There is a moment that sticks out in my career that sparked my love of Number Talks. I was in maybe my fourth year of teaching and a student answered a difficult multiplication problem really quickly and I was amazed. I asked, ‘how did you get your answer so quickly?’ and as this student described how he got there, he truly opened my eyes to a new way of thinking about how to get to our answer. I grew up and took math courses before the Common Core Curriculum was widely put into place, however I also did not take the traditional math pathway of - Algebra 1, Geometry, Algebra 2, etc. My school taught Integrated Math where a lot of those concepts are blended as well as presented through a real life task and then broken down into smaller skills needed to complete the problem. While I feel that this helps me think about math in more creative ways I still had missed out on some really interesting approaches to numeracy skills. The Common Core Curriculum has a list of Mathematical Practices that hope to promote creative problem solving, make sense of problems and persevere in solving them, define and structure skills that students need to become mathematically proficient that push students to work hard at math. While parents sometimes complain that there are too many steps involved in some of these common core processes, the long-term benefits of showing students subtraction with rounding or multiplication with addition cannot be overstated. One could also argue that taking five or so minutes in my high school math classroom to talk about a simple subtraction problem is a waste of time with all the curriculum I have to teach, but I would argue that these are the skills I want my students to leave high school with perhaps more than memorizing the quadratic formula (however cool and handy that formula is). This fast multiplying student of mine had really changed my thinking about how to come to answers quickly and my love for talking about how you multiplied or how you subtracted became a staple in my classroom. Another really important concept in math that is both part of Common Core’s mathematical practices and National Council of Mathematics Teacher’s process standards is making mathematical connections. For a long time in my career I thought students would just make them. Seeing how the clues to graphing a quadratic are just staring at you in the equation, or finding patterns when looking at slope in a table vs. a graph vs. an equation, was so obvious to me but let me tell you, students don’t just see those things. A small thing I began doing is giving my students ‘helpful hints’ to guide them in the right direction. A major thing I did was rearrange the textbook provided curriculum to follow what I felt was a more ‘linear path’ that connected new big ideas back to smaller ideas we had previously discussed, and I would spell out that connection. Another thing that I feel like has really helped my students is to make them practice being ‘pattern seekers’. This way of thinking came from the amazing Jo Boaler and her Youcubed website (see video below) that has amazing videos to share with your students to give them pep talks concerning how mistakes are good, math is all about patterns, and how making connections is so powerful for brain growth! Rearranging curriculum was not easy, and again taking time away from that curriculum to show videos is hard, but doing this set up at the beginning of the year sets my students up for success for the rest of the year. They look for patterns all the time, they take more chances in class and on homework assignments, and because they understand that mistakes help our brain grow they are more likely to succeed overall because they are less anxious in class about getting the right answer quickly and instead think through the process more carefully. Resources:
Common Core State Standards Initiative (CCSSI). (n.d.). Standards for Mathematical Practice. Mathematics Standards | Common Core State Standards Initiative. http://www.corestandards.org/Math/Practice/ National Council of Teachers of Mathematics (NCTM). (2000). Principles and Standards for School Mathematics. The National Council of Teachers of Mathematics. Youcubed at Stanford. (2021, July 30). Math and Patterns [Video]. Youtube. https://youtu.be/DZ9kXRLvSZU Park, J. (2017). Common Core Infographic [Image]. Bearing News. https://bearingnews.org/293575/features/common-core-math-provides-crucial-problem-solving-skills/#
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The first few years of every teacher's career is mostly about surviving. Creating lessons, materials, resources, assessments is an overwhelming and daunting task that I remember all too well. By the time I was in my fourth or fifth year of teaching (after only 10 years they all start to blend together but I blame the fact that covid and teaching virtually felt like adding 10 more years onto my career) I realized that I needed to start making more connections between mathematical topics for my students to be able to fall back on or relate new math concepts to. So, I started the creating process all over again. This led me first down the path of ditching calculators in my class almost all together. I came up with the line ‘your Spanish teacher wouldn’t let you use a Spanish to English dictionary on your test, why should I let you use a calculator?’. I don’t know if that totally correlates but it felt right to me. I also wanted a better way to organize and present each mathematical concept to my students so next, I ditched my textbook and began creating Interactive Student Notebooks (ISN) with my students, essentially building our own math textbook. I have stuck to these two important notions each year that I have taught since:
Below is an example of my completed Geometry ISN from two school years ago. I love to have discussions with students about how they subtracted a number quickly or multiplied some really big numbers together. This allows students to gain a deeper knowledge of if their answer makes sense. I love giving students or discussing with students multiple ways to represent their thinking about how to solve a problem. Both of these are crucially important to ways I can bring a MKT, Mathematical Knowledge for Teaching into my classroom (Hill & Ball, 2009). By having our number talks in class to confirm that their answer makes sense we are tapping into our common content knowledge - making sense of answers in all classes is vitally important. We are also using our specialized mathematical knowledge by representing our answers or work in different ways. Skemp (1978) explained how important it is to push students to have a deep understanding of relationships between numbers, which helps me feel validated in my choices to push students to work harder to think in different ways, not use a calculator and sometimes dwell on a seemingly simple math problem for a while. He argues that using tricks and rules to help students get through math are detrimental to their true understanding of the beauty and interactivity of mathematics. However I am absolutely not perfect in this and I love a good math trick whenever I see it and it is difficult to avoid using them when there is so much content to cover in each of the courses I teach. Two big weaknesses that I have tried to tackle over the past few years are teaching logarithms and permutations and combinations. While I definitely feel I have developed in myself a deeper understanding of logarithms and can pass that onto my students I still struggle with how to apply logarithms to something that they can ground themselves in - some sort of real life situation that can make logarithms feel more real. Next I have to tackle my lack of confidence with counting really large numbers and gain a deeper understanding of overcounting and explaining differences in permutations and combinations. Resources:
Misura, M. (2019 August). Interactive Notebooks. [Image]. Hill, H., & Ball, D. L. (2009). [Image]. The curious - and crucial - case of Mathematical Knowledge for Teaching. Skemp, R. R. (1978). Relational understanding and instrumental understanding. The Arithmetic Teacher, 26(3), 9-15. |
AuthorMarissa McGregor, high school math teacher extraordinaire. I love my husband, daughter, and family dearly. Archives
August 2022
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