Tag Archives: How to Learn Math

Learning On My Own

Since I’ve been out of the classroom, I’ve been on my own to seek out professional development. As a classroom teacher, I had numerous district-mandated PD sessions before school started every year, not to mention in-service days throughout the school year. My district also offered summer PD for a couple of weeks every July, and because I’m a big nerd when it comes to learning more about teaching, I always took advantage of what they offered me.

Nowadays I don’t have a district planning and offering PD to me on a regular basis, so I have to do a bit more legwork to make it happen. However, the benefit is that I tend to seek out and find things I’m personally interested in rather than doing something dictated by my district or principal. (Not that I disliked what they offered, mind you, but there is something be said for making your own PD choices.)

Last year, for example, I took @joboaler’s online course How To Learn Math. Considering I have 8 years of experience teaching math, I felt silly telling my co-workers that I was taking a course with that title. However, the course was fantastic and I’m glad I took part. As a foster parent, it was particularly beneficial because it helped me think more about how I’m talking math with a kiddo every single day and how different that is from talking math with a class of students for about 60 minutes per day, 5 days per week, for only 36 weeks a year.

Right now I’m taking a course called Teaching Middle School Math with Sketchpad. I was kind of hoping to take the elementary school version of the course because I have an elementary school teaching background, but unfortunately there weren’t enough people signed up for that course to make. I’m actually happy I’m taking the middle school version because I think the elementary one would have been too much reaffirming of things I already know, whereas the middle school course is making me re-examine content that I took in school but I haven’t personally taught to kids.

Basically, I get to learn all the cool ways to teach these concepts and feel jealous that way back when I was in middle school, I was taught a very traditional approach where everything was step-by-step with little to no exploration. I won’t go so far as to say kids have it easy these days, but they sure do have a lot of tools available to make it interesting work!

In addition to learning how Sketchpad can help students explore math concepts in interactive, visual ways, I’m also learning how to apply it to my job. As I mentioned yesterday, I’m leading a team that is converting grade 4 and 5 curriculum into our new Digital Teaching Platform. Unfortunately, as part of the upgrade to the new platform, we did change some functionality, and I have had to periodically do some problem solving to ensure that lessons aren’t adversely affected.

Sketchpad can’t solve all of my problems, but it has come in handy already. For example, in a lesson on multiplying by 10 and 100, the original lesson included two Excel spreadsheets that acted as calculation machines. Students could use a slider to change the first factor while the second was locked at 10 or 100. They explored changing the first factor and watching what happened to the product. In our new DTP, we don’t want to have to send students out to a program like Excel, so I had to think of a way to recreate these calculating machines in another way.

We just so happen to have an applet in our content generation studio that lets us embed interactive sketches within lesson screens. What perfect timing that I’m in the middle of this 6-week Sketchpad course! I’m no pro at Sketchpad, but I was pretty proud of myself that in about 15 minutes I was able to create a sketch that exactly mimics the functionality of the Excel file from the original lesson.

Well, that’s a lie. It was more like 30 minutes. I encountered a problem with the slider I made. Despite everything *looking* like it worked, the value of the first factor multiplied by 10 or 100 was not giving me the correct product. (I can only imagine the conclusions students would draw if they used this version of my sketch.) To make a long story short, and because it really is hard to explain with no pictures, I called my co-worker Meredith and together we brainstormed and figured out that Sketchpad has an ability to truncate values which solved my problem. Yay!

I’m not sure I’ll have the time to become a Sketchpad expert, but I am happy to have it in my toolbox now. As new design challenges arise in my job, even if I can’t personally make something I want, I’m learning enough about the program to know if it could provide the right solution and to be able to talk to an expert to get it made.

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How to Learn Math Part 4: Teaching for a Growth Mindset

I’m halfway through Jo Boaler’s online course “How to Learn Math”. Normally I would devour a course like this as quickly as I can, but as a foster parent to a 3-month old baby, I find that I don’t have as much free time as I used to. (What a surprise.) In this session in particular, I could tell I didn’t have as much energy as I wrote my reflections, so I’m not quite as proud of my work. Nonetheless, I’m determined to finish the second half of the course by the end of September. It’s been such a valuable learning experience, and I don’t want to miss anything.

