Visible Learning for Mathematics

It has been a year since I have visited my site. In that time, I have read and used a number of texts but just haven’t had time to share. My goal this Easter was to re-visit my blog and share a current text. A copy of  Visible Learning for Literacy by Hattie, Fisher and Frey was given to all Queensland State School principals at a recent conference. As pre-reading for a professional learning session that I was conducting, I was given a couple of chapters to read and make reference. This pre-reading whetted my appetite and when I found that these authors had also produced a text called Visible Learning for Mathematics, I immediately bought both.

Visible learning in Mathematics

In preparation for another professional learning session, I  began reading the mathematics one. It was such an easy read and I read it from front to back. It cleverly combines the effect size work of Hattie and the brilliant pedagogical instruction of Fisher and Frey. Although it is written for mathematics, I can see the ideas transferring to other learning areas.

In the preface to the book, Hattie, Fisher and Frey (2017) write that the denominator of great mathematicians is that they knew how to struggle. Students need to be persistent, enjoy the struggle, and continue to try. As I reflect on my own mathematics education, it is interesting that although I never received brilliant grades, I always enjoyed the challenge of mathematics (and still do to this day). I especially love seeing how other people (including students and my own children) solve problems and have learnt so much by engaging in rich and rigorous discussion about these different techniques. As a teacher, this discussion has transferred to effective mathematics pedagogy and how to enhance mathematics engagement for students.

The authors mention four things that make mathematics teaching more powerful: appropriately challenging learning intentions and success criteria; learning that is embedded in discourse and collaboration; students doing more of the thinking and talking than their teacher and; students owning their learning. Hattie adds that greater than an effect size of 0.4 allows students to learn at an appropriate rate. The effect size for classroom discussion is 0.82. Students should be invited to talk in collaborative groups, working to solve complex and rich tasks.

Why am I taking the time to share my insights? People who understand mathematics have a higher quality of life. Our moral imperative is every student succeeding. Therefore, to be successful, students must receive high quality instruction (Boaler, 2015).

There are seven chapters in this book and over the next series of blogs, I will share my insights under the following chapters.

Chapter 1: Making learning visible in mathematics.

Chapter 2: Teacher clarity.

Chapter 3: Mathematical tasks and talk that guides learning.

Chapters 4-6: The three phases of learning, surface, deep and transfer, one chapter for each phase.

Chapter 7: Assessment, feedback and meeting the needs of all learners.


Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics What works best to optimize student learning Grades K-12. Thousand Oaks, CA: Corwin.

More on vocabulary

Marzano’s Six Step Process to Teaching Academic Vocabulary

Six is the magic number. My last blog looked at Konza’s take on vocabulary as a part of her big 6 in understanding the reading process. Today’s blog will look at Marzano’s six step process to teaching academic vocabulary. I love this process for the explicit and deep teaching of vocabulary as it can be used in any year level and for any learning area.

This is an example of selecting an essential word from a year 8/9 biology unit. I deliberately chose a year level and learning area that were unfamiliar to me.

Teaching the word: mutualism

1. Provide a description, explanation, or example of the new term. (Include a non-linguistic representation of the term for ESL kids.)

The idea is to make this interesting and something that students will remember.

Telling a joke: What’s worse than a giraffe with a sore throat? Why, a crocodile with a toothache, of course.

crocodile and plover

Nile crocodile and Egyptian plover

Showing a photograph and telling  an interesting story. Introduce the Nile crocodile, an animal that is a messy eater and cannot move food from side to side as its jaws only open and close. It also can’t move food around with its tongue and this results in food becoming caught in its teeth which attracts parasites, bacteria and leeches. Thankfully the Egyptian plover works mutually with the crocodile. The crocodile beeches itself, remains very still and leaves its mouth open. The plover enters the crocodile’s mouth and eats the scraps and parasites – a win win situation. The plover gets to eat and the crocodile gets it teeth cleaned.  (This would also be a great teaching moment to show where Egypt and the Nile river are located).

The crocodile and plover have a mutualistic relationship – one where both of them benefit. In science, we call this mutualism. This is one of the three types of symbiotic relationships, the other two being commensalism and parasitism.

Using a video clip: I found a short video clip that added to the description.

2. Ask students to restate the description, explanation, or example in their own words. (Allow students whose primary existing knowledge base is still in their native language to write in it.)


Recording sheet example

Oral rehearsal: give students time to discuss what mutualism is (pairs or small groups) and then have them rewrite the definition in their own words. If this is a second language, let students write the word and its meaning in their first language.

3. Ask students to construct a picture, symbol, or graphic representing the word.

Model first: Some students will be reluctant to draw but the idea is to have some kind of graphic representation  that will help them remember the word.

Share: Sharing the different representations helps students see others ways the word can be represented.

4. Engage students periodically in activities that help them add to their knowledge of the terms in their notebooks.


Instagrok word map

There are lots of ways students can practise using the word and broadening their understanding of it. Writing synonyms or antonyms. Finding similarities or differences. Completing an analogy – Mutualism is to symbiosis as ____________ is to  ___________.  (As mutualism is one of the types of symbiosis, I wrote …as an edge is to a triangle).

5. Periodically ask students to discuss the terms with one another. (Allow in native language when appropriate).

Oral rehearsal is vital and giving students time to talk about the word and all its accompanying vocabulary enhances student knowledge. This can be done as Think-Pair-Shares. As you have seen this step has been done when students rewrote their definitions and drew their graphic representations. They could explain any new learning or clarify confusions or disagreements. Allow time for students to make any revisions.

6. Involve students periodically in games that allow them to play with terms.

There are so many games that can be played: Pictionary, memory, jeopardy, charades, talk a mile a minute, bingo, create a skit…

students helping

Students studying together

sea anenome

Sea anemone and clown fish

For the word mutualism I made a simple game that matched images of animals in symbiotic relationships to a human representation  – mutualism  – sea anemone and clown fish – students helping one another with an assignment, commensalism –  barnacles on a whale and student copying neighbour’s homework, parasitism – flea on a dog and someone taking another child’s tuckshop money.

Look at all the words the students would have learned by examining this one word in detail. Mutualism – mutual, mutualistic, commensalism, parasitism, plover, Nile, Egyptian, relationship and the list goes on.

It would also be an opportunity to explore the meaning and use of the suffix –ism.

All of these steps should be modelled first (I would use the gradual release of responsibility – I do, we do, you do together, you do alone).  The steps can also be done in any order and can be completed over a period of days as students explore the word deeper. Some steps can overlap. 

Selecting a word for study

To select a word for study, I like Fisher and Frey’s guidelines:

  1. Is the word representative of an essential idea or concept?
  2. Will the word be used repeatedly within and across units of instruction?
  3. Is the word transportable across other disciplines?
  4. Does the use of the word invite contextual analysis?
  5. Does the word offer an opportunity for structural analysis?
  6. Do the selected word honour the learner’s cognitive load?

From: The Value of Intentional Vocabulary Instruction in the Middle Grades