each lesson plan and video, try them out! Jo calls them ‘low floor, high ceiling tasks’, which simply means that students of ALL abilities can access and be challenged by the same task. These tasks replace the worksheets and endless textbook questions that focus primarily on procedural fluency and instead offer students a chance to: engage in problem solving, communicate about their understanding, develop conceptual understanding and search for patterns and generalizations as well practice procedures. Some teachers are skeptical as to how one question can be used for an entire block – but trust me, it can be done and the results are very positive as it achieves so much more than procedural practice. Furthermore if you haven’t tried using groups in math class, I highly recommend it!
Before I read ‘Mathematical Mindsets’ I read ‘Math Expressions’ by Cathy Marks Krpan, which is also a great resource as it focuses on communication in learning mathematics. Both books recommend students working in groups or teams, but in slightly different ways. Both also encourage spending the appropriate time ‘training’ the students how to communicate and collaborate effectively in groups. This is a way bigger challenge than the math! These are such important skills for our students to develop and in my experience, when the groups are able to collaborate and communicate effectively, magic happens! Their engagement is through the roof, they are reasoning mathematically, debating about different approaches and answers and digging really deep into the math. It also frees up the teacher to facilitate the groups as only ONE person in the group is allowed to ask for help from the teacher and that is only after they’ve ensured that no one in the group can help. Check out some of our grade 8 students’ reflections about their experiences with working in groups. These are responses to the following three questions: 1) What do you enjoy about solving problems in a group setting? 2) What are your personal challenges working in a group? 3) Explain whether you think you had a growth or fixed mindset in today’s class.
This approach completely changes the whole class dynamics, especially if you pair it with teaching about a growth mindset as we’ve been doing. Jo’s book focuses on developing a growth mindset in mathematics and discusses that how we teach as well as what we teach can lead to either growth or fixed mindsets. I have found that all the best teaching practices in the world are ineffective when dealing with a student with a fixed mindset- and now the science backs this up! A fixed mindset student’s brain actually functions differently than a growth mindset student’s brain. If you’ve never heard of these terms – I recommend watching this TED" class="redactor-linkify-object">https://www.ted.com/talks/carol_dweck_the_power_of... talk (it’s just over 10 minutes long and well worth the time).
One of the suggestions made in the book really resonated with me as I’ve used this technique a lot but not as consistently as I could be. The suggestion was to give the students a problem before teaching them the concept or any methods of solving it. For example: last week I was working with a grade 6 class that has been working solely in teams for 3 weeks. We were about to move them into the concept of multiplying with decimals – something they’ve never encountered before. We had one of those awesome teachable moments where a student actually gave us the problem. He wrote on the board 2.76 x 0.21 = 57.96. As soon as he wrote it, the class started murmuring about what the heck he was doing but then they looked at it and one student said “that can’t be true”, and I asked him why not and he responded “it’s common sense”, to which I replied “Ok, you’re using your intuition but explain what about it doesn’t make sense to you?”, he said “you can’t multiply 2 and a bit by a number less than one and end up with over 57”. We then spent the rest of the block getting out of their way and facilitating the discussion about this single question. The whole class was engaged. We’d stop and ask them to discuss in their groups the reasonableness of the answer. Then they started trying to solve using their own methods. We shared the methods on the board (many were on the right track but had incorrect place values). In their groups they discussed the merits of each method.
They were literally begging to be taught the method; what better way can you imagine for introducing a concept! I didn’t give any clue as to the correctness of their conjectures or methods, but rather kept asking them questions allowing them to make more conjectures and then test them out. This is what real mathematicians do! It was amazing! At the very end a student took out his calculator because he was dying to know the ‘real answer’ and when it came out as 0.5796 they were dumbfounded! They recognized that the digits were correct but they were so confused about where the wholes went! I, of course, didn’t explain it but simply stated that we’ll be exploring this concept more tomorrow with base 10 blocks and will solve the great mystery as to “what happened to the two wholes?”
Another example of the introductory problem is in a grade 8 class, we are about to start division of fractions and so I posed the question: Andrew is making gym bags for his friend and bought 1 4/5 meters of material that was on sale. If each bag requires 2/3 meter of material, how many bags can he make? I asked them to show their solution visually as well as to explain how they know their solution is reasonable and to explain what they did to solve and why. I haven’t taught them how to do this method. I want them to explore, thus exploring this concept, which is equal grouping. Essentially they need to understand that 1/45 ÷ 2/3 means how many 2/3 can I fit into 1/45? I want them to develop conceptual understanding rather than learning a procedure that they won’t know when to apply in a given context.
So, keep your eyes on the video pages to look for these introductory problems (many of our lessons already have these) that would allow students a chance to engage in thinking and perhaps even discovering the methods for themselves for the concepts. Research shows that students who have this opportunity are more successful than those who are just taught the methods/concepts without the introductory problem. From what I’ve seen so far- I believe it!
Educating Now was created due to teacher requests to have Nikki as their daily math coach. The site has lesson by lesson video tutorials for teachers to help them prep for their next math class and incorporate manipulatives, differentiated tasks, games and specific language into their class. Teachers who use the site can improve student engagement and understanding, in addition to saving prep time, by watching a 10 minute video tutorial and downloading a detailed lesson plan.