You can download the report here…
If you have any questions, comments, or suggestions, please Tweet us at @dhabecker and @pelelover1
Show notes at http://theothermath.com
Question #5: Can I help my students learn how to learn mathematics?
A teacher’s role is to recommend or encourage the use of specific learning strategies that are the most beneficial for individual learners or the problem at hand. While no one learning strategy is perfect for all learners and all situations, PISA results indicate that students benefit throughout their schooling when they control their learning. Students who approach mathematics learning strategically are shown to have higher success rates on all types of mathematics problems, regardless of their difficulty.
WHAT ARE CONTROL STRATEGIES IN MATHEMATICS?
Control strategies are part of the METACOGNITION family of general approaches to learning. The two other strategy members of the metacognition family are memorization and elaboration. We talked about memorization strategies in Episode 12 and will talk about elaboration next episode.
Control strategies are learning strategies that allow students to set their own goals and track their own learning progress. In other words, control strategies are methods that help learners control their own learning.
This approach includes activities such as:
PISA asked students questions that measured their use of control strategies in mathematics and found that students around the world tend to use control strategies to learn mathematics more than memorization or elaboration strategies (which is looked at in detail in Question 6).
The questions to measure a student’s tendency to use control strategies were:
Students in the United States reported using control strategies less than the OECD average…although it was just a little less than average. Countries at the top of the list (high use of control strategies) were China, Japan, Hong Kong…mathematical powerhouses. Countries at the bottom of the list were Jordan, Tunisia, Qatar. I’m not sure what to make of this because it is more complicated than just a matter of “good math teaching utilizes lots of control strategies”, because Korea, another math powerhouse, reported using control strategies far less than average.
WHAT IS THE BENEFIT OF STRATEGIC LEARNING IN MATHEMATICS?
Students who use control strategies more frequently score higher in mathematics than students who use other learning strategies such as memorization. What’s more, control strategies work equally well for nearly all mathematics problems, EXCEPT the most difficult ones.
I don’t understand this from the report:
Control strategies might not be as effective for solving the most complex mathematics problems because too much control and strategic learning might hinder students from tapping their creativity and engaging in the deep thinking needed to solve them.
I thought control strategies allow the student to be in control of his own learning and goals. So…I’m not too sure what this means.
What is taught in mathematics class and how learning is assessed might also limit the effectiveness of these strategies. Research suggests that the success of these strategies depends on what is being asked of students by their teachers, schools and education systems. For example, when students are only being assessed on surface-level knowledge of concepts, they won’t venture into deeper learning of mathematics on their own. Control strategies need to be practiced by the student on complex problems as well as easy ones.
IF CONTROL STRATEGIES ARE SO SUCCESSFUL, WHY SHOULDN’T I ENCOURAGE STUDENTS TO USE ONLY THESE LEARNING STRATEGIES AND NOTHING ELSE?
One size does not fit all!
Just as one teaching strategy doesn’t work for every student or every mathematical concept, control strategies aren’t appropriate for every student or every problem all the time.
Also, the use of control strategies is somewhat associated with great anxiety towards mathematics and less self-confidence.
Despite the negative characteristic associated with control strategies, students who report the highest use of control strategies score the highest on the PISA.
WHAT CAN TEACHERS DO?
But don’t take our word for it. Please read the report in its entirety.
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]]>Math anxiety is a real thing, a new brain imaging study by Stanford researchers has confirmed. In a study in the journal Psychological Science, researchers found that there is increased activity in the brain region linked with fear in the brains of second and third graders with math anxiety.
from Pocket
]]>If Levi Vaughan, a 5-year-old kindergartner in Braidwood, Ill., makes it through math class without a meltdown, it’s a good day. The transition to school has been tough in other ways for Levi, said Stefanie Vaughan, his mother, but math has been uniquely challenging.
from Pocket
]]>You can download the report here…
If you have any questions, comments, or suggestions, please Tweet us at @dhabecker and @pelelover1
Show notes at http://theothermath.com
Question #4: What do we know about memorization and learning mathematics?
Every mathematics course involves some level of memorization. PISA data suggest that the way teachers require students to use their memory makes a difference. Data indicate that students who rely on memorization alone may be successful with the easiest mathematics problems, but may find that a deeper understanding of mathematics concepts is necessary to tackle more difficult or non-routine problems.
Let’s dig a little deeper into this subject…
HOW PREVALENT IS MEMORIZATION AS A LEARNING STRATEGY IN MATHEMATICS?
To find out how students around the world learn mathematics, PISA asked those students whether they agreed with statements that corresponded to memorization strategies.