So in session 4, we returned to the idea of a growth mindset. Whereas in the earlier session we compared and contrasted fixed and growth mindsets, this session focused specifically on activities and actions we can take to help develop a growth mindset in our students/children. Here are my reflections from this session:

The lesson opened with a clip from a classroom. A teacher poses the problem 1 ÷ 2/3. There’s no context or story, just a bare naked math problem. She asks the students to solve the problem in a way that makes sense to them. How long does she spend on this computation problem? 15 minutes. That’s right, she keeps her class engaged and thinking for 15 minutes. It was enjoyable to watch.

After the video was over, we were asked to discuss what the teacher did to support her students’ learning:

She held off confirming if anyone was correct or incorrect. Every answer was considered valid, the students just had to explain why it made sense to them. Her classroom discussion showed that it is okay to show your work in a variety of ways, as long as it makes sense to you. In this problem we saw a circle, a number line, and an equation as three different representations of the solution. She never made anyone feel like they had done something bad because they made a mistake. Even though three people had come up and shown that 1 1/2 is the answer, she still had someone who thought it was 6 come up to the board to show their work. This actually helped some students solidify their belief that the answer was 1 1/2 because they realized that 6 was too large of an answer.

I liked this video clip especially because it acknowledged that there is an algorithm that can be used to solve the problem, but the teacher showed that what she valued more was understanding why you were doing what you were doing, whether it was using the algorithm or some other method. What she has done is create a classroom culture that values sense making, as messy and dirty as that may be. I’m sure it’s scary for many teachers because they feel the pressure to teach all the content before the test, but as Jo Boaler points out, students who have been taught to problem solve rather than answer endless test questions actually performed very well on the dreaded standardized assessments.

Next we reviewed an activity from Fawn Nguyen’s blog. I’ve been following Fawn on Twitter (@fawnpnguyen) for the past year, so I couldn’t help but smile when I saw that I was being directed to her blog. I admire her work, and I’m thrilled that everyone in the course got a little taste of it. You can (and should!) check out Fawn’s activity by clicking the link above. We were asked what ideas this task gave us about what goes into a quality math task.

Quality tasks are open-ended. There were some basic constraints that everyone had to work within, but the students were able to personalize the problem. It also wasn’t clear what would be needed in order to solve the problem. Just like in real-life problems, the students needed to analyze it and figure out what tools, strategies, and skills were going to be needed.

After analyzing a quality task, we had to actually do a task and analyze it. We were shown the following image:

Image of 3 different groups of stacked blocks. The first group has, from left to right, 1 block, 2 blocks, and 1 block. The second group has, from left to right, 1 block, 2 blocks, 3 blocks, 2 blocks, and 1 block. The third group has, from left to right, 1 block, 2 blocks, 3 blocks, 4 blocks, 3 blocks, 2 blocks, 1 block.

and asked these questions:

How do you see this shape growing?

How many cubes are in case 100?

In every case, the outside towers are kept. The inside tower is duplicated and a new inside tower is added whose height is one greater than the previous inside tower. In case 100 there is going to be a series of towers on the left that are 1 + 2 + 3 + 4 + 5… + 98 + 99. This same sum will appear on the right. The middle tower will be 100. So the total number of cubes is double the sum of all numbers 1-99 and then add 100 to that amount.

Then we had to analyze the task using a Growth Mindset Task Framework that Jo Boaler presented.

1. Openness

The question asking how this shape is growing is a very open task. There isn’t just one correct answer. The question about the 100th case is more of a closed task because there is one correct answer.

2. Different ways of seeing

In the growth question we saw two different descriptions and both were different from the description I gave, but they all described the growth in a way that was happening. In the 100th case question, there are different ways of seeing. Some students may use/see a quadratic equation, but it is not required. I saw it as the sum of the numbers from 1-99 doubled plus 100. I think this is the problem some teachers have, especially in middle and high school, where they get what skill a problem is meant to utilize so they automatically jump to that when solving it. As an elementary school teacher, I didn’t think about quadratics. I talked about it in terms of addition.

3. Multiple entry points

In the growth problem, someone could start by drawing more cases or they might start by getting out some blocks and making models of the cases. Others may not even make any more. They might just analyze the ones they are given. In the 100th case question, there is less variety in entry points. You need to get from case 3 to case 100. Chances are students will need to start solving more cases. Some might continue solving until they get to case 100 while others might stop to look for a pattern to save them the work of solving to case 100.

4. Multiple paths/strategies

I think this is related to the previous item. When solving the growth problem, there are lots of different ways to describe how the pattern is growing. Students can use words, pictures, real-world objects, and/or numbers to make their explanations. In the 100th case problem there are multiple strategies as well. You could solve every case up to 100. Or you could solve some, look for a pattern, and try to generate a rule to help you find case 100 without having to solve all of the ones up to that case.

5. Clear learning goals and opportunities for feedback

The growth problem seems like it’s trying to teach me about how to analyze and describe growth patterns. The 100th case problem is trying to get me to show how to find a specific case, but it isn’t clear that it really wants me to generate a rule so that I don’t actually find all the cases from 1 to 100. Some students might just see that they are being given a big number, not that they are purposefully being given a big number to discourage them from finding every case from 1 to 100. Since there is more personal expression involved in describing the growth pattern, it seems like that problem has more opportunities for feedback.

Now that we had analyzed a task that fosters a growth mindset, we were asked to take a closed task that encourages fixed mindset thinking and revise it so that it becomes a growth-mindset task.

I’d take a problem that says so-and-so has a recipe for something. The students are given the recipe and told they need to double/triple it. How much of x ingredient will they use? To make it more open I would tell the students to each find a recipe in a cookbook or online. Then they have to determine how much of each ingredient in the recipe would be needed to make enough of the recipe so each person in class gets one serving.

Later we learned about assessments for learning. This is an interesting idea I wish I had been able to explore more while I was teaching. We had a choice of three tasks to review. I reviewed one geared towards 6th grade students. The problem has to do with optimizing the location of a security camera in a shop. We were asked to note what features we saw in the task that would support our work as teachers and how the task would support a growth mindset.

The structure of the assessment is a resource. First, students are given a chance to try out the task on their own. Then the teacher reviews each students’ work and provides guiding questions, but there is no grade. On another day, the students have time to reflect on their work and the questions the teacher asked them in the feedback before attempting a final, group solution. The assessment ends with reflection for students to think about their learning from this experience.

The sample questions for the teacher is an excellent resource. Not all teachers are going to know about the common mistakes students will make, so not only does this activity provide a list of those, but also a list of accompany questions to help students who are making one of those mistakes. This is great modeling for a teacher so they can ask similar questions in the future when they are conferring with students on other tasks. The assessment supports a growth mindset because the activity is set over three time periods. It’s not a “done in one” assessment. Students wouldn’t even think they were being assessed. Instead they are revisiting an activity and attempting to grow and improve every time they interact with it.

I like the emphasis on reflection and growth. When students first get the assignment back, they are asked to reflect on the questions they were given to think about how they could improve their response. There is no judgment that what they did is right or wrong. I also like the focus on the idea that the work they do with their group is for the purpose of creating something better than any one of them could have made individually. Finally, I liked the reflection as they compare their work to sample student work, and then their final reflections. Both of these reflections help students see how they have grown through the course of this activity.

After analyzing activities, we learned about the harmful effects of tracking/grouping students by ability level. We were asked to reflect on why we think tracking results in lower achievement for students.