Not too surprisingly, PISA findings reveal that memorization in mathematics is commonplace around the world. In almost every country, when students were asked about the learning strategies they use, they identified some sort of memorization-related strategy. Essentially, there is international consensus that memorization leads to mathematics “automaticity”, speeding up basic arithmetic computations and leaving more time for deeper mathematical reasoning.
WHO USES MEMORIZATION THE MOST?
Of the 35 OECD countries participating in the PISA, students in the United States use memorization strategies more often than the OECD average. Many countries that are amongst the highest performing countries on the PISA are less likely to report memorization as a learning strategy.
PISA data suggest that students who mainly use memorization, drilling, or repetitive learning, likely are using that memorization to avoid intense mental effort, especially if they do not feel confident with mathematics. Rote memorization is used in lieu of – not in addition to – conceptual understanding.
PISA results also show that, across OECD countries, students with positive attitudes about math, who have little or no anxiety towards mathematics are somewhat less likely to use memorization strategies.
In a nutshell, students who rely on memorization are more likely to exhibit math anxiety. The less a student relies on memorization, the more likely that students will have high self-efficacy in mathematics.
WILL MEMORIZATION HELP OR HURT MY STUDENTS’ PERFORMANCE IN MATHEMATICS?
Memorization seems to be a somewhat effective way to learn the easiest math problems, but as the problems get more difficult, students relying on memorization are less likely to be successful. No matter the level of difficulty of the math problems, students who rely on memorization alone are never more successful in solving math problems compared to students relying on strategies other than memorization.
Let’s be clear…there is a difference between rote memorization and learning through repetition. When students practice through repetition (repetition is a form of memorization), math anxiety goes down, procedures become less challenging, and students are more successful in solving difficult problems.
WHAT CAN TEACHERS DO?
ALL KINDS OF MINDS:
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You can download the report here.
If you have any questions, comments, or suggestions, please Tweet us at @dhabecker and @pelelover1
Show notes at http://theothermath.com
Today’s Question is Question #3: As a mathematics teacher, how important is the relationship I have with my students?
It is almost a silly question. Really? We actually need to ask whether the teacher’s relationship with his/her students has an impact on student achievement? Okay, here’s the answer.
Relationship is really important!
Of course, that would make an entirely too short podcast, so let dig into the PISA data to learn more.
Think about your classroom climate. Specifically, think back to your last great teaching day:
Things probably went amazingly smooth that day. You probably noticed some things:
WHAT IS A GOOD CLASSROOM ENVIRONMENT FOR MATHEMATICS TEACHING AND LEARNING?
Jim Knight’s Big Four of Coaching begins with classroom environment before moving to content, instructional strategies, and formative assessment. The OECD report clearly agrees with Knight’s Big Four. They identify the prerequisites for high-quality instruction as:
Why are those prerequisites?
To begin with, more teaching and hopefully more learning occurs in a positive school environment. Disciplinary climate of the classroom is related to what/how teachers are able to teach (Less disruptions = more ability to use cognitive-activation strategies).
PISA data suggest a clear link between the behavior of students in a class and their overall familiarity with mathematics in general. In most countries, a better disciplinary climate is related to greater familiarity with mathematics, even after comparing students and schools with similar socioeconomic profiles. (Disciplinary climate in math lessons and student performance go hand-in-hand.)
This finding is especially important as students’ familiarity with mathematics and their access to mathematics content at school can affect not only their performance in school but also their social and economic situation later in life.
HOW DOES THE LEARNING ENVIRONMENT IN MY CLASSROOM INFLUENCE MY TEACHING AND MY STUDENTS’ LEARNING?
According to PISA data, students say that their teachers are more likely to use all teaching practices if there is
TALIS 2013 asked teachers about both the climate of their classroom and their relationships with their students. Their responses revealed important connections between the quality of the learning environment and teachers’ job satisfaction, as well as their confidence in their own abilities as teachers. (Basically, a teacher is confident, satisfied, and happy when they are able to teach a well-behaved/well-managed classroom. When they feel that way, their teaching is more effective/successful and the performance of the student is positively affected.)
WHAT CAN TEACHERS DO?
Thoughts? Leave comments below!
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]]>Maggie and I have been loving recording our episodes of Infinite Insights and we wanted to keep the momentum going during our Winter Breaks. Here is a repeat of one of our favorite episodes about Math Anxiety. The original blog post with TONS of references is here.
Please visit our website at http://theothermath.com or Tweet us at
@pelelover1 (Maggie)
@dhabecker (Duane)
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Here’s the link to the original blog post that went with this episode…
Infinite Insights Episode 2 – What is UDL in the math classroom?
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http://www.oecd.org/publications/ten-questions-for-mathematics-teachers-and-how-pisa-can-help-answer-them-9789264265387-en.htm
We focus on Question #2: Are some mathematics teaching methods more effective than others? Really, this episode focuses specifically on cognitive-action strategies. What are they? How effective are they? When should they be used?