First, it leads to fixed mindset thinking – “I’m dumb” or “I’m smart” – and either way that hampers achievement. Second, once students are tracked, teachers claim that students can move up to a higher group if/when they’re ready, but the trouble is that the higher groups have continued moving at a faster pace, so the students in lower groups will always have a gap between them and the higher groups. They’re stuck! I saw this quite frequently as a teacher in elementary school. Once students were identified as needing academic intervention, they always needed academic intervention from then on.

It also keeps students from encountering different points of view that can help everyone grow and achieve more. The assumption is that lower level students can’t handle the same math as higher level students. However, I’ve personally witnessed a mixed ability classroom all work on the same activity and learn a lot together. It was amazing because the “low” kids actually provided more thoughtful explanations of their work than the “high” kids because it was truly a challenge for them. They had to think and reason, and the end product for them was great learning. The “high” kids were not as challenged and so their solutions and explanations weren’t as interesting. However, they got to learn from listening to the “low” kids share their correct and thoughtful solutions to the problem.

The session ended asking us to design something we will do to foster a growth mindset in our students/children.

As a foster parent, I am taking the information about the growth mindset to heart. I have children who can come to me at a variety of ages with a variety of backgrounds. The last thing they need to feel is that anything about them is “fixed” or “stuck”. They have the ability to grow intellectually and emotionally.

One thing I can do with these children to help them develop a growth mindset is create a lifebook together. Once they arrive at our home, we can start documenting their life through words and pictures (theirs and ours). By revisiting the book together regularly, we can talk about the ways they have grown and changed since they arrived. The growth won’t be something they’ll have to “trust” me about. They’ll have the lifebook as a tangible reminder of who they have been at every step of their journey, and they can identify in exactly what ways they have grown.

How to Learn Math Part 3: Mistakes and Persistence

In session 3 of “How to Learn Math” we learned about the importance of making mistakes and the importance of developing persistence.

If you have heard the news about the “controversy” that the new Common Core State Standards will give students credit for saying 3 x 4 = 11, and you are worried, then this session is right up your alley! If you haven’t heard about this controversy, let me assure you it’s a completely manufactured controversy and total crap. My advice is to avoid reading anything about it. I especially implore you to avoid watching any news shows where they talk to “experts” for their reactions. Utter crap.

Anyway, back to another awesome session in this FREE online course. (I’m excited I’m getting such great information at no cost to me. I love it.)

The session started by talking about how when we make mistakes we are actually putting ourselves in a great position to learn. Here’s my first reflection:

It makes sense that greater learning happens when students make mistakes. This puts them in a state of disequilibrium, a great place to be in for learning.

The questions it leaves me with are how do the students realize they’ve made a mistake? Do they have to realize the mistake for themselves? Or is it as powerful if someone tells them they’ve made a mistake? Also, once the mistake is out in the open, what are they or someone else doing to help the student learn from the mistake? Basically what I’m getting at is what are the actions a teacher and/or student should do upon discovery of a mistake to take advantage of the student’s brain being at a prime spot for learning.

It also makes sense that people with growth mindsets would benefit greatest from this learning. A person with a fixed mindset would see a mistake as a reflection on their natural ability to learn math, which cannot change. A person with a growth mindset would see a mistake as a sign that this is an area where they can learn and grow from.

Next, Carol Dweck spoke about the importance of making mistakes and we were asked to jot down a list of key ideas we heard.

1. Everyone can get better at math when they work on it.

2. It is very important to show students the progress they have made.

3. Show students how to use the feedback they’re given to become better at problems like the one they are solving and at math in general.

4. When things are easy and you get everything right, you shouldn’t feel good. You probably weren’t learning or growing. Challenge should be the new comfort zone.

5. Mistakes are our friend. They are a natural part of learning. Often they give us clues about what we’ve done wrong, what we need to learn, and what we need to do next.

6. Teaching a growth mindset is very important for math achievement because it’s a subject where lots of students have a fixed mindset and have difficulty. When these students encounter problems that require effort, they think they’re dumb.

7. Students need to learn that they can grow their math brain through hard work, practice, and good mentoring.

Later we had to respond to a quote from the New York Times columnist Peter Sims. The part of his quote that stuck out to me was about being willing to be misunderstood, despite the conventional wisdom, possibly for long periods of time.

The idea that struck me most was the one about being okay with being misunderstood. I’ve had several experiences in my career where I have been known to go along to the beat of my own drum. I’m not doing it to be contrary, but because I feel strongly about doing things a certain way that I believe is best for my students. Just because other people disagree with me doesn’t mean that I should stop what I am doing.

When I taught 4th grade at my last school before taking a job as a curriculum developer, I did not fit in with my team because my teaching methods were so different from theirs. They thought I wasn’t a team player or that I was doing it to be put on a pedestal, when really I was just using the methods that research said would be good for my students. I tried sharing them with my team, but they didn’t want to hear it. At one point I even asked my principal if it would be better if I gave up what I was doing in order to follow my team to create a better team atmosphere, but she advised me not to. She said she would rather I keep doing what I was doing, which she thought was great for my kids, and hopefully eventually my team would see the value. It was difficult to do, and it didn’t endear me to my team, but I believed it was important to keep teaching the way that I was despite it not fitting with the “conventional wisdom” of my team.

We also responded to a quote from a mathematician, Laurent Schwartz. His quote speaks about how someone’s ability to answer questions quickly does not correlate to their intelligence in a subject. Rather, the important thing is being able to deeply understand things and their relation to one another.

The part about being called out as an imposter resonates with me. I felt that I was a fraud starting in algebra and up through my second semester of college calculus. I was able to follow along in lectures and to repeat the procedures I learned to earn As on my assignments, but I never really felt like I understood what I was doing. For example, in algebra, I knew that the equations we were working with were related to the way a graph looked, but because I couldn’t tell right away myself, and other people could, I thought I was secretly bad at math. This continued through the years. As we rushed through the curriculum, I could tell that I wasn’t fully developing an understanding of what we were doing and why. My ability to regurgitate steps helped me succeed, but that just made me feel worse about possibly being found out. Looking back, I think I had the capacity to do well AND understand what I was doing, but I probably needed more time to make sense of it for myself.

At the very end of the lesson we had to design a motivational poster related to what we learned in the session. I wasn’t feeling particularly artsy, so I only made up the slogan:

Mistakes are moments for learning.

Make a mistake today – Show your friends!

How to Learn Math Part 2: Mindsets

In the second session of “How to Learn Math”, we learned about mindsets: fixed vs. growth. Unfortunately, our society and teaching methods often lead students to develop fixed mindsets with regards to learning math.

If you want to know more about mindsets yourself, check out Carol Dweck’s book  Mindset: The New Psychology of Success, or if you only have about 4 minutes 32 seconds, you can watch this video.

Here are some of my reflections from this session:

Question: Where do you think kids hear negative messages about math?

Unfortunately, they get it from adults in their lives. Either a parent shares their own stories about how they were bad at math, too, when they were a kid, or they might even hear about it from a teacher. I was at an end-of-year awards ceremony for 5th graders and my jaw dropped when the teacher announcing the awards for best math students said, “The next award is for one of those subjects you either love or hate. Most of the students hate it. The award for best math student goes to….” (True story!)

Question: If schools took on mindset evidence seriously, what would they need to change?

Giving out awards for grades would need to change. At the end of every grading period, when they are handed out, it gives the message that these are the “smart” kids in every subject, but there isn’t necessarily a correlation between their grades and the effort expended. I was one of those kids who got all As on my report card and the pressure mounted every grading period because I was afraid I would eventually let everyone down by getting a B in something. I focused less on learning and understanding and more on doing what I had to do to maintain my As, including cheating on assignments. If schools can focus instead on effort, and maybe even do away with awards assemblies altogether, students will start to see that everyone struggles at one time or another. I would go so far as to say that grades should be done away with because they are too abstract in relation to the learning students are doing on a daily basis.

Question: How has the information in this session changed your thinking about your own ability in any way?

It made me think about how I had a fixed mindset as a child. I thought I was naturally good at school. I didn’t feel like I had to put in that much effort and I still got all As. The trouble is that I had to maintain those grades for fear of letting my parents down. If I could do it before, why couldn’t I do it now? As a result, I was one of those students who avoided taking risks. I chose the easiest path, and I got upset if anything was challenging. I know now that I have the capacity to learn and change based on the effort I put into my work. I wish I had that mindset all along. Who knows what I would have done differently in my life as a result.

How to Learn Math Part 1

Back in July I started a free online course offered by Jo Boaler at Stanford. The course is called “How to Learn Math”. It’s sort of a trial run with teachers and parents. The course is ultimately being designed for students to help them understand better how to learn math and how to overcome obstacles to their learning. I don’t mind being a guinea pig. Apparently, neither do 35,000 other people! I can’t believe so many people signed up for the course. It’s been a great experience, and it makes me happy so many others are taking part.

I have no idea if I’ll be able to access my work once the course ends at the end of September, so I’m going to capture some of my responses to the course here.

The first session of the course introduced some initial data and anecdotes to help us understand just how much fear and trauma math causes our students. We were asked to reflect on this information to say if it surprised us at all and how it connected with our experiences with our students. Here’s my response (with some edits now that I’m rereading this):

I wasn’t really surprised by the data. As a former classroom teacher, I’m all too aware of the troubles students have with math starting at an early age and the trauma it causes. I’m even aware of the trauma it’s caused some of the teachers I’ve worked with. One summer, I attended a math training whose focus was on helping teachers feel more confident about math, and it worked! They realized that they could make sense of math problems, and they didn’t have to hide their reasoning because they felt like they were “cheating” by not solving problems the “right” way. By the end of the training, several of the teachers cried and got extremely angry. They felt cheated. All these years they had thought of themselves as bad at math. But now that they realized that math is a subject they can make sense of, they questioned whether they could have taken a different path in their careers had they been taught math in a more nurturing way growing up.

As for myself, I was always “good” at math, but I never really understood what I was doing. I stayed “good” at math all the way through two semesters of college calculus, but it was way back in 8th grade algebra that I really stopped “understanding” what I was doing. I could follow the steps I was shown, but I never knew why I was doing them. I had a sense that there was something “behind the curtain”, but I never got a peek. It left me feeling like a fraud because I was an A student who could only solve procedural problems. I lacked the ability to reason, problem solve, and think creatively with math. It didn’t help that our curriculum materials obviously favored practicing procedural skills. I could get 22 basic skills problems correct with ease, flounder on the 2 word problems at the end of a worksheet, and still get an A.

Ever since I started teaching math, and more specifically, after I started attending quality professional development about teaching math, I learned that I could make sense of math. I had a capacity I never believed or knew I had. It was an empowering experience, but I know I have room to grow.

I was an elementary school teacher for 8 years, and I felt great about teaching math up to grade 5; however, I still got nervous anytime I had to approach anything remotely close to algebra. Currently I design digital math curriculum, and for the first time I’m having to develop lessons for grades 6 and 7 (I’m getting that much closer to algebra!). When I first started designing middle school math lessons, I was extremely nervous and self-conscious about it, and admittedly I still worry from time to time, but I’ve learned through designing lessons with my team that everything I learned about thinking mathematically when approaching elementary school math still applies to middle school math.

My team has said it’s been good working with me because I don’t just rely on the “tried and true” procedures and methods. I question why we are doing things a certain way, especially if it doesn’t make sense. Is it because of a lack of knowledge on my part? If so, they help me understand it. Or are we doing it out of habit and it’s not really best for the students? In which case I’ve opened their minds to thinking about it differently.

So, long story short, learning about math is important to me professionally because I design learning experiences for students to learn math, but it’s also important to me personally as I continue to develop as a math learner myself.