If you have any questions, comments, or suggestions, please contact us on Twitter: Maggie (@pelelover1) and Duane (@dhabecker)
Maggie and Duane continue digging into the OECD report “Ten Questions for Mathematics Teachers…and how PISA can help answer them.”
You can download their report here…
Today’s Question is Question #2: Are some mathematics teaching methods more effective than others?
What we will find is that this chapter seems to focus on Cognitive Activation.
First a little intro: The role of a teacher is exhausting. We make thousands of decisions daily. We are bombarded by emails, meetings, report cards, committees…oh…AND we teach! Because of this “putting out the next fire” reality, teachers are rarely afforded the opportunity of taking a step back and reflecting on whether the teaching methods they are using are really the best for student learning. It’s time for all of us to stop and think.
As the previous chapter discusses, using a variety of teaching strategies is particularly important when teaching mathematics to students with different abilities, motivation and interests. But student data indicate that, on average across PISA-participating countries, the use of cognitive-activation strategies has the greatest positive association with students’ mean mathematics scores.1
Cognitive-activation strategies give students a chance to think deeply about problems, discuss solution methods and mistakes with others, and reflect on their own learning.
WHAT IS COGNITIVE ACTIVATION IN MATHEMATICS TEACHING?
Cognitive activation is, in essence, making the students do the heavy lifting/heavy thinking. It is about teaching pupils strategies, such as summarizing, questioning and predicting, which they can call upon when solving mathematics problems.
Some of these strategies will require pupils to link new information to information they have already learned, apply their skills to a new context, solve challenging mathematics problems that require extended thought and that could have either multiple solutions or an answer that is not immediately obvious.
These strategies enhance learning and lead to a deeper understanding of the concepts. They encourage students to think more deeply to find solutions and focus on the method they used instead of just answer getting.
HOW WIDELY USED ARE COGNITIVE-ACTIVATION STRATEGIES?
The good news is that, across countries, cognitive-activation strategies are frequently used in mathematics teaching. The most commonly cited cognitive-activation strategy – the teacher asking students to explain how they solved a problem – was reported by 70% of the students as occurring in most lessons.
Examples of cognitive-activation strategies are remarkably mundane and commonplace teacher moves.
Since cognitive-activation strategies themselves are not earth-shattering, perhaps what we should take from this is the need for teachers to deliberately do these things.
HOW CAN THE USE OF COGNITIVE-ACTIVATION STRATEGIES BENEFIT STUDENT ACHIEVEMENT?
Students who reported that their teachers use cognitive-activation strategies in their mathematics classes also have higher mean mathematics scores on the PISA. When we take into account teachers’ use of other teaching strategies in the students’ mathematics classes, the strength of the relationship between cognitive-activation teaching and student achievement is even stronger.
The use of cognitive-activation teaching strategies makes a difference no matter how difficult the mathematics problem. In fact, the odds of student success are even greater for more challenging problems. Students who are more frequently exposed to cognitive-activation teaching methods are about 10% more likely to answer easier items correctly and about 50% more likely to answer more difficult items correctly.
In our last podcast episode where we discussed teacher-directed versus student-directed classrooms, we learned that teacher-directed was more successful with low-level questions, while student-directed instruction benefitted students the most with high-level questions.
In this episode, we learn that cognitive-activation strategies benefit students no matter the difficulty of the problems.
IN WHAT ENVIRONMENT DOES COGNITIVE ACTIVATION FLOURISH?
Socio-economically advantaged students reported more exposure to these strategies than disadvantaged students; and when cognitive-activation strategies are used, the association with student performance is stronger in advantaged schools than in disadvantaged schools.
If these strategies are so beneficial, why isn’t every teacher using them more frequently? PISA data suggest that certain school and student characteristics might be more conducive to using cognitive-activation strategies. These types of teaching strategies emphasize thinking and reasoning for extended periods of time, which may take time away from covering the fundamentals of mathematics.
Thus, using cognitive-activation strategies might be easier in schools or classes in which students don’t spend as much time focusing on basic concepts.
Student behavior in the classroom also impacts a teacher’s ability to use cognitive-activation strategies with students.
The OECD teacher survey, TALIS, also suggests that teachers who collaborate with their colleagues are more inclined to then incorporate cognitive-activation strategies in the classroom.
WHAT CAN TEACHERS DO?
The big take away: Learning mathematics should not be a passive endeavor. Teachers can activate student thinking simply by asking provocative questions, allowing students to struggle, and encouraging students to explain their thinking.
Let’s get on it.
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]]>This episode is special because it is our first episode topic that was suggested by a listener. A huge shoutout to our friend Erick Lee (Twitter handle @TheErickLee) who suggested this great report published by OECD. If you are on Twitter, please give Erick a follow!
Here is the link to the report:
Every three years the OECD administers and publishes the Programme for International Student Assessment, better known as PISA, which evaluates 15 year-old students around the world to determine how well their education system has prepared them for life after compulsory schooling. This test is important because it allows the performance of educational systems to be examined and compared on a common measure across countries. Currently 70 countries participated in the latest PISA.
Ten Questions for Mathematics Teachers… and How PISA Can Help Answer Them is a report that takes the findings from analyses of the 2012 PISA and organizes them into ten questions that discuss what we know about mathematics teaching and learning around the world – and how these data might help you in your mathematics classes right now.
The questions encompass four broad categories:
Each question concludes with concrete, evidence-based suggestions to help teachers develop their mathematics teaching practice.
For the next several weeks, Maggie and I will tackle one new question from this report. Of course, we begin with Question #1: How much should I direct student learning in my mathematics classes?
WHERE DOES MATHEMATICS TEACHING FALL IN THE TEACHER- VS. STUDENT DIRECTED LEARNING DEBATE?
For years, the most common teaching strategy has been teacher directed with a small – but vocal – contingent calling for a more student-oriente
d teaching. Which one is better? Unfortunately, it is not a simple “either/or” proposition. It would have been so nice if the data simply said “do THIS and not THAT”. Rather, it is a bit more nuanced.
It depends on the the content and students being taught.
It is a given that most teachers are directly teaching. Student-centered practices are most commonly used within the context of differentiating instruction. The PISA survey indicates that students may be exposed to different teaching strategies based on their socio-economic status or gender. Girls reported being less frequently exposed to student-oriented instruction in mathematics class than boys did. Disadvantaged students, who are from the bottom quarter of the socio-economic distribution in their countries, reported more frequent exposure to these student-oriented strategies than advantaged students did.
The data show that as the instruction becomes more teacher-directed the more student learning relies upon using memorization skills. Conversely, the more student-oriented the instruction, the less students rely upon memorization and are increasingly able to elaborate upon their thinking.
WHICH TEACHERS USE ACTIVE-LEARNING TEACHING PRACTICES IN MATHEMATICS?
From the Teaching and Learning International Study (TALIS) – a different OECD-led survey – four active-learning (student-oriented) teaching practices are identified:
These practices have been shown by many research studies to have positive effects on student learning and motivation. TALIS data show that teachers who are confident in their own abilities are more likely to engage in active-teaching practices – which is the bottom line, really. If a teacher feels comfortable with the necessary pedagogy, content knowledge, and classroom management, then they will be able to flexibly think about how to teach it in a manner other than direct instruction.
If this doesn’t scream “WE NEED MATH COACHES!!!”, then nothing does.
HOW CAN A VARIETY OF TEACHING STRATEGIES BENEFIT STUDENT ACHIEVEMENT?
As stated above, as the instruction becomes more teacher-directed the student learning becomes more reliant upon memorization. Conversely, the more student-oriented the instruction, the more students are able to elaborate upon their thinking.
The data indicate that students are slightly more successful in solving the easiest mathematics problems in PISA when teachers direct student learning. Yet as the problems become more difficult, students with more exposure to direct instruction no longer have a better chance of success. Students exposed to greater amounts of student-oriented teaching are more likely to solve the difficult problems on PISA.
This means that one teaching method is not sufficient to teach all math problems; teaching complex math skills might require different instructions strategies than those used to teach basic math skills. In fact, rather than succumbing to an “either/or” mentality (or a direct-instruction versus constructivist debate), Singapore is using this research to require teachers to use a variety of teaching methods depending on the complexity of the mathematics being learned.
Teacher-directed and student-oriented instruction must work in tandem.
WHAT CAN TEACHERS DO?
So, let’s wrap this up. What are teachers supposed to take from Question 1? Three things…
Make sure each lesson/unit has extension activities for those who can go deeper. (This is the low-floor/high-ceiling concept that Jo Boaler talks about.) Offer support for the struggling learner. And provide a variety of activities and roles for students with different abilities/interests
This requires that the teacher move beyond the textbook provided lessons and homework and add new activities to lessons that allow students to work together or use new tools (technology or games).
Reserve your teacher-directed lessons for simpler math concepts and research other strategies for teaching more difficult concepts.
Please read the actual report! Here…
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When do you know it’s time to try something different in your math lesson? For me, I knew the moment I read this word problem to my fifth-grade summer school students: “On average, the sun’s energy density reaching Earth’s upper atmosphere is 1,350 watts per square meter.
from Pocket
